%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP208^2 : 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 : n013.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 09:03:52 AM UTC 2026
% Result : Theorem 87.48s 29.63s
% Output : Refutation 87.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP208^2 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.13/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n013.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Thu Feb 26 01:51:53 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP208^2 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer problem Array_Time 00247_008755
% 0.20/0.46 % Version : [Des22] axioms.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des22]
% 0.20/0.46 % Names : 0019_Array_Time_00247_008755 [Des22]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : ? v9.2.1
% 0.20/0.46 % Syntax : Number of formulae : 8580 (2574 unt; 713 typ; 0 def)
% 0.20/0.46 % Number of atoms : 23193 (8297 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 20 ( 2 avg)
% 0.20/0.46 % Number of connectives : 127328 (1596 ~; 278 |;1294 &;114130 @)
% 0.20/0.46 % ( 0 <=>;10030 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 27 ( 7 avg)
% 0.20/0.46 % Number of types : 12 ( 11 usr)
% 0.20/0.46 % Number of type conns : 4623 (4623 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 705 ( 702 usr; 21 con; 0-8 aty)
% 0.20/0.46 % Number of variables : 23756 (2422 ^;20238 !; 419 ?;23756 :)
% 0.20/0.46 % ( 677 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.20/0.46 % from the van Emde Boas Trees session in the Archive of Formal
% 0.20/0.46 % proofs -
% 0.20/0.46 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.20/0.46 % 2022-02-17 14:11:04.542
% 0.20/0.46 % Bugfixes : Added ()s around negated sides of equations.
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (24)
% 0.20/0.46 thf(ty_t_Record_Otuple__isomorphism,type,
% 0.20/0.46 tuple_isomorphism: $tType > $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Heap__Time__Monad_OHeap,type,
% 0.20/0.46 heap_Time_Heap: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Code__Numeral_Onatural,type,
% 0.20/0.46 code_natural: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Code__Numeral_Ointeger,type,
% 0.20/0.46 code_integer: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Heap_Oheap_Oheap__ext,type,
% 0.20/0.46 heap_ext: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Product__Type_Ounit,type,
% 0.20/0.46 product_unit: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.20/0.46 product_prod: $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Old__Datatype_Onode,type,
% 0.20/0.46 old_node: $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nitpick_Ofun__box,type,
% 0.20/0.46 fun_box: $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Typerep_Otyperep,type,
% 0.20/0.46 typerep: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_String_Oliteral,type,
% 0.20/0.46 literal: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Sum__Type_Osum,type,
% 0.20/0.46 sum_sum: $tType > $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Option_Ooption,type,
% 0.20/0.46 option: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nitpick_Oword,type,
% 0.20/0.46 word: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_String_Ochar,type,
% 0.20/0.46 char: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Heap_Oarray,type,
% 0.20/0.46 array: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_List_Olist,type,
% 0.20/0.46 list: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Rat_Orat,type,
% 0.20/0.46 rat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Num_Onum,type,
% 0.20/0.46 num: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nat_Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Int_Oint,type,
% 0.20/0.46 int: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_itself,type,
% 0.20/0.46 itself: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_tf_a,type,
% 0.20/0.46 a: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (689)
% 0.20/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.20/0.46 finite_finite:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.20/0.46 bounded_lattice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Osize,type,
% 0.20/0.46 size:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Heap_Oheap,type,
% 0.20/0.46 heap:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Odvd,type,
% 0.20/0.46 dvd:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oone,type,
% 0.20/0.46 one:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oidom,type,
% 0.20/0.46 idom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring,type,
% 0.20/0.46 ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oplus,type,
% 0.20/0.46 plus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.46 zero:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Num_Onumeral,type,
% 0.20/0.46 numeral:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Power_Opower,type,
% 0.20/0.46 power:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.20/0.46 field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_GCD_Oring__gcd,type,
% 0.20/0.46 ring_gcd:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.20/0.46 minus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oabs__if,type,
% 0.20/0.46 abs_if:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring__1,type,
% 0.20/0.46 ring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ouminus,type,
% 0.20/0.46 uminus:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.20/0.46 bot:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.46 ord:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.20/0.46 top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemidom,type,
% 0.20/0.46 semidom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Oinverse,type,
% 0.20/0.46 inverse:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring,type,
% 0.20/0.46 semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Oring__char__0,type,
% 0.20/0.46 ring_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.20/0.46 neg_numeral:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.46 order:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__ring,type,
% 0.20/0.46 comm_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Omult__zero,type,
% 0.20/0.46 mult_zero:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Typerep_Otyperep,type,
% 0.20/0.46 typerep2:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_GCD_Osemiring__Gcd,type,
% 0.20/0.46 semiring_Gcd:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_GCD_Osemiring__gcd,type,
% 0.20/0.46 semiring_gcd:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.46 group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.20/0.46 lattice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.20/0.46 no_bot:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.20/0.46 no_top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__0,type,
% 0.20/0.46 semiring_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1,type,
% 0.20/0.46 semiring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Omonoid__add,type,
% 0.20/0.46 monoid_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.20/0.46 comm_ring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oidom__divide,type,
% 0.20/0.46 idom_divide:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oidom__modulo,type,
% 0.20/0.46 idom_modulo:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.20/0.46 monoid_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.46 linorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.46 preorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Parity_Oring__parity,type,
% 0.20/0.46 ring_parity:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 0.20/0.46 idom_abs_sgn:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oordered__ring,type,
% 0.20/0.46 ordered_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.20/0.46 zero_neq_one:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Countable_Ocountable,type,
% 0.20/0.46 countable:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.20/0.46 field_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.20/0.46 ab_group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.20/0.46 semiring_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.20/0.46 order_bot:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.20/0.46 order_top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.20/0.46 wellorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring,type,
% 0.20/0.46 comm_semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.20/0.46 zero_less_one:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.20/0.46 division_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Ofield__abs__sgn,type,
% 0.20/0.46 field_abs_sgn:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.20/0.46 semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Num_Osemiring__numeral,type,
% 0.20/0.46 semiring_numeral:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.20/0.46 semidom_divide:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemidom__modulo,type,
% 0.20/0.46 semidom_modulo:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.20/0.46 semigroup_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.20/0.46 dense_order:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.20/0.46 comm_semiring_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.20/0.46 comm_semiring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.20/0.46 linordered_idom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.20/0.46 linordered_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__modulo,type,
% 0.20/0.46 semiring_modulo:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.20/0.46 comm_monoid_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Parity_Osemiring__parity,type,
% 0.20/0.46 semiring_parity:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oordered__ring__abs,type,
% 0.20/0.46 ordered_ring_abs:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oordered__semiring,type,
% 0.20/0.46 ordered_semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.20/0.46 linordered_field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.20/0.46 ab_semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.20/0.46 comm_monoid_diff:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.20/0.46 comm_monoid_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 0.20/0.46 algebraic_semidom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 0.20/0.46 semiring_1_cancel:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.20/0.46 ab_semigroup_mult:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.20/0.46 distrib_lattice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.20/0.46 semilattice_inf:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.20/0.46 semilattice_sup:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.20/0.46 dense_linorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.20/0.46 linordered_semidom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 0.20/0.46 ordered_semiring_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__semiring,type,
% 0.20/0.46 linordered_semiring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 0.20/0.46 comple9053668089753744459l_ccpo:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
% 0.20/0.46 finite8700451911770168679attice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.20/0.46 cancel_semigroup_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.20/0.46 ordered_ab_group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__semiring__1,type,
% 0.20/0.46 linord6961819062388156250ring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Onormalization__semidom,type,
% 0.20/0.46 normal8620421768224518004emidom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oordered__comm__semiring,type,
% 0.20/0.46 ordere2520102378445227354miring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Bit__Operations_Osemiring__bits,type,
% 0.20/0.46 bit_semiring_bits:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.20/0.46 bounded_lattice_bot:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.20/0.46 bounded_lattice_top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 0.20/0.46 comm_s4317794764714335236cancel:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 0.20/0.46 linord4710134922213307826strict:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.20/0.46 cancel1802427076303600483id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 0.20/0.46 ring_15535105094025558882visors:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.20/0.46 cancel2418104881723323429up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.20/0.46 linord5086331880401160121up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.20/0.46 ordere6911136660526730532id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.20/0.46 archim2362893244070406136eiling:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
% 0.20/0.46 semiri6843258321239162965malize:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 0.20/0.46 ordere166539214618696060dd_abs:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.20/0.46 ordere6658533253407199908up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.20/0.46 semiri3467727345109120633visors:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
% 0.20/0.46 boolea8198339166811842893lgebra:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__semiring__strict,type,
% 0.20/0.46 linord8928482502909563296strict:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
% 0.20/0.46 semido2269285787275462019factor:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
% 0.20/0.46 euclid5891614535332579305n_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.46 linord181362715937106298miring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.20/0.46 semiri2026040879449505780visors:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
% 0.20/0.46 bit_ri3973907225187159222ations:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.20/0.46 comple6319245703460814977attice:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
% 0.20/0.46 linord4140545234300271783up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
% 0.20/0.46 linord715952674999750819strict:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.20/0.46 archim462609752435547400_field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.20/0.46 comple5582772986160207858norder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 0.20/0.46 bounde4346867609351753570nf_top:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.20/0.46 bounde4967611905675639751up_bot:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
% 0.20/0.46 euclid3725896446679973847miring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.46 canoni5634975068530333245id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 0.20/0.46 ordere8940638589300402666id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 0.20/0.47 strict7427464778891057005id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
% 0.20/0.47 linord2810124833399127020strict:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
% 0.20/0.47 bit_se359711467146920520ations:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.20/0.47 ordere2412721322843649153imp_le:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.20/0.47 ordere580206878836729694up_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 0.20/0.47 ordere1170586879665033532d_diff:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.20/0.47 strict9044650504122735259up_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.20/0.47 semiri6575147826004484403cancel:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
% 0.20/0.47 euclid8851590272496341667cancel:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.20/0.47 unique1627219031080169319umeral:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
% 0.20/0.47 comple489889107523837845lgebra:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
% 0.20/0.47 comple592849572758109894attice:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
% 0.20/0.47 normal6328177297339901930cative:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
% 0.20/0.47 euclid4440199948858584721cancel:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
% 0.20/0.47 euclid3128863361964157862miring:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.20/0.47 ordere1937475149494474687imp_le:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.20/0.47 condit5016429287641298734tinuum:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
% 0.20/0.47 euclid8789492081693882211th_nat:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
% 0.20/0.47 euclid5411537665997757685th_nat:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 0.20/0.47 semiri1453513574482234551roduct:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
% 0.20/0.47 bit_un5681908812861735899ations:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 0.20/0.47 condit1219197933456340205attice:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.20/0.47 condit6923001295902523014norder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Archimedean__Field_Oceiling,type,
% 0.20/0.47 archimedean_ceiling:
% 0.20/0.47 !>[A: $tType] : ( A > int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
% 0.20/0.47 archim6421214686448440834_floor:
% 0.20/0.47 !>[A: $tType] : ( A > int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Archimedean__Field_Ofrac,type,
% 0.20/0.47 archimedean_frac:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Archimedean__Field_Oround,type,
% 0.20/0.47 archimedean_round:
% 0.20/0.47 !>[A: $tType] : ( A > int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oalloc,type,
% 0.20/0.47 array_alloc:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Ofreeze,type,
% 0.20/0.47 array_freeze:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oget,type,
% 0.20/0.47 array_get:
% 0.20/0.47 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Olen,type,
% 0.20/0.47 array_len:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Olength,type,
% 0.20/0.47 array_length:
% 0.20/0.47 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Omake,type,
% 0.20/0.47 array_make:
% 0.20/0.47 !>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Omap__entry,type,
% 0.20/0.47 array_map_entry:
% 0.20/0.47 !>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Onew,type,
% 0.20/0.47 array_new:
% 0.20/0.47 !>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Onoteq,type,
% 0.20/0.47 array_noteq:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Onth,type,
% 0.20/0.47 array_nth:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oof__list,type,
% 0.20/0.47 array_of_list:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Opresent,type,
% 0.20/0.47 array_present:
% 0.20/0.47 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oset,type,
% 0.20/0.47 array_set:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oswap,type,
% 0.20/0.47 array_swap:
% 0.20/0.47 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oupd,type,
% 0.20/0.47 array_upd:
% 0.20/0.47 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Array__Time_Oupdate,type,
% 0.20/0.47 array_update:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 0.20/0.47 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
% 0.20/0.47 bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.20/0.47 bNF_Ca3754400796208372196lChain:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.20/0.47 bNF_rel_fun:
% 0.20/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
% 0.20/0.47 bNF_Wellorder_bsqr:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
% 0.20/0.47 bNF_We413866401316099525erIncl:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
% 0.20/0.47 bNF_Wellorder_ordIso:
% 0.20/0.47 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
% 0.20/0.47 bNF_We4044943003108391690rdLess:
% 0.20/0.47 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
% 0.20/0.47 bNF_Wellorder_embed:
% 0.20/0.47 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
% 0.20/0.47 bNF_Wellorder_embedS:
% 0.20/0.47 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
% 0.20/0.47 bNF_Wellorder_iso:
% 0.20/0.47 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
% 0.20/0.47 bNF_Wellorder_wo_rel:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 0.20/0.47 basic_BNF_size_prod:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
% 0.20/0.47 basic_BNF_size_sum:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Basic__BNFs_Opred__prod,type,
% 0.20/0.47 basic_pred_prod:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Binomial_Obinomial,type,
% 0.20/0.47 binomial: nat > nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Binomial_Ogbinomial,type,
% 0.20/0.47 gbinomial:
% 0.20/0.47 !>[A: $tType] : ( A > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.20/0.47 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.20/0.47 bit_and_not_num: num > num > ( option @ num ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.20/0.47 bit_concat_bit: nat > int > int > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.20/0.47 bit_or_not_num_neg: num > num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
% 0.20/0.47 bit_ri4277139882892585799ns_not:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
% 0.20/0.47 bit_ri4674362597316999326ke_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
% 0.20/0.47 bit_se5824344872417868541ns_and:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
% 0.20/0.47 bit_se4197421643247451524op_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
% 0.20/0.47 bit_se8732182000553998342ip_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
% 0.20/0.47 bit_se2239418461657761734s_mask:
% 0.20/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
% 0.20/0.47 bit_se1065995026697491101ons_or:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
% 0.20/0.47 bit_se4730199178511100633sh_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
% 0.20/0.47 bit_se5668285175392031749et_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
% 0.20/0.47 bit_se2584673776208193580ke_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
% 0.20/0.47 bit_se2638667681897837118et_bit:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
% 0.20/0.47 bit_se5824344971392196577ns_xor:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
% 0.20/0.47 bit_se5641148757651400278ts_bit:
% 0.20/0.47 !>[A: $tType] : ( A > nat > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
% 0.20/0.47 bit_se6407376104438227557le_bit:
% 0.20/0.47 !>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 0.20/0.47 bit_take_bit_num: nat > num > ( option @ num ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 0.20/0.47 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 0.20/0.47 bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
% 0.20/0.47 boolea2506097494486148201lgebra:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__axioms,type,
% 0.20/0.47 boolea6902313364301356556axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
% 0.20/0.47 boolea3799213064322606851m_diff:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
% 0.20/0.47 boolea5476839437570043046axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_ONat,type,
% 0.20/0.47 code_Nat: code_integer > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_ONeg,type,
% 0.20/0.47 code_Neg: num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_OPos,type,
% 0.20/0.47 code_Pos: num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_OSuc,type,
% 0.20/0.47 code_Suc: code_natural > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 0.20/0.47 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ocr__integer,type,
% 0.20/0.47 code_cr_integer: int > code_integer > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ocr__natural,type,
% 0.20/0.47 code_cr_natural: nat > code_natural > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 0.20/0.47 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 0.20/0.47 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Odup,type,
% 0.20/0.47 code_dup: code_integer > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 0.20/0.47 code_int_of_integer: code_integer > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 0.20/0.47 code_integer_of_int: int > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 0.20/0.47 code_integer_of_nat: nat > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
% 0.20/0.47 code_i5400310926305786745atural: code_natural > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 0.20/0.47 code_integer_of_num: num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 0.20/0.47 code_nat_of_integer: code_integer > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 0.20/0.47 code_case_natural:
% 0.20/0.47 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
% 0.20/0.47 code_nat_of_natural: code_natural > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
% 0.20/0.47 code_natural_of_nat: nat > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 0.20/0.47 code_rec_natural:
% 0.20/0.47 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
% 0.20/0.47 code_rec_set_natural:
% 0.20/0.47 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onatural__of__integer,type,
% 0.20/0.47 code_n4118661773612635043nteger: code_integer > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onegative,type,
% 0.20/0.47 code_negative: num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 0.20/0.47 code_num_of_integer: code_integer > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Opcr__integer,type,
% 0.20/0.47 code_pcr_integer: int > code_integer > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Opcr__natural,type,
% 0.20/0.47 code_pcr_natural: nat > code_natural > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Opositive,type,
% 0.20/0.47 code_positive: num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Osize__nat,type,
% 0.20/0.47 code_size_nat: code_natural > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Code__Numeral_Osub,type,
% 0.20/0.47 code_sub: num > num > code_integer ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 0.20/0.47 complete_Inf_Inf:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.20/0.47 complete_Sup_Sup:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
% 0.20/0.47 comple1908693960933563346ssible:
% 0.20/0.47 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
% 0.20/0.47 comple187402453842119260l_fixp:
% 0.20/0.47 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 0.20/0.47 comple115746919287870866o_fixp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
% 0.20/0.47 comple6359979572994053840erates:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
% 0.20/0.47 comple7512665784863727008ratesp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 0.20/0.47 comple1602240252501008431_chain:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 0.20/0.47 comple7038119648293358887notone:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
% 0.20/0.47 condit941137186595557371_above:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
% 0.20/0.47 condit1013018076250108175_below:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
% 0.20/0.47 condit622319405099724424ng_bdd:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
% 0.20/0.47 condit16957441358409770ng_bdd:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Countable_Ofrom__nat,type,
% 0.20/0.47 from_nat:
% 0.20/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Countable_Onth__item,type,
% 0.20/0.47 nth_item:
% 0.20/0.47 !>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Countable_Onth__item__rel,type,
% 0.20/0.47 nth_item_rel: nat > nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Oadjust__div,type,
% 0.20/0.47 adjust_div: ( product_prod @ int @ int ) > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Oadjust__mod,type,
% 0.20/0.47 adjust_mod: int > int > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Odivmod__nat,type,
% 0.20/0.47 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Oeucl__rel__int,type,
% 0.20/0.47 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 0.20/0.47 unique5940410009612947441es_aux:
% 0.20/0.47 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
% 0.20/0.47 unique8689654367752047608divmod:
% 0.20/0.47 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
% 0.20/0.47 unique1321980374590559556d_step:
% 0.20/0.47 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
% 0.20/0.47 euclid6346220572633701492n_size:
% 0.20/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
% 0.20/0.47 euclid7384307370059645450egment:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
% 0.20/0.47 comm_s3205402744901411588hammer:
% 0.20/0.47 !>[A: $tType] : ( A > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
% 0.20/0.47 semiring_char_0_fact:
% 0.20/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.20/0.47 inverse_inverse:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ocard,type,
% 0.20/0.47 finite_card:
% 0.20/0.47 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite2:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofold,type,
% 0.20/0.47 finite_fold:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofolding__on,type,
% 0.20/0.47 finite_folding_on:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofolding__on_OF,type,
% 0.20/0.47 finite_folding_F:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Obij__betw,type,
% 0.20/0.47 bij_betw:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Ocomp,type,
% 0.20/0.47 comp:
% 0.20/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Ofun__upd,type,
% 0.20/0.47 fun_upd:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Oid,type,
% 0.20/0.47 id:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Oinj__on,type,
% 0.20/0.47 inj_on:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun_Omap__fun,type,
% 0.20/0.47 map_fun:
% 0.20/0.47 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Ois__measure,type,
% 0.20/0.47 fun_is_measure:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Omax__strict,type,
% 0.20/0.47 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Omax__weak,type,
% 0.20/0.47 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Omin__strict,type,
% 0.20/0.47 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Omin__weak,type,
% 0.20/0.47 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Opair__leq,type,
% 0.20/0.47 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Opair__less,type,
% 0.20/0.47 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.20/0.47 fun_reduction_pair:
% 0.20/0.47 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.20/0.47 fun_rp_inv_image:
% 0.20/0.47 !>[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 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_OGcd__class_OGcd,type,
% 0.20/0.47 gcd_Gcd:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_OGcd__class_OLcm,type,
% 0.20/0.47 gcd_Lcm:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Obezw,type,
% 0.20/0.47 bezw: nat > nat > ( product_prod @ int @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Obezw__rel,type,
% 0.20/0.47 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Obounded__quasi__semilattice,type,
% 0.20/0.47 bounde8507323023520639062attice:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
% 0.20/0.47 bounde6485984586167503788ce_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
% 0.20/0.47 bounde2362111253966948842tice_F:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Ogcd__class_Ogcd,type,
% 0.20/0.47 gcd_gcd:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Ogcd__class_Olcm,type,
% 0.20/0.47 gcd_lcm:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 0.20/0.47 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
% 0.20/0.47 semiri4206861660011772517g_char:
% 0.20/0.47 !>[A: $tType] : ( ( itself @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
% 0.20/0.47 semiring_gcd_Gcd_fin:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
% 0.20/0.47 semiring_gcd_Lcm_fin:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oabel__semigroup,type,
% 0.20/0.47 abel_semigroup:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 0.20/0.47 abel_s757365448890700780axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.20/0.47 abs_abs:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ocomm__monoid,type,
% 0.20/0.47 comm_monoid:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ocomm__monoid__axioms,type,
% 0.20/0.47 comm_monoid_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ogroup,type,
% 0.20/0.47 group:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ogroup__axioms,type,
% 0.20/0.47 group_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.20/0.47 minus_minus:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Omonoid,type,
% 0.20/0.47 monoid:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Omonoid__axioms,type,
% 0.20/0.47 monoid_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.20/0.47 one_one:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.20/0.47 plus_plus:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Osemigroup,type,
% 0.20/0.47 semigroup:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 0.20/0.47 sgn_sgn:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.20/0.47 times_times:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.20/0.47 uminus_uminus:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.47 zero_zero:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 0.20/0.47 groups7311177749621191930dd_sum:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
% 0.20/0.47 groups1027152243600224163dd_sum:
% 0.20/0.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
% 0.20/0.47 groups7121269368397514597t_prod:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
% 0.20/0.47 groups1962203154675924110t_prod:
% 0.20/0.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
% 0.20/0.47 groups778175481326437816id_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
% 0.20/0.47 groups_comm_monoid_F:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
% 0.20/0.47 groups_comm_monoid_G:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Ocomm__monoid__list,type,
% 0.20/0.47 groups1828464146339083142d_list:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
% 0.20/0.47 groups4802862169904069756st_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
% 0.20/0.47 groups4207007520872428315er_sum:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 0.20/0.47 groups8242544230860333062m_list:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Omonoid__list,type,
% 0.20/0.47 groups_monoid_list:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Omonoid__list_OF,type,
% 0.20/0.47 groups_monoid_F:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
% 0.20/0.47 groups5270119922927024881d_list:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_HOL_ONO__MATCH,type,
% 0.20/0.47 nO_MATCH:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_HOL_OThe,type,
% 0.20/0.47 the:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_HOL_OUniq,type,
% 0.20/0.47 uniq:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_HOL_Oundefined,type,
% 0.20/0.47 undefined:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oaddr__of__array,type,
% 0.20/0.47 addr_of_array:
% 0.20/0.47 !>[A: $tType] : ( ( array @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oarray_OArray,type,
% 0.20/0.47 array2:
% 0.20/0.47 !>[A: $tType] : ( nat > ( array @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oarray_Osize__array,type,
% 0.20/0.47 size_array:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oempty,type,
% 0.20/0.47 empty: heap_ext @ product_unit ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oheap_Oarrays,type,
% 0.20/0.47 arrays:
% 0.20/0.47 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oheap_Oheap__ext,type,
% 0.20/0.47 heap_ext2:
% 0.20/0.47 !>[Z: $tType] : ( ( typerep > nat > ( list @ nat ) ) > ( typerep > nat > nat ) > nat > Z > ( heap_ext @ Z ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oheap_Olim,type,
% 0.20/0.47 lim:
% 0.20/0.47 !>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap_Oheap_Olim__update,type,
% 0.20/0.47 lim_update:
% 0.20/0.47 !>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
% 0.20/0.47 heap_Time_Heap2:
% 0.20/0.47 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
% 0.20/0.47 heap_Time_case_Heap:
% 0.20/0.47 !>[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 ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
% 0.20/0.47 heap_Time_map_Heap:
% 0.20/0.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
% 0.20/0.47 heap_Time_pred_Heap:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
% 0.20/0.47 heap_Time_rec_Heap:
% 0.20/0.47 !>[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 ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
% 0.20/0.47 heap_Time_rel_Heap:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
% 0.20/0.47 heap_Time_set_Heap:
% 0.20/0.47 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
% 0.20/0.47 heap_Time_size_Heap:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
% 0.20/0.47 heap_Time_Heap_lub:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
% 0.20/0.47 heap_Time_Heap_ord:
% 0.20/0.47 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oassert,type,
% 0.20/0.47 heap_Time_assert:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Obind,type,
% 0.20/0.47 heap_Time_bind:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oeffect,type,
% 0.20/0.47 heap_Time_effect:
% 0.20/0.47 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oexecute,type,
% 0.20/0.47 heap_Time_execute:
% 0.20/0.47 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oguard,type,
% 0.20/0.47 heap_Time_guard:
% 0.20/0.47 !>[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 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oheap,type,
% 0.20/0.47 heap_Time_heap:
% 0.20/0.47 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Olift,type,
% 0.20/0.47 heap_Time_lift:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oraise,type,
% 0.20/0.47 heap_Time_raise:
% 0.20/0.47 !>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oreturn,type,
% 0.20/0.47 heap_Time_return:
% 0.20/0.47 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Osuccess,type,
% 0.20/0.47 heap_Time_success:
% 0.20/0.47 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Otap,type,
% 0.20/0.47 heap_Time_tap:
% 0.20/0.47 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
% 0.20/0.47 heap_Time_timeFrame:
% 0.20/0.47 !>[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 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
% 0.20/0.47 heap_T5500966940807335491me_rel:
% 0.20/0.47 !>[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 ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Oureturn,type,
% 0.20/0.47 heap_Time_ureturn:
% 0.20/0.47 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Heap__Time__Monad_Owait,type,
% 0.20/0.47 heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Hilbert__Choice_Obijection,type,
% 0.20/0.47 hilbert_bijection:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 0.20/0.47 hilbert_inv_into:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_If,type,
% 0.20/0.47 if:
% 0.20/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
% 0.20/0.47 complete_lattice_gfp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
% 0.20/0.47 complete_lattice_lfp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_OAbs__Integ,type,
% 0.20/0.47 abs_Integ: ( product_prod @ nat @ nat ) > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_ONeg,type,
% 0.20/0.47 neg: num > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_OPos,type,
% 0.20/0.47 pos: num > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_ORep__Integ,type,
% 0.20/0.47 rep_Integ: int > ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Ocr__int,type,
% 0.20/0.47 cr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Odup,type,
% 0.20/0.47 dup: int > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oint_OAbs__int,type,
% 0.20/0.47 abs_int: ( set @ ( product_prod @ nat @ nat ) ) > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oint_ORep__int,type,
% 0.20/0.47 rep_int: int > ( set @ ( product_prod @ nat @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oint__ge__less__than,type,
% 0.20/0.47 int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oint__ge__less__than2,type,
% 0.20/0.47 int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Ointrel,type,
% 0.20/0.47 intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Onat,type,
% 0.20/0.47 nat2: int > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Opcr__int,type,
% 0.20/0.47 pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Opower__int,type,
% 0.20/0.47 power_int:
% 0.20/0.47 !>[A: $tType] : ( A > int > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oring__1__class_OInts,type,
% 0.20/0.47 ring_1_Ints:
% 0.20/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.20/0.47 ring_1_of_int:
% 0.20/0.47 !>[A: $tType] : ( int > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Int_Osub,type,
% 0.20/0.47 sub: num > num > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.20/0.47 inf_inf:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice,type,
% 0.20/0.47 semilattice:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice__axioms,type,
% 0.20/0.47 semilattice_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 0.20/0.47 semilattice_neutr:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice__neutr__order,type,
% 0.20/0.47 semila1105856199041335345_order:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice__order,type,
% 0.20/0.47 semilattice_order:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osemilattice__order__axioms,type,
% 0.20/0.47 semila6385135966242565138axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.20/0.47 sup_sup:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Olinorder_OMax,type,
% 0.20/0.47 lattices_Max:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Olinorder_OMin,type,
% 0.20/0.47 lattices_Min:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 0.20/0.47 lattic643756798349783984er_Max:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 0.20/0.47 lattic643756798350308766er_Min:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
% 0.20/0.47 lattices_ord_arg_max:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
% 0.20/0.47 lattic1883929316492267755max_on:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
% 0.20/0.47 lattices_ord_arg_min:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 0.20/0.47 lattic7623131987881927897min_on:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
% 0.20/0.47 lattic501386751176901750rg_max:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
% 0.20/0.47 lattic501386751177426532rg_min:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 0.20/0.47 lattic7752659483105999362nf_fin:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
% 0.20/0.47 lattic5652469242046573047tr_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
% 0.20/0.47 lattic5214292709420241887eutr_F:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__order__neutr__set,type,
% 0.20/0.47 lattic3600114342068043075tr_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 0.20/0.47 lattic4895041142388067077er_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 0.20/0.47 lattic149705377957585745ce_set:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
% 0.20/0.47 lattic1715443433743089157tice_F:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 0.20/0.47 lattic5882676163264333800up_fin:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Lifting_OQuotient,type,
% 0.20/0.47 quotient:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oappend,type,
% 0.20/0.47 append:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oarg__min__list,type,
% 0.20/0.47 arg_min_list:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oarg__min__list__rel,type,
% 0.20/0.47 arg_min_list_rel:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Obutlast,type,
% 0.20/0.47 butlast:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oconcat,type,
% 0.20/0.47 concat:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ocoset,type,
% 0.20/0.47 coset:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ocount__list,type,
% 0.20/0.47 count_list:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Odistinct,type,
% 0.20/0.47 distinct:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Odrop,type,
% 0.20/0.47 drop:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_OdropWhile,type,
% 0.20/0.47 dropWhile:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oenumerate,type,
% 0.20/0.47 enumerate:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ofilter,type,
% 0.20/0.47 filter:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ofind,type,
% 0.20/0.47 find:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ofold,type,
% 0.20/0.47 fold:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ofolding__insort__key,type,
% 0.20/0.47 folding_insort_key:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ofoldr,type,
% 0.20/0.47 foldr:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ogen__length,type,
% 0.20/0.47 gen_length:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olast,type,
% 0.20/0.47 last:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olenlex,type,
% 0.20/0.47 lenlex:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olex,type,
% 0.20/0.47 lex:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olexord,type,
% 0.20/0.47 lexord:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
% 0.20/0.47 linord329482645794927042rt_key:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olinorder__class_Oinsort__key,type,
% 0.20/0.47 linorder_insort_key:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olinorder__class_Osort__key,type,
% 0.20/0.47 linorder_sort_key:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 0.20/0.47 linord4507533701916653071of_set:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_OCons,type,
% 0.20/0.47 cons:
% 0.20/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_ONil,type,
% 0.20/0.47 nil:
% 0.20/0.47 !>[A: $tType] : ( list @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Ohd,type,
% 0.20/0.47 hd:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Olist__all,type,
% 0.20/0.47 list_all:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Olist__all2,type,
% 0.20/0.47 list_all2:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Omap,type,
% 0.20/0.47 map:
% 0.20/0.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Oset,type,
% 0.20/0.47 set2:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Osize__list,type,
% 0.20/0.47 size_list:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist_Otl,type,
% 0.20/0.47 tl:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist__ex,type,
% 0.20/0.47 list_ex:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olist__update,type,
% 0.20/0.47 list_update:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olistrel,type,
% 0.20/0.47 listrel:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Olistrel1,type,
% 0.20/0.47 listrel1:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Omeasures,type,
% 0.20/0.47 measures:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_On__lists,type,
% 0.20/0.47 n_lists:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Onth,type,
% 0.20/0.47 nth:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Onths,type,
% 0.20/0.47 nths:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oord__class_Olexordp,type,
% 0.20/0.47 ord_lexordp:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oord__class_Olexordp__eq,type,
% 0.20/0.47 ord_lexordp_eq:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oproduct,type,
% 0.20/0.47 product:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oproduct__lists,type,
% 0.20/0.47 product_lists:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oremdups,type,
% 0.20/0.47 remdups:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oremdups__adj,type,
% 0.20/0.47 remdups_adj:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oremove1,type,
% 0.20/0.47 remove1:
% 0.20/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_OremoveAll,type,
% 0.20/0.47 removeAll:
% 0.20/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oreplicate,type,
% 0.20/0.47 replicate:
% 0.20/0.47 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Orev,type,
% 0.20/0.47 rev:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Orotate,type,
% 0.20/0.47 rotate:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Orotate1,type,
% 0.20/0.47 rotate1:
% 0.20/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Osorted__wrt,type,
% 0.20/0.47 sorted_wrt:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Otake,type,
% 0.20/0.47 take:
% 0.20/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_OtakeWhile,type,
% 0.20/0.47 takeWhile:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Otranspose,type,
% 0.20/0.47 transpose:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oupt,type,
% 0.20/0.47 upt: nat > nat > ( list @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oupto,type,
% 0.20/0.47 upto: int > int > ( list @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oupto__aux,type,
% 0.20/0.47 upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Oupto__rel,type,
% 0.20/0.47 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_List_Ozip,type,
% 0.20/0.47 zip:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Map_Omap__of,type,
% 0.20/0.47 map_of:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Map_Omap__upds,type,
% 0.20/0.47 map_upds:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Map_Orestrict__map,type,
% 0.20/0.47 restrict_map:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_OSuc,type,
% 0.20/0.47 suc: nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Ocompow,type,
% 0.20/0.47 compow:
% 0.20/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Ofunpow,type,
% 0.20/0.47 funpow:
% 0.20/0.47 !>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 0.20/0.47 case_nat:
% 0.20/0.47 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Onat_Opred,type,
% 0.20/0.47 pred: nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 0.20/0.47 rec_nat:
% 0.20/0.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
% 0.20/0.47 rec_set_nat:
% 0.20/0.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Osemiring__1__class_ONats,type,
% 0.20/0.47 semiring_1_Nats:
% 0.20/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.20/0.47 semiring_1_of_nat:
% 0.20/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.20/0.47 semiri8178284476397505188at_aux:
% 0.20/0.47 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.20/0.47 size_size:
% 0.20/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 0.20/0.47 nat_int_decode: nat > int ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 0.20/0.47 nat_int_encode: int > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 0.20/0.47 nat_list_decode: nat > ( list @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 0.20/0.47 nat_list_decode_rel: nat > nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.20/0.47 nat_list_encode: ( list @ nat ) > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.20/0.47 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 0.20/0.47 nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.20/0.47 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.20/0.47 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.20/0.47 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.20/0.47 nat_set_decode: nat > ( set @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.20/0.47 nat_set_encode: ( set @ nat ) > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 0.20/0.47 nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Osum__encode,type,
% 0.20/0.47 nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.20/0.47 nat_triangle: nat > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_OBitM,type,
% 0.20/0.47 bitM: num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oinc,type,
% 0.20/0.47 inc: num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onat__of__num,type,
% 0.20/0.47 nat_of_num: num > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.20/0.47 neg_numeral_dbl:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
% 0.20/0.47 neg_numeral_dbl_dec:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 0.20/0.47 neg_numeral_dbl_inc:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
% 0.20/0.47 neg_numeral_is_num:
% 0.20/0.47 !>[A: $tType] : ( A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 0.20/0.47 neg_numeral_sub:
% 0.20/0.47 !>[A: $tType] : ( num > num > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_OBit0,type,
% 0.20/0.47 bit0: num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_OBit1,type,
% 0.20/0.47 bit1: num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_OOne,type,
% 0.20/0.47 one2: num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_Ocase__num,type,
% 0.20/0.47 case_num:
% 0.20/0.47 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_Orec__num,type,
% 0.20/0.47 rec_num:
% 0.20/0.47 !>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum_Osize__num,type,
% 0.20/0.47 size_num: num > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onum__of__nat,type,
% 0.20/0.47 num_of_nat: nat > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.20/0.47 numeral_numeral:
% 0.20/0.47 !>[A: $tType] : ( num > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Opow,type,
% 0.20/0.47 pow: num > num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Opred__numeral,type,
% 0.20/0.47 pred_numeral: num > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oring__1__class_Oiszero,type,
% 0.20/0.47 ring_1_iszero:
% 0.20/0.47 !>[A: $tType] : ( A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Osqr,type,
% 0.20/0.47 sqr: num > num ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OAtom,type,
% 0.20/0.47 old_Atom:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OIn0,type,
% 0.20/0.47 old_In0:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OIn1,type,
% 0.20/0.47 old_In1:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OLeaf,type,
% 0.20/0.47 old_Leaf:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_ONode,type,
% 0.20/0.47 old_Node:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_ONumb,type,
% 0.20/0.47 old_Numb:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OPush,type,
% 0.20/0.47 old_Push:
% 0.20/0.47 !>[B: $tType] : ( ( sum_sum @ B @ nat ) > ( nat > ( sum_sum @ B @ nat ) ) > nat > ( sum_sum @ B @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OPush__Node,type,
% 0.20/0.47 old_Push_Node:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_OScons,type,
% 0.20/0.47 old_Scons:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_Ondepth,type,
% 0.20/0.47 old_ndepth:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
% 0.20/0.47 old_Abs_Node:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_Onode_ORep__Node,type,
% 0.20/0.47 old_Rep_Node:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Old__Datatype_Ontrunc,type,
% 0.20/0.47 old_ntrunc:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Option_Ooption_ONone,type,
% 0.20/0.47 none:
% 0.20/0.47 !>[A: $tType] : ( option @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Option_Ooption_OSome,type,
% 0.20/0.47 some:
% 0.20/0.47 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Option_Ooption_Ocase__option,type,
% 0.20/0.47 case_option:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Option_Ooption_Osize__option,type,
% 0.20/0.47 size_option:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Option_Ooption_Othe,type,
% 0.20/0.47 the2:
% 0.20/0.47 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_Oofilter,type,
% 0.20/0.47 order_ofilter:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_Ounder,type,
% 0.20/0.47 order_under:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_OunderS,type,
% 0.20/0.47 order_underS:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Order__Relation_Owell__order__on,type,
% 0.20/0.47 order_well_order_on:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.47 bot_bot:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord_OLeast,type,
% 0.20/0.47 least:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord_Omax,type,
% 0.20/0.47 max:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord_Omin,type,
% 0.20/0.47 min:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 0.20/0.47 ord_Least:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.47 ord_less:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.47 ord_less_eq:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Omax,type,
% 0.20/0.47 ord_max:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Omin,type,
% 0.20/0.47 ord_min:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.20/0.47 order_Greatest:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.20/0.47 order_antimono:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 0.20/0.47 order_mono:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oordering,type,
% 0.20/0.47 ordering:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oordering__axioms,type,
% 0.20/0.47 ordering_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oordering__top,type,
% 0.20/0.47 ordering_top:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oordering__top__axioms,type,
% 0.20/0.47 ordering_top_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Opartial__preordering,type,
% 0.20/0.47 partial_preordering:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Opreordering,type,
% 0.20/0.47 preordering:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Opreordering__axioms,type,
% 0.20/0.47 preordering_axioms:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.20/0.47 top_top:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Partial__Function_Ofun__lub,type,
% 0.20/0.47 partial_fun_lub:
% 0.20/0.47 !>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Partial__Function_Ofun__ord,type,
% 0.20/0.47 partial_fun_ord:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Partial__Function_Opartial__function__definitions,type,
% 0.20/0.47 partia7178651479351089652itions:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Power_Opower_Opower,type,
% 0.20/0.47 power2:
% 0.20/0.47 !>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Power_Opower__class_Opower,type,
% 0.20/0.47 power_power:
% 0.20/0.47 !>[A: $tType] : ( A > nat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.20/0.47 product_Pair:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_OSigma,type,
% 0.20/0.47 product_Sigma:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_OUnity,type,
% 0.20/0.47 product_Unity: product_unit ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oapfst,type,
% 0.20/0.47 product_apfst:
% 0.20/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oapsnd,type,
% 0.20/0.47 product_apsnd:
% 0.20/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.20/0.47 product_case_prod:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.20/0.47 product_fst:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.20/0.47 product_snd:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Pure_Otype,type,
% 0.20/0.47 type2:
% 0.20/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Quotient_OQuotient3,type,
% 0.20/0.47 quotient3:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Quotient_Oquot__type_Oabs,type,
% 0.20/0.47 quot_abs:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > A > B ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Quotient_Oquot__type_Orep,type,
% 0.20/0.47 quot_rep:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > B > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Olog,type,
% 0.20/0.47 log: code_natural > code_natural > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Olog__rel,type,
% 0.20/0.47 log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Ominus__shift,type,
% 0.20/0.47 minus_shift: code_natural > code_natural > code_natural > code_natural ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Opick,type,
% 0.20/0.47 pick:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Orange,type,
% 0.20/0.47 range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Random_Oselect__weight,type,
% 0.20/0.47 select_weight:
% 0.20/0.47 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_OAbs__Rat,type,
% 0.20/0.47 abs_Rat: ( product_prod @ int @ int ) > rat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_OFract,type,
% 0.20/0.47 fract: int > int > rat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_OFrct,type,
% 0.20/0.47 frct: ( product_prod @ int @ int ) > rat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_ORep__Rat,type,
% 0.20/0.47 rep_Rat: rat > ( product_prod @ int @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 0.20/0.47 field_char_0_Rats:
% 0.20/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
% 0.20/0.47 field_char_0_of_rat:
% 0.20/0.47 !>[A: $tType] : ( rat > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Onormalize,type,
% 0.20/0.47 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Oof__int,type,
% 0.20/0.47 of_int: int > rat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Opcr__rat,type,
% 0.20/0.47 pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Opositive,type,
% 0.20/0.47 positive: rat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Oquotient__of,type,
% 0.20/0.47 quotient_of: rat > ( product_prod @ int @ int ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rat_Oratrel,type,
% 0.20/0.47 ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
% 0.20/0.47 tuple_1188178415141063261rphism:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( ( product_prod @ B @ C ) > A ) > ( tuple_isomorphism @ A @ B @ C ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
% 0.20/0.47 tuple_9181185373184732606rphism:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( tuple_isomorphism @ A @ B @ C ) > nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_ODomainp,type,
% 0.20/0.47 domainp:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_OField,type,
% 0.20/0.47 field2:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_OId,type,
% 0.20/0.47 id2:
% 0.20/0.47 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_OPowp,type,
% 0.20/0.47 powp:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Oinv__image,type,
% 0.20/0.47 inv_image:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Oirreflp,type,
% 0.20/0.47 irreflp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Orelcomp,type,
% 0.20/0.47 relcomp:
% 0.20/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Relation_Otransp,type,
% 0.20/0.47 transp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
% 0.20/0.47 algebr8660921524188924756oprime:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.20/0.47 divide_divide:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.20/0.47 dvd_dvd:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Omodulo__class_Omodulo,type,
% 0.20/0.47 modulo_modulo:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
% 0.20/0.47 normal6383669964737779283malize:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
% 0.20/0.47 unit_f5069060285200089521factor:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 0.20/0.47 zero_neq_one_of_bool:
% 0.20/0.47 !>[A: $tType] : ( $o > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OBall,type,
% 0.20/0.47 ball:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OBex,type,
% 0.20/0.47 bex:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OCollect,type,
% 0.20/0.47 collect:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OPow,type,
% 0.20/0.47 pow2:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Odisjnt,type,
% 0.20/0.47 disjnt:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Oimage,type,
% 0.20/0.47 image:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Oinsert,type,
% 0.20/0.47 insert:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.20/0.47 is_singleton:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_Ovimage,type,
% 0.20/0.47 vimage:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
% 0.20/0.47 set_fo6178422350223883121st_nat:
% 0.20/0.47 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
% 0.20/0.47 set_fo1817059534552279752at_rel:
% 0.20/0.47 !>[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 ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 0.20/0.47 set_ord_atLeast:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.20/0.47 set_or1337092689740270186AtMost:
% 0.20/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 0.20/0.47 set_or7035219750837199246ssThan:
% 0.20/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 0.20/0.47 set_ord_atMost:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 0.20/0.47 set_ord_greaterThan:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 0.20/0.47 set_or3652927894154168847AtMost:
% 0.20/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 0.20/0.47 set_or5935395276787703475ssThan:
% 0.20/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.20/0.47 set_ord_lessThan:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_String_OLiteral,type,
% 0.20/0.47 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_String_Ochar_OChar,type,
% 0.20/0.47 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_String_Ochar_Osize__char,type,
% 0.20/0.47 size_char: char > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
% 0.20/0.47 comm_s6883823935334413003f_char:
% 0.20/0.47 !>[A: $tType] : ( char > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
% 0.20/0.47 unique5772411509450598832har_of:
% 0.20/0.47 !>[A: $tType] : ( A > char ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sum__Type_OInl,type,
% 0.20/0.47 sum_Inl:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sum__Type_OInr,type,
% 0.20/0.47 sum_Inr:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sum__Type_OPlus,type,
% 0.20/0.47 sum_Plus:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 0.20/0.47 sum_case_sum:
% 0.20/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transfer_Obi__total,type,
% 0.20/0.47 bi_total:
% 0.20/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transitive__Closure_Ontrancl,type,
% 0.20/0.47 transitive_ntrancl:
% 0.20/0.47 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 0.20/0.47 transitive_rtrancl:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transitive__Closure_Otrancl,type,
% 0.20/0.47 transitive_trancl:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Transitive__Closure_Otranclp,type,
% 0.20/0.47 transitive_tranclp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Typedef_Otype__definition,type,
% 0.20/0.47 type_definition:
% 0.20/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Typerep_Otyperep_OTyperep,type,
% 0.20/0.47 typerep3: literal > ( list @ typerep ) > typerep ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
% 0.20/0.47 size_typerep: typerep > nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
% 0.20/0.47 typerep_typerep:
% 0.20/0.47 !>[A: $tType] : ( ( itself @ A ) > typerep ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.20/0.47 accp:
% 0.20/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Ofinite__psubset,type,
% 0.20/0.47 finite_psubset:
% 0.20/0.47 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Oless__than,type,
% 0.20/0.47 less_than: set @ ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.20/0.47 lex_prod:
% 0.20/0.47 !>[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 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Omax__ext,type,
% 0.20/0.47 max_ext:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Omeasure,type,
% 0.20/0.47 measure:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Omin__ext,type,
% 0.20/0.47 min_ext:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Omlex__prod,type,
% 0.20/0.47 mlex_prod:
% 0.20/0.47 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Opred__nat,type,
% 0.20/0.47 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Wellfounded_Owf,type,
% 0.20/0.47 wf:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Ochain__subset,type,
% 0.20/0.47 chain_subset:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Ochains,type,
% 0.20/0.47 chains:
% 0.20/0.47 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 0.20/0.47 pred_chain:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Opred__on_Omaxchain,type,
% 0.20/0.47 pred_maxchain:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Opred__on_Osuc,type,
% 0.20/0.47 pred_suc:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
% 0.20/0.47 pred_s596693808085603175closed:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
% 0.20/0.47 pred_s7749564232668923593losedp:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_fChoice,type,
% 0.20/0.47 fChoice:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member,type,
% 0.20/0.47 member:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_a,type,
% 0.20/0.47 a2: array @ a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_h,type,
% 0.20/0.47 h: heap_ext @ product_unit ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_h_H,type,
% 0.20/0.47 h2: heap_ext @ product_unit ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_n,type,
% 0.20/0.47 n: nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_r,type,
% 0.20/0.47 r: nat ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_thesis,type,
% 0.20/0.47 thesis: $o ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (7200)
% 0.20/0.47 thf(fact_0_assms,axiom,
% 0.20/0.47 heap_Time_effect @ nat @ ( array_len @ a @ a2 ) @ h @ h2 @ r @ n ).
% 0.20/0.47
% 0.20/0.47 % assms
% 0.20/0.47 thf(fact_1_that,axiom,
% 0.20/0.47 ( ( r
% 0.20/0.47 = ( array_length @ a @ h2 @ a2 ) )
% 0.20/0.47 => ( ( h2 = h )
% 0.20/0.47 => ( ( n
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 => thesis ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % that
% 0.20/0.47 thf(fact_2_effect__lengthI,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( heap @ A )
% 0.20/0.47 => ! [H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
% 0.20/0.47 ( ( H = H2 )
% 0.20/0.47 => ( ( R
% 0.20/0.47 = ( array_length @ A @ H2 @ A3 ) )
% 0.20/0.47 => ( ( N
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 => ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_lengthI
% 0.20/0.47 thf(fact_3_one__natural_Orsp,axiom,
% 0.20/0.47 ( ( one_one @ nat )
% 0.20/0.47 = ( one_one @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % one_natural.rsp
% 0.20/0.47 thf(fact_4_one__reorient,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( one @ A )
% 0.20/0.47 => ! [X: A] :
% 0.20/0.47 ( ( ( one_one @ A )
% 0.20/0.47 = X )
% 0.20/0.47 = ( X
% 0.20/0.47 = ( one_one @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % one_reorient
% 0.20/0.47 thf(fact_5_length__update,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( heap @ A )
% 0.20/0.47 & ( heap @ B ) )
% 0.20/0.47 => ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
% 0.20/0.47 ( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
% 0.20/0.47 = ( array_length @ A @ H2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % length_update
% 0.20/0.47 thf(fact_6_euclidean__size__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
% 0.20/0.47 = ( one_one @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % euclidean_size_1
% 0.20/0.47 thf(fact_7_fact__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fact_1
% 0.20/0.47 thf(fact_8_of__nat__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_1 @ A )
% 0.20/0.47 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_1
% 0.20/0.47 thf(fact_9_of__nat__1__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ( one_one @ A )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_1_eq_iff
% 0.20/0.47 thf(fact_10_of__nat__eq__1__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ A @ N )
% 0.20/0.47 = ( one_one @ A ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_eq_1_iff
% 0.20/0.47 thf(fact_11_dbl__dec__simps_I3_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( neg_numeral @ A )
% 0.20/0.47 => ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dbl_dec_simps(3)
% 0.20/0.47 thf(fact_12_round__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( archim2362893244070406136eiling @ A )
% 0.20/0.47 => ( ( archimedean_round @ A @ ( one_one @ A ) )
% 0.20/0.47 = ( one_one @ int ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % round_1
% 0.20/0.47 thf(fact_13_of__nat__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( M = N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_eq_iff
% 0.20/0.47 thf(fact_14_of__nat__fact,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
% 0.20/0.47 = ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_fact
% 0.20/0.47 thf(fact_15_euclidean__size__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = N ) ) ).
% 0.20/0.47
% 0.20/0.47 % euclidean_size_of_nat
% 0.20/0.47 thf(fact_16_round__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( archim2362893244070406136eiling @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % round_of_nat
% 0.20/0.47 thf(fact_17_one__integer_Orsp,axiom,
% 0.20/0.47 ( ( one_one @ int )
% 0.20/0.47 = ( one_one @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 % one_integer.rsp
% 0.20/0.47 thf(fact_18_update__swap__neqIndex,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( heap @ A )
% 0.20/0.47 => ! [I: nat,I2: nat,A3: array @ A,V: A,V2: A,H2: heap_ext @ product_unit] :
% 0.20/0.47 ( ( I != I2 )
% 0.20/0.47 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I2 @ V2 @ H2 ) )
% 0.20/0.47 = ( array_update @ A @ A3 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % update_swap_neqIndex
% 0.20/0.47 thf(fact_19_int__ops_I2_J,axiom,
% 0.20/0.47 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 0.20/0.47 = ( one_one @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_ops(2)
% 0.20/0.47 thf(fact_20_effect__deterministic_I3_J,axiom,
% 0.20/0.47 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 0.20/0.47 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 0.20/0.47 => ( N = N2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_deterministic(3)
% 0.20/0.47 thf(fact_21_effect__deterministic_I2_J,axiom,
% 0.20/0.47 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 0.20/0.47 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 0.20/0.47 => ( H = H3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_deterministic(2)
% 0.20/0.47 thf(fact_22_effect__deterministic_I1_J,axiom,
% 0.20/0.47 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 0.20/0.47 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 0.20/0.47 => ( A3 = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_deterministic(1)
% 0.20/0.47 thf(fact_23_effect__ifE,axiom,
% 0.20/0.47 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N )
% 0.20/0.47 => ( ( C2
% 0.20/0.47 => ~ ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
% 0.20/0.47 => ~ ( ~ C2
% 0.20/0.47 => ~ ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_ifE
% 0.20/0.47 thf(fact_24_effect__ifI,axiom,
% 0.20/0.47 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,E: heap_Time_Heap @ A] :
% 0.20/0.47 ( ( C2
% 0.20/0.47 => ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
% 0.20/0.47 => ( ( ~ C2
% 0.20/0.47 => ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) )
% 0.20/0.47 => ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_ifI
% 0.20/0.47 thf(fact_25_division__segment__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_of_nat
% 0.20/0.47 thf(fact_26_present__update,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( heap @ A )
% 0.20/0.47 & ( heap @ B ) )
% 0.20/0.47 => ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
% 0.20/0.47 ( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
% 0.20/0.47 = ( array_present @ A @ H2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % present_update
% 0.20/0.47 thf(fact_27_fact__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fact_0
% 0.20/0.47 thf(fact_28_effect__tapI,axiom,
% 0.20/0.47 ! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
% 0.20/0.47 ( ( H = H2 )
% 0.20/0.47 => ( ( R
% 0.20/0.47 = ( F @ H2 ) )
% 0.20/0.47 => ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ ( one_one @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_tapI
% 0.20/0.47 thf(fact_29_effect__tapE,axiom,
% 0.20/0.47 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ N )
% 0.20/0.47 => ~ ( ( H = H2 )
% 0.20/0.47 => ( ( R
% 0.20/0.47 = ( F @ H2 ) )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( one_one @ nat ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_tapE
% 0.20/0.47 thf(fact_30_update__swap__neq,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( heap @ A )
% 0.20/0.47 & ( heap @ B ) )
% 0.20/0.47 => ! [A3: array @ A,A4: array @ B,I: nat,V: A,I2: nat,V2: B,H2: heap_ext @ product_unit] :
% 0.20/0.47 ( ( array_noteq @ A @ B @ A3 @ A4 )
% 0.20/0.47 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A4 @ I2 @ V2 @ H2 ) )
% 0.20/0.47 = ( array_update @ B @ A4 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % update_swap_neq
% 0.20/0.47 thf(fact_31_unequal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( heap @ A )
% 0.20/0.47 => ! [A3: array @ A,A4: array @ A] :
% 0.20/0.47 ( ( A3 != A4 )
% 0.20/0.47 = ( array_noteq @ A @ A @ A3 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % unequal
% 0.20/0.47 thf(fact_32_division__segment__1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_1
% 0.20/0.47 thf(fact_33_round__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( archim2362893244070406136eiling @ A )
% 0.20/0.47 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ int ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % round_0
% 0.20/0.47 thf(fact_34_of__nat__eq__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [M: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( M
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_eq_0_iff
% 0.20/0.47 thf(fact_35_of__nat__0__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( ( zero_zero @ nat )
% 0.20/0.47 = N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_0_eq_iff
% 0.20/0.47 thf(fact_36_of__nat__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_1 @ A )
% 0.20/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_0
% 0.20/0.47 thf(fact_37_euclidean__size__eq__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid3725896446679973847miring @ A )
% 0.20/0.47 => ! [B2: A] :
% 0.20/0.47 ( ( ( euclid6346220572633701492n_size @ A @ B2 )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( B2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % euclidean_size_eq_0_iff
% 0.20/0.47 thf(fact_38_size__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid3725896446679973847miring @ A )
% 0.20/0.47 => ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_0
% 0.20/0.47 thf(fact_39_nat__int__comparison_I1_J,axiom,
% 0.20/0.47 ( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.20/0.47 = ( ^ [A5: nat,B3: nat] :
% 0.20/0.47 ( ( semiring_1_of_nat @ int @ A5 )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_int_comparison(1)
% 0.20/0.47 thf(fact_40_zero__natural_Orsp,axiom,
% 0.20/0.47 ( ( zero_zero @ nat )
% 0.20/0.47 = ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_natural.rsp
% 0.20/0.47 thf(fact_41_division__segment__not__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid3128863361964157862miring @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( euclid7384307370059645450egment @ A @ A3 )
% 0.20/0.47 != ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_not_0
% 0.20/0.47 thf(fact_42_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.20/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.20/0.47 = ( P @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_43_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A6: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 0.20/0.47 = A6 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_44_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( P @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collect @ A @ P )
% 0.20/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_45_ext,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( F @ X3 )
% 0.20/0.47 = ( G @ X3 ) )
% 0.20/0.47 => ( F = G ) ) ).
% 0.20/0.47
% 0.20/0.47 % ext
% 0.20/0.47 thf(fact_46_int__if,axiom,
% 0.20/0.47 ! [P: $o,A3: nat,B2: nat] :
% 0.20/0.47 ( ( P
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ A3 ) ) )
% 0.20/0.47 & ( ~ P
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_if
% 0.20/0.47 thf(fact_47_zero__reorient,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( zero @ A )
% 0.20/0.47 => ! [X: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = X )
% 0.20/0.47 = ( X
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_48_noteq__sym,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( ( heap @ A )
% 0.20/0.47 & ( heap @ B ) )
% 0.20/0.47 => ! [A3: array @ A,B2: array @ B] :
% 0.20/0.47 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 0.20/0.47 => ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % noteq_sym
% 0.20/0.47 thf(fact_49_noteq__irrefl,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( heap @ A )
% 0.20/0.47 => ! [R: array @ A] :
% 0.20/0.47 ~ ( array_noteq @ A @ A @ R @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % noteq_irrefl
% 0.20/0.47 thf(fact_50_fact__nonzero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ( semiring_char_0 @ A )
% 0.20/0.47 & ( semiri3467727345109120633visors @ A ) )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( semiring_char_0_fact @ A @ N )
% 0.20/0.47 != ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fact_nonzero
% 0.20/0.47 thf(fact_51_division__segment__nat__def,axiom,
% 0.20/0.47 ( ( euclid7384307370059645450egment @ nat )
% 0.20/0.47 = ( ^ [N3: nat] : ( one_one @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_nat_def
% 0.20/0.47 thf(fact_52_division__segment__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ! [A3: A,B2: A] :
% 0.20/0.47 ( ( ( euclid7384307370059645450egment @ A @ A3 )
% 0.20/0.47 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.20/0.47 => ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.20/0.47 = ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.20/0.47 => ( A3 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_eq_iff
% 0.20/0.47 thf(fact_53_zero__neq__one,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( zero_neq_one @ A )
% 0.20/0.47 => ( ( zero_zero @ A )
% 0.20/0.47 != ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_neq_one
% 0.20/0.47 thf(fact_54_dbl__inc__simps_I2_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( neg_numeral @ A )
% 0.20/0.47 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dbl_inc_simps(2)
% 0.20/0.47 thf(fact_55_effect__ureturnI,axiom,
% 0.20/0.47 ! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
% 0.20/0.47 ( ( H2 = H )
% 0.20/0.47 => ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ X @ ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_ureturnI
% 0.20/0.47 thf(fact_56_effect__ureturnE,axiom,
% 0.20/0.47 ! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ R @ N )
% 0.20/0.47 => ~ ( ( R = X )
% 0.20/0.47 => ( ( H = H2 )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_ureturnE
% 0.20/0.47 thf(fact_57_effect__assertE,axiom,
% 0.20/0.47 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N )
% 0.20/0.47 => ~ ( ( P @ X )
% 0.20/0.47 => ( ( R = X )
% 0.20/0.47 => ( ( H = H2 )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( one_one @ nat ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_assertE
% 0.20/0.47 thf(fact_58_effect__assertI,axiom,
% 0.20/0.47 ! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( P @ X )
% 0.20/0.47 => ( ( H = H2 )
% 0.20/0.47 => ( ( R = X )
% 0.20/0.47 => ( ( N
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 => ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_assertI
% 0.20/0.47 thf(fact_59_dbl__dec__simps_I2_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( neg_numeral @ A )
% 0.20/0.47 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % dbl_dec_simps(2)
% 0.20/0.47 thf(fact_60_division__segment__euclidean__size,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid5411537665997757685th_nat @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
% 0.20/0.47 = A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % division_segment_euclidean_size
% 0.20/0.47 thf(fact_61_euclidean__size__greater__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( euclid3725896446679973847miring @ A )
% 0.20/0.47 => ! [B2: A] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.20/0.47 = ( B2
% 0.20/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % euclidean_size_greater_0_iff
% 0.20/0.47 thf(fact_62_fact__Suc__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 0.20/0.47 = ( one_one @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % fact_Suc_0
% 0.20/0.47 thf(fact_63_of__nat__le__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord181362715937106298miring @ A )
% 0.20/0.47 => ! [M: nat] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( M
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_le_0_iff
% 0.20/0.47 thf(fact_64_effect__returnE,axiom,
% 0.20/0.47 ! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.20/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ R @ N )
% 0.20/0.47 => ~ ( ( R = X )
% 0.20/0.47 => ( ( H = H2 )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( one_one @ nat ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % effect_returnE
% 0.20/0.47 thf(fact_65_verit__minus__simplify_I4_J,axiom,
% 0.20/0.47 ! [B: $tType] :
% 0.20/0.47 ( ( group_add @ B )
% 0.20/0.47 => ! [B2: B] :
% 0.20/0.47 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 0.20/0.47 = B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % verit_minus_simplify(4)
% 0.20/0.47 thf(fact_66_neg__equal__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A3: A,B2: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A3 )
% 0.20/0.47 = ( uminus_uminus @ A @ B2 ) )
% 0.20/0.47 = ( A3 = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_iff_equal
% 0.20/0.47 thf(fact_67_add_Oinverse__inverse,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.20/0.47 = A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.inverse_inverse
% 0.20/0.47 thf(fact_68_Suc__le__mono,axiom,
% 0.20/0.47 ! [N: nat,M: nat] :
% 0.20/0.47 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.20/0.47 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.20/0.47
% 0.20/0.47 % Suc_le_mono
% 0.20/0.47 thf(fact_69_old_Onat_Oinject,axiom,
% 0.20/0.47 ! [Nat: nat,Nat2: nat] :
% 0.20/0.47 ( ( ( suc @ Nat )
% 0.20/0.47 = ( suc @ Nat2 ) )
% 0.20/0.47 = ( Nat = Nat2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.nat.inject
% 0.20/0.47 thf(fact_70_nat_Oinject,axiom,
% 0.20/0.47 ! [X22: nat,Y2: nat] :
% 0.20/0.47 ( ( ( suc @ X22 )
% 0.20/0.47 = ( suc @ Y2 ) )
% 0.20/0.47 = ( X22 = Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat.inject
% 0.20/0.47 thf(fact_71_mult__cancel2,axiom,
% 0.20/0.47 ! [M: nat,K: nat,N: nat] :
% 0.20/0.47 ( ( ( times_times @ nat @ M @ K )
% 0.20/0.47 = ( times_times @ nat @ N @ K ) )
% 0.20/0.47 = ( ( M = N )
% 0.20/0.47 | ( K
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_cancel2
% 0.20/0.47 thf(fact_72_mult__cancel1,axiom,
% 0.20/0.47 ! [K: nat,M: nat,N: nat] :
% 0.20/0.47 ( ( ( times_times @ nat @ K @ M )
% 0.20/0.47 = ( times_times @ nat @ K @ N ) )
% 0.20/0.47 = ( ( M = N )
% 0.20/0.47 | ( K
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_cancel1
% 0.20/0.47 thf(fact_73_mult__0__right,axiom,
% 0.20/0.47 ! [M: nat] :
% 0.20/0.47 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_0_right
% 0.20/0.47 thf(fact_74_mult__is__0,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( times_times @ nat @ M @ N )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( ( M
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 | ( N
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_is_0
% 0.20/0.47 thf(fact_75_le0,axiom,
% 0.20/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.20/0.47
% 0.20/0.47 % le0
% 0.20/0.47 thf(fact_76_bot__nat__0_Oextremum,axiom,
% 0.20/0.47 ! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum
% 0.20/0.47 thf(fact_77_nat__mult__eq__1__iff,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( times_times @ nat @ M @ N )
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 = ( ( M
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 & ( N
% 0.20/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_mult_eq_1_iff
% 0.20/0.47 thf(fact_78_nat__1__eq__mult__iff,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( one_one @ nat )
% 0.20/0.47 = ( times_times @ nat @ M @ N ) )
% 0.20/0.47 = ( ( M
% 0.20/0.47 = ( one_one @ nat ) )
% 0.20/0.47 & ( N
% 0.20/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_1_eq_mult_iff
% 0.20/0.47 thf(fact_79_le__zero__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni5634975068530333245id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_zero_eq
% 0.20/0.47 thf(fact_80_not__gr__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni5634975068530333245id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_gr_zero
% 0.20/0.47 thf(fact_81_mult__zero__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( mult_zero @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_zero_left
% 0.20/0.47 thf(fact_82_mult__zero__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( mult_zero @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_zero_right
% 0.20/0.47 thf(fact_83_mult__eq__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiri3467727345109120633visors @ A )
% 0.20/0.47 => ! [A3: A,B2: A] :
% 0.20/0.47 ( ( ( times_times @ A @ A3 @ B2 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( ( A3
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 | ( B2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_eq_0_iff
% 0.20/0.47 thf(fact_84_mult__cancel__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiri6575147826004484403cancel @ A )
% 0.20/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.20/0.47 ( ( ( times_times @ A @ C2 @ A3 )
% 0.20/0.47 = ( times_times @ A @ C2 @ B2 ) )
% 0.20/0.47 = ( ( C2
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 | ( A3 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_cancel_left
% 0.20/0.47 thf(fact_85_mult__cancel__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiri6575147826004484403cancel @ A )
% 0.20/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.20/0.47 ( ( ( times_times @ A @ A3 @ C2 )
% 0.20/0.47 = ( times_times @ A @ B2 @ C2 ) )
% 0.20/0.47 = ( ( C2
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 | ( A3 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mult_cancel_right
% 0.20/0.47 thf(fact_86_neg__le__iff__le,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [B2: A,A3: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_le_iff_le
% 0.20/0.47 thf(fact_87_neg__equal__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A3 )
% 0.20/0.47 = A3 )
% 0.20/0.47 = ( A3
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_zero
% 0.20/0.47 thf(fact_88_equal__neg__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( A3
% 0.20/0.47 = ( uminus_uminus @ A @ A3 ) )
% 0.20/0.47 = ( A3
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equal_neg_zero
% 0.20/0.47 thf(fact_89_neg__equal__0__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A3 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( A3
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_0_iff_equal
% 0.20/0.47 thf(fact_90_neg__0__equal__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A3: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.30/0.47 = ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ( zero_zero @ A )
% 0.30/0.47 = A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_0_equal_iff_equal
% 0.30/0.47 thf(fact_91_add_Oinverse__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.30/0.47 = ( zero_zero @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % add.inverse_neutral
% 0.30/0.47 thf(fact_92_mult__1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( monoid_mult @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.47 = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_1
% 0.30/0.47 thf(fact_93_mult_Oright__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( monoid_mult @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.47 = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.right_neutral
% 0.30/0.47 thf(fact_94_neg__less__iff__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [B2: A,A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_less_iff_less
% 0.30/0.47 thf(fact_95_mult__minus__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.47 = ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_minus_left
% 0.30/0.47 thf(fact_96_minus__mult__minus,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( times_times @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % minus_mult_minus
% 0.30/0.47 thf(fact_97_mult__minus__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.47 = ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_minus_right
% 0.30/0.47 thf(fact_98_of__nat__le__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord181362715937106298miring @ A )
% 0.30/0.47 => ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_le_iff
% 0.30/0.47 thf(fact_99_one__le__mult__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
% 0.30/0.47 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.30/0.47 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_le_mult_iff
% 0.30/0.47 thf(fact_100_one__eq__mult__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.30/0.47 = ( times_times @ nat @ M @ N ) )
% 0.30/0.47 = ( ( M
% 0.30/0.47 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.47 & ( N
% 0.30/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_eq_mult_iff
% 0.30/0.47 thf(fact_101_mult__eq__1__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ( times_times @ nat @ M @ N )
% 0.30/0.47 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.47 = ( ( M
% 0.30/0.47 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.47 & ( N
% 0.30/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_eq_1_iff
% 0.30/0.47 thf(fact_102_of__nat__mult,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semiring_1 @ A )
% 0.30/0.47 => ! [M: nat,N: nat] :
% 0.30/0.47 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
% 0.30/0.47 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_mult
% 0.30/0.47 thf(fact_103_Suc__less__eq,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_less_eq
% 0.30/0.47 thf(fact_104_Suc__mono,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ M @ N )
% 0.30/0.47 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Suc_mono
% 0.30/0.47 thf(fact_105_lessI,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % lessI
% 0.30/0.47 thf(fact_106_mult__le__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel2
% 0.30/0.47 thf(fact_107_nat__0__less__mult__iff,axiom,
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_0_less_mult_iff
% 0.30/0.47 thf(fact_108_mult__less__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel2
% 0.30/0.47 thf(fact_109_less__nat__zero__code,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nat_zero_code
% 0.30/0.47 thf(fact_110_neq0__conv,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( N
% 0.30/0.47 != ( zero_zero @ nat ) )
% 0.30/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq0_conv
% 0.30/0.47 thf(fact_111_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.47 ! [A3: nat] :
% 0.30/0.47 ( ( A3
% 0.30/0.47 != ( zero_zero @ nat ) )
% 0.30/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.not_eq_extremum
% 0.30/0.47 thf(fact_112_neg__less__eq__nonneg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.30/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_less_eq_nonneg
% 0.30/0.47 thf(fact_113_less__eq__neg__nonpos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_neg_nonpos
% 0.30/0.47 thf(fact_114_neg__le__0__iff__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_le_0_iff_le
% 0.30/0.47 thf(fact_115_neg__0__le__iff__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_0_le_iff_le
% 0.30/0.47 thf(fact_116_mult__cancel__right2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.47 => ! [A3: A,C2: A] :
% 0.30/0.47 ( ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.47 = C2 )
% 0.30/0.47 = ( ( C2
% 0.30/0.47 = ( zero_zero @ A ) )
% 0.30/0.47 | ( A3
% 0.30/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_cancel_right2
% 0.30/0.47 thf(fact_117_mult__cancel__right1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.47 => ! [C2: A,B2: A] :
% 0.30/0.47 ( ( C2
% 0.30/0.47 = ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( ( C2
% 0.30/0.47 = ( zero_zero @ A ) )
% 0.30/0.47 | ( B2
% 0.30/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_cancel_right1
% 0.30/0.47 thf(fact_118_mult__cancel__left2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.47 => ! [C2: A,A3: A] :
% 0.30/0.47 ( ( ( times_times @ A @ C2 @ A3 )
% 0.30/0.47 = C2 )
% 0.30/0.47 = ( ( C2
% 0.30/0.47 = ( zero_zero @ A ) )
% 0.30/0.47 | ( A3
% 0.30/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_cancel_left2
% 0.30/0.47 thf(fact_119_mult__cancel__left1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.47 => ! [C2: A,B2: A] :
% 0.30/0.47 ( ( C2
% 0.30/0.47 = ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ( C2
% 0.30/0.47 = ( zero_zero @ A ) )
% 0.30/0.47 | ( B2
% 0.30/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_cancel_left1
% 0.30/0.47 thf(fact_120_neg__less__0__iff__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_less_0_iff_less
% 0.30/0.47 thf(fact_121_neg__0__less__iff__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_0_less_iff_less
% 0.30/0.47 thf(fact_122_neg__less__pos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.30/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neg_less_pos
% 0.30/0.47 thf(fact_123_less__neg__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.47 => ! [A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.47 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_neg_neg
% 0.30/0.47 thf(fact_124_mult__minus1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_1 @ A )
% 0.30/0.47 => ! [Z3: A] :
% 0.30/0.47 ( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z3 )
% 0.30/0.47 = ( uminus_uminus @ A @ Z3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_minus1
% 0.30/0.47 thf(fact_125_mult__minus1__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring_1 @ A )
% 0.30/0.47 => ! [Z3: A] :
% 0.30/0.47 ( ( times_times @ A @ Z3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.47 = ( uminus_uminus @ A @ Z3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_minus1_right
% 0.30/0.47 thf(fact_126_of__nat__less__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord181362715937106298miring @ A )
% 0.30/0.47 => ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_less_iff
% 0.30/0.47 thf(fact_127_zero__less__Suc,axiom,
% 0.30/0.47 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_Suc
% 0.30/0.47 thf(fact_128_less__Suc0,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.47 = ( N
% 0.30/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_Suc0
% 0.30/0.47 thf(fact_129_less__one,axiom,
% 0.30/0.47 ! [N: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.30/0.47 = ( N
% 0.30/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_one
% 0.30/0.47 thf(fact_130_dbl__inc__simps_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( neg_numeral @ A )
% 0.30/0.47 => ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.47 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dbl_inc_simps(4)
% 0.30/0.47 thf(fact_131_fact__Suc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semiring_char_0 @ A )
% 0.30/0.47 => ! [N: nat] :
% 0.30/0.47 ( ( semiring_char_0_fact @ A @ ( suc @ N ) )
% 0.30/0.47 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fact_Suc
% 0.30/0.47 thf(fact_132_of__nat__0__less__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord181362715937106298miring @ A )
% 0.30/0.47 => ! [N: nat] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_nat_0_less_iff
% 0.30/0.47 thf(fact_133_zero__integer_Orsp,axiom,
% 0.30/0.47 ( ( zero_zero @ int )
% 0.30/0.47 = ( zero_zero @ int ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_integer.rsp
% 0.30/0.47 thf(fact_134_one__less__mult,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.47 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.30/0.47 => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_less_mult
% 0.30/0.47 thf(fact_135_n__less__m__mult__n,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.47 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.30/0.47 => ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % n_less_m_mult_n
% 0.30/0.47 thf(fact_136_n__less__n__mult__m,axiom,
% 0.30/0.47 ! [N: nat,M: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.47 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.30/0.47 => ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % n_less_n_mult_m
% 0.30/0.47 thf(fact_137_ex__least__nat__less,axiom,
% 0.30/0.47 ! [P: nat > $o,N: nat] :
% 0.30/0.47 ( ( P @ N )
% 0.30/0.47 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.47 => ? [K2: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ K2 @ N )
% 0.30/0.47 & ! [I3: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ I3 @ K2 )
% 0.30/0.47 => ~ ( P @ I3 ) )
% 0.30/0.47 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ex_least_nat_less
% 0.30/0.47 thf(fact_138_mult__le__cancel__iff1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A )
% 0.30/0.47 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_iff1
% 0.30/0.47 thf(fact_139_mult__le__cancel__iff2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A )
% 0.30/0.47 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y3 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_iff2
% 0.30/0.47 thf(fact_140_division__segment__int__def,axiom,
% 0.30/0.47 ( ( euclid7384307370059645450egment @ int )
% 0.30/0.47 = ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % division_segment_int_def
% 0.30/0.47 thf(fact_141_verit__la__disequality,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 | ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.47 | ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % verit_la_disequality
% 0.30/0.47 thf(fact_142_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord2810124833399127020strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.47 thf(fact_143_mult__less__cancel__right__disj,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 & ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.47 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_right_disj
% 0.30/0.47 thf(fact_144_mult__strict__right__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_right_mono
% 0.30/0.47 thf(fact_145_mult__strict__right__mono__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_right_mono_neg
% 0.30/0.47 thf(fact_146_mult__less__cancel__left__disj,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 & ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.47 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_left_disj
% 0.30/0.47 thf(fact_147_mult__strict__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_left_mono
% 0.30/0.47 thf(fact_148_mult__less__le__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_le_imp_less
% 0.30/0.47 thf(fact_149_mult__le__less__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_less_imp_less
% 0.30/0.47 thf(fact_150_mult__strict__left__mono__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_left_mono_neg
% 0.30/0.47 thf(fact_151_mult__less__cancel__left__pos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_left_pos
% 0.30/0.47 thf(fact_152_mult__less__cancel__left__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_left_neg
% 0.30/0.47 thf(fact_153_mult__right__le__imp__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_right_le_imp_le
% 0.30/0.47 thf(fact_154_zero__less__mult__pos2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [B2: A,A3: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_mult_pos2
% 0.30/0.47 thf(fact_155_mult__left__le__imp__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_left_le_imp_le
% 0.30/0.47 thf(fact_156_mult__le__cancel__left__pos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_left_pos
% 0.30/0.47 thf(fact_157_mult__le__cancel__left__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_left_neg
% 0.30/0.47 thf(fact_158_zero__less__mult__pos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_mult_pos
% 0.30/0.47 thf(fact_159_mult__less__cancel__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.47 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_right
% 0.30/0.47 thf(fact_160_mult__strict__mono_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_mono'
% 0.30/0.47 thf(fact_161_mult__right__less__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semiring @ A )
% 0.30/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_right_less_imp_less
% 0.30/0.47 thf(fact_162_mult__less__cancel__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.47 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel_left
% 0.30/0.47 thf(fact_163_mult__strict__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_strict_mono
% 0.30/0.47 thf(fact_164_mult__left__less__imp__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_semiring @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_left_less_imp_less
% 0.30/0.47 thf(fact_165_mult__le__cancel__right,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.47 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_right
% 0.30/0.47 thf(fact_166_mult__le__cancel__left,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.47 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel_left
% 0.30/0.47 thf(fact_167_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordere2520102378445227354miring @ A )
% 0.30/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.47 thf(fact_168_zero__less__mult__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord4710134922213307826strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.30/0.47 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_mult_iff
% 0.30/0.47 thf(fact_169_mult__pos__neg2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_pos_neg2
% 0.30/0.47 thf(fact_170_mult__pos__pos,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_pos_pos
% 0.30/0.47 thf(fact_171_mult__pos__neg,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord8928482502909563296strict @ A )
% 0.30/0.47 => ! [A3: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_pos_neg
% 0.30/0.48 thf(fact_172_mult__neg__pos,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord8928482502909563296strict @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.48 => ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_neg_pos
% 0.30/0.48 thf(fact_173_zero__le__mult__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord4710134922213307826strict @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.30/0.48 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_le_mult_iff
% 0.30/0.48 thf(fact_174_mult__nonneg__nonpos2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring_0 @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_nonneg_nonpos2
% 0.30/0.48 thf(fact_175_mult__less__0__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord4710134922213307826strict @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.30/0.48 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_0_iff
% 0.30/0.48 thf(fact_176_mult__nonpos__nonneg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring_0 @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_nonpos_nonneg
% 0.30/0.48 thf(fact_177_mult__nonneg__nonpos,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring_0 @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_nonneg_nonpos
% 0.30/0.48 thf(fact_178_mult__nonneg__nonneg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring_0 @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_nonneg_nonneg
% 0.30/0.48 thf(fact_179_split__mult__neg__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring_0 @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.30/0.48 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % split_mult_neg_le
% 0.30/0.48 thf(fact_180_mult__le__0__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord4710134922213307826strict @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.30/0.48 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_0_iff
% 0.30/0.48 thf(fact_181_not__square__less__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_ring @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_square_less_zero
% 0.30/0.48 thf(fact_182_mult__neg__neg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord4710134922213307826strict @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_neg_neg
% 0.30/0.48 thf(fact_183_mult__right__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_right_mono
% 0.30/0.48 thf(fact_184_mult__right__mono__neg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ring @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_right_mono_neg
% 0.30/0.48 thf(fact_185_mult__left__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_left_mono
% 0.30/0.48 thf(fact_186_mult__nonpos__nonpos,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_nonpos_nonpos
% 0.30/0.48 thf(fact_187_mult__left__mono__neg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ring @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_left_mono_neg
% 0.30/0.48 thf(fact_188_split__mult__pos__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.30/0.48 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % split_mult_pos_le
% 0.30/0.48 thf(fact_189_zero__le__square,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_ring @ A )
% 0.30/0.48 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_le_square
% 0.30/0.48 thf(fact_190_mult__mono_H,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_mono'
% 0.30/0.48 thf(fact_191_mult__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_semiring @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_mono
% 0.30/0.48 thf(fact_192_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_negate_coefficient(2)
% 0.30/0.48 thf(fact_193_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3 = B2 )
% 0.30/0.48 => ( ( uminus_uminus @ A @ A3 )
% 0.30/0.48 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_negate_coefficient(3)
% 0.30/0.48 thf(fact_194_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_comp_simplify1(1)
% 0.30/0.48 thf(fact_195_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_comp_simplify1(2)
% 0.30/0.48 thf(fact_196_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48 ! [B: $tType] :
% 0.30/0.48 ( ( linorder @ B )
% 0.30/0.48 => ! [B4: B,A4: B] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
% 0.30/0.48 = ( ord_less @ B @ A4 @ B4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_comp_simplify1(3)
% 0.30/0.48 thf(fact_197_square__eq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( idom @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( times_times @ A @ A3 @ A3 )
% 0.30/0.48 = ( times_times @ A @ B2 @ B2 ) )
% 0.30/0.48 = ( ( A3 = B2 )
% 0.30/0.48 | ( A3
% 0.30/0.48 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % square_eq_iff
% 0.30/0.48 thf(fact_198_minus__mult__commute,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.48 = ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minus_mult_commute
% 0.30/0.48 thf(fact_199_linorder__neqE__linordered__idom,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( X != Y3 )
% 0.30/0.48 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE_linordered_idom
% 0.30/0.48 thf(fact_200_infinite__descent__measure,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,V3: A > nat,X: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ~ ( P @ X3 )
% 0.30/0.48 => ? [Y4: A] :
% 0.30/0.48 ( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
% 0.30/0.48 & ~ ( P @ Y4 ) ) )
% 0.30/0.48 => ( P @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent_measure
% 0.30/0.48 thf(fact_201_transitive__stepwise__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 0.30/0.48 => ( ! [X3: nat,Y5: nat,Z4: nat] :
% 0.30/0.48 ( ( R2 @ X3 @ Y5 )
% 0.30/0.48 => ( ( R2 @ Y5 @ Z4 )
% 0.30/0.48 => ( R2 @ X3 @ Z4 ) ) )
% 0.30/0.48 => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
% 0.30/0.48 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % transitive_stepwise_le
% 0.30/0.48 thf(fact_202_less__mono__imp__le__mono,axiom,
% 0.30/0.48 ! [F: nat > nat,I: nat,J: nat] :
% 0.30/0.48 ( ! [I4: nat,J2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I4 @ J2 )
% 0.30/0.48 => ( ord_less @ nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_mono_imp_le_mono
% 0.30/0.48 thf(fact_203_Suc__mult__less__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
% 0.30/0.48 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_mult_less_cancel1
% 0.30/0.48 thf(fact_204_not__less__less__Suc__eq,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.48 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.30/0.48 = ( N = M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_less_Suc_eq
% 0.30/0.48 thf(fact_205_nat__induct__at__least,axiom,
% 0.30/0.48 ! [M: nat,N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ( P @ M )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N4 )
% 0.30/0.48 => ( ( P @ N4 )
% 0.30/0.48 => ( P @ ( suc @ N4 ) ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_induct_at_least
% 0.30/0.48 thf(fact_206_measure__induct__rule,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( wellorder @ B )
% 0.30/0.48 => ! [F: A > B,P: A > $o,A3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ! [Y4: A] :
% 0.30/0.48 ( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
% 0.30/0.48 => ( P @ Y4 ) )
% 0.30/0.48 => ( P @ X3 ) )
% 0.30/0.48 => ( P @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % measure_induct_rule
% 0.30/0.48 thf(fact_207_le__neq__implies__less,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ( M != N )
% 0.30/0.48 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_implies_less
% 0.30/0.48 thf(fact_208_Suc__mult__le__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
% 0.30/0.48 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_mult_le_cancel1
% 0.30/0.48 thf(fact_209_strict__inc__induct,axiom,
% 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J )
% 0.30/0.48 => ( ! [I4: nat] :
% 0.30/0.48 ( ( J
% 0.30/0.48 = ( suc @ I4 ) )
% 0.30/0.48 => ( P @ I4 ) )
% 0.30/0.48 => ( ! [I4: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I4 @ J )
% 0.30/0.48 => ( ( P @ ( suc @ I4 ) )
% 0.30/0.48 => ( P @ I4 ) ) )
% 0.30/0.48 => ( P @ I ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_inc_induct
% 0.30/0.48 thf(fact_210_linorder__neqE__nat,axiom,
% 0.30/0.48 ! [X: nat,Y3: nat] :
% 0.30/0.48 ( ( X != Y3 )
% 0.30/0.48 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 0.30/0.48 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE_nat
% 0.30/0.48 thf(fact_211_less__or__eq__imp__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 | ( M = N ) )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_or_eq_imp_le
% 0.30/0.48 thf(fact_212_le__eq__less__or__eq,axiom,
% 0.30/0.48 ( ( ord_less_eq @ nat )
% 0.30/0.48 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M2 @ N3 )
% 0.30/0.48 | ( M2 = N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_eq_less_or_eq
% 0.30/0.48 thf(fact_213_infinite__descent,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( ~ ( P @ N4 )
% 0.30/0.48 => ? [M3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M3 @ N4 )
% 0.30/0.48 & ~ ( P @ M3 ) ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent
% 0.30/0.48 thf(fact_214_Suc__mult__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ( times_times @ nat @ ( suc @ K ) @ M )
% 0.30/0.48 = ( times_times @ nat @ ( suc @ K ) @ N ) )
% 0.30/0.48 = ( M = N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_mult_cancel1
% 0.30/0.48 thf(fact_215_nat__less__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( ! [M3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M3 @ N4 )
% 0.30/0.48 => ( P @ M3 ) )
% 0.30/0.48 => ( P @ N4 ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_induct
% 0.30/0.48 thf(fact_216_less__irrefl__nat,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less @ nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_irrefl_nat
% 0.30/0.48 thf(fact_217_less__imp__le__nat,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le_nat
% 0.30/0.48 thf(fact_218_less__Suc__induct,axiom,
% 0.30/0.48 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J )
% 0.30/0.48 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
% 0.30/0.48 => ( ! [I4: nat,J2: nat,K2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I4 @ J2 )
% 0.30/0.48 => ( ( ord_less @ nat @ J2 @ K2 )
% 0.30/0.48 => ( ( P @ I4 @ J2 )
% 0.30/0.48 => ( ( P @ J2 @ K2 )
% 0.30/0.48 => ( P @ I4 @ K2 ) ) ) ) )
% 0.30/0.48 => ( P @ I @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_induct
% 0.30/0.48 thf(fact_219_le__imp__less__Suc,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_imp_less_Suc
% 0.30/0.48 thf(fact_220_full__nat__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( ! [M3: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N4 )
% 0.30/0.48 => ( P @ M3 ) )
% 0.30/0.48 => ( P @ N4 ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % full_nat_induct
% 0.30/0.48 thf(fact_221_not__less__eq__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.30/0.48 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_eq_eq
% 0.30/0.48 thf(fact_222_measure__induct,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( wellorder @ B )
% 0.30/0.48 => ! [F: A > B,P: A > $o,A3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ! [Y4: A] :
% 0.30/0.48 ( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
% 0.30/0.48 => ( P @ Y4 ) )
% 0.30/0.48 => ( P @ X3 ) )
% 0.30/0.48 => ( P @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % measure_induct
% 0.30/0.48 thf(fact_223_less__trans__Suc,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less @ nat @ J @ K )
% 0.30/0.48 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_trans_Suc
% 0.30/0.48 thf(fact_224_less__not__refl3,axiom,
% 0.30/0.48 ! [S: nat,T2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ S @ T2 )
% 0.30/0.48 => ( S != T2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl3
% 0.30/0.48 thf(fact_225_less__not__refl2,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ N @ M )
% 0.30/0.48 => ( M != N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl2
% 0.30/0.48 thf(fact_226_less__eq__Suc__le,axiom,
% 0.30/0.48 ( ( ord_less @ nat )
% 0.30/0.48 = ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_eq_Suc_le
% 0.30/0.48 thf(fact_227_less__Suc__eq__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq_le
% 0.30/0.48 thf(fact_228_le__less__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.30/0.48 = ( N = M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_Suc_eq
% 0.30/0.48 thf(fact_229_Suc__n__not__le__n,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_n_not_le_n
% 0.30/0.48 thf(fact_230_less__not__refl,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less @ nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl
% 0.30/0.48 thf(fact_231_Suc__less__SucD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.48 => ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_less_SucD
% 0.30/0.48 thf(fact_232_less__antisym,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.48 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_antisym
% 0.30/0.48 thf(fact_233_Suc__less__eq2,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.48 = ( ? [M4: nat] :
% 0.30/0.48 ( ( M
% 0.30/0.48 = ( suc @ M4 ) )
% 0.30/0.48 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_less_eq2
% 0.30/0.48 thf(fact_234_Suc__le__lessD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_lessD
% 0.30/0.48 thf(fact_235_All__less__Suc,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ! [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 0.30/0.48 => ( P @ I5 ) ) )
% 0.30/0.48 = ( ( P @ N )
% 0.30/0.48 & ! [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ N )
% 0.30/0.48 => ( P @ I5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % All_less_Suc
% 0.30/0.48 thf(fact_236_not__less__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.30/0.48 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_eq
% 0.30/0.48 thf(fact_237_nat__neq__iff,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( M != N )
% 0.30/0.48 = ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_neq_iff
% 0.30/0.48 thf(fact_238_nat__less__le,axiom,
% 0.30/0.48 ( ( ord_less @ nat )
% 0.30/0.48 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 0.30/0.48 & ( M2 != N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_le
% 0.30/0.48 thf(fact_239_n__not__Suc__n,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 != ( suc @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % n_not_Suc_n
% 0.30/0.48 thf(fact_240_less__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 | ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq
% 0.30/0.48 thf(fact_241_Ex__less__Suc,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ? [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 0.30/0.48 & ( P @ I5 ) ) )
% 0.30/0.48 = ( ( P @ N )
% 0.30/0.48 | ? [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ N )
% 0.30/0.48 & ( P @ I5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Ex_less_Suc
% 0.30/0.48 thf(fact_242_inc__induct,axiom,
% 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ( P @ J )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ N4 )
% 0.30/0.48 => ( ( ord_less @ nat @ N4 @ J )
% 0.30/0.48 => ( ( P @ ( suc @ N4 ) )
% 0.30/0.48 => ( P @ N4 ) ) ) )
% 0.30/0.48 => ( P @ I ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inc_induct
% 0.30/0.48 thf(fact_243_dec__induct,axiom,
% 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ( P @ I )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ N4 )
% 0.30/0.48 => ( ( ord_less @ nat @ N4 @ J )
% 0.30/0.48 => ( ( P @ N4 )
% 0.30/0.48 => ( P @ ( suc @ N4 ) ) ) ) )
% 0.30/0.48 => ( P @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dec_induct
% 0.30/0.48 thf(fact_244_Suc__inject,axiom,
% 0.30/0.48 ! [X: nat,Y3: nat] :
% 0.30/0.48 ( ( ( suc @ X )
% 0.30/0.48 = ( suc @ Y3 ) )
% 0.30/0.48 => ( X = Y3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_inject
% 0.30/0.48 thf(fact_245_less__SucI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_SucI
% 0.30/0.48 thf(fact_246_less__SucE,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 => ( ~ ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_SucE
% 0.30/0.48 thf(fact_247_le__Suc__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 | ( M
% 0.30/0.48 = ( suc @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_Suc_eq
% 0.30/0.48 thf(fact_248_Suc__lessI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ( ( suc @ M )
% 0.30/0.48 != N )
% 0.30/0.48 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessI
% 0.30/0.48 thf(fact_249_Suc__lessE,axiom,
% 0.30/0.48 ! [I: nat,K: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( suc @ I ) @ K )
% 0.30/0.48 => ~ ! [J2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J2 )
% 0.30/0.48 => ( K
% 0.30/0.48 != ( suc @ J2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessE
% 0.30/0.48 thf(fact_250_Suc__lessD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_lessD
% 0.30/0.48 thf(fact_251_Suc__le__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.30/0.48 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_eq
% 0.30/0.48 thf(fact_252_Suc__le__D,axiom,
% 0.30/0.48 ! [N: nat,M5: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 0.30/0.48 => ? [M6: nat] :
% 0.30/0.48 ( M5
% 0.30/0.48 = ( suc @ M6 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_le_D
% 0.30/0.48 thf(fact_253_le__SucI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_SucI
% 0.30/0.48 thf(fact_254_le__SucE,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( M
% 0.30/0.48 = ( suc @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_SucE
% 0.30/0.48 thf(fact_255_Suc__leI,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_leI
% 0.30/0.48 thf(fact_256_Suc__leD,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_leD
% 0.30/0.48 thf(fact_257_Nat_OlessE,axiom,
% 0.30/0.48 ! [I: nat,K: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ K )
% 0.30/0.48 => ( ( K
% 0.30/0.48 != ( suc @ I ) )
% 0.30/0.48 => ~ ! [J2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J2 )
% 0.30/0.48 => ( K
% 0.30/0.48 != ( suc @ J2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Nat.lessE
% 0.30/0.48 thf(fact_258_square__eq__1__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ( ( ( times_times @ A @ X @ X )
% 0.30/0.48 = ( one_one @ A ) )
% 0.30/0.48 = ( ( X
% 0.30/0.48 = ( one_one @ A ) )
% 0.30/0.48 | ( X
% 0.30/0.48 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % square_eq_1_iff
% 0.30/0.48 thf(fact_259_minus__less__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.48 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minus_less_iff
% 0.30/0.48 thf(fact_260_less__minus__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.48 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_minus_iff
% 0.30/0.48 thf(fact_261_mult_Oleft__commute,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ab_semigroup_mult @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
% 0.30/0.48 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult.left_commute
% 0.30/0.48 thf(fact_262_le__imp__neg__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_imp_neg_le
% 0.30/0.48 thf(fact_263_minus__le__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.48 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minus_le_iff
% 0.30/0.48 thf(fact_264_le__minus__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ordered_ab_group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.48 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_minus_iff
% 0.30/0.48 thf(fact_265_mult_Ocommute,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ab_semigroup_mult @ A )
% 0.30/0.48 => ( ( times_times @ A )
% 0.30/0.48 = ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult.commute
% 0.30/0.48 thf(fact_266_less__1__mult,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [M: A,N: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( one_one @ A ) @ M )
% 0.30/0.48 => ( ( ord_less @ A @ ( one_one @ A ) @ N )
% 0.30/0.48 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_1_mult
% 0.30/0.48 thf(fact_267_minus__equation__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( uminus_uminus @ A @ A3 )
% 0.30/0.48 = B2 )
% 0.30/0.48 = ( ( uminus_uminus @ A @ B2 )
% 0.30/0.48 = A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minus_equation_iff
% 0.30/0.48 thf(fact_268_equation__minus__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( group_add @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.48 = ( B2
% 0.30/0.48 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % equation_minus_iff
% 0.30/0.48 thf(fact_269_lift__Suc__mono__less__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [F: nat > A,N: nat,M: nat] :
% 0.30/0.48 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.48 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.48 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_less_iff
% 0.30/0.48 thf(fact_270_mult_Oassoc,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semigroup_mult @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.48 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult.assoc
% 0.30/0.48 thf(fact_271_lift__Suc__antimono__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [F: nat > A,N: nat,N2: nat] :
% 0.30/0.48 ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ N @ N2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_antimono_le
% 0.30/0.48 thf(fact_272_lift__Suc__mono__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [F: nat > A,N: nat,N2: nat] :
% 0.30/0.48 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.48 => ( ( ord_less @ nat @ N @ N2 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_less
% 0.30/0.48 thf(fact_273_lift__Suc__mono__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [F: nat > A,N: nat,N2: nat] :
% 0.30/0.48 ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ N @ N2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_Suc_mono_le
% 0.30/0.48 thf(fact_274_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ab_semigroup_mult @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.48 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.48 thf(fact_275_fold__if__return,axiom,
% 0.30/0.48 ! [A: $tType,B2: $o,C2: A,D2: A] :
% 0.30/0.48 ( ( B2
% 0.30/0.48 => ( ( heap_Time_return @ A @ C2 )
% 0.30/0.48 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) )
% 0.30/0.48 & ( ~ B2
% 0.30/0.48 => ( ( heap_Time_return @ A @ D2 )
% 0.30/0.48 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fold_if_return
% 0.30/0.48 thf(fact_276_mult__less__cancel__right2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 0.30/0.48 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_cancel_right2
% 0.30/0.48 thf(fact_277_mult__less__cancel__right1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 0.30/0.48 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_cancel_right1
% 0.30/0.48 thf(fact_278_mult__less__cancel__left2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,A3: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 0.30/0.48 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_cancel_left2
% 0.30/0.48 thf(fact_279_mult__less__cancel__left1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 0.30/0.48 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_cancel_left1
% 0.30/0.48 thf(fact_280_mult__le__cancel__right2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 0.30/0.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 0.30/0.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_cancel_right2
% 0.30/0.48 thf(fact_281_mult__le__cancel__right1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 0.30/0.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_cancel_right1
% 0.30/0.48 thf(fact_282_mult__le__cancel__left2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,A3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 0.30/0.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 0.30/0.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_cancel_left2
% 0.30/0.48 thf(fact_283_mult__le__cancel__left1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [C2: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 0.30/0.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_cancel_left1
% 0.30/0.48 thf(fact_284_mult__less__iff1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 0.30/0.48 => ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
% 0.30/0.48 = ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_iff1
% 0.30/0.48 thf(fact_285_less__minus__one__simps_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_minus_one_simps(4)
% 0.30/0.48 thf(fact_286_less__minus__one__simps_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_minus_one_simps(2)
% 0.30/0.48 thf(fact_287_le__minus__one__simps_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_minus_one_simps(4)
% 0.30/0.48 thf(fact_288_le__minus__one__simps_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_minus_one_simps(2)
% 0.30/0.48 thf(fact_289_less__Suc__eq__0__disj,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.30/0.48 = ( ( M
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 | ? [J3: nat] :
% 0.30/0.48 ( ( M
% 0.30/0.48 = ( suc @ J3 ) )
% 0.30/0.48 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_Suc_eq_0_disj
% 0.30/0.48 thf(fact_290_gr0__implies__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.48 => ? [M6: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M6 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0_implies_Suc
% 0.30/0.48 thf(fact_291_All__less__Suc2,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ! [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 0.30/0.48 => ( P @ I5 ) ) )
% 0.30/0.48 = ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 & ! [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ N )
% 0.30/0.48 => ( P @ ( suc @ I5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % All_less_Suc2
% 0.30/0.48 thf(fact_292_gr0__conv__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.48 = ( ? [M2: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0_conv_Suc
% 0.30/0.48 thf(fact_293_Ex__less__Suc2,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ? [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 0.30/0.48 & ( P @ I5 ) ) )
% 0.30/0.48 = ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 | ? [I5: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I5 @ N )
% 0.30/0.48 & ( P @ ( suc @ I5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Ex_less_Suc2
% 0.30/0.48 thf(fact_294_mult__less__mono2,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_mono2
% 0.30/0.48 thf(fact_295_mult__less__mono1,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_less_mono1
% 0.30/0.48 thf(fact_296_ex__least__nat__le,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ N )
% 0.30/0.48 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 => ? [K2: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.30/0.48 & ! [I3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I3 @ K2 )
% 0.30/0.48 => ~ ( P @ I3 ) )
% 0.30/0.48 & ( P @ K2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_least_nat_le
% 0.30/0.48 thf(fact_297_less__imp__of__nat__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_of_nat_less
% 0.30/0.48 thf(fact_298_of__nat__less__imp__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.48 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % of_nat_less_imp_less
% 0.30/0.48 thf(fact_299_of__nat__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [I: nat,J: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % of_nat_mono
% 0.30/0.48 thf(fact_300_int__ops_I7_J,axiom,
% 0.30/0.48 ! [A3: nat,B2: nat] :
% 0.30/0.48 ( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B2 ) )
% 0.30/0.48 = ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_ops(7)
% 0.30/0.48 thf(fact_301_nat__int__comparison_I3_J,axiom,
% 0.30/0.48 ( ( ord_less_eq @ nat )
% 0.30/0.48 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_int_comparison(3)
% 0.30/0.48 thf(fact_302_mult__not__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( mult_zero @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( A3
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 & ( B2
% 0.30/0.48 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_not_zero
% 0.30/0.48 thf(fact_303_divisors__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiri3467727345109120633visors @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.48 = ( zero_zero @ A ) )
% 0.30/0.48 => ( ( A3
% 0.30/0.48 = ( zero_zero @ A ) )
% 0.30/0.48 | ( B2
% 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % divisors_zero
% 0.30/0.48 thf(fact_304_no__zero__divisors,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiri3467727345109120633visors @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( B2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.48 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % no_zero_divisors
% 0.30/0.48 thf(fact_305_mult__left__cancel,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiri6575147826004484403cancel @ A )
% 0.30/0.48 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.48 ( ( C2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ( times_times @ A @ C2 @ A3 )
% 0.30/0.48 = ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.48 = ( A3 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_left_cancel
% 0.30/0.48 thf(fact_306_mult__right__cancel,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiri6575147826004484403cancel @ A )
% 0.30/0.48 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.48 ( ( C2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.48 = ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.48 = ( A3 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_right_cancel
% 0.30/0.48 thf(fact_307_fact__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_mono
% 0.30/0.48 thf(fact_308_euclidean__size__mult,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.48 = ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % euclidean_size_mult
% 0.30/0.48 thf(fact_309_round__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % round_mono
% 0.30/0.48 thf(fact_310_mult__left__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [C2: A,A3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_left_le
% 0.30/0.48 thf(fact_311_mult__le__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_one
% 0.30/0.48 thf(fact_312_mult__right__le__one__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y3 ) @ X ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_right_le_one_le
% 0.30/0.48 thf(fact_313_mult__left__le__one__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ Y3 @ X ) @ X ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_left_le_one_le
% 0.30/0.48 thf(fact_314_less__minus__one__simps_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_minus_one_simps(1)
% 0.30/0.48 thf(fact_315_less__minus__one__simps_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_minus_one_simps(3)
% 0.30/0.48 thf(fact_316_le__minus__one__simps_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_minus_one_simps(1)
% 0.30/0.48 thf(fact_317_le__minus__one__simps_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_minus_one_simps(3)
% 0.30/0.48 thf(fact_318_ex__less__of__nat__mult,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim462609752435547400_field @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.48 => ? [N4: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_less_of_nat_mult
% 0.30/0.48 thf(fact_319_nat__induct__non__zero,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.48 => ( ( P @ ( one_one @ nat ) )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.48 => ( ( P @ N4 )
% 0.30/0.48 => ( P @ ( suc @ N4 ) ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_induct_non_zero
% 0.30/0.48 thf(fact_320_size__mult__mono_H,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( B2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % size_mult_mono'
% 0.30/0.48 thf(fact_321_size__mult__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( B2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % size_mult_mono
% 0.30/0.48 thf(fact_322_fact__less__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.48 => ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_less_mono
% 0.30/0.48 thf(fact_323_fact__ge__Suc__0__nat,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_ge_Suc_0_nat
% 0.30/0.48 thf(fact_324_one__neq__neg__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ring_char_0 @ A )
% 0.30/0.48 => ( ( one_one @ A )
% 0.30/0.48 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_neq_neg_one
% 0.30/0.48 thf(fact_325_zero__less__iff__neq__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.48 => ! [N: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.48 = ( N
% 0.30/0.48 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_iff_neq_zero
% 0.30/0.48 thf(fact_326_gr__implies__not__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.48 => ! [M: A,N: A] :
% 0.30/0.48 ( ( ord_less @ A @ M @ N )
% 0.30/0.48 => ( N
% 0.30/0.48 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr_implies_not_zero
% 0.30/0.48 thf(fact_327_not__less__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.48 => ! [N: A] :
% 0.30/0.48 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_zero
% 0.30/0.48 thf(fact_328_gr__zeroI,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.48 => ! [N: A] :
% 0.30/0.48 ( ( N
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr_zeroI
% 0.30/0.48 thf(fact_329_less__numeral__extra_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_numeral_extra(3)
% 0.30/0.48 thf(fact_330_mult_Ocomm__neutral,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comm_monoid_mult @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.48 = A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult.comm_neutral
% 0.30/0.48 thf(fact_331_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comm_monoid_mult @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.48 = A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % comm_monoid_mult_class.mult_1
% 0.30/0.48 thf(fact_332_zero__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.48 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_le
% 0.30/0.48 thf(fact_333_le__numeral__extra_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_numeral_extra(3)
% 0.30/0.48 thf(fact_334_less__numeral__extra_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_numeral_extra(4)
% 0.30/0.48 thf(fact_335_exists__least__lemma,axiom,
% 0.30/0.48 ! [P: nat > $o] :
% 0.30/0.48 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.48 => ? [N4: nat] :
% 0.30/0.48 ( ~ ( P @ N4 )
% 0.30/0.48 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % exists_least_lemma
% 0.30/0.48 thf(fact_336_not0__implies__Suc,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( N
% 0.30/0.48 != ( zero_zero @ nat ) )
% 0.30/0.48 => ? [M6: nat] :
% 0.30/0.48 ( N
% 0.30/0.48 = ( suc @ M6 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not0_implies_Suc
% 0.30/0.48 thf(fact_337_Zero__not__Suc,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( zero_zero @ nat )
% 0.30/0.48 != ( suc @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % Zero_not_Suc
% 0.30/0.48 thf(fact_338_Zero__neq__Suc,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( zero_zero @ nat )
% 0.30/0.48 != ( suc @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % Zero_neq_Suc
% 0.30/0.48 thf(fact_339_Suc__neq__Zero,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( suc @ M )
% 0.30/0.48 != ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % Suc_neq_Zero
% 0.30/0.48 thf(fact_340_zero__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,K: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( P @ ( suc @ N4 ) )
% 0.30/0.48 => ( P @ N4 ) )
% 0.30/0.48 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_induct
% 0.30/0.48 thf(fact_341_diff__induct,axiom,
% 0.30/0.48 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.48 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 0.30/0.48 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 0.30/0.48 => ( ! [X3: nat,Y5: nat] :
% 0.30/0.48 ( ( P @ X3 @ Y5 )
% 0.30/0.48 => ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
% 0.30/0.48 => ( P @ M @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % diff_induct
% 0.30/0.48 thf(fact_342_nat__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( P @ N4 )
% 0.30/0.48 => ( P @ ( suc @ N4 ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_induct
% 0.30/0.48 thf(fact_343_old_Onat_Oexhaust,axiom,
% 0.30/0.48 ! [Y3: nat] :
% 0.30/0.48 ( ( Y3
% 0.30/0.48 != ( zero_zero @ nat ) )
% 0.30/0.48 => ~ ! [Nat3: nat] :
% 0.30/0.48 ( Y3
% 0.30/0.48 != ( suc @ Nat3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.exhaust
% 0.30/0.48 thf(fact_344_nat_OdiscI,axiom,
% 0.30/0.48 ! [Nat: nat,X22: nat] :
% 0.30/0.48 ( ( Nat
% 0.30/0.48 = ( suc @ X22 ) )
% 0.30/0.48 => ( Nat
% 0.30/0.48 != ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat.discI
% 0.30/0.48 thf(fact_345_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.48 ! [Nat2: nat] :
% 0.30/0.48 ( ( zero_zero @ nat )
% 0.30/0.48 != ( suc @ Nat2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.distinct(1)
% 0.30/0.48 thf(fact_346_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.48 ! [Nat4: nat] :
% 0.30/0.48 ( ( suc @ Nat4 )
% 0.30/0.48 != ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % old.nat.distinct(2)
% 0.30/0.48 thf(fact_347_nat_Odistinct_I1_J,axiom,
% 0.30/0.48 ! [X22: nat] :
% 0.30/0.48 ( ( zero_zero @ nat )
% 0.30/0.48 != ( suc @ X22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat.distinct(1)
% 0.30/0.48 thf(fact_348_le__numeral__extra_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_numeral_extra(4)
% 0.30/0.48 thf(fact_349_mult__0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.48 = ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_0
% 0.30/0.48 thf(fact_350_mult__of__nat__commute,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiring_1 @ A )
% 0.30/0.48 => ! [X: nat,Y3: A] :
% 0.30/0.48 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y3 )
% 0.30/0.48 = ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_of_nat_commute
% 0.30/0.48 thf(fact_351_infinite__descent0__measure,axiom,
% 0.30/0.48 ! [A: $tType,V3: A > nat,P: A > $o,X: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ( ( V3 @ X3 )
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 => ( P @ X3 ) )
% 0.30/0.48 => ( ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X3 ) )
% 0.30/0.48 => ( ~ ( P @ X3 )
% 0.30/0.48 => ? [Y4: A] :
% 0.30/0.48 ( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
% 0.30/0.48 & ~ ( P @ Y4 ) ) ) )
% 0.30/0.48 => ( P @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent0_measure
% 0.30/0.48 thf(fact_352_infinite__descent0,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.48 => ( ~ ( P @ N4 )
% 0.30/0.48 => ? [M3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M3 @ N4 )
% 0.30/0.48 & ~ ( P @ M3 ) ) ) )
% 0.30/0.48 => ( P @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent0
% 0.30/0.48 thf(fact_353_gr__implies__not0,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( N
% 0.30/0.48 != ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr_implies_not0
% 0.30/0.48 thf(fact_354_less__zeroE,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_zeroE
% 0.30/0.48 thf(fact_355_not__less0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less0
% 0.30/0.48 thf(fact_356_not__gr0,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.48 = ( N
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_gr0
% 0.30/0.48 thf(fact_357_gr0I,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( N
% 0.30/0.48 != ( zero_zero @ nat ) )
% 0.30/0.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % gr0I
% 0.30/0.48 thf(fact_358_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.48 ! [A3: nat] :
% 0.30/0.48 ~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_strict
% 0.30/0.48 thf(fact_359_reals__Archimedean2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim462609752435547400_field @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [N4: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % reals_Archimedean2
% 0.30/0.48 thf(fact_360_le__0__eq,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/0.48 = ( N
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_0_eq
% 0.30/0.48 thf(fact_361_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.48 ! [A3: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.48 => ( A3
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_uniqueI
% 0.30/0.48 thf(fact_362_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.48 ! [A3: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.48 = ( A3
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % bot_nat_0.extremum_unique
% 0.30/0.48 thf(fact_363_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_eq_nat.simps(1)
% 0.30/0.48 thf(fact_364_real__arch__simple,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim462609752435547400_field @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [N4: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % real_arch_simple
% 0.30/0.48 thf(fact_365_nat__mult__1__right,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( times_times @ nat @ N @ ( one_one @ nat ) )
% 0.30/0.48 = N ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_1_right
% 0.30/0.48 thf(fact_366_nat__mult__1,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ( ( times_times @ nat @ ( one_one @ nat ) @ N )
% 0.30/0.48 = N ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_1
% 0.30/0.48 thf(fact_367_not__one__less__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_one_less_zero
% 0.30/0.48 thf(fact_368_zero__less__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( zero_less_one @ A )
% 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_one
% 0.30/0.48 thf(fact_369_not__one__le__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_one_le_zero
% 0.30/0.48 thf(fact_370_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.48 thf(fact_371_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( zero_less_one @ A )
% 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_one_class.zero_le_one
% 0.30/0.48 thf(fact_372_fact__mono__nat,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_mono_nat
% 0.30/0.48 thf(fact_373_fact__ge__self,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_ge_self
% 0.30/0.48 thf(fact_374_zero__neq__neg__one,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ring_char_0 @ A )
% 0.30/0.48 => ( ( zero_zero @ A )
% 0.30/0.48 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_neq_neg_one
% 0.30/0.48 thf(fact_375_less__numeral__extra_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_numeral_extra(1)
% 0.30/0.48 thf(fact_376_of__nat__neq__0,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiring_char_0 @ A )
% 0.30/0.48 => ! [N: nat] :
% 0.30/0.48 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.30/0.48 != ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % of_nat_neq_0
% 0.30/0.48 thf(fact_377_of__nat__less__0__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [M: nat] :
% 0.30/0.48 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % of_nat_less_0_iff
% 0.30/0.48 thf(fact_378_of__nat__0__le__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linord181362715937106298miring @ A )
% 0.30/0.48 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % of_nat_0_le_iff
% 0.30/0.48 thf(fact_379_One__nat__def,axiom,
% 0.30/0.48 ( ( one_one @ nat )
% 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % One_nat_def
% 0.30/0.48 thf(fact_380_mult__eq__self__implies__10,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( M
% 0.30/0.48 = ( times_times @ nat @ M @ N ) )
% 0.30/0.48 => ( ( N
% 0.30/0.48 = ( one_one @ nat ) )
% 0.30/0.48 | ( M
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_eq_self_implies_10
% 0.30/0.48 thf(fact_381_fact__gt__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_gt_zero
% 0.30/0.48 thf(fact_382_fact__not__neg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [N: nat] :
% 0.30/0.48 ~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_not_neg
% 0.30/0.48 thf(fact_383_fact__ge__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_ge_zero
% 0.30/0.48 thf(fact_384_division__segment__mult,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( B2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.48 = ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % division_segment_mult
% 0.30/0.48 thf(fact_385_fact__ge__1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_semidom @ A )
% 0.30/0.48 => ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_ge_1
% 0.30/0.48 thf(fact_386_int__ops_I1_J,axiom,
% 0.30/0.48 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 0.30/0.48 = ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_ops(1)
% 0.30/0.48 thf(fact_387_fact__less__mono__nat,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.48 => ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 => ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_less_mono_nat
% 0.30/0.48 thf(fact_388_effect__returnI,axiom,
% 0.30/0.48 ! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
% 0.30/0.48 ( ( H2 = H )
% 0.30/0.48 => ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ X @ ( one_one @ nat ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % effect_returnI
% 0.30/0.48 thf(fact_389_negative__eq__positive,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.48 = ( semiring_1_of_nat @ int @ M ) )
% 0.30/0.48 = ( ( N
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 & ( M
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % negative_eq_positive
% 0.30/0.48 thf(fact_390_nat__mult__less__cancel__disj,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_less_cancel_disj
% 0.30/0.48 thf(fact_391_nat__mult__le__cancel__disj,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_le_cancel_disj
% 0.30/0.48 thf(fact_392_compl__less__compl__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.48 = ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_less_compl_iff
% 0.30/0.48 thf(fact_393_compl__le__compl__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.48 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_le_compl_iff
% 0.30/0.48 thf(fact_394_field__le__mult__one__interval,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_field @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ! [Z4: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
% 0.30/0.48 => ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
% 0.30/0.48 => ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X ) @ Y3 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % field_le_mult_one_interval
% 0.30/0.48 thf(fact_395_int__cases3,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( K
% 0.30/0.48 != ( zero_zero @ int ) )
% 0.30/0.48 => ( ! [N4: nat] :
% 0.30/0.48 ( ( K
% 0.30/0.48 = ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( ( K
% 0.30/0.48 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
% 0.30/0.48 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_cases3
% 0.30/0.48 thf(fact_396_not__zle__0__negative,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_zle_0_negative
% 0.30/0.48 thf(fact_397_int__cases4,axiom,
% 0.30/0.48 ! [M: int] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( M
% 0.30/0.48 != ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.48 => ( M
% 0.30/0.48 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_cases4
% 0.30/0.48 thf(fact_398_negative__zle,axiom,
% 0.30/0.48 ! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % negative_zle
% 0.30/0.48 thf(fact_399_lift__collapse,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( heap_Time_lift @ B @ A )
% 0.30/0.48 = ( ^ [F2: B > A,X2: B] : ( heap_Time_return @ A @ ( F2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lift_collapse
% 0.30/0.48 thf(fact_400_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.48 = X ) ) ).
% 0.30/0.48
% 0.30/0.48 % boolean_algebra_class.boolean_algebra.double_compl
% 0.30/0.48 thf(fact_401_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ( uminus_uminus @ A @ X )
% 0.30/0.48 = ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.48 = ( X = Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
% 0.30/0.48 thf(fact_402_negative__zless,axiom,
% 0.30/0.48 ! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % negative_zless
% 0.30/0.48 thf(fact_403_verit__la__generic,axiom,
% 0.30/0.48 ! [A3: int,X: int] :
% 0.30/0.48 ( ( ord_less_eq @ int @ A3 @ X )
% 0.30/0.48 | ( A3 = X )
% 0.30/0.48 | ( ord_less_eq @ int @ X @ A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % verit_la_generic
% 0.30/0.48 thf(fact_404_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.48 ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [Y5: nat] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 0.30/0.48 => ? [X3: nat] :
% 0.30/0.48 ( ( P @ X3 )
% 0.30/0.48 & ! [Y4: nat] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Nat.ex_has_greatest_nat
% 0.30/0.48 thf(fact_405_nat__le__linear,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_le_linear
% 0.30/0.48 thf(fact_406_mult__le__mono2,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_mono2
% 0.30/0.48 thf(fact_407_mult__le__mono1,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_mono1
% 0.30/0.48 thf(fact_408_mult__le__mono,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ K @ L )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_le_mono
% 0.30/0.48 thf(fact_409_le__antisym,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.48 => ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_antisym
% 0.30/0.48 thf(fact_410_le__square,axiom,
% 0.30/0.48 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_square
% 0.30/0.48 thf(fact_411_eq__imp__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( M = N )
% 0.30/0.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_imp_le
% 0.30/0.48 thf(fact_412_le__trans,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ J @ K )
% 0.30/0.48 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_trans
% 0.30/0.48 thf(fact_413_le__refl,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % le_refl
% 0.30/0.48 thf(fact_414_le__cube,axiom,
% 0.30/0.48 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_cube
% 0.30/0.48 thf(fact_415_less__int__code_I1_J,axiom,
% 0.30/0.48 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_int_code(1)
% 0.30/0.48 thf(fact_416_zmult__zless__mono2,axiom,
% 0.30/0.48 ! [I: int,J: int,K: int] :
% 0.30/0.48 ( ( ord_less @ int @ I @ J )
% 0.30/0.48 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.48 => ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zmult_zless_mono2
% 0.30/0.48 thf(fact_417_not__int__zless__negative,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_int_zless_negative
% 0.30/0.48 thf(fact_418_int__one__le__iff__zero__less,axiom,
% 0.30/0.48 ! [Z3: int] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 )
% 0.30/0.48 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_one_le_iff_zero_less
% 0.30/0.48 thf(fact_419_pos__zmult__eq__1__iff,axiom,
% 0.30/0.48 ! [M: int,N: int] :
% 0.30/0.48 ( ( ord_less @ int @ ( zero_zero @ int ) @ M )
% 0.30/0.48 => ( ( ( times_times @ int @ M @ N )
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 = ( ( M
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 & ( N
% 0.30/0.48 = ( one_one @ int ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pos_zmult_eq_1_iff
% 0.30/0.48 thf(fact_420_zmult__zless__mono2__lemma,axiom,
% 0.30/0.48 ! [I: int,J: int,K: nat] :
% 0.30/0.48 ( ( ord_less @ int @ I @ J )
% 0.30/0.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zmult_zless_mono2_lemma
% 0.30/0.48 thf(fact_421_negative__zless__0,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % negative_zless_0
% 0.30/0.48 thf(fact_422_negD,axiom,
% 0.30/0.48 ! [X: int] :
% 0.30/0.48 ( ( ord_less @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.48 => ? [N4: nat] :
% 0.30/0.48 ( X
% 0.30/0.48 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % negD
% 0.30/0.48 thf(fact_423_nat__int__comparison_I2_J,axiom,
% 0.30/0.48 ( ( ord_less @ nat )
% 0.30/0.48 = ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_int_comparison(2)
% 0.30/0.48 thf(fact_424_pos__int__cases,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( ( K
% 0.30/0.48 = ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pos_int_cases
% 0.30/0.48 thf(fact_425_zero__less__imp__eq__int,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.48 => ? [N4: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.48 & ( K
% 0.30/0.48 = ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_less_imp_eq_int
% 0.30/0.48 thf(fact_426_neg__int__cases,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( ( K
% 0.30/0.48 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
% 0.30/0.48 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neg_int_cases
% 0.30/0.48 thf(fact_427_linordered__field__no__ub,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_field @ A )
% 0.30/0.48 => ! [X4: A] :
% 0.30/0.48 ? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% 0.30/0.48
% 0.30/0.48 % linordered_field_no_ub
% 0.30/0.48 thf(fact_428_linordered__field__no__lb,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_field @ A )
% 0.30/0.48 => ! [X4: A] :
% 0.30/0.48 ? [Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ).
% 0.30/0.48
% 0.30/0.48 % linordered_field_no_lb
% 0.30/0.48 thf(fact_429_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ( times_times @ nat @ K @ M )
% 0.30/0.48 = ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( ( K
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 | ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_eq_cancel_disj
% 0.30/0.48 thf(fact_430_times__int__code_I2_J,axiom,
% 0.30/0.48 ! [L: int] :
% 0.30/0.48 ( ( times_times @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.48 = ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % times_int_code(2)
% 0.30/0.48 thf(fact_431_times__int__code_I1_J,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( times_times @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.48 = ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % times_int_code(1)
% 0.30/0.48 thf(fact_432_uminus__int__code_I1_J,axiom,
% 0.30/0.48 ( ( uminus_uminus @ int @ ( zero_zero @ int ) )
% 0.30/0.48 = ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % uminus_int_code(1)
% 0.30/0.48 thf(fact_433_less__eq__int__code_I1_J,axiom,
% 0.30/0.48 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.30/0.48
% 0.30/0.48 % less_eq_int_code(1)
% 0.30/0.48 thf(fact_434_zmult__eq__1__iff,axiom,
% 0.30/0.48 ! [M: int,N: int] :
% 0.30/0.48 ( ( ( times_times @ int @ M @ N )
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 = ( ( ( M
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 & ( N
% 0.30/0.48 = ( one_one @ int ) ) )
% 0.30/0.48 | ( ( M
% 0.30/0.48 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.48 & ( N
% 0.30/0.48 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zmult_eq_1_iff
% 0.30/0.48 thf(fact_435_pos__zmult__eq__1__iff__lemma,axiom,
% 0.30/0.48 ! [M: int,N: int] :
% 0.30/0.48 ( ( ( times_times @ int @ M @ N )
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 => ( ( M
% 0.30/0.48 = ( one_one @ int ) )
% 0.30/0.48 | ( M
% 0.30/0.48 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pos_zmult_eq_1_iff_lemma
% 0.30/0.48 thf(fact_436_int__int__eq,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( semiring_1_of_nat @ int @ M )
% 0.30/0.48 = ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.48 = ( M = N ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_int_eq
% 0.30/0.48 thf(fact_437_int__cases2,axiom,
% 0.30/0.48 ! [Z3: int] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( Z3
% 0.30/0.48 != ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( Z3
% 0.30/0.48 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_cases2
% 0.30/0.48 thf(fact_438_zle__int,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.48 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % zle_int
% 0.30/0.48 thf(fact_439_compl__le__swap2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 0.30/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_le_swap2
% 0.30/0.48 thf(fact_440_compl__le__swap1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_le_swap1
% 0.30/0.48 thf(fact_441_compl__mono,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_mono
% 0.30/0.48 thf(fact_442_compl__less__swap2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 0.30/0.48 => ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_less_swap2
% 0.30/0.48 thf(fact_443_compl__less__swap1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 0.30/0.48 => ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % compl_less_swap1
% 0.30/0.48 thf(fact_444_nat__mult__eq__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ( ( times_times @ nat @ K @ M )
% 0.30/0.48 = ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( M = N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_eq_cancel1
% 0.30/0.48 thf(fact_445_nat__mult__le__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_le_cancel1
% 0.30/0.48 thf(fact_446_nat__mult__less__cancel1,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.48 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.48 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_mult_less_cancel1
% 0.30/0.48 thf(fact_447_int__of__nat__induct,axiom,
% 0.30/0.48 ! [P: int > $o,Z3: int] :
% 0.30/0.48 ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ( ! [N4: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) )
% 0.30/0.48 => ( P @ Z3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_of_nat_induct
% 0.30/0.48 thf(fact_448_int__cases,axiom,
% 0.30/0.48 ! [Z3: int] :
% 0.30/0.48 ( ! [N4: nat] :
% 0.30/0.48 ( Z3
% 0.30/0.48 != ( semiring_1_of_nat @ int @ N4 ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( Z3
% 0.30/0.48 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_cases
% 0.30/0.48 thf(fact_449_int__zle__neg,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
% 0.30/0.48 = ( ( N
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 & ( M
% 0.30/0.48 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % int_zle_neg
% 0.30/0.48 thf(fact_450_zero__le__imp__eq__int,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.48 => ? [N4: nat] :
% 0.30/0.48 ( K
% 0.30/0.48 = ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_le_imp_eq_int
% 0.30/0.48 thf(fact_451_nonpos__int__cases,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( K
% 0.30/0.48 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nonpos_int_cases
% 0.30/0.48 thf(fact_452_nonneg__int__cases,axiom,
% 0.30/0.48 ! [K: int] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( K
% 0.30/0.48 != ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nonneg_int_cases
% 0.30/0.48 thf(fact_453_negative__zle__0,axiom,
% 0.30/0.48 ! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% 0.30/0.48
% 0.30/0.48 % negative_zle_0
% 0.30/0.48 thf(fact_454_uminus__apply,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( uminus @ B )
% 0.30/0.48 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.48 = ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % uminus_apply
% 0.30/0.48 thf(fact_455_dual__order_Orefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.refl
% 0.30/0.48 thf(fact_456_order__refl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_refl
% 0.30/0.48 thf(fact_457_imp__le__cong,axiom,
% 0.30/0.48 ! [X: int,X5: int,P: $o,P2: $o] :
% 0.30/0.48 ( ( X = X5 )
% 0.30/0.48 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.30/0.48 => ( P = P2 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.48 => P )
% 0.30/0.48 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.30/0.48 => P2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % imp_le_cong
% 0.30/0.48 thf(fact_458_conj__le__cong,axiom,
% 0.30/0.48 ! [X: int,X5: int,P: $o,P2: $o] :
% 0.30/0.48 ( ( X = X5 )
% 0.30/0.48 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.30/0.48 => ( P = P2 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.48 & P )
% 0.30/0.48 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.30/0.48 & P2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % conj_le_cong
% 0.30/0.48 thf(fact_459_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 0.30/0.48 => ? [X3: A] :
% 0.30/0.48 ( ( P @ X3 )
% 0.30/0.48 & ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Lattices_Big.ex_has_greatest_nat
% 0.30/0.48 thf(fact_460_nat__descend__induct,axiom,
% 0.30/0.48 ! [N: nat,P: nat > $o,M: nat] :
% 0.30/0.48 ( ! [K2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ N @ K2 )
% 0.30/0.48 => ( P @ K2 ) )
% 0.30/0.48 => ( ! [K2: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.30/0.48 => ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ K2 @ I3 )
% 0.30/0.48 => ( P @ I3 ) )
% 0.30/0.48 => ( P @ K2 ) ) )
% 0.30/0.48 => ( P @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_descend_induct
% 0.30/0.48 thf(fact_461_dependent__nat__choice,axiom,
% 0.30/0.48 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 0.30/0.48 ( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
% 0.30/0.48 => ( ! [X3: A,N4: nat] :
% 0.30/0.48 ( ( P @ N4 @ X3 )
% 0.30/0.48 => ? [Y4: A] :
% 0.30/0.48 ( ( P @ ( suc @ N4 ) @ Y4 )
% 0.30/0.48 & ( Q @ N4 @ X3 @ Y4 ) ) )
% 0.30/0.48 => ? [F3: nat > A] :
% 0.30/0.48 ! [N5: nat] :
% 0.30/0.48 ( ( P @ N5 @ ( F3 @ N5 ) )
% 0.30/0.48 & ( Q @ N5 @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dependent_nat_choice
% 0.30/0.48 thf(fact_462_list__decode_Ocases,axiom,
% 0.30/0.48 ! [X: nat] :
% 0.30/0.48 ( ( X
% 0.30/0.48 != ( zero_zero @ nat ) )
% 0.30/0.48 => ~ ! [N4: nat] :
% 0.30/0.48 ( X
% 0.30/0.48 != ( suc @ N4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % list_decode.cases
% 0.30/0.48 thf(fact_463_order__le__imp__less__or__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 | ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_imp_less_or_eq
% 0.30/0.48 thf(fact_464_nle__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.48 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 & ( B2 != A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nle_le
% 0.30/0.48 thf(fact_465_le__cases3,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.30/0.48 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_cases3
% 0.30/0.48 thf(fact_466_order__class_Oorder__eq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.48 = ( ^ [X2: A,Y6: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.30/0.48 & ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_class.order_eq_iff
% 0.30/0.48 thf(fact_467_ord__eq__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( A3 = B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_le_trans
% 0.30/0.48 thf(fact_468_ord__le__eq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( B2 = C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_le_eq_trans
% 0.30/0.48 thf(fact_469_order__antisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.48 => ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_antisym
% 0.30/0.48 thf(fact_470_order_Otrans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.trans
% 0.30/0.48 thf(fact_471_order__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_trans
% 0.30/0.48 thf(fact_472_linorder__wlog,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.30/0.48 ( ! [A8: A,B5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A8 @ B5 )
% 0.30/0.48 => ( P @ A8 @ B5 ) )
% 0.30/0.48 => ( ! [A8: A,B5: A] :
% 0.30/0.48 ( ( P @ B5 @ A8 )
% 0.30/0.48 => ( P @ A8 @ B5 ) )
% 0.30/0.48 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_wlog
% 0.30/0.48 thf(fact_473_dual__order_Oeq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.48 = ( ^ [A5: A,B3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.30/0.48 & ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.eq_iff
% 0.30/0.48 thf(fact_474_dual__order_Oantisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( A3 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.antisym
% 0.30/0.48 thf(fact_475_dual__order_Otrans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.trans
% 0.30/0.48 thf(fact_476_antisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( A3 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym
% 0.30/0.48 thf(fact_477_le__funD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B )
% 0.30/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funD
% 0.30/0.48 thf(fact_478_le__funE,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B )
% 0.30/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funE
% 0.30/0.48 thf(fact_479_le__funI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B )
% 0.30/0.48 => ! [F: A > B,G: A > B] :
% 0.30/0.48 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.48 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funI
% 0.30/0.48 thf(fact_480_le__fun__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B )
% 0.30/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.48 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.48 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_fun_def
% 0.30/0.48 thf(fact_481_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.48 = ( ^ [A5: A,B3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.30/0.48 & ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Orderings.order_eq_iff
% 0.30/0.48 thf(fact_482_order__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_subst1
% 0.30/0.48 thf(fact_483_order__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType] :
% 0.30/0.48 ( ( ( order @ C )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_subst2
% 0.30/0.48 thf(fact_484_order__eq__refl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( X = Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_eq_refl
% 0.30/0.48 thf(fact_485_linorder__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_linear
% 0.30/0.48 thf(fact_486_ord__eq__le__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B )
% 0.30/0.48 & ( ord @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 = ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_le_subst
% 0.30/0.48 thf(fact_487_ord__le__eq__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B )
% 0.30/0.48 & ( ord @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ( F @ B2 )
% 0.30/0.48 = C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_le_eq_subst
% 0.30/0.48 thf(fact_488_linorder__le__cases,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_le_cases
% 0.30/0.48 thf(fact_489_order__antisym__conv,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 = ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_antisym_conv
% 0.30/0.48 thf(fact_490_minf_I11_J,axiom,
% 0.30/0.48 ! [C: $tType,D: $tType] :
% 0.30/0.48 ( ( ord @ C )
% 0.30/0.48 => ! [F4: D] :
% 0.30/0.48 ? [Z4: C] :
% 0.30/0.48 ! [X4: C] :
% 0.30/0.48 ( ( ord_less @ C @ X4 @ Z4 )
% 0.30/0.48 => ( F4 = F4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(11)
% 0.30/0.48 thf(fact_491_minf_I7_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(7)
% 0.30/0.48 thf(fact_492_minf_I5_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(5)
% 0.30/0.48 thf(fact_493_minf_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( X4 != T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(4)
% 0.30/0.48 thf(fact_494_minf_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( X4 != T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(3)
% 0.30/0.48 thf(fact_495_minf_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Z5 )
% 0.30/0.48 => ( ( P @ X3 )
% 0.30/0.48 = ( P2 @ X3 ) ) )
% 0.30/0.48 => ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Z5 )
% 0.30/0.48 => ( ( Q @ X3 )
% 0.30/0.48 = ( Q2 @ X3 ) ) )
% 0.30/0.48 => ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( ( ( P @ X4 )
% 0.30/0.48 | ( Q @ X4 ) )
% 0.30/0.48 = ( ( P2 @ X4 )
% 0.30/0.48 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(2)
% 0.30/0.48 thf(fact_496_minf_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Z5 )
% 0.30/0.48 => ( ( P @ X3 )
% 0.30/0.48 = ( P2 @ X3 ) ) )
% 0.30/0.48 => ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Z5 )
% 0.30/0.48 => ( ( Q @ X3 )
% 0.30/0.48 = ( Q2 @ X3 ) ) )
% 0.30/0.48 => ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( ( ( P @ X4 )
% 0.30/0.48 & ( Q @ X4 ) )
% 0.30/0.48 = ( ( P2 @ X4 )
% 0.30/0.48 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(1)
% 0.30/0.48 thf(fact_497_pinf_I11_J,axiom,
% 0.30/0.48 ! [C: $tType,D: $tType] :
% 0.30/0.48 ( ( ord @ C )
% 0.30/0.48 => ! [F4: D] :
% 0.30/0.48 ? [Z4: C] :
% 0.30/0.48 ! [X4: C] :
% 0.30/0.48 ( ( ord_less @ C @ Z4 @ X4 )
% 0.30/0.48 => ( F4 = F4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(11)
% 0.30/0.48 thf(fact_498_pinf_I7_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(7)
% 0.30/0.48 thf(fact_499_pinf_I5_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(5)
% 0.30/0.48 thf(fact_500_pinf_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( X4 != T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(4)
% 0.30/0.48 thf(fact_501_pinf_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( X4 != T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(3)
% 0.30/0.48 thf(fact_502_pinf_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z5 @ X3 )
% 0.30/0.48 => ( ( P @ X3 )
% 0.30/0.48 = ( P2 @ X3 ) ) )
% 0.30/0.48 => ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z5 @ X3 )
% 0.30/0.48 => ( ( Q @ X3 )
% 0.30/0.48 = ( Q2 @ X3 ) ) )
% 0.30/0.48 => ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( ( ( P @ X4 )
% 0.30/0.48 | ( Q @ X4 ) )
% 0.30/0.48 = ( ( P2 @ X4 )
% 0.30/0.48 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(2)
% 0.30/0.48 thf(fact_503_pinf_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z5 @ X3 )
% 0.30/0.48 => ( ( P @ X3 )
% 0.30/0.48 = ( P2 @ X3 ) ) )
% 0.30/0.48 => ( ? [Z5: A] :
% 0.30/0.48 ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z5 @ X3 )
% 0.30/0.48 => ( ( Q @ X3 )
% 0.30/0.48 = ( Q2 @ X3 ) ) )
% 0.30/0.48 => ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( ( ( P @ X4 )
% 0.30/0.48 & ( Q @ X4 ) )
% 0.30/0.48 = ( ( P2 @ X4 )
% 0.30/0.48 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(1)
% 0.30/0.48 thf(fact_504_lt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_bot @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % lt_ex
% 0.30/0.48 thf(fact_505_gt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_top @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% 0.30/0.48
% 0.30/0.48 % gt_ex
% 0.30/0.48 thf(fact_506_dense,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ? [Z4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Z4 )
% 0.30/0.48 & ( ord_less @ A @ Z4 @ Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense
% 0.30/0.48 thf(fact_507_less__imp__neq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( X != Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_neq
% 0.30/0.48 thf(fact_508_order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.asym
% 0.30/0.48 thf(fact_509_ord__eq__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( A3 = B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_trans
% 0.30/0.48 thf(fact_510_ord__less__eq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( B2 = C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_trans
% 0.30/0.48 thf(fact_511_less__induct,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A )
% 0.30/0.48 => ! [P: A > $o,A3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ! [Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y4 @ X3 )
% 0.30/0.48 => ( P @ Y4 ) )
% 0.30/0.48 => ( P @ X3 ) )
% 0.30/0.48 => ( P @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_induct
% 0.30/0.48 thf(fact_512_antisym__conv3,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ Y3 @ X )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.30/0.48 = ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv3
% 0.30/0.48 thf(fact_513_linorder__cases,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ( X != Y3 )
% 0.30/0.48 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_cases
% 0.30/0.48 thf(fact_514_dual__order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.48 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.asym
% 0.30/0.48 thf(fact_515_dual__order_Oirrefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.irrefl
% 0.30/0.48 thf(fact_516_exists__least__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A )
% 0.30/0.48 => ( ( ^ [P3: A > $o] :
% 0.30/0.48 ? [X6: A] : ( P3 @ X6 ) )
% 0.30/0.48 = ( ^ [P4: A > $o] :
% 0.30/0.48 ? [N3: A] :
% 0.30/0.48 ( ( P4 @ N3 )
% 0.30/0.48 & ! [M2: A] :
% 0.30/0.48 ( ( ord_less @ A @ M2 @ N3 )
% 0.30/0.48 => ~ ( P4 @ M2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % exists_least_iff
% 0.30/0.48 thf(fact_517_linorder__less__wlog,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.30/0.48 ( ! [A8: A,B5: A] :
% 0.30/0.48 ( ( ord_less @ A @ A8 @ B5 )
% 0.30/0.48 => ( P @ A8 @ B5 ) )
% 0.30/0.48 => ( ! [A8: A] : ( P @ A8 @ A8 )
% 0.30/0.48 => ( ! [A8: A,B5: A] :
% 0.30/0.48 ( ( P @ B5 @ A8 )
% 0.30/0.48 => ( P @ A8 @ B5 ) )
% 0.30/0.48 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_less_wlog
% 0.30/0.48 thf(fact_518_order_Ostrict__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans
% 0.30/0.48 thf(fact_519_not__less__iff__gr__or__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.30/0.48 = ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.48 | ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_iff_gr_or_eq
% 0.30/0.48 thf(fact_520_dual__order_Ostrict__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less @ A @ C2 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans
% 0.30/0.48 thf(fact_521_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( A3 != B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_implies_not_eq
% 0.30/0.48 thf(fact_522_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.48 => ( A3 != B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_implies_not_eq
% 0.30/0.48 thf(fact_523_linorder__neqE,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( X != Y3 )
% 0.30/0.48 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE
% 0.30/0.48 thf(fact_524_order__less__asym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_asym
% 0.30/0.48 thf(fact_525_linorder__neq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( X != Y3 )
% 0.30/0.48 = ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 | ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neq_iff
% 0.30/0.48 thf(fact_526_order__less__asym_H,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_asym'
% 0.30/0.48 thf(fact_527_order__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less @ A @ Y3 @ Z3 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_trans
% 0.30/0.48 thf(fact_528_ord__eq__less__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B )
% 0.30/0.48 & ( ord @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 = ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_subst
% 0.30/0.48 thf(fact_529_ord__less__eq__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B )
% 0.30/0.48 & ( ord @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ( F @ B2 )
% 0.30/0.48 = C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_subst
% 0.30/0.48 thf(fact_530_order__less__irrefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_irrefl
% 0.30/0.48 thf(fact_531_order__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst1
% 0.30/0.48 thf(fact_532_order__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType] :
% 0.30/0.48 ( ( ( order @ C )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst2
% 0.30/0.48 thf(fact_533_order__less__not__sym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_not_sym
% 0.30/0.48 thf(fact_534_order__less__imp__triv,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,P: $o] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.48 => P ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_imp_triv
% 0.30/0.48 thf(fact_535_linorder__less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 | ( X = Y3 )
% 0.30/0.48 | ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_less_linear
% 0.30/0.48 thf(fact_536_order__less__imp__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( X != Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_imp_not_eq
% 0.30/0.48 thf(fact_537_order__less__imp__not__eq2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( Y3 != X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_imp_not_eq2
% 0.30/0.48 thf(fact_538_order__less__imp__not__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_imp_not_less
% 0.30/0.48 thf(fact_539_fun__Compl__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( uminus @ B )
% 0.30/0.48 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.48 = ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fun_Compl_def
% 0.30/0.48 thf(fact_540_ex__has__least__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ? [X3: A] :
% 0.30/0.48 ( ( P @ X3 )
% 0.30/0.48 & ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_has_least_nat
% 0.30/0.48 thf(fact_541_minf_I8_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(8)
% 0.30/0.48 thf(fact_542_minf_I6_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X4 @ Z4 )
% 0.30/0.48 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(6)
% 0.30/0.48 thf(fact_543_pinf_I8_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(8)
% 0.30/0.48 thf(fact_544_pinf_I6_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [T2: A] :
% 0.30/0.48 ? [Z4: A] :
% 0.30/0.48 ! [X4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X4 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(6)
% 0.30/0.48 thf(fact_545_leD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.48 => ~ ( ord_less @ A @ X @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % leD
% 0.30/0.48 thf(fact_546_leI,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % leI
% 0.30/0.48 thf(fact_547_nless__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.48 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nless_le
% 0.30/0.48 thf(fact_548_antisym__conv1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 = ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv1
% 0.30/0.48 thf(fact_549_antisym__conv2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.30/0.48 = ( X = Y3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv2
% 0.30/0.48 thf(fact_550_dense__ge,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A )
% 0.30/0.48 => ! [Z3: A,Y3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X3 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge
% 0.30/0.48 thf(fact_551_dense__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A )
% 0.30/0.48 => ! [Y3: A,Z3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le
% 0.30/0.48 thf(fact_552_less__le__not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X2: A,Y6: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.30/0.48 & ~ ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le_not_le
% 0.30/0.48 thf(fact_553_not__le__imp__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [Y3: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.48 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_le_imp_less
% 0.30/0.48 thf(fact_554_order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [A5: A,B3: A] :
% 0.30/0.48 ( ( ord_less @ A @ A5 @ B3 )
% 0.30/0.48 | ( A5 = B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.order_iff_strict
% 0.30/0.48 thf(fact_555_order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [A5: A,B3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.30/0.48 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_iff_order
% 0.30/0.48 thf(fact_556_order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans1
% 0.30/0.48 thf(fact_557_order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans2
% 0.30/0.48 thf(fact_558_order_Ostrict__iff__not,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [A5: A,B3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.30/0.48 & ~ ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_iff_not
% 0.30/0.48 thf(fact_559_dense__ge__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A )
% 0.30/0.48 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ X )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge_bounded
% 0.30/0.48 thf(fact_560_dense__le__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le_bounded
% 0.30/0.48 thf(fact_561_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [B3: A,A5: A] :
% 0.30/0.48 ( ( ord_less @ A @ B3 @ A5 )
% 0.30/0.48 | ( A5 = B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.order_iff_strict
% 0.30/0.48 thf(fact_562_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [B3: A,A5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.30/0.48 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_iff_order
% 0.30/0.48 thf(fact_563_dual__order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less @ A @ C2 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans1
% 0.30/0.48 thf(fact_564_dual__order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans2
% 0.30/0.48 thf(fact_565_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [B3: A,A5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B3 @ A5 )
% 0.30/0.48 & ~ ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_iff_not
% 0.30/0.48 thf(fact_566_order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_implies_order
% 0.30/0.48 thf(fact_567_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [B2: A,A3: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.48 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_implies_order
% 0.30/0.48 thf(fact_568_order__le__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [X2: A,Y6: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y6 )
% 0.30/0.48 | ( X2 = Y6 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less
% 0.30/0.48 thf(fact_569_order__less__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X2: A,Y6: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.30/0.48 & ( X2 != Y6 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le
% 0.30/0.48 thf(fact_570_linorder__not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ A @ X @ Y3 ) )
% 0.30/0.48 = ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_not_le
% 0.30/0.48 thf(fact_571_linorder__not__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.30/0.48 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_not_less
% 0.30/0.48 thf(fact_572_order__less__imp__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_imp_le
% 0.30/0.48 thf(fact_573_order__le__neq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( A3 != B2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_neq_trans
% 0.30/0.48 thf(fact_574_order__neq__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3 != B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_neq_le_trans
% 0.30/0.48 thf(fact_575_order__le__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less @ A @ Y3 @ Z3 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_trans
% 0.30/0.48 thf(fact_576_order__less__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_trans
% 0.30/0.48 thf(fact_577_order__le__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst1
% 0.30/0.48 thf(fact_578_order__le__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType] :
% 0.30/0.48 ( ( ( order @ C )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst2
% 0.30/0.48 thf(fact_579_order__less__le__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.30/0.48 => ( ! [X3: B,Y5: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst1
% 0.30/0.48 thf(fact_580_order__less__le__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C: $tType] :
% 0.30/0.48 ( ( ( order @ C )
% 0.30/0.48 & ( order @ A ) )
% 0.30/0.48 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.30/0.48 => ( ! [X3: A,Y5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.48 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.48 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst2
% 0.30/0.48 thf(fact_581_linorder__le__less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A )
% 0.30/0.48 => ! [X: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.48 | ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_le_less_linear
% 0.30/0.48 thf(fact_582_complete__interval,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( condit6923001295902523014norder @ A )
% 0.30/0.48 => ! [A3: A,B2: A,P: A > $o] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.48 => ( ( P @ A3 )
% 0.30/0.48 => ( ~ ( P @ B2 )
% 0.30/0.48 => ? [C3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ C3 )
% 0.30/0.48 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.30/0.48 & ! [X4: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ A3 @ X4 )
% 0.30/0.48 & ( ord_less @ A @ X4 @ C3 ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 & ! [D3: A] :
% 0.30/0.48 ( ! [X3: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.30/0.48 & ( ord_less @ A @ X3 @ D3 ) )
% 0.30/0.48 => ( P @ X3 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % complete_interval
% 0.30/0.48 thf(fact_583_fact__reduce,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiring_char_0 @ A )
% 0.30/0.48 => ! [N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.48 => ( ( semiring_char_0_fact @ A @ N )
% 0.30/0.48 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_reduce
% 0.30/0.48 thf(fact_584_fact__num__eq__if,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( semiring_char_0 @ A )
% 0.30/0.48 => ( ( semiring_char_0_fact @ A )
% 0.30/0.48 = ( ^ [M2: nat] :
% 0.30/0.48 ( if @ A
% 0.30/0.48 @ ( M2
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 @ ( one_one @ A )
% 0.30/0.48 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % fact_num_eq_if
% 0.30/0.48 thf(fact_585_zero__le__ceiling,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.48 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % zero_le_ceiling
% 0.30/0.48 thf(fact_586_ceiling__less__zero,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.30/0.48 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ceiling_less_zero
% 0.30/0.48 thf(fact_587_one__less__nat__eq,axiom,
% 0.30/0.48 ! [Z3: int] :
% 0.30/0.48 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z3 ) )
% 0.30/0.48 = ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_less_nat_eq
% 0.30/0.48 thf(fact_588_split__div_H,axiom,
% 0.30/0.48 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.48 ( ( P @ ( divide_divide @ nat @ M @ N ) )
% 0.30/0.48 = ( ( ( N
% 0.30/0.48 = ( zero_zero @ nat ) )
% 0.30/0.48 & ( P @ ( zero_zero @ nat ) ) )
% 0.30/0.48 | ? [Q3: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
% 0.30/0.48 & ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
% 0.30/0.48 & ( P @ Q3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % split_div'
% 0.30/0.48 thf(fact_589_frac__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ( ( ( archimedean_frac @ A @ X )
% 0.30/0.48 = X )
% 0.30/0.48 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.48 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % frac_eq
% 0.30/0.48 thf(fact_590_euclidean__size__times__nonunit,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.48 => ! [A3: A,B2: A] :
% 0.30/0.48 ( ( A3
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ( B2
% 0.30/0.48 != ( zero_zero @ A ) )
% 0.30/0.48 => ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.48 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % euclidean_size_times_nonunit
% 0.30/0.48 thf(fact_591_sgn__neg,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linordered_idom @ A )
% 0.30/0.48 => ! [A3: A] :
% 0.30/0.48 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_neg
% 0.30/0.49 thf(fact_592_Suc__times__mod__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
% 0.30/0.49 = ( one_one @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_times_mod_eq
% 0.30/0.49 thf(fact_593_minus__apply,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( minus @ B )
% 0.30/0.49 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.49 = ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_apply
% 0.30/0.49 thf(fact_594_mod__mod__trivial,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mod_trivial
% 0.30/0.49 thf(fact_595_nat__dvd__1__iff__1,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 0.30/0.49 = ( M
% 0.30/0.49 = ( one_one @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_dvd_1_iff_1
% 0.30/0.49 thf(fact_596_int__dvd__int__iff,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.49 = ( dvd_dvd @ nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % int_dvd_int_iff
% 0.30/0.49 thf(fact_597_sgn__sgn,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.49 = ( sgn_sgn @ A @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_sgn
% 0.30/0.49 thf(fact_598_frac__frac,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
% 0.30/0.49 = ( archimedean_frac @ A @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_frac
% 0.30/0.49 thf(fact_599_diff__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_self
% 0.30/0.49 thf(fact_600_diff__0__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 = A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_0_right
% 0.30/0.49 thf(fact_601_zero__diff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_diff @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_diff
% 0.30/0.49 thf(fact_602_diff__zero,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 = A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_zero
% 0.30/0.49 thf(fact_603_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.49 thf(fact_604_dvd__0__left__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_semiring_1 @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_0_left_iff
% 0.30/0.49 thf(fact_605_dvd__0__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_semiring_1 @ A )
% 0.30/0.49 => ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_0_right
% 0.30/0.49 thf(fact_606_division__ring__divide__zero,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % division_ring_divide_zero
% 0.30/0.49 thf(fact_607_divide__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ A3 @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_cancel_right
% 0.30/0.49 thf(fact_608_divide__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ C2 @ A3 )
% 0.30/0.49 = ( divide_divide @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_cancel_left
% 0.30/0.49 thf(fact_609_divide__eq__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( ( A3
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 | ( B2
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_eq_0_iff
% 0.30/0.49 thf(fact_610_div__by__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_by_0
% 0.30/0.49 thf(fact_611_div__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_0
% 0.30/0.49 thf(fact_612_times__divide__eq__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % times_divide_eq_right
% 0.30/0.49 thf(fact_613_divide__divide__eq__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_divide_eq_right
% 0.30/0.49 thf(fact_614_divide__divide__eq__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_divide_eq_left
% 0.30/0.49 thf(fact_615_times__divide__eq__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % times_divide_eq_left
% 0.30/0.49 thf(fact_616_minus__diff__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_diff_eq
% 0.30/0.49 thf(fact_617_div__by__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 = A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_by_1
% 0.30/0.49 thf(fact_618_mod__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ A3 @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_self
% 0.30/0.49 thf(fact_619_mod__by__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 = A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_by_0
% 0.30/0.49 thf(fact_620_mod__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_0
% 0.30/0.49 thf(fact_621_minus__dvd__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_ring_1 @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y3 )
% 0.30/0.49 = ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_dvd_iff
% 0.30/0.49 thf(fact_622_dvd__minus__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_ring_1 @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y3 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_minus_iff
% 0.30/0.49 thf(fact_623_div__minus__minus,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_minus_minus
% 0.30/0.49 thf(fact_624_dvd__1__left,axiom,
% 0.30/0.49 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_1_left
% 0.30/0.49 thf(fact_625_dvd__1__iff__1,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.49 = ( M
% 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_1_iff_1
% 0.30/0.49 thf(fact_626_div__dvd__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( divide_divide @ A @ C2 @ A3 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_dvd_div
% 0.30/0.49 thf(fact_627_minus__mod__self2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_mod_self2
% 0.30/0.49 thf(fact_628_mod__minus__minus,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_minus_minus
% 0.30/0.49 thf(fact_629_diff__Suc__Suc,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.49 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_Suc_Suc
% 0.30/0.49 thf(fact_630_Suc__diff__diff,axiom,
% 0.30/0.49 ! [M: nat,N: nat,K: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.30/0.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_diff_diff
% 0.30/0.49 thf(fact_631_diff__0__eq__0,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 = ( zero_zero @ nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_0_eq_0
% 0.30/0.49 thf(fact_632_diff__self__eq__0,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ M @ M )
% 0.30/0.49 = ( zero_zero @ nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_self_eq_0
% 0.30/0.49 thf(fact_633_nat__mult__dvd__cancel__disj,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.49 = ( ( K
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 | ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_mult_dvd_cancel_disj
% 0.30/0.49 thf(fact_634_diff__diff__cancel,axiom,
% 0.30/0.49 ! [I: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ I @ N )
% 0.30/0.49 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.30/0.49 = I ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_diff_cancel
% 0.30/0.49 thf(fact_635_sgn__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_0
% 0.30/0.49 thf(fact_636_sgn__mult__dvd__iff,axiom,
% 0.30/0.49 ! [R: int,L: int,K: int] :
% 0.30/0.49 ( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ L ) @ K )
% 0.30/0.49 = ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.49 & ( ( R
% 0.30/0.49 = ( zero_zero @ int ) )
% 0.30/0.49 => ( K
% 0.30/0.49 = ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_mult_dvd_iff
% 0.30/0.49 thf(fact_637_mult__sgn__dvd__iff,axiom,
% 0.30/0.49 ! [L: int,R: int,K: int] :
% 0.30/0.49 ( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R ) ) @ K )
% 0.30/0.49 = ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.49 & ( ( R
% 0.30/0.49 = ( zero_zero @ int ) )
% 0.30/0.49 => ( K
% 0.30/0.49 = ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_sgn_dvd_iff
% 0.30/0.49 thf(fact_638_dvd__sgn__mult__iff,axiom,
% 0.30/0.49 ! [L: int,R: int,K: int] :
% 0.30/0.49 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ K ) )
% 0.30/0.49 = ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.49 | ( R
% 0.30/0.49 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_sgn_mult_iff
% 0.30/0.49 thf(fact_639_dvd__mult__sgn__iff,axiom,
% 0.30/0.49 ! [L: int,K: int,R: int] :
% 0.30/0.49 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R ) ) )
% 0.30/0.49 = ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.49 | ( R
% 0.30/0.49 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_sgn_iff
% 0.30/0.49 thf(fact_640_sgn__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ( ( sgn_sgn @ A @ ( one_one @ A ) )
% 0.30/0.49 = ( one_one @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_1
% 0.30/0.49 thf(fact_641_sgn__minus,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.49 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_minus
% 0.30/0.49 thf(fact_642_sgn__divide,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field_abs_sgn @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_divide
% 0.30/0.49 thf(fact_643_mod__less,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ M @ N )
% 0.30/0.49 = M ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_less
% 0.30/0.49 thf(fact_644_nat__int,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.49 = N ) ).
% 0.30/0.49
% 0.30/0.49 % nat_int
% 0.30/0.49 thf(fact_645_diff__ge__0__iff__ge,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_ge_0_iff_ge
% 0.30/0.49 thf(fact_646_diff__gt__0__iff__gt,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_gt_0_iff_gt
% 0.30/0.49 thf(fact_647_diff__numeral__special_I9_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( neg_numeral @ A )
% 0.30/0.49 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_numeral_special(9)
% 0.30/0.49 thf(fact_648_dvd__mult__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 | ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_cancel_left
% 0.30/0.49 thf(fact_649_dvd__mult__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 | ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_cancel_right
% 0.30/0.49 thf(fact_650_dvd__times__left__cancel__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_times_left_cancel_iff
% 0.30/0.49 thf(fact_651_dvd__times__right__cancel__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_times_right_cancel_iff
% 0.30/0.49 thf(fact_652_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( zero_zero @ A ) ) )
% 0.30/0.49 & ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_divide_mult_cancel_left_if
% 0.30/0.49 thf(fact_653_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.49 thf(fact_654_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.49 thf(fact_655_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.49 thf(fact_656_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.49 thf(fact_657_div__mult__mult1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_mult1
% 0.30/0.49 thf(fact_658_div__mult__mult2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_mult2
% 0.30/0.49 thf(fact_659_div__mult__mult1__if,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( zero_zero @ A ) ) )
% 0.30/0.49 & ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_mult1_if
% 0.30/0.49 thf(fact_660_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
% 0.30/0.49 = B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_div_cancel_left
% 0.30/0.49 thf(fact_661_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.49 = A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_mult_div_cancel_right
% 0.30/0.49 thf(fact_662_diff__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_0
% 0.30/0.49 thf(fact_663_verit__minus__simplify_I3_J,axiom,
% 0.30/0.49 ! [B: $tType] :
% 0.30/0.49 ( ( group_add @ B )
% 0.30/0.49 => ! [B2: B] :
% 0.30/0.49 ( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
% 0.30/0.49 = ( uminus_uminus @ B @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % verit_minus_simplify(3)
% 0.30/0.49 thf(fact_664_divide__eq__1__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( one_one @ A ) )
% 0.30/0.49 = ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 & ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_eq_1_iff
% 0.30/0.49 thf(fact_665_one__eq__divide__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( one_one @ A )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 & ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_divide_iff
% 0.30/0.49 thf(fact_666_divide__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.30/0.49 = ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_self
% 0.30/0.49 thf(fact_667_divide__self__if,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ( A3
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) )
% 0.30/0.49 & ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.30/0.49 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_self_if
% 0.30/0.49 thf(fact_668_divide__eq__eq__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.30/0.49 = ( one_one @ A ) )
% 0.30/0.49 = ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 & ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_eq_eq_1
% 0.30/0.49 thf(fact_669_eq__divide__eq__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( ( one_one @ A )
% 0.30/0.49 = ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 & ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_divide_eq_1
% 0.30/0.49 thf(fact_670_one__divide__eq__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_divide_eq_0_iff
% 0.30/0.49 thf(fact_671_zero__eq__1__divide__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ( zero_zero @ A )
% 0.30/0.49 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_eq_1_divide_iff
% 0.30/0.49 thf(fact_672_div__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.30/0.49 = ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_self
% 0.30/0.49 thf(fact_673_unit__prod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_prod
% 0.30/0.49 thf(fact_674_mod__mult__self1__is__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_self1_is_0
% 0.30/0.49 thf(fact_675_mod__mult__self2__is__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_self2_is_0
% 0.30/0.49 thf(fact_676_mod__by__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_by_1
% 0.30/0.49 thf(fact_677_divide__minus1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.49 = ( uminus_uminus @ A @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_minus1
% 0.30/0.49 thf(fact_678_div__minus1__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.49 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_minus1_right
% 0.30/0.49 thf(fact_679_dvd__mult__div__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_div_cancel
% 0.30/0.49 thf(fact_680_dvd__div__mult__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 0.30/0.49 = B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_mult_self
% 0.30/0.49 thf(fact_681_unit__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div
% 0.30/0.49 thf(fact_682_unit__div__1__unit,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_1_unit
% 0.30/0.49 thf(fact_683_unit__div__1__div__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.30/0.49 = A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_1_div_1
% 0.30/0.49 thf(fact_684_div__diff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_modulo @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_diff
% 0.30/0.49 thf(fact_685_dvd__imp__mod__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( modulo_modulo @ A @ B2 @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_imp_mod_0
% 0.30/0.49 thf(fact_686_mod__div__trivial,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_div_trivial
% 0.30/0.49 thf(fact_687_minus__mod__self1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_mod_self1
% 0.30/0.49 thf(fact_688_sgn__less,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.49 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_less
% 0.30/0.49 thf(fact_689_sgn__greater,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.49 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_greater
% 0.30/0.49 thf(fact_690_zero__less__diff,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.49 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_less_diff
% 0.30/0.49 thf(fact_691_diff__is__0__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ( minus_minus @ nat @ M @ N )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_is_0_eq
% 0.30/0.49 thf(fact_692_diff__is__0__eq_H,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ( minus_minus @ nat @ M @ N )
% 0.30/0.49 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_is_0_eq'
% 0.30/0.49 thf(fact_693_divide__sgn,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
% 0.30/0.49 = ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_sgn
% 0.30/0.49 thf(fact_694_diff__Suc__1,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
% 0.30/0.49 = N ) ).
% 0.30/0.49
% 0.30/0.49 % diff_Suc_1
% 0.30/0.49 thf(fact_695_mod__by__Suc__0,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.49 = ( zero_zero @ nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_by_Suc_0
% 0.30/0.49 thf(fact_696_div__by__Suc__0,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.49 = M ) ).
% 0.30/0.49
% 0.30/0.49 % div_by_Suc_0
% 0.30/0.49 thf(fact_697_div__less,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 => ( ( divide_divide @ nat @ M @ N )
% 0.30/0.49 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_less
% 0.30/0.49 thf(fact_698_ceiling__zero,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 0.30/0.49 = ( zero_zero @ int ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_zero
% 0.30/0.49 thf(fact_699_nat__mult__div__cancel__disj,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( ( K
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.49 = ( zero_zero @ nat ) ) )
% 0.30/0.49 & ( ( K
% 0.30/0.49 != ( zero_zero @ nat ) )
% 0.30/0.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.49 = ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_mult_div_cancel_disj
% 0.30/0.49 thf(fact_700_ceiling__one,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 0.30/0.49 = ( one_one @ int ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_one
% 0.30/0.49 thf(fact_701_ceiling__of__nat,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [N: nat] :
% 0.30/0.49 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.49 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_of_nat
% 0.30/0.49 thf(fact_702_divide__le__0__1__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.49 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_le_0_1_iff
% 0.30/0.49 thf(fact_703_zero__le__divide__1__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.30/0.49 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_le_divide_1_iff
% 0.30/0.49 thf(fact_704_divide__less__0__1__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.49 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_0_1_iff
% 0.30/0.49 thf(fact_705_divide__less__eq__1__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_eq_1_neg
% 0.30/0.49 thf(fact_706_divide__less__eq__1__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_eq_1_pos
% 0.30/0.49 thf(fact_707_less__divide__eq__1__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_divide_eq_1_neg
% 0.30/0.49 thf(fact_708_less__divide__eq__1__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_divide_eq_1_pos
% 0.30/0.49 thf(fact_709_zero__less__divide__1__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.30/0.49 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_less_divide_1_iff
% 0.30/0.49 thf(fact_710_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_divide_mult_cancel_left
% 0.30/0.49 thf(fact_711_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_divide_mult_cancel_right
% 0.30/0.49 thf(fact_712_diff__numeral__special_I12_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( neg_numeral @ A )
% 0.30/0.49 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_numeral_special(12)
% 0.30/0.49 thf(fact_713_unit__mult__div__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.30/0.49 = ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_mult_div_div
% 0.30/0.49 thf(fact_714_unit__div__mult__self,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 0.30/0.49 = B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_mult_self
% 0.30/0.49 thf(fact_715_mod__minus1__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_minus1_right
% 0.30/0.49 thf(fact_716_sgn__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 = ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_pos
% 0.30/0.49 thf(fact_717_Suc__pred,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.49 = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_pred
% 0.30/0.49 thf(fact_718_nat__1,axiom,
% 0.30/0.49 ( ( nat2 @ ( one_one @ int ) )
% 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_1
% 0.30/0.49 thf(fact_719_nat__le__0,axiom,
% 0.30/0.49 ! [Z3: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 0.30/0.49 => ( ( nat2 @ Z3 )
% 0.30/0.49 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_le_0
% 0.30/0.49 thf(fact_720_nat__0__iff,axiom,
% 0.30/0.49 ! [I: int] :
% 0.30/0.49 ( ( ( nat2 @ I )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_0_iff
% 0.30/0.49 thf(fact_721_zless__nat__conj,axiom,
% 0.30/0.49 ! [W2: int,Z3: int] :
% 0.30/0.49 ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 0.30/0.49 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 & ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zless_nat_conj
% 0.30/0.49 thf(fact_722_div__mult__self1__is__m,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
% 0.30/0.49 = M ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_self1_is_m
% 0.30/0.49 thf(fact_723_div__mult__self__is__m,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
% 0.30/0.49 = M ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_self_is_m
% 0.30/0.49 thf(fact_724_ceiling__diff__one,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 0.30/0.49 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_diff_one
% 0.30/0.49 thf(fact_725_int__nat__eq,axiom,
% 0.30/0.49 ! [Z3: int] :
% 0.30/0.49 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.30/0.49 = Z3 ) )
% 0.30/0.49 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.30/0.49 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % int_nat_eq
% 0.30/0.49 thf(fact_726_nat__zminus__int,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.49 = ( zero_zero @ nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_zminus_int
% 0.30/0.49 thf(fact_727_divide__le__eq__1__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_le_eq_1_neg
% 0.30/0.49 thf(fact_728_divide__le__eq__1__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_le_eq_1_pos
% 0.30/0.49 thf(fact_729_le__divide__eq__1__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_divide_eq_1_neg
% 0.30/0.49 thf(fact_730_le__divide__eq__1__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_divide_eq_1_pos
% 0.30/0.49 thf(fact_731_Suc__diff__1,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.49 = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_diff_1
% 0.30/0.49 thf(fact_732_ceiling__le__zero,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.30/0.49 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_le_zero
% 0.30/0.49 thf(fact_733_zero__less__ceiling,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.49 = ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_less_ceiling
% 0.30/0.49 thf(fact_734_zero__less__nat__eq,axiom,
% 0.30/0.49 ! [Z3: int] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z3 ) )
% 0.30/0.49 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_less_nat_eq
% 0.30/0.49 thf(fact_735_ceiling__less__one,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 0.30/0.49 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_less_one
% 0.30/0.49 thf(fact_736_one__le__ceiling,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.49 = ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_ceiling
% 0.30/0.49 thf(fact_737_ceiling__le__one,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 0.30/0.49 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_le_one
% 0.30/0.49 thf(fact_738_one__less__ceiling,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] :
% 0.30/0.49 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.49 = ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_ceiling
% 0.30/0.49 thf(fact_739_less__fun__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( ord @ B )
% 0.30/0.49 => ( ( ord_less @ ( A > B ) )
% 0.30/0.49 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.49 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.30/0.49 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_fun_def
% 0.30/0.49 thf(fact_740_fun__diff__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( minus @ B )
% 0.30/0.49 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.49 = ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % fun_diff_def
% 0.30/0.49 thf(fact_741_mod__eq__0__iff__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_eq_0_iff_dvd
% 0.30/0.49 thf(fact_742_dvd__eq__mod__eq__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ( ( dvd_dvd @ A )
% 0.30/0.49 = ( ^ [A5: A,B3: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ B3 @ A5 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_eq_mod_eq_0
% 0.30/0.49 thf(fact_743_mod__0__imp__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_0_imp_dvd
% 0.30/0.49 thf(fact_744_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.49 = A3 )
% 0.30/0.49 = ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_eq_self_iff_div_eq_0
% 0.30/0.49 thf(fact_745_dvd__div__eq__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_eq_0_iff
% 0.30/0.49 thf(fact_746_dvd__diff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_ring_1 @ A )
% 0.30/0.49 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ X @ Y3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ X @ Z3 )
% 0.30/0.49 => ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_diff
% 0.30/0.49 thf(fact_747_dvd__refl,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_refl
% 0.30/0.49 thf(fact_748_dvd__trans,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ C2 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_trans
% 0.30/0.49 thf(fact_749_dvd__mod__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mod_iff
% 0.30/0.49 thf(fact_750_dvd__minus__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_minus_mod
% 0.30/0.49 thf(fact_751_dvd__div__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( ( divide_divide @ A @ A3 @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( A3 = B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_eq_iff
% 0.30/0.49 thf(fact_752_dvd__mod__imp__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_modulo @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mod_imp_dvd
% 0.30/0.49 thf(fact_753_dvd__div__eq__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semidom_divide @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ A3 @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( A3 = B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_eq_cancel
% 0.30/0.49 thf(fact_754_div__div__div__same,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [D2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ D2 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_div_div_same
% 0.30/0.49 thf(fact_755_diff__eq__diff__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.49 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.30/0.49 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.30/0.49 => ( ( A3 = B2 )
% 0.30/0.49 = ( C2 = D2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_eq_diff_eq
% 0.30/0.49 thf(fact_756_diff__right__commute,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 0.30/0.49 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_right_commute
% 0.30/0.49 thf(fact_757_dvd__diff__commute,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid5891614535332579305n_ring @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_diff_commute
% 0.30/0.49 thf(fact_758_mod__diff__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_diff_eq
% 0.30/0.49 thf(fact_759_dvd__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [K: A,M: A,N: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ K @ M )
% 0.30/0.49 => ( ( dvd_dvd @ A @ K @ N )
% 0.30/0.49 => ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mod
% 0.30/0.49 thf(fact_760_mod__diff__cong,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 0.30/0.49 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 0.30/0.49 => ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_diff_cong
% 0.30/0.49 thf(fact_761_mod__eq__dvd__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_eq_dvd_iff
% 0.30/0.49 thf(fact_762_mod__diff__left__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_diff_left_eq
% 0.30/0.49 thf(fact_763_mod__diff__right__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_diff_right_eq
% 0.30/0.49 thf(fact_764_mod__mod__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mod_cancel
% 0.30/0.49 thf(fact_765_diff__commute,axiom,
% 0.30/0.49 ! [I: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.30/0.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_commute
% 0.30/0.49 thf(fact_766_dvd__diff__nat,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ K @ M )
% 0.30/0.49 => ( ( dvd_dvd @ nat @ K @ N )
% 0.30/0.49 => ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_diff_nat
% 0.30/0.49 thf(fact_767_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field_char_0 @ A )
% 0.30/0.49 => ! [M: nat,N: nat] :
% 0.30/0.49 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % field_char_0_class.of_nat_div
% 0.30/0.49 thf(fact_768_diff__divide__distrib,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_divide_distrib
% 0.30/0.49 thf(fact_769_modulo__nat__def,axiom,
% 0.30/0.49 ( ( modulo_modulo @ nat )
% 0.30/0.49 = ( ^ [M2: nat,N3: nat] : ( minus_minus @ nat @ M2 @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % modulo_nat_def
% 0.30/0.49 thf(fact_770_minus__mult__div__eq__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 0.30/0.49 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_mult_div_eq_mod
% 0.30/0.49 thf(fact_771_minus__mod__eq__mult__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_mod_eq_mult_div
% 0.30/0.49 thf(fact_772_minus__mod__eq__div__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_mod_eq_div_mult
% 0.30/0.49 thf(fact_773_minus__div__mult__eq__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_modulo @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 0.30/0.49 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_div_mult_eq_mod
% 0.30/0.49 thf(fact_774_div__mult__div__if__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,D2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ D2 @ C2 )
% 0.30/0.49 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_div_if_dvd
% 0.30/0.49 thf(fact_775_dvd__mult__imp__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_imp_div
% 0.30/0.49 thf(fact_776_dvd__div__mult2__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_mult2_eq
% 0.30/0.49 thf(fact_777_div__div__eq__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_div_eq_right
% 0.30/0.49 thf(fact_778_div__mult__swap,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_swap
% 0.30/0.49 thf(fact_779_dvd__div__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_mult
% 0.30/0.49 thf(fact_780_unit__div__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.30/0.49 = ( divide_divide @ A @ C2 @ A3 ) )
% 0.30/0.49 = ( B2 = C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_cancel
% 0.30/0.49 thf(fact_781_div__unit__dvd__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_unit_dvd_iff
% 0.30/0.49 thf(fact_782_dvd__div__unit__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_unit_iff
% 0.30/0.49 thf(fact_783_of__nat__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.49 => ! [M: nat,N: nat] :
% 0.30/0.49 ( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
% 0.30/0.49 = ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % of_nat_mod
% 0.30/0.49 thf(fact_784_dvd__neg__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_divide @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.49 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_neg_div
% 0.30/0.49 thf(fact_785_dvd__div__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_divide @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_neg
% 0.30/0.49 thf(fact_786_mod__if,axiom,
% 0.30/0.49 ( ( modulo_modulo @ nat )
% 0.30/0.49 = ( ^ [M2: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N3 ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_if
% 0.30/0.49 thf(fact_787_le__mod__geq,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ M @ N )
% 0.30/0.49 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_mod_geq
% 0.30/0.49 thf(fact_788_dvd__minus__self,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.49 = ( ( ord_less @ nat @ N @ M )
% 0.30/0.49 | ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_minus_self
% 0.30/0.49 thf(fact_789_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.49 => ! [M: nat,N: nat] :
% 0.30/0.49 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 0.30/0.49 = ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.30/0.49 thf(fact_790_of__nat__dvd__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.49 => ! [M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.49 = ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % of_nat_dvd_iff
% 0.30/0.49 thf(fact_791_less__eq__dvd__minus,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ( dvd_dvd @ nat @ M @ N )
% 0.30/0.49 = ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_eq_dvd_minus
% 0.30/0.49 thf(fact_792_dvd__diffD1,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.49 => ( ( dvd_dvd @ nat @ K @ M )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_diffD1
% 0.30/0.49 thf(fact_793_dvd__diffD,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.49 => ( ( dvd_dvd @ nat @ K @ N )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_diffD
% 0.30/0.49 thf(fact_794_mod__nat__eqI,axiom,
% 0.30/0.49 ! [R: nat,N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ R @ N )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ R @ M )
% 0.30/0.49 => ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R ) )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ M @ N )
% 0.30/0.49 = R ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_nat_eqI
% 0.30/0.49 thf(fact_795_unit__imp__mod__eq__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_imp_mod_eq_0
% 0.30/0.49 thf(fact_796_of__nat__ceiling,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % of_nat_ceiling
% 0.30/0.49 thf(fact_797_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( Z3
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 0.30/0.49 = A3 ) )
% 0.30/0.49 & ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_divide_eq_if_simps(4)
% 0.30/0.49 thf(fact_798_diff__frac__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 0.30/0.49 ( ( Y3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_frac_eq
% 0.30/0.49 thf(fact_799_diff__divide__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.49 ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_divide_eq_iff
% 0.30/0.49 thf(fact_800_divide__diff__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.49 ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_diff_eq_iff
% 0.30/0.49 thf(fact_801_dvd__div__eq__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.30/0.49 = C2 )
% 0.30/0.49 = ( B2
% 0.30/0.49 = ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_eq_mult
% 0.30/0.49 thf(fact_802_div__dvd__iff__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_dvd_iff_mult
% 0.30/0.49 thf(fact_803_dvd__div__iff__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_iff_mult
% 0.30/0.49 thf(fact_804_dvd__div__div__eq__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ D2 )
% 0.30/0.49 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.30/0.49 = ( divide_divide @ A @ D2 @ C2 ) )
% 0.30/0.49 = ( ( times_times @ A @ B2 @ C2 )
% 0.30/0.49 = ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_div_div_eq_mult
% 0.30/0.49 thf(fact_805_unit__div__eq__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_eq_0_iff
% 0.30/0.49 thf(fact_806_unit__eq__div1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = C2 )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_eq_div1
% 0.30/0.49 thf(fact_807_unit__eq__div2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( A3
% 0.30/0.49 = ( divide_divide @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.49 = C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_eq_div2
% 0.30/0.49 thf(fact_808_div__mult__unit2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult_unit2
% 0.30/0.49 thf(fact_809_unit__div__commute,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_commute
% 0.30/0.49 thf(fact_810_unit__div__mult__swap,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_div_mult_swap
% 0.30/0.49 thf(fact_811_is__unit__div__mult2__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unit_div_mult2_eq
% 0.30/0.49 thf(fact_812_div__less__mono,axiom,
% 0.30/0.49 ! [A6: nat,B7: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ A6 @ B7 )
% 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( ( modulo_modulo @ nat @ A6 @ N )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 => ( ( ( modulo_modulo @ nat @ B7 @ N )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 => ( ord_less @ nat @ ( divide_divide @ nat @ A6 @ N ) @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_less_mono
% 0.30/0.49 thf(fact_813_mod__greater__zero__iff__not__dvd,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
% 0.30/0.49 = ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_greater_zero_iff_not_dvd
% 0.30/0.49 thf(fact_814_of__nat__diff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( semiring_1_cancel @ A )
% 0.30/0.49 => ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.49 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % of_nat_diff
% 0.30/0.49 thf(fact_815_division__segment__mod,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ~ ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % division_segment_mod
% 0.30/0.49 thf(fact_816_mod__mult__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_eq
% 0.30/0.49 thf(fact_817_mod__mult__cong,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 0.30/0.49 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 0.30/0.49 => ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_cong
% 0.30/0.49 thf(fact_818_mod__mult__mult2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_mult2
% 0.30/0.49 thf(fact_819_mult__mod__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_mod_right
% 0.30/0.49 thf(fact_820_mod__mult__left__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_left_eq
% 0.30/0.49 thf(fact_821_mod__mult__right__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_mult_right_eq
% 0.30/0.49 thf(fact_822_diff__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,D2: A,C2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_mono
% 0.30/0.49 thf(fact_823_diff__left__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_left_mono
% 0.30/0.49 thf(fact_824_diff__right__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_right_mono
% 0.30/0.49 thf(fact_825_diff__eq__diff__less__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.49 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.30/0.49 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.49 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_eq_diff_less_eq
% 0.30/0.49 thf(fact_826_mod__minus__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_minus_eq
% 0.30/0.49 thf(fact_827_mod__minus__cong,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A,A4: A] :
% 0.30/0.49 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ A4 @ B2 ) )
% 0.30/0.49 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.49 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A4 ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_minus_cong
% 0.30/0.49 thf(fact_828_mod__minus__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_minus_right
% 0.30/0.49 thf(fact_829_eq__iff__diff__eq__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( group_add @ A )
% 0.30/0.49 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.49 = ( ^ [A5: A,B3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ A5 @ B3 )
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_iff_diff_eq_0
% 0.30/0.49 thf(fact_830_diff__strict__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,D2: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less @ A @ D2 @ C2 )
% 0.30/0.49 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_strict_mono
% 0.30/0.49 thf(fact_831_diff__eq__diff__less,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.49 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.30/0.49 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.30/0.49 => ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.49 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_eq_diff_less
% 0.30/0.49 thf(fact_832_diff__strict__left__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.49 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_strict_left_mono
% 0.30/0.49 thf(fact_833_diff__strict__right__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.49 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_strict_right_mono
% 0.30/0.49 thf(fact_834_right__diff__distrib_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % right_diff_distrib'
% 0.30/0.49 thf(fact_835_left__diff__distrib_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_diff_distrib'
% 0.30/0.49 thf(fact_836_right__diff__distrib,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % right_diff_distrib
% 0.30/0.49 thf(fact_837_left__diff__distrib,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_diff_distrib
% 0.30/0.49 thf(fact_838_inf__period_I1_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ( comm_ring @ A )
% 0.30/0.49 & ( dvd @ A ) )
% 0.30/0.49 => ! [P: A > $o,D4: A,Q: A > $o] :
% 0.30/0.49 ( ! [X3: A,K2: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 0.30/0.49 => ( ! [X3: A,K2: A] :
% 0.30/0.49 ( ( Q @ X3 )
% 0.30/0.49 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 0.30/0.49 => ! [X4: A,K4: A] :
% 0.30/0.49 ( ( ( P @ X4 )
% 0.30/0.49 & ( Q @ X4 ) )
% 0.30/0.49 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
% 0.30/0.49 & ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inf_period(1)
% 0.30/0.49 thf(fact_839_inf__period_I2_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ( comm_ring @ A )
% 0.30/0.49 & ( dvd @ A ) )
% 0.30/0.49 => ! [P: A > $o,D4: A,Q: A > $o] :
% 0.30/0.49 ( ! [X3: A,K2: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 0.30/0.49 => ( ! [X3: A,K2: A] :
% 0.30/0.49 ( ( Q @ X3 )
% 0.30/0.49 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 0.30/0.49 => ! [X4: A,K4: A] :
% 0.30/0.49 ( ( ( P @ X4 )
% 0.30/0.49 | ( Q @ X4 ) )
% 0.30/0.49 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
% 0.30/0.49 | ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inf_period(2)
% 0.30/0.49 thf(fact_840_dvd__field__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ( ( dvd_dvd @ A )
% 0.30/0.49 = ( ^ [A5: A,B3: A] :
% 0.30/0.49 ( ( A5
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( B3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_field_iff
% 0.30/0.49 thf(fact_841_dvd__0__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_semiring_1 @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 => ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_0_left
% 0.30/0.49 thf(fact_842_mod__Suc__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
% 0.30/0.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_Suc_eq
% 0.30/0.49 thf(fact_843_mod__Suc__Suc__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
% 0.30/0.49 = ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_Suc_Suc_eq
% 0.30/0.49 thf(fact_844_minus__diff__commute,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ab_group_add @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
% 0.30/0.49 = ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_diff_commute
% 0.30/0.49 thf(fact_845_dvd__triv__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_triv_right
% 0.30/0.49 thf(fact_846_dvd__mult__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 => ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_right
% 0.30/0.49 thf(fact_847_mult__dvd__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ C2 @ D2 )
% 0.30/0.49 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_dvd_mono
% 0.30/0.49 thf(fact_848_dvd__triv__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_triv_left
% 0.30/0.49 thf(fact_849_dvd__mult__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_left
% 0.30/0.49 thf(fact_850_dvd__mult2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult2
% 0.30/0.49 thf(fact_851_dvd__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult
% 0.30/0.49 thf(fact_852_dvd__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( dvd @ A )
% 0.30/0.49 => ( ( dvd_dvd @ A )
% 0.30/0.49 = ( ^ [B3: A,A5: A] :
% 0.30/0.49 ? [K3: A] :
% 0.30/0.49 ( A5
% 0.30/0.49 = ( times_times @ A @ B3 @ K3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_def
% 0.30/0.49 thf(fact_853_dvdI,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( dvd @ A )
% 0.30/0.49 => ! [A3: A,B2: A,K: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( times_times @ A @ B2 @ K ) )
% 0.30/0.49 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvdI
% 0.30/0.49 thf(fact_854_dvdE,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( dvd @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ~ ! [K2: A] :
% 0.30/0.49 ( A3
% 0.30/0.49 != ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvdE
% 0.30/0.49 thf(fact_855_one__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comm_monoid_mult @ A )
% 0.30/0.49 => ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_dvd
% 0.30/0.49 thf(fact_856_unit__imp__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_imp_dvd
% 0.30/0.49 thf(fact_857_dvd__unit__imp__unit,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_unit_imp_unit
% 0.30/0.49 thf(fact_858_divide__divide__eq__left_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_divide_eq_left'
% 0.30/0.49 thf(fact_859_divide__divide__times__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [X: A,Y3: A,Z3: A,W2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_divide_times_eq
% 0.30/0.49 thf(fact_860_times__divide__times__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [X: A,Y3: A,Z3: A,W2: A] :
% 0.30/0.49 ( ( times_times @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ W2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % times_divide_times_eq
% 0.30/0.49 thf(fact_861_mod__less__eq__dividend,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % mod_less_eq_dividend
% 0.30/0.49 thf(fact_862_minus__divide__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_divide_left
% 0.30/0.49 thf(fact_863_minus__divide__divide,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_divide_divide
% 0.30/0.49 thf(fact_864_minus__divide__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_divide_right
% 0.30/0.49 thf(fact_865_div__minus__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_minus_right
% 0.30/0.49 thf(fact_866_zero__induct__lemma,axiom,
% 0.30/0.49 ! [P: nat > $o,K: nat,I: nat] :
% 0.30/0.49 ( ( P @ K )
% 0.30/0.49 => ( ! [N4: nat] :
% 0.30/0.49 ( ( P @ ( suc @ N4 ) )
% 0.30/0.49 => ( P @ N4 ) )
% 0.30/0.49 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_induct_lemma
% 0.30/0.49 thf(fact_867_minus__nat_Odiff__0,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.30/0.49 = M ) ).
% 0.30/0.49
% 0.30/0.49 % minus_nat.diff_0
% 0.30/0.49 thf(fact_868_diffs0__imp__equal,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ( minus_minus @ nat @ M @ N )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 => ( ( ( minus_minus @ nat @ N @ M )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 => ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diffs0_imp_equal
% 0.30/0.49 thf(fact_869_sgn__eq__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_eq_0_iff
% 0.30/0.49 thf(fact_870_sgn__0__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_0_0
% 0.30/0.49 thf(fact_871_diff__less__mono2,axiom,
% 0.30/0.49 ! [M: nat,N: nat,L: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 => ( ( ord_less @ nat @ M @ L )
% 0.30/0.49 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_less_mono2
% 0.30/0.49 thf(fact_872_less__imp__diff__less,axiom,
% 0.30/0.49 ! [J: nat,K: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ J @ K )
% 0.30/0.49 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_imp_diff_less
% 0.30/0.49 thf(fact_873_div__le__dividend,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % div_le_dividend
% 0.30/0.49 thf(fact_874_div__le__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_le_mono
% 0.30/0.49 thf(fact_875_diff__le__mono2,axiom,
% 0.30/0.49 ! [M: nat,N: nat,L: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_le_mono2
% 0.30/0.49 thf(fact_876_le__diff__iff_H,axiom,
% 0.30/0.49 ! [A3: nat,C2: nat,B2: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ A3 @ C2 )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 0.30/0.49 = ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_diff_iff'
% 0.30/0.49 thf(fact_877_diff__le__self,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % diff_le_self
% 0.30/0.49 thf(fact_878_diff__le__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat,L: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_le_mono
% 0.30/0.49 thf(fact_879_Nat_Odiff__diff__eq,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.49 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.30/0.49 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Nat.diff_diff_eq
% 0.30/0.49 thf(fact_880_le__diff__iff,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.30/0.49 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_diff_iff
% 0.30/0.49 thf(fact_881_eq__diff__iff,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.49 => ( ( ( minus_minus @ nat @ M @ K )
% 0.30/0.49 = ( minus_minus @ nat @ N @ K ) )
% 0.30/0.49 = ( M = N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_diff_iff
% 0.30/0.49 thf(fact_882_sgn__mult,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_mult
% 0.30/0.49 thf(fact_883_div__mult2__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.49 ( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
% 0.30/0.49 = ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult2_eq
% 0.30/0.49 thf(fact_884_diff__mult__distrib2,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.49 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_mult_distrib2
% 0.30/0.49 thf(fact_885_diff__mult__distrib,axiom,
% 0.30/0.49 ! [M: nat,N: nat,K: nat] :
% 0.30/0.49 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 0.30/0.49 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_mult_distrib
% 0.30/0.49 thf(fact_886_uminus__dvd__conv_I2_J,axiom,
% 0.30/0.49 ( ( dvd_dvd @ int )
% 0.30/0.49 = ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % uminus_dvd_conv(2)
% 0.30/0.49 thf(fact_887_uminus__dvd__conv_I1_J,axiom,
% 0.30/0.49 ( ( dvd_dvd @ int )
% 0.30/0.49 = ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % uminus_dvd_conv(1)
% 0.30/0.49 thf(fact_888_nat__dvd__iff,axiom,
% 0.30/0.49 ! [Z3: int,M: nat] :
% 0.30/0.49 ( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M )
% 0.30/0.49 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M ) ) )
% 0.30/0.49 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( M
% 0.30/0.49 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_dvd_iff
% 0.30/0.49 thf(fact_889_frac__le__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 0.30/0.49 ( ( Y3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 0.30/0.49 = ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_le_eq
% 0.30/0.49 thf(fact_890_frac__less__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 0.30/0.49 ( ( Y3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 0.30/0.49 = ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_less_eq
% 0.30/0.49 thf(fact_891_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( Z3
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 0.30/0.49 = ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.49 & ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_divide_eq_if_simps(6)
% 0.30/0.49 thf(fact_892_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ( Z3
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 0.30/0.49 = ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.49 & ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_divide_eq_if_simps(5)
% 0.30/0.49 thf(fact_893_minus__divide__diff__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.49 ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
% 0.30/0.49 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minus_divide_diff_eq_iff
% 0.30/0.49 thf(fact_894_is__unitE,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ~ ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ! [B5: A] :
% 0.30/0.49 ( ( B5
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.49 = B5 )
% 0.30/0.49 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
% 0.30/0.49 = A3 )
% 0.30/0.49 => ( ( ( times_times @ A @ A3 @ B5 )
% 0.30/0.49 = ( one_one @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ C2 @ A3 )
% 0.30/0.49 != ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unitE
% 0.30/0.49 thf(fact_895_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unit_div_mult_cancel_left
% 0.30/0.49 thf(fact_896_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unit_div_mult_cancel_right
% 0.30/0.49 thf(fact_897_div__if,axiom,
% 0.30/0.49 ( ( divide_divide @ nat )
% 0.30/0.49 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.49 ( if @ nat
% 0.30/0.49 @ ( ( ord_less @ nat @ M2 @ N3 )
% 0.30/0.49 | ( N3
% 0.30/0.49 = ( zero_zero @ nat ) ) )
% 0.30/0.49 @ ( zero_zero @ nat )
% 0.30/0.49 @ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_if
% 0.30/0.49 thf(fact_898_le__div__geq,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( ( divide_divide @ nat @ M @ N )
% 0.30/0.49 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_div_geq
% 0.30/0.49 thf(fact_899_nat__zero__as__int,axiom,
% 0.30/0.49 ( ( zero_zero @ nat )
% 0.30/0.49 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_zero_as_int
% 0.30/0.49 thf(fact_900_le__iff__diff__le__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ( ( ord_less_eq @ A )
% 0.30/0.49 = ( ^ [A5: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_iff_diff_le_0
% 0.30/0.49 thf(fact_901_less__iff__diff__less__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordered_ab_group_add @ A )
% 0.30/0.49 => ( ( ord_less @ A )
% 0.30/0.49 = ( ^ [A5: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_iff_diff_less_0
% 0.30/0.49 thf(fact_902_int__sgnE,axiom,
% 0.30/0.49 ! [K: int] :
% 0.30/0.49 ~ ! [N4: nat,L2: int] :
% 0.30/0.49 ( K
% 0.30/0.49 != ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % int_sgnE
% 0.30/0.49 thf(fact_903_ex__nat,axiom,
% 0.30/0.49 ( ( ^ [P3: nat > $o] :
% 0.30/0.49 ? [X6: nat] : ( P3 @ X6 ) )
% 0.30/0.49 = ( ^ [P4: nat > $o] :
% 0.30/0.49 ? [X2: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
% 0.30/0.49 & ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_nat
% 0.30/0.49 thf(fact_904_all__nat,axiom,
% 0.30/0.49 ( ( ^ [P3: nat > $o] :
% 0.30/0.49 ! [X6: nat] : ( P3 @ X6 ) )
% 0.30/0.49 = ( ^ [P4: nat > $o] :
% 0.30/0.49 ! [X2: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
% 0.30/0.49 => ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nat
% 0.30/0.49 thf(fact_905_eq__nat__nat__iff,axiom,
% 0.30/0.49 ! [Z3: int,Z6: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 0.30/0.49 => ( ( ( nat2 @ Z3 )
% 0.30/0.49 = ( nat2 @ Z6 ) )
% 0.30/0.49 = ( Z3 = Z6 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_nat_nat_iff
% 0.30/0.49 thf(fact_906_divide__le__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.30/0.49 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_le_0_iff
% 0.30/0.49 thf(fact_907_divide__right__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_right_mono
% 0.30/0.49 thf(fact_908_zero__le__divide__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.30/0.49 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_le_divide_iff
% 0.30/0.49 thf(fact_909_divide__nonneg__nonneg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonneg_nonneg
% 0.30/0.49 thf(fact_910_divide__nonneg__nonpos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonneg_nonpos
% 0.30/0.49 thf(fact_911_divide__nonpos__nonneg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonpos_nonneg
% 0.30/0.49 thf(fact_912_divide__nonpos__nonpos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonpos_nonpos
% 0.30/0.49 thf(fact_913_divide__right__mono__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_right_mono_neg
% 0.30/0.49 thf(fact_914_nat__mono,axiom,
% 0.30/0.49 ! [X: int,Y3: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ X @ Y3 )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_mono
% 0.30/0.49 thf(fact_915_divide__neg__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_neg_neg
% 0.30/0.49 thf(fact_916_divide__neg__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_neg_pos
% 0.30/0.49 thf(fact_917_divide__pos__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_pos_neg
% 0.30/0.49 thf(fact_918_divide__pos__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_pos_pos
% 0.30/0.49 thf(fact_919_divide__less__0__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.30/0.49 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_0_iff
% 0.30/0.49 thf(fact_920_divide__less__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ B2 @ A3 ) )
% 0.30/0.49 & ( C2
% 0.30/0.49 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_cancel
% 0.30/0.49 thf(fact_921_zero__less__divide__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.30/0.49 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zero_less_divide_iff
% 0.30/0.49 thf(fact_922_divide__strict__right__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_strict_right_mono
% 0.30/0.49 thf(fact_923_divide__strict__right__mono__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_strict_right_mono_neg
% 0.30/0.49 thf(fact_924_not__is__unit__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_is_unit_0
% 0.30/0.49 thf(fact_925_frac__eq__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( field @ A )
% 0.30/0.49 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 0.30/0.49 ( ( Y3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( Z3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ X @ Y3 )
% 0.30/0.49 = ( divide_divide @ A @ W2 @ Z3 ) )
% 0.30/0.49 = ( ( times_times @ A @ X @ Z3 )
% 0.30/0.49 = ( times_times @ A @ W2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_eq_eq
% 0.30/0.49 thf(fact_926_divide__eq__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( ( divide_divide @ A @ B2 @ C2 )
% 0.30/0.49 = A3 )
% 0.30/0.49 = ( ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( B2
% 0.30/0.49 = ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.49 & ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_eq_eq
% 0.30/0.49 thf(fact_927_eq__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.49 = B2 ) )
% 0.30/0.49 & ( ( C2
% 0.30/0.49 = ( zero_zero @ A ) )
% 0.30/0.49 => ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_divide_eq
% 0.30/0.49 thf(fact_928_divide__eq__imp,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( B2
% 0.30/0.49 = ( times_times @ A @ A3 @ C2 ) )
% 0.30/0.49 => ( ( divide_divide @ A @ B2 @ C2 )
% 0.30/0.49 = A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_eq_imp
% 0.30/0.49 thf(fact_929_eq__divide__imp,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.49 = B2 )
% 0.30/0.49 => ( A3
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_divide_imp
% 0.30/0.49 thf(fact_930_nonzero__divide__eq__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ B2 @ C2 )
% 0.30/0.49 = A3 )
% 0.30/0.49 = ( B2
% 0.30/0.49 = ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_divide_eq_eq
% 0.30/0.49 thf(fact_931_nonzero__eq__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( A3
% 0.30/0.49 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.49 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_eq_divide_eq
% 0.30/0.49 thf(fact_932_right__inverse__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.49 = ( one_one @ A ) )
% 0.30/0.49 = ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % right_inverse_eq
% 0.30/0.49 thf(fact_933_mod__Suc,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
% 0.30/0.49 = N )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( zero_zero @ nat ) ) )
% 0.30/0.49 & ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
% 0.30/0.49 != N )
% 0.30/0.49 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_Suc
% 0.30/0.49 thf(fact_934_is__unit__mult__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unit_mult_iff
% 0.30/0.49 thf(fact_935_dvd__mult__unit__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_unit_iff
% 0.30/0.49 thf(fact_936_mult__unit__dvd__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_unit_dvd_iff
% 0.30/0.49 thf(fact_937_dvd__mult__unit__iff_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_mult_unit_iff'
% 0.30/0.49 thf(fact_938_mult__unit__dvd__iff_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_unit_dvd_iff'
% 0.30/0.49 thf(fact_939_unit__mult__left__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.49 = ( times_times @ A @ A3 @ C2 ) )
% 0.30/0.49 = ( B2 = C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_mult_left_cancel
% 0.30/0.49 thf(fact_940_unit__mult__right__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( ( times_times @ A @ B2 @ A3 )
% 0.30/0.49 = ( times_times @ A @ C2 @ A3 ) )
% 0.30/0.49 = ( B2 = C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_mult_right_cancel
% 0.30/0.49 thf(fact_941_nonzero__minus__divide__right,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_minus_divide_right
% 0.30/0.49 thf(fact_942_nonzero__minus__divide__divide,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( B2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.49 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nonzero_minus_divide_divide
% 0.30/0.49 thf(fact_943_mod__induct,axiom,
% 0.30/0.49 ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 0.30/0.49 ( ( P @ N )
% 0.30/0.49 => ( ( ord_less @ nat @ N @ P5 )
% 0.30/0.49 => ( ( ord_less @ nat @ M @ P5 )
% 0.30/0.49 => ( ! [N4: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ N4 @ P5 )
% 0.30/0.49 => ( ( P @ N4 )
% 0.30/0.49 => ( P @ ( modulo_modulo @ nat @ ( suc @ N4 ) @ P5 ) ) ) )
% 0.30/0.49 => ( P @ M ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_induct
% 0.30/0.49 thf(fact_944_nat__one__as__int,axiom,
% 0.30/0.49 ( ( one_one @ nat )
% 0.30/0.49 = ( nat2 @ ( one_one @ int ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_one_as_int
% 0.30/0.49 thf(fact_945_mod__less__divisor,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % mod_less_divisor
% 0.30/0.49 thf(fact_946_mod__Suc__le__divisor,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.30/0.49
% 0.30/0.49 % mod_Suc_le_divisor
% 0.30/0.49 thf(fact_947_ceiling__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [Y3: A,X: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.49 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y3 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_mono
% 0.30/0.49 thf(fact_948_ceiling__less__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) )
% 0.30/0.49 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_less_cancel
% 0.30/0.49 thf(fact_949_div__mult2__eq_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.49 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.49 ( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 0.30/0.49 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_mult2_eq'
% 0.30/0.49 thf(fact_950_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ( divide_divide @ nat @ M @ N )
% 0.30/0.49 = ( zero_zero @ nat ) )
% 0.30/0.49 = ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 | ( N
% 0.30/0.49 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Euclidean_Division.div_eq_0_iff
% 0.30/0.49 thf(fact_951_Suc__diff__Suc,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ N @ M )
% 0.30/0.49 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.49 = ( minus_minus @ nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_diff_Suc
% 0.30/0.49 thf(fact_952_diff__less__Suc,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_less_Suc
% 0.30/0.49 thf(fact_953_nat__dvd__not__less,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.49 => ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 => ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_dvd_not_less
% 0.30/0.49 thf(fact_954_Suc__div__le__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_div_le_mono
% 0.30/0.49 thf(fact_955_diff__less,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.49 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_less
% 0.30/0.49 thf(fact_956_Suc__diff__le,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_diff_le
% 0.30/0.49 thf(fact_957_less__diff__iff,axiom,
% 0.30/0.49 ! [K: nat,M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.49 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.30/0.49 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_diff_iff
% 0.30/0.49 thf(fact_958_diff__less__mono,axiom,
% 0.30/0.49 ! [A3: nat,B2: nat,C2: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ C2 @ A3 )
% 0.30/0.49 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_less_mono
% 0.30/0.49 thf(fact_959_zdvd__antisym__nonneg,axiom,
% 0.30/0.49 ! [M: int,N: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
% 0.30/0.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.49 => ( ( dvd_dvd @ int @ M @ N )
% 0.30/0.49 => ( ( dvd_dvd @ int @ N @ M )
% 0.30/0.49 => ( M = N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zdvd_antisym_nonneg
% 0.30/0.49 thf(fact_960_sgn__not__eq__imp,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( ( sgn_sgn @ A @ B2 )
% 0.30/0.49 != ( sgn_sgn @ A @ A3 ) )
% 0.30/0.49 => ( ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( sgn_sgn @ A @ B2 )
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( sgn_sgn @ A @ A3 )
% 0.30/0.49 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_not_eq_imp
% 0.30/0.49 thf(fact_961_zdvd__not__zless,axiom,
% 0.30/0.49 ! [M: int,N: int] :
% 0.30/0.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ M )
% 0.30/0.49 => ( ( ord_less @ int @ M @ N )
% 0.30/0.49 => ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zdvd_not_zless
% 0.30/0.49 thf(fact_962_less__mult__imp__div__less,axiom,
% 0.30/0.49 ! [M: nat,I: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
% 0.30/0.49 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_mult_imp_div_less
% 0.30/0.49 thf(fact_963_times__div__less__eq__dividend,axiom,
% 0.30/0.49 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % times_div_less_eq_dividend
% 0.30/0.49 thf(fact_964_div__times__less__eq__dividend,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % div_times_less_eq_dividend
% 0.30/0.49 thf(fact_965_zdvd__mult__cancel,axiom,
% 0.30/0.49 ! [K: int,M: int,N: int] :
% 0.30/0.49 ( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) )
% 0.30/0.49 => ( ( K
% 0.30/0.49 != ( zero_zero @ int ) )
% 0.30/0.49 => ( dvd_dvd @ int @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zdvd_mult_cancel
% 0.30/0.49 thf(fact_966_zdvd__mono,axiom,
% 0.30/0.49 ! [K: int,M: int,T2: int] :
% 0.30/0.49 ( ( K
% 0.30/0.49 != ( zero_zero @ int ) )
% 0.30/0.49 => ( ( dvd_dvd @ int @ M @ T2 )
% 0.30/0.49 = ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % zdvd_mono
% 0.30/0.49 thf(fact_967_diff__Suc__eq__diff__pred,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( minus_minus @ nat @ M @ ( suc @ N ) )
% 0.30/0.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % diff_Suc_eq_diff_pred
% 0.30/0.49 thf(fact_968_sgn__minus__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( idom_abs_sgn @ A )
% 0.30/0.49 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.49 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sgn_minus_1
% 0.30/0.49 thf(fact_969_dvd__euclidean__size__eq__imp__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.30/0.49 = ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.49 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.49 => ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dvd_euclidean_size_eq_imp_dvd
% 0.30/0.49 thf(fact_970_euclidean__size__unit,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.49 => ! [A3: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.30/0.49 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % euclidean_size_unit
% 0.30/0.49 thf(fact_971_fact__dvd,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_semidom @ A )
% 0.30/0.49 => ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.49 => ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % fact_dvd
% 0.30/0.49 thf(fact_972_is__unit__division__segment,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.49 => ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_unit_division_segment
% 0.30/0.49 thf(fact_973_division__segment__eq__sgn,axiom,
% 0.30/0.49 ! [K: int] :
% 0.30/0.49 ( ( K
% 0.30/0.49 != ( zero_zero @ int ) )
% 0.30/0.49 => ( ( euclid7384307370059645450egment @ int @ K )
% 0.30/0.49 = ( sgn_sgn @ int @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % division_segment_eq_sgn
% 0.30/0.49 thf(fact_974_ceiling__ge__round,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ceiling_ge_round
% 0.30/0.49 thf(fact_975_frac__ge__0,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_ge_0
% 0.30/0.49 thf(fact_976_frac__lt__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.49 => ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_lt_1
% 0.30/0.49 thf(fact_977_nat__mono__iff,axiom,
% 0.30/0.49 ! [Z3: int,W2: int] :
% 0.30/0.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 0.30/0.49 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_mono_iff
% 0.30/0.49 thf(fact_978_frac__le,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [Y3: A,X: A,W2: A,Z3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ W2 @ Z3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_le
% 0.30/0.49 thf(fact_979_frac__less,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A,W2: A,Z3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 0.30/0.49 => ( ( ord_less_eq @ A @ W2 @ Z3 )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_less
% 0.30/0.49 thf(fact_980_frac__less2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A,W2: A,Z3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 0.30/0.49 => ( ( ord_less @ A @ W2 @ Z3 )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % frac_less2
% 0.30/0.49 thf(fact_981_divide__le__cancel,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_le_cancel
% 0.30/0.49 thf(fact_982_divide__nonneg__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonneg_neg
% 0.30/0.49 thf(fact_983_divide__nonneg__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonneg_pos
% 0.30/0.49 thf(fact_984_divide__nonpos__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonpos_neg
% 0.30/0.49 thf(fact_985_divide__nonpos__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [X: A,Y3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_nonpos_pos
% 0.30/0.49 thf(fact_986_divide__less__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 0.30/0.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_eq
% 0.30/0.49 thf(fact_987_less__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 0.30/0.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_divide_eq
% 0.30/0.49 thf(fact_988_neg__divide__less__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % neg_divide_less_eq
% 0.30/0.49 thf(fact_989_neg__less__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % neg_less_divide_eq
% 0.30/0.49 thf(fact_990_pos__divide__less__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.49 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pos_divide_less_eq
% 0.30/0.49 thf(fact_991_pos__less__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.49 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pos_less_divide_eq
% 0.30/0.49 thf(fact_992_mult__imp__div__pos__less,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [Y3: A,X: A,Z3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_imp_div_pos_less
% 0.30/0.49 thf(fact_993_mult__imp__less__div__pos,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
% 0.30/0.49 => ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mult_imp_less_div_pos
% 0.30/0.49 thf(fact_994_divide__strict__left__mono,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_strict_left_mono
% 0.30/0.49 thf(fact_995_divide__strict__left__mono__neg,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.49 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.49 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_strict_left_mono_neg
% 0.30/0.49 thf(fact_996_unit__dvdE,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( algebraic_semidom @ A )
% 0.30/0.49 => ! [A3: A,B2: A] :
% 0.30/0.49 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.49 => ~ ( ( A3
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ! [C3: A] :
% 0.30/0.49 ( B2
% 0.30/0.49 != ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % unit_dvdE
% 0.30/0.49 thf(fact_997_divide__less__eq__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less @ A @ B2 @ A3 ) )
% 0.30/0.49 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.49 | ( A3
% 0.30/0.49 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % divide_less_eq_1
% 0.30/0.49 thf(fact_998_less__divide__eq__1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A )
% 0.30/0.49 => ! [B2: A,A3: A] :
% 0.30/0.49 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.49 & ( ord_less @ A @ A3 @ B2 ) )
% 0.30/0.49 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.49 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_divide_eq_1
% 0.30/0.49 thf(fact_999_zless__nat__eq__int__zless,axiom,
% 0.30/0.49 ! [M: nat,Z3: int] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ ( nat2 @ Z3 ) )
% 0.30/0.49 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % zless_nat_eq_int_zless
% 0.30/0.49 thf(fact_1000_int__eq__iff,axiom,
% 0.30/0.49 ! [M: nat,Z3: int] :
% 0.30/0.49 ( ( ( semiring_1_of_nat @ int @ M )
% 0.30/0.49 = Z3 )
% 0.30/0.49 = ( ( M
% 0.30/0.49 = ( nat2 @ Z3 ) )
% 0.30/0.49 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % int_eq_iff
% 0.30/0.49 thf(fact_1001_nat__0__le,axiom,
% 0.30/0.49 ! [Z3: int] :
% 0.30/0.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.49 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.30/0.49 = Z3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_0_le
% 0.30/0.49 thf(fact_1002_nat__le__iff,axiom,
% 0.30/0.49 ! [X: int,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 0.30/0.49 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_le_iff
% 0.30/0.49 thf(fact_1003_eq__minus__divide__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( division_ring @ A )
% 0.30/0.49 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.49 = ( ( ( C2
% 0.30/0.49 != ( zero_zero @ A ) )
% 0.30/0.49 => ( ( times_times @ A @ A3 @ C2 )
% 0.30/0.50 = ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.50 & ( ( C2
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 => ( A3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_minus_divide_eq
% 0.30/0.50 thf(fact_1004_minus__divide__eq__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( division_ring @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.50 = A3 )
% 0.30/0.50 = ( ( ( C2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( uminus_uminus @ A @ B2 )
% 0.30/0.50 = ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ( C2
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 => ( A3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_divide_eq_eq
% 0.30/0.50 thf(fact_1005_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( division_ring @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.50 = C2 )
% 0.30/0.50 = ( ( uminus_uminus @ A @ A3 )
% 0.30/0.50 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nonzero_neg_divide_eq_eq
% 0.30/0.50 thf(fact_1006_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( division_ring @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( C2
% 0.30/0.50 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 0.30/0.50 = ( ( times_times @ A @ C2 @ B2 )
% 0.30/0.50 = ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nonzero_neg_divide_eq_eq2
% 0.30/0.50 thf(fact_1007_divide__eq__minus__1__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.50 = ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 & ( A3
% 0.30/0.50 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_eq_minus_1_iff
% 0.30/0.50 thf(fact_1008_mod__le__divisor,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_le_divisor
% 0.30/0.50 thf(fact_1009_fact__mod,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ( linordered_semidom @ A )
% 0.30/0.50 & ( semidom_modulo @ A ) )
% 0.30/0.50 => ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.50 => ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % fact_mod
% 0.30/0.50 thf(fact_1010_mod__size__less,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_size_less
% 0.30/0.50 thf(fact_1011_diff__Suc__less,axiom,
% 0.30/0.50 ! [N: nat,I: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_less
% 0.30/0.50 thf(fact_1012_div__greater__zero__iff,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 0.30/0.50 = ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_greater_zero_iff
% 0.30/0.50 thf(fact_1013_div__le__mono2,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.50 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_le_mono2
% 0.30/0.50 thf(fact_1014_sgn__1__pos,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ( sgn_sgn @ A @ A3 )
% 0.30/0.50 = ( one_one @ A ) )
% 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_1_pos
% 0.30/0.50 thf(fact_1015_dvd__imp__le,axiom,
% 0.30/0.50 ! [K: nat,N: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ K @ N )
% 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_imp_le
% 0.30/0.50 thf(fact_1016_nat__mult__div__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.50 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.50 = ( divide_divide @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_div_cancel1
% 0.30/0.50 thf(fact_1017_div__less__iff__less__mult,axiom,
% 0.30/0.50 ! [Q4: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
% 0.30/0.50 => ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N )
% 0.30/0.50 = ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q4 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_less_iff_less_mult
% 0.30/0.50 thf(fact_1018_nat__mult__dvd__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.50 = ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_dvd_cancel1
% 0.30/0.50 thf(fact_1019_dvd__mult__cancel,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.50 => ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_mult_cancel
% 0.30/0.50 thf(fact_1020_unit__iff__euclidean__size,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.50 = ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.30/0.50 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
% 0.30/0.50 & ( A3
% 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unit_iff_euclidean_size
% 0.30/0.50 thf(fact_1021_div__less__dividend,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_less_dividend
% 0.30/0.50 thf(fact_1022_div__eq__dividend__iff,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 => ( ( ( divide_divide @ nat @ M @ N )
% 0.30/0.50 = M )
% 0.30/0.50 = ( N
% 0.30/0.50 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_eq_dividend_iff
% 0.30/0.50 thf(fact_1023_euclidean__size__times__unit,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.50 => ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % euclidean_size_times_unit
% 0.30/0.50 thf(fact_1024_zdvd__imp__le,axiom,
% 0.30/0.50 ! [Z3: int,N: int] :
% 0.30/0.50 ( ( dvd_dvd @ int @ Z3 @ N )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.50 => ( ord_less_eq @ int @ Z3 @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdvd_imp_le
% 0.30/0.50 thf(fact_1025_dvd__proper__imp__size__less,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.50 => ( ~ ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.50 => ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_proper_imp_size_less
% 0.30/0.50 thf(fact_1026_dvd__imp__size__le,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_imp_size_le
% 0.30/0.50 thf(fact_1027_ex__gt__or__lt,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( condit5016429287641298734tinuum @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ? [B5: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B5 )
% 0.30/0.50 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ex_gt_or_lt
% 0.30/0.50 thf(fact_1028_of__nat__euclidean__size,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) )
% 0.30/0.50 = ( divide_divide @ A @ A3 @ ( euclid7384307370059645450egment @ A @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_nat_euclidean_size
% 0.30/0.50 thf(fact_1029_dvd__fact,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.50 => ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_fact
% 0.30/0.50 thf(fact_1030_zsgn__def,axiom,
% 0.30/0.50 ( ( sgn_sgn @ int )
% 0.30/0.50 = ( ^ [I5: int] :
% 0.30/0.50 ( if @ int
% 0.30/0.50 @ ( I5
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 @ ( zero_zero @ int )
% 0.30/0.50 @ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I5 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zsgn_def
% 0.30/0.50 thf(fact_1031_nat__less__eq__zless,axiom,
% 0.30/0.50 ! [W2: int,Z3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 0.30/0.50 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_eq_zless
% 0.30/0.50 thf(fact_1032_nat__le__eq__zle,axiom,
% 0.30/0.50 ! [W2: int,Z3: int] :
% 0.30/0.50 ( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 0.30/0.50 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_eq_zle
% 0.30/0.50 thf(fact_1033_nat__eq__iff,axiom,
% 0.30/0.50 ! [W2: int,M: nat] :
% 0.30/0.50 ( ( ( nat2 @ W2 )
% 0.30/0.50 = M )
% 0.30/0.50 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( W2
% 0.30/0.50 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.30/0.50 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( M
% 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_iff
% 0.30/0.50 thf(fact_1034_nat__eq__iff2,axiom,
% 0.30/0.50 ! [M: nat,W2: int] :
% 0.30/0.50 ( ( M
% 0.30/0.50 = ( nat2 @ W2 ) )
% 0.30/0.50 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( W2
% 0.30/0.50 = ( semiring_1_of_nat @ int @ M ) ) )
% 0.30/0.50 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( M
% 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_iff2
% 0.30/0.50 thf(fact_1035_divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_le_eq
% 0.30/0.50 thf(fact_1036_le__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_divide_eq
% 0.30/0.50 thf(fact_1037_divide__left__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_left_mono
% 0.30/0.50 thf(fact_1038_neg__divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.50 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_divide_le_eq
% 0.30/0.50 thf(fact_1039_neg__le__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_le_divide_eq
% 0.30/0.50 thf(fact_1040_pos__divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.50 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_divide_le_eq
% 0.30/0.50 thf(fact_1041_pos__le__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_le_divide_eq
% 0.30/0.50 thf(fact_1042_mult__imp__div__pos__le,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [Y3: A,X: A,Z3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_imp_div_pos_le
% 0.30/0.50 thf(fact_1043_mult__imp__le__div__pos,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
% 0.30/0.50 => ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_imp_le_div_pos
% 0.30/0.50 thf(fact_1044_divide__left__mono__neg,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_left_mono_neg
% 0.30/0.50 thf(fact_1045_split__nat,axiom,
% 0.30/0.50 ! [P: nat > $o,I: int] :
% 0.30/0.50 ( ( P @ ( nat2 @ I ) )
% 0.30/0.50 = ( ! [N3: nat] :
% 0.30/0.50 ( ( I
% 0.30/0.50 = ( semiring_1_of_nat @ int @ N3 ) )
% 0.30/0.50 => ( P @ N3 ) )
% 0.30/0.50 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
% 0.30/0.50 => ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % split_nat
% 0.30/0.50 thf(fact_1046_divide__le__eq__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 & ( ord_less_eq @ A @ B2 @ A3 ) )
% 0.30/0.50 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.50 | ( A3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_le_eq_1
% 0.30/0.50 thf(fact_1047_le__divide__eq__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.30/0.50 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_divide_eq_1
% 0.30/0.50 thf(fact_1048_le__nat__iff,axiom,
% 0.30/0.50 ! [K: int,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 0.30/0.50 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_nat_iff
% 0.30/0.50 thf(fact_1049_nat__mult__distrib,axiom,
% 0.30/0.50 ! [Z3: int,Z6: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.50 => ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
% 0.30/0.50 = ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_distrib
% 0.30/0.50 thf(fact_1050_less__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_minus_divide_eq
% 0.30/0.50 thf(fact_1051_minus__divide__less__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_divide_less_eq
% 0.30/0.50 thf(fact_1052_neg__less__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_less_minus_divide_eq
% 0.30/0.50 thf(fact_1053_neg__minus__divide__less__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_minus_divide_less_eq
% 0.30/0.50 thf(fact_1054_pos__less__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_less_minus_divide_eq
% 0.30/0.50 thf(fact_1055_pos__minus__divide__less__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_minus_divide_less_eq
% 0.30/0.50 thf(fact_1056_div__nat__eqI,axiom,
% 0.30/0.50 ! [N: nat,Q4: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M )
% 0.30/0.50 => ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
% 0.30/0.50 => ( ( divide_divide @ nat @ M @ N )
% 0.30/0.50 = Q4 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_nat_eqI
% 0.30/0.50 thf(fact_1057_less__eq__div__iff__mult__less__eq,axiom,
% 0.30/0.50 ! [Q4: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q4 ) )
% 0.30/0.50 = ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_eq_div_iff_mult_less_eq
% 0.30/0.50 thf(fact_1058_sgn__if,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ( ( sgn_sgn @ A )
% 0.30/0.50 = ( ^ [X2: A] :
% 0.30/0.50 ( if @ A
% 0.30/0.50 @ ( X2
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 @ ( zero_zero @ A )
% 0.30/0.50 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_if
% 0.30/0.50 thf(fact_1059_sgn__1__neg,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ( sgn_sgn @ A @ A3 )
% 0.30/0.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.50 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_1_neg
% 0.30/0.50 thf(fact_1060_Suc__pred_H,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( N
% 0.30/0.50 = ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_pred'
% 0.30/0.50 thf(fact_1061_Suc__diff__eq__diff__pred,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 0.30/0.50 = ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_eq_diff_pred
% 0.30/0.50 thf(fact_1062_dvd__mult__cancel1,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
% 0.30/0.50 = ( N
% 0.30/0.50 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_mult_cancel1
% 0.30/0.50 thf(fact_1063_dvd__mult__cancel2,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
% 0.30/0.50 = ( N
% 0.30/0.50 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_mult_cancel2
% 0.30/0.50 thf(fact_1064_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ( euclid7384307370059645450egment @ A @ A3 )
% 0.30/0.50 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.30/0.50 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.50 | ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_class.div_eq_0_iff
% 0.30/0.50 thf(fact_1065_nat__less__iff,axiom,
% 0.30/0.50 ! [W2: int,M: nat] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 0.30/0.50 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
% 0.30/0.50 = ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_iff
% 0.30/0.50 thf(fact_1066_le__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_minus_divide_eq
% 0.30/0.50 thf(fact_1067_minus__divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_divide_le_eq
% 0.30/0.50 thf(fact_1068_neg__le__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_le_minus_divide_eq
% 0.30/0.50 thf(fact_1069_neg__minus__divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_minus_divide_le_eq
% 0.30/0.50 thf(fact_1070_pos__le__minus__divide__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_le_minus_divide_eq
% 0.30/0.50 thf(fact_1071_pos__minus__divide__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_field @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 0.30/0.50 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_minus_divide_le_eq
% 0.30/0.50 thf(fact_1072_nat__mult__distrib__neg,axiom,
% 0.30/0.50 ! [Z3: int,Z6: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
% 0.30/0.50 = ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_distrib_neg
% 0.30/0.50 thf(fact_1073_nat__approx__posE,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.50 => ! [E: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.30/0.50 => ~ ! [N4: nat] :
% 0.30/0.50 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ E ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_approx_posE
% 0.30/0.50 thf(fact_1074_mult__ceiling__le,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_ceiling_le
% 0.30/0.50 thf(fact_1075_bits__mod__div__trivial,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_mod_div_trivial
% 0.30/0.50 thf(fact_1076_bits__mod__by__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_mod_by_1
% 0.30/0.50 thf(fact_1077_choose__dvd,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [K: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.50 => ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % choose_dvd
% 0.30/0.50 thf(fact_1078_div__geq,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ~ ( ord_less @ nat @ M @ N )
% 0.30/0.50 => ( ( divide_divide @ nat @ M @ N )
% 0.30/0.50 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_geq
% 0.30/0.50 thf(fact_1079_bits__mod__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_mod_0
% 0.30/0.50 thf(fact_1080_bits__div__by__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_div_by_1
% 0.30/0.50 thf(fact_1081_bits__div__by__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_div_by_0
% 0.30/0.50 thf(fact_1082_bits__div__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( bit_semiring_bits @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bits_div_0
% 0.30/0.50 thf(fact_1083_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.50 = A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.50 thf(fact_1084_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.50 thf(fact_1085_mod__neg__neg__trivial,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ L @ K )
% 0.30/0.50 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.50 = K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_neg_neg_trivial
% 0.30/0.50 thf(fact_1086_mod__pos__pos__trivial,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 => ( ( ord_less @ int @ K @ L )
% 0.30/0.50 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.50 = K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_pos_pos_trivial
% 0.30/0.50 thf(fact_1087_div__neg__neg__trivial,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ L @ K )
% 0.30/0.50 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.50 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_neg_neg_trivial
% 0.30/0.50 thf(fact_1088_div__pos__pos__trivial,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 => ( ( ord_less @ int @ K @ L )
% 0.30/0.50 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.50 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_pos_pos_trivial
% 0.30/0.50 thf(fact_1089_zle__diff1__eq,axiom,
% 0.30/0.50 ! [W2: int,Z3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
% 0.30/0.50 = ( ord_less @ int @ W2 @ Z3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % zle_diff1_eq
% 0.30/0.50 thf(fact_1090_dvd__antisym,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ M @ N )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ N @ M )
% 0.30/0.50 => ( M = N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_antisym
% 0.30/0.50 thf(fact_1091_zdvd__zdiffD,axiom,
% 0.30/0.50 ! [K: int,M: int,N: int] :
% 0.30/0.50 ( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
% 0.30/0.50 => ( ( dvd_dvd @ int @ K @ N )
% 0.30/0.50 => ( dvd_dvd @ int @ K @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdvd_zdiffD
% 0.30/0.50 thf(fact_1092_zmod__zminus1__eq__if,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 0.30/0.50 = ( zero_zero @ int ) ) )
% 0.30/0.50 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 0.30/0.50 = ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_zminus1_eq_if
% 0.30/0.50 thf(fact_1093_zmod__zminus2__eq__if,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 0.30/0.50 = ( zero_zero @ int ) ) )
% 0.30/0.50 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 0.30/0.50 = ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_zminus2_eq_if
% 0.30/0.50 thf(fact_1094_zdiv__zminus1__eq__if,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 0.30/0.50 = ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
% 0.30/0.50 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 0.30/0.50 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_zminus1_eq_if
% 0.30/0.50 thf(fact_1095_zdiv__zminus2__eq__if,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 0.30/0.50 = ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
% 0.30/0.50 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 0.30/0.50 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_zminus2_eq_if
% 0.30/0.50 thf(fact_1096_zdiv__mono__strict,axiom,
% 0.30/0.50 ! [A6: int,B7: int,N: int] :
% 0.30/0.50 ( ( ord_less @ int @ A6 @ B7 )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.50 => ( ( ( modulo_modulo @ int @ A6 @ N )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ( modulo_modulo @ int @ B7 @ N )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ( ord_less @ int @ ( divide_divide @ int @ A6 @ N ) @ ( divide_divide @ int @ B7 @ N ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_mono_strict
% 0.30/0.50 thf(fact_1097_mod__pos__geq,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 => ( ( ord_less_eq @ int @ L @ K )
% 0.30/0.50 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.50 = ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_pos_geq
% 0.30/0.50 thf(fact_1098_sgn__mod,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( L
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ~ ( dvd_dvd @ int @ L @ K )
% 0.30/0.50 => ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
% 0.30/0.50 = ( sgn_sgn @ int @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_mod
% 0.30/0.50 thf(fact_1099_minus__int__code_I1_J,axiom,
% 0.30/0.50 ! [K: int] :
% 0.30/0.50 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.50 = K ) ).
% 0.30/0.50
% 0.30/0.50 % minus_int_code(1)
% 0.30/0.50 thf(fact_1100_int__distrib_I3_J,axiom,
% 0.30/0.50 ! [Z1: int,Z22: int,W2: int] :
% 0.30/0.50 ( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
% 0.30/0.50 = ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_distrib(3)
% 0.30/0.50 thf(fact_1101_int__distrib_I4_J,axiom,
% 0.30/0.50 ! [W2: int,Z1: int,Z22: int] :
% 0.30/0.50 ( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
% 0.30/0.50 = ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_distrib(4)
% 0.30/0.50 thf(fact_1102_int__diff__cases,axiom,
% 0.30/0.50 ! [Z3: int] :
% 0.30/0.50 ~ ! [M6: nat,N4: nat] :
% 0.30/0.50 ( Z3
% 0.30/0.50 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M6 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_diff_cases
% 0.30/0.50 thf(fact_1103_zmod__le__nonneg__dividend,axiom,
% 0.30/0.50 ! [M: int,K: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
% 0.30/0.50 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_le_nonneg_dividend
% 0.30/0.50 thf(fact_1104_zmod__eq__0__iff,axiom,
% 0.30/0.50 ! [M: int,D2: int] :
% 0.30/0.50 ( ( ( modulo_modulo @ int @ M @ D2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 = ( ? [Q3: int] :
% 0.30/0.50 ( M
% 0.30/0.50 = ( times_times @ int @ D2 @ Q3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_eq_0_iff
% 0.30/0.50 thf(fact_1105_zmod__eq__0D,axiom,
% 0.30/0.50 ! [M: int,D2: int] :
% 0.30/0.50 ( ( ( modulo_modulo @ int @ M @ D2 )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 => ? [Q5: int] :
% 0.30/0.50 ( M
% 0.30/0.50 = ( times_times @ int @ D2 @ Q5 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_eq_0D
% 0.30/0.50 thf(fact_1106_zmod__zminus2__not__zero,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.50 != ( zero_zero @ int ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_zminus2_not_zero
% 0.30/0.50 thf(fact_1107_zmod__zminus1__not__zero,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
% 0.30/0.50 != ( zero_zero @ int ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.50 != ( zero_zero @ int ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_zminus1_not_zero
% 0.30/0.50 thf(fact_1108_div__neg__pos__less0,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_neg_pos_less0
% 0.30/0.50 thf(fact_1109_neg__imp__zdiv__neg__iff,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 0.30/0.50 = ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_imp_zdiv_neg_iff
% 0.30/0.50 thf(fact_1110_pos__imp__zdiv__neg__iff,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 0.30/0.50 = ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_imp_zdiv_neg_iff
% 0.30/0.50 thf(fact_1111_zmod__int,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B2 ) )
% 0.30/0.50 = ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_int
% 0.30/0.50 thf(fact_1112_zdiv__int,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B2 ) )
% 0.30/0.50 = ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_int
% 0.30/0.50 thf(fact_1113_minus__mod__int__eq,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
% 0.30/0.50 = ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_mod_int_eq
% 0.30/0.50 thf(fact_1114_zmod__minus1,axiom,
% 0.30/0.50 ! [B2: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
% 0.30/0.50 = ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_minus1
% 0.30/0.50 thf(fact_1115_neg__mod__conj,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 0.30/0.50 & ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_mod_conj
% 0.30/0.50 thf(fact_1116_pos__mod__conj,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
% 0.30/0.50 & ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_mod_conj
% 0.30/0.50 thf(fact_1117_zmod__trivial__iff,axiom,
% 0.30/0.50 ! [I: int,K: int] :
% 0.30/0.50 ( ( ( modulo_modulo @ int @ I @ K )
% 0.30/0.50 = I )
% 0.30/0.50 = ( ( K
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 0.30/0.50 & ( ord_less @ int @ I @ K ) )
% 0.30/0.50 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 0.30/0.50 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zmod_trivial_iff
% 0.30/0.50 thf(fact_1118_mod__int__pos__iff,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
% 0.30/0.50 = ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.50 | ( ( L
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
% 0.30/0.50 | ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_int_pos_iff
% 0.30/0.50 thf(fact_1119_neg__mod__bound,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_mod_bound
% 0.30/0.50 thf(fact_1120_Euclidean__Division_Opos__mod__bound,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 => ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% 0.30/0.50
% 0.30/0.50 % Euclidean_Division.pos_mod_bound
% 0.30/0.50 thf(fact_1121_zdiv__mono1,axiom,
% 0.30/0.50 ! [A3: int,A4: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ A3 @ A4 )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A4 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_mono1
% 0.30/0.50 thf(fact_1122_zdiv__mono2,axiom,
% 0.30/0.50 ! [A3: int,B4: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B4 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_mono2
% 0.30/0.50 thf(fact_1123_zdiv__eq__0__iff,axiom,
% 0.30/0.50 ! [I: int,K: int] :
% 0.30/0.50 ( ( ( divide_divide @ int @ I @ K )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 = ( ( K
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 0.30/0.50 & ( ord_less @ int @ I @ K ) )
% 0.30/0.50 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 0.30/0.50 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_eq_0_iff
% 0.30/0.50 thf(fact_1124_zdiv__mono1__neg,axiom,
% 0.30/0.50 ! [A3: int,A4: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ A3 @ A4 )
% 0.30/0.50 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A4 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_mono1_neg
% 0.30/0.50 thf(fact_1125_zdiv__mono2__neg,axiom,
% 0.30/0.50 ! [A3: int,B4: int,B2: int] :
% 0.30/0.50 ( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B4 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_mono2_neg
% 0.30/0.50 thf(fact_1126_div__int__pos__iff,axiom,
% 0.30/0.50 ! [K: int,L: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
% 0.30/0.50 = ( ( K
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( L
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
% 0.30/0.50 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.50 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_int_pos_iff
% 0.30/0.50 thf(fact_1127_div__positive__int,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ L @ K )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_positive_int
% 0.30/0.50 thf(fact_1128_div__nonneg__neg__le0,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_nonneg_neg_le0
% 0.30/0.50 thf(fact_1129_div__nonpos__pos__le0,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_nonpos_pos_le0
% 0.30/0.50 thf(fact_1130_pos__imp__zdiv__pos__iff,axiom,
% 0.30/0.50 ! [K: int,I: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
% 0.30/0.50 = ( ord_less_eq @ int @ K @ I ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_imp_zdiv_pos_iff
% 0.30/0.50 thf(fact_1131_neg__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 0.30/0.50 = ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_imp_zdiv_nonneg_iff
% 0.30/0.50 thf(fact_1132_pos__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.50 ! [B2: int,A3: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 0.30/0.50 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_imp_zdiv_nonneg_iff
% 0.30/0.50 thf(fact_1133_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.30/0.50 ! [A3: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 0.30/0.50 = ( ( ord_less_eq @ int @ B2 @ A3 )
% 0.30/0.50 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nonneg1_imp_zdiv_pos_iff
% 0.30/0.50 thf(fact_1134_int__div__less__self,axiom,
% 0.30/0.50 ! [X: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.50 => ( ( ord_less @ int @ ( one_one @ int ) @ K )
% 0.30/0.50 => ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_div_less_self
% 0.30/0.50 thf(fact_1135_minus__int__code_I2_J,axiom,
% 0.30/0.50 ! [L: int] :
% 0.30/0.50 ( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 = ( uminus_uminus @ int @ L ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_int_code(2)
% 0.30/0.50 thf(fact_1136_int__le__induct,axiom,
% 0.30/0.50 ! [I: int,K: int,P: int > $o] :
% 0.30/0.50 ( ( ord_less_eq @ int @ I @ K )
% 0.30/0.50 => ( ( P @ K )
% 0.30/0.50 => ( ! [I4: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ I4 @ K )
% 0.30/0.50 => ( ( P @ I4 )
% 0.30/0.50 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.50 => ( P @ I ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_le_induct
% 0.30/0.50 thf(fact_1137_int__less__induct,axiom,
% 0.30/0.50 ! [I: int,K: int,P: int > $o] :
% 0.30/0.50 ( ( ord_less @ int @ I @ K )
% 0.30/0.50 => ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
% 0.30/0.50 => ( ! [I4: int] :
% 0.30/0.50 ( ( ord_less @ int @ I4 @ K )
% 0.30/0.50 => ( ( P @ I4 )
% 0.30/0.50 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.50 => ( P @ I ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_less_induct
% 0.30/0.50 thf(fact_1138_div__eq__minus1,axiom,
% 0.30/0.50 ! [B2: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.50 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
% 0.30/0.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_eq_minus1
% 0.30/0.50 thf(fact_1139_neg__mod__sign,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.30/0.50 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % neg_mod_sign
% 0.30/0.50 thf(fact_1140_Euclidean__Division_Opos__mod__sign,axiom,
% 0.30/0.50 ! [L: int,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Euclidean_Division.pos_mod_sign
% 0.30/0.50 thf(fact_1141_plusinfinity,axiom,
% 0.30/0.50 ! [D2: int,P2: int > $o,P: int > $o] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.50 => ( ! [X3: int,K2: int] :
% 0.30/0.50 ( ( P2 @ X3 )
% 0.30/0.50 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 0.30/0.50 => ( ? [Z5: int] :
% 0.30/0.50 ! [X3: int] :
% 0.30/0.50 ( ( ord_less @ int @ Z5 @ X3 )
% 0.30/0.50 => ( ( P @ X3 )
% 0.30/0.50 = ( P2 @ X3 ) ) )
% 0.30/0.50 => ( ? [X_1: int] : ( P2 @ X_1 )
% 0.30/0.50 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % plusinfinity
% 0.30/0.50 thf(fact_1142_minusinfinity,axiom,
% 0.30/0.50 ! [D2: int,P1: int > $o,P: int > $o] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.50 => ( ! [X3: int,K2: int] :
% 0.30/0.50 ( ( P1 @ X3 )
% 0.30/0.50 = ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 0.30/0.50 => ( ? [Z5: int] :
% 0.30/0.50 ! [X3: int] :
% 0.30/0.50 ( ( ord_less @ int @ X3 @ Z5 )
% 0.30/0.50 => ( ( P @ X3 )
% 0.30/0.50 = ( P1 @ X3 ) ) )
% 0.30/0.50 => ( ? [X_1: int] : ( P1 @ X_1 )
% 0.30/0.50 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minusinfinity
% 0.30/0.50 thf(fact_1143_zdiv__zmult2__eq,axiom,
% 0.30/0.50 ! [C2: int,A3: int,B2: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 0.30/0.50 => ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 0.30/0.50 = ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiv_zmult2_eq
% 0.30/0.50 thf(fact_1144_int__minus,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.50 = ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_minus
% 0.30/0.50 thf(fact_1145_nat__mod__distrib,axiom,
% 0.30/0.50 ! [X: int,Y3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.50 => ( ( nat2 @ ( modulo_modulo @ int @ X @ Y3 ) )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mod_distrib
% 0.30/0.50 thf(fact_1146_decr__mult__lemma,axiom,
% 0.30/0.50 ! [D2: int,P: int > $o,K: int] :
% 0.30/0.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.50 => ( ! [X3: int] :
% 0.30/0.50 ( ( P @ X3 )
% 0.30/0.50 => ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.50 => ! [X4: int] :
% 0.30/0.50 ( ( P @ X4 )
% 0.30/0.50 => ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % decr_mult_lemma
% 0.30/0.50 thf(fact_1147_int__ops_I6_J,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
% 0.30/0.50 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.30/0.50 = ( zero_zero @ int ) ) )
% 0.30/0.50 & ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
% 0.30/0.50 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.30/0.50 = ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_ops(6)
% 0.30/0.50 thf(fact_1148_nat__diff__distrib,axiom,
% 0.30/0.50 ! [Z6: int,Z3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ Z6 @ Z3 )
% 0.30/0.50 => ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) )
% 0.30/0.50 = ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_distrib
% 0.30/0.50 thf(fact_1149_nat__diff__distrib_H,axiom,
% 0.30/0.50 ! [X: int,Y3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.50 => ( ( nat2 @ ( minus_minus @ int @ X @ Y3 ) )
% 0.30/0.50 = ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_distrib'
% 0.30/0.50 thf(fact_1150_verit__less__mono__div__int2,axiom,
% 0.30/0.50 ! [A6: int,B7: int,N: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ A6 @ B7 )
% 0.30/0.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
% 0.30/0.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ B7 @ N ) @ ( divide_divide @ int @ A6 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % verit_less_mono_div_int2
% 0.30/0.50 thf(fact_1151_nat__div__distrib_H,axiom,
% 0.30/0.50 ! [Y3: int,X: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.50 => ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
% 0.30/0.50 = ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_div_distrib'
% 0.30/0.50 thf(fact_1152_nat__div__distrib,axiom,
% 0.30/0.50 ! [X: int,Y3: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.50 => ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
% 0.30/0.50 = ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_div_distrib
% 0.30/0.50 thf(fact_1153_zdiff__int__split,axiom,
% 0.30/0.50 ! [P: int > $o,X: nat,Y3: nat] :
% 0.30/0.50 ( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y3 ) ) )
% 0.30/0.50 = ( ( ( ord_less_eq @ nat @ Y3 @ X )
% 0.30/0.50 => ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) )
% 0.30/0.50 & ( ( ord_less @ nat @ X @ Y3 )
% 0.30/0.50 => ( P @ ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zdiff_int_split
% 0.30/0.50 thf(fact_1154_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.50 thf(fact_1155_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.50 thf(fact_1156_mod__eq__0D,axiom,
% 0.30/0.50 ! [M: nat,D2: nat] :
% 0.30/0.50 ( ( ( modulo_modulo @ nat @ M @ D2 )
% 0.30/0.50 = ( zero_zero @ nat ) )
% 0.30/0.50 => ? [Q5: nat] :
% 0.30/0.50 ( M
% 0.30/0.50 = ( times_times @ nat @ D2 @ Q5 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_eq_0D
% 0.30/0.50 thf(fact_1157_mod__geq,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ~ ( ord_less @ nat @ M @ N )
% 0.30/0.50 => ( ( modulo_modulo @ nat @ M @ N )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_geq
% 0.30/0.50 thf(fact_1158_mod__eq__dvd__iff__nat,axiom,
% 0.30/0.50 ! [N: nat,M: nat,Q4: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.50 => ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 0.30/0.50 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 0.30/0.50 = ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_eq_dvd_iff_nat
% 0.30/0.50 thf(fact_1159_div__positive,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_positive
% 0.30/0.50 thf(fact_1160_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.50 thf(fact_1161_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.50 thf(fact_1162_bezout1__nat,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ? [D6: nat,X3: nat,Y5: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 0.30/0.50 & ( dvd_dvd @ nat @ D6 @ B2 )
% 0.30/0.50 & ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
% 0.30/0.50 = D6 )
% 0.30/0.50 | ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
% 0.30/0.50 = D6 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bezout1_nat
% 0.30/0.50 thf(fact_1163_gcd__nat__induct,axiom,
% 0.30/0.50 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.50 ( ! [M6: nat] : ( P @ M6 @ ( zero_zero @ nat ) )
% 0.30/0.50 => ( ! [M6: nat,N4: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.50 => ( ( P @ N4 @ ( modulo_modulo @ nat @ M6 @ N4 ) )
% 0.30/0.50 => ( P @ M6 @ N4 ) ) )
% 0.30/0.50 => ( P @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gcd_nat_induct
% 0.30/0.50 thf(fact_1164_dvd__pos__nat,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ M @ N )
% 0.30/0.50 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_pos_nat
% 0.30/0.50 thf(fact_1165_gbinomial__absorption_H,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [K: nat,A3: A] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.50 => ( ( gbinomial @ A @ A3 @ K )
% 0.30/0.50 = ( 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 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_absorption'
% 0.30/0.50 thf(fact_1166_modulo__int__unfold,axiom,
% 0.30/0.50 ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.50 ( ( ( ( ( sgn_sgn @ int @ L )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( ( sgn_sgn @ int @ K )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( N
% 0.30/0.50 = ( zero_zero @ nat ) ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.50 = ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
% 0.30/0.50 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( ( sgn_sgn @ int @ K )
% 0.30/0.50 = ( zero_zero @ int ) )
% 0.30/0.50 | ( N
% 0.30/0.50 = ( zero_zero @ nat ) ) )
% 0.30/0.50 => ( ( ( ( sgn_sgn @ int @ K )
% 0.30/0.50 = ( sgn_sgn @ int @ L ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.50 = ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
% 0.30/0.50 & ( ( ( sgn_sgn @ int @ K )
% 0.30/0.50 != ( sgn_sgn @ int @ L ) )
% 0.30/0.50 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.50 = ( times_times @ int @ ( sgn_sgn @ int @ L )
% 0.30/0.50 @ ( minus_minus @ int
% 0.30/0.50 @ ( semiring_1_of_nat @ int
% 0.30/0.50 @ ( times_times @ nat @ N
% 0.30/0.50 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.50 @ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
% 0.30/0.50 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % modulo_int_unfold
% 0.30/0.50 thf(fact_1167_divmod__cases,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 => ( A3
% 0.30/0.50 != ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
% 0.30/0.50 => ( ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ! [Q5: A,R3: A] :
% 0.30/0.50 ( ( ( euclid7384307370059645450egment @ A @ R3 )
% 0.30/0.50 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.30/0.50 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.50 => ( ( R3
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.50 = Q5 )
% 0.30/0.50 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.50 = R3 )
% 0.30/0.50 => ( A3
% 0.30/0.50 != ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R3 ) ) ) ) ) ) ) )
% 0.30/0.50 => ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divmod_cases
% 0.30/0.50 thf(fact_1168_gcd__nat_Oextremum,axiom,
% 0.30/0.50 ! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % gcd_nat.extremum
% 0.30/0.50 thf(fact_1169_add__left__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel_semigroup_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.30/0.50 = ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_left_cancel
% 0.30/0.50 thf(fact_1170_add__right__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel_semigroup_add @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.30/0.50 = ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_right_cancel
% 0.30/0.50 thf(fact_1171_add__Suc__right,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 0.30/0.50 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc_right
% 0.30/0.50 thf(fact_1172_add__is__0,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ( plus_plus @ nat @ M @ N )
% 0.30/0.50 = ( zero_zero @ nat ) )
% 0.30/0.50 = ( ( M
% 0.30/0.50 = ( zero_zero @ nat ) )
% 0.30/0.50 & ( N
% 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_is_0
% 0.30/0.50 thf(fact_1173_Nat_Oadd__0__right,axiom,
% 0.30/0.50 ! [M: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.add_0_right
% 0.30/0.50 thf(fact_1174_nat__add__left__cancel__less,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.30/0.50 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_add_left_cancel_less
% 0.30/0.50 thf(fact_1175_nat__add__left__cancel__le,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.30/0.50 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_add_left_cancel_le
% 0.30/0.50 thf(fact_1176_diff__diff__left,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.30/0.50 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_left
% 0.30/0.50 thf(fact_1177_add__le__cancel__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_cancel_left
% 0.30/0.50 thf(fact_1178_add__le__cancel__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_cancel_right
% 0.30/0.50 thf(fact_1179_double__eq__0__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( A3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % double_eq_0_iff
% 0.30/0.50 thf(fact_1180_add_Oright__neutral,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( monoid_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.right_neutral
% 0.30/0.50 thf(fact_1181_double__zero__sym,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ( zero_zero @ A )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ A3 ) )
% 0.30/0.50 = ( A3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % double_zero_sym
% 0.30/0.50 thf(fact_1182_add__cancel__left__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.30/0.50 = A3 )
% 0.30/0.50 = ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_cancel_left_left
% 0.30/0.50 thf(fact_1183_add__cancel__left__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.50 = A3 )
% 0.30/0.50 = ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_cancel_left_right
% 0.30/0.50 thf(fact_1184_add__cancel__right__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( A3
% 0.30/0.50 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_cancel_right_left
% 0.30/0.50 thf(fact_1185_add__cancel__right__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( A3
% 0.30/0.50 = ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( B2
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_cancel_right_right
% 0.30/0.50 thf(fact_1186_add__eq__0__iff__both__eq__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.50 => ! [X: A,Y3: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ X @ Y3 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( ( X
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 & ( Y3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_eq_0_iff_both_eq_0
% 0.30/0.50 thf(fact_1187_zero__eq__add__iff__both__eq__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.50 => ! [X: A,Y3: A] :
% 0.30/0.50 ( ( ( zero_zero @ A )
% 0.30/0.50 = ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.50 = ( ( X
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 & ( Y3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_eq_add_iff_both_eq_0
% 0.30/0.50 thf(fact_1188_add__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( monoid_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_0
% 0.30/0.50 thf(fact_1189_add__less__cancel__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_cancel_left
% 0.30/0.50 thf(fact_1190_add__less__cancel__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_cancel_right
% 0.30/0.50 thf(fact_1191_add__diff__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel
% 0.30/0.50 thf(fact_1192_diff__add__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_cancel
% 0.30/0.50 thf(fact_1193_add__diff__cancel__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left
% 0.30/0.50 thf(fact_1194_add__diff__cancel__left_H,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
% 0.30/0.50 = B2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left'
% 0.30/0.50 thf(fact_1195_add__diff__cancel__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right
% 0.30/0.50 thf(fact_1196_add__diff__cancel__right_H,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right'
% 0.30/0.50 thf(fact_1197_add__minus__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
% 0.30/0.50 = B2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_minus_cancel
% 0.30/0.50 thf(fact_1198_minus__add__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = B2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_add_cancel
% 0.30/0.50 thf(fact_1199_minus__add__distrib,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % minus_add_distrib
% 0.30/0.50 thf(fact_1200_dvd__add__triv__left__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_triv_left_iff
% 0.30/0.50 thf(fact_1201_dvd__add__triv__right__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_triv_right_iff
% 0.30/0.50 thf(fact_1202_of__nat__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring_1 @ A )
% 0.30/0.50 => ! [M: nat,N: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.50 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_nat_add
% 0.30/0.50 thf(fact_1203_mod__add__self1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_self1
% 0.30/0.50 thf(fact_1204_mod__add__self2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_self2
% 0.30/0.50 thf(fact_1205_add__gr__0,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.50 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_gr_0
% 0.30/0.50 thf(fact_1206_mult__Suc__right,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( times_times @ nat @ M @ ( suc @ N ) )
% 0.30/0.50 = ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_Suc_right
% 0.30/0.50 thf(fact_1207_Nat_Odiff__diff__right,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.30/0.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_diff_right
% 0.30/0.50 thf(fact_1208_Nat_Oadd__diff__assoc2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.50 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.30/0.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.add_diff_assoc2
% 0.30/0.50 thf(fact_1209_Nat_Oadd__diff__assoc,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.50 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.30/0.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.add_diff_assoc
% 0.30/0.50 thf(fact_1210_of__bool__less__eq__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord181362715937106298miring @ A )
% 0.30/0.50 => ! [P: $o,Q: $o] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
% 0.30/0.50 = ( P
% 0.30/0.50 => Q ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_less_eq_iff
% 0.30/0.50 thf(fact_1211_of__bool__eq__0__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( ( zero_neq_one_of_bool @ A @ P )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( ~ P ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_eq_0_iff
% 0.30/0.50 thf(fact_1212_of__bool__eq_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ( ( zero_neq_one_of_bool @ A @ $false )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_eq(1)
% 0.30/0.50 thf(fact_1213_of__bool__less__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord181362715937106298miring @ A )
% 0.30/0.50 => ! [P: $o,Q: $o] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
% 0.30/0.50 = ( ~ P
% 0.30/0.50 & Q ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_less_iff
% 0.30/0.50 thf(fact_1214_of__bool__eq_I2_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ( ( zero_neq_one_of_bool @ A @ $true )
% 0.30/0.50 = ( one_one @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_eq(2)
% 0.30/0.50 thf(fact_1215_of__bool__eq__1__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( ( zero_neq_one_of_bool @ A @ P )
% 0.30/0.50 = ( one_one @ A ) )
% 0.30/0.50 = P ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_eq_1_iff
% 0.30/0.50 thf(fact_1216_gbinomial__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ( semiring_char_0 @ A )
% 0.30/0.50 & ( semidom_divide @ A ) )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_1
% 0.30/0.50 thf(fact_1217_add__le__same__cancel1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_same_cancel1
% 0.30/0.50 thf(fact_1218_add__le__same__cancel2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_same_cancel2
% 0.30/0.50 thf(fact_1219_le__add__same__cancel1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_same_cancel1
% 0.30/0.50 thf(fact_1220_le__add__same__cancel2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_same_cancel2
% 0.30/0.50 thf(fact_1221_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.50 thf(fact_1222_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.50 thf(fact_1223_add__less__same__cancel1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.50 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_same_cancel1
% 0.30/0.50 thf(fact_1224_add__less__same__cancel2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_same_cancel2
% 0.30/0.50 thf(fact_1225_less__add__same__cancel1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_same_cancel1
% 0.30/0.50 thf(fact_1226_less__add__same__cancel2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1937475149494474687imp_le @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_same_cancel2
% 0.30/0.50 thf(fact_1227_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.50 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.50 thf(fact_1228_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.50 thf(fact_1229_le__add__diff__inverse2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.50 => ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.50 = A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_diff_inverse2
% 0.30/0.50 thf(fact_1230_le__add__diff__inverse,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.50 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.50 = A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_diff_inverse
% 0.30/0.50 thf(fact_1231_diff__add__zero,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_monoid_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_zero
% 0.30/0.50 thf(fact_1232_add_Oright__inverse,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.right_inverse
% 0.30/0.50 thf(fact_1233_ab__left__minus,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_left_minus
% 0.30/0.50 thf(fact_1234_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_times_triv_right_iff
% 0.30/0.50 thf(fact_1235_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_times_triv_left_iff
% 0.30/0.50 thf(fact_1236_diff__minus__eq__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_minus_eq_add
% 0.30/0.50 thf(fact_1237_uminus__add__conv__diff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.50 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % uminus_add_conv_diff
% 0.30/0.50 thf(fact_1238_div__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( algebraic_semidom @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.50 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_add
% 0.30/0.50 thf(fact_1239_mod__mult__self1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_mult_self1
% 0.30/0.50 thf(fact_1240_mod__mult__self2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_mult_self2
% 0.30/0.50 thf(fact_1241_mod__mult__self3,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_mult_self3
% 0.30/0.50 thf(fact_1242_mod__mult__self4,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_mult_self4
% 0.30/0.50 thf(fact_1243_zero__less__of__bool__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
% 0.30/0.50 = P ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_less_of_bool_iff
% 0.30/0.50 thf(fact_1244_diff__Suc__diff__eq1,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
% 0.30/0.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_diff_eq1
% 0.30/0.50 thf(fact_1245_diff__Suc__diff__eq2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
% 0.30/0.50 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_diff_eq2
% 0.30/0.50 thf(fact_1246_of__bool__less__one__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
% 0.30/0.50 = ( ~ P ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_less_one_iff
% 0.30/0.50 thf(fact_1247_of__bool__not__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ring_1 @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( zero_neq_one_of_bool @ A @ ~ P )
% 0.30/0.50 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_not_iff
% 0.30/0.50 thf(fact_1248_gbinomial__0_I2_J,axiom,
% 0.30/0.50 ! [B: $tType] :
% 0.30/0.50 ( ( ( semiring_char_0 @ B )
% 0.30/0.50 & ( semidom_divide @ B ) )
% 0.30/0.50 => ! [K: nat] :
% 0.30/0.50 ( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
% 0.30/0.50 = ( zero_zero @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_0(2)
% 0.30/0.50 thf(fact_1249_zle__add1__eq__le,axiom,
% 0.30/0.50 ! [W2: int,Z3: int] :
% 0.30/0.50 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
% 0.30/0.50 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % zle_add1_eq_le
% 0.30/0.50 thf(fact_1250_gbinomial__0_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ( semiring_char_0 @ A )
% 0.30/0.50 & ( semidom_divide @ A ) )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.50 = ( one_one @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_0(1)
% 0.30/0.50 thf(fact_1251_Suc__mod__mult__self1,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mod_mult_self1
% 0.30/0.50 thf(fact_1252_Suc__mod__mult__self2,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mod_mult_self2
% 0.30/0.50 thf(fact_1253_Suc__mod__mult__self3,axiom,
% 0.30/0.50 ! [K: nat,N: nat,M: nat] :
% 0.30/0.50 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mod_mult_self3
% 0.30/0.50 thf(fact_1254_Suc__mod__mult__self4,axiom,
% 0.30/0.50 ! [N: nat,K: nat,M: nat] :
% 0.30/0.50 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
% 0.30/0.50 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mod_mult_self4
% 0.30/0.50 thf(fact_1255_gbinomial__Suc0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ( semiring_char_0 @ A )
% 0.30/0.50 & ( semidom_divide @ A ) )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_Suc0
% 0.30/0.50 thf(fact_1256_of__nat__of__bool,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring_1 @ A )
% 0.30/0.50 => ! [P: $o] :
% 0.30/0.50 ( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
% 0.30/0.50 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_nat_of_bool
% 0.30/0.50 thf(fact_1257_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( neg_numeral @ A )
% 0.30/0.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_neg_numeral_special(8)
% 0.30/0.50 thf(fact_1258_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( neg_numeral @ A )
% 0.30/0.50 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.50 = ( zero_zero @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_neg_numeral_special(7)
% 0.30/0.50 thf(fact_1259_div__mult__self4,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mult_self4
% 0.30/0.50 thf(fact_1260_div__mult__self3,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mult_self3
% 0.30/0.50 thf(fact_1261_div__mult__self2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mult_self2
% 0.30/0.50 thf(fact_1262_div__mult__self1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 != ( zero_zero @ A ) )
% 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mult_self1
% 0.30/0.50 thf(fact_1263_of__nat__Suc,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring_1 @ A )
% 0.30/0.50 => ! [M: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ A @ ( suc @ M ) )
% 0.30/0.50 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_nat_Suc
% 0.30/0.50 thf(fact_1264_sgn__mult__self__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.50 = ( zero_neq_one_of_bool @ A
% 0.30/0.50 @ ( A3
% 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_mult_self_eq
% 0.30/0.50 thf(fact_1265_ceiling__add__one,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.50 => ! [X: A] :
% 0.30/0.50 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 0.30/0.50 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ceiling_add_one
% 0.30/0.50 thf(fact_1266_Suc__0__mod__eq,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.50 = ( zero_neq_one_of_bool @ nat
% 0.30/0.50 @ ( N
% 0.30/0.50 != ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_0_mod_eq
% 0.30/0.50 thf(fact_1267_sgn__of__nat,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [N: nat] :
% 0.30/0.50 ( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.50 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sgn_of_nat
% 0.30/0.50 thf(fact_1268_of__nat__gbinomial,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [N: nat,K: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
% 0.30/0.50 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_nat_gbinomial
% 0.30/0.50 thf(fact_1269_int__plus,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
% 0.30/0.50 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_plus
% 0.30/0.50 thf(fact_1270_int__ops_I5_J,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
% 0.30/0.50 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_ops(5)
% 0.30/0.50 thf(fact_1271_zadd__int__left,axiom,
% 0.30/0.50 ! [M: nat,N: nat,Z3: int] :
% 0.30/0.50 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z3 ) )
% 0.30/0.50 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % zadd_int_left
% 0.30/0.50 thf(fact_1272_Euclid__induct,axiom,
% 0.30/0.50 ! [P: nat > nat > $o,A3: nat,B2: nat] :
% 0.30/0.50 ( ! [A8: nat,B5: nat] :
% 0.30/0.50 ( ( P @ A8 @ B5 )
% 0.30/0.50 = ( P @ B5 @ A8 ) )
% 0.30/0.50 => ( ! [A8: nat] : ( P @ A8 @ ( zero_zero @ nat ) )
% 0.30/0.50 => ( ! [A8: nat,B5: nat] :
% 0.30/0.50 ( ( P @ A8 @ B5 )
% 0.30/0.50 => ( P @ A8 @ ( plus_plus @ nat @ A8 @ B5 ) ) )
% 0.30/0.50 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Euclid_induct
% 0.30/0.50 thf(fact_1273_of__bool__eq__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ! [P5: $o,Q4: $o] :
% 0.30/0.50 ( ( ( zero_neq_one_of_bool @ A @ P5 )
% 0.30/0.50 = ( zero_neq_one_of_bool @ A @ Q4 ) )
% 0.30/0.50 = ( P5 = Q4 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_eq_iff
% 0.30/0.50 thf(fact_1274_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_semigroup_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.50 thf(fact_1275_is__num__normalize_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( neg_numeral @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % is_num_normalize(1)
% 0.30/0.50 thf(fact_1276_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( I = J )
% 0.30/0.50 & ( K = L ) )
% 0.30/0.50 => ( ( plus_plus @ A @ I @ K )
% 0.30/0.50 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_semiring(4)
% 0.30/0.50 thf(fact_1277_group__cancel_Oadd1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_monoid_add @ A )
% 0.30/0.50 => ! [A6: A,K: A,A3: A,B2: A] :
% 0.30/0.50 ( ( A6
% 0.30/0.50 = ( plus_plus @ A @ K @ A3 ) )
% 0.30/0.50 => ( ( plus_plus @ A @ A6 @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.add1
% 0.30/0.50 thf(fact_1278_group__cancel_Oadd2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_monoid_add @ A )
% 0.30/0.50 => ! [B7: A,K: A,B2: A,A3: A] :
% 0.30/0.50 ( ( B7
% 0.30/0.50 = ( plus_plus @ A @ K @ B2 ) )
% 0.30/0.50 => ( ( plus_plus @ A @ A3 @ B7 )
% 0.30/0.50 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.add2
% 0.30/0.50 thf(fact_1279_add_Oassoc,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semigroup_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.assoc
% 0.30/0.50 thf(fact_1280_add_Oleft__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.30/0.50 = ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.left_cancel
% 0.30/0.50 thf(fact_1281_add_Oright__cancel,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.30/0.50 = ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.right_cancel
% 0.30/0.50 thf(fact_1282_add_Ocommute,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_semigroup_add @ A )
% 0.30/0.50 => ( ( plus_plus @ A )
% 0.30/0.50 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.commute
% 0.30/0.50 thf(fact_1283_add_Oleft__commute,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_semigroup_add @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.left_commute
% 0.30/0.50 thf(fact_1284_add__left__imp__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel_semigroup_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.30/0.50 => ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_left_imp_eq
% 0.30/0.50 thf(fact_1285_add__right__imp__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel_semigroup_add @ A )
% 0.30/0.50 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.30/0.50 => ( B2 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_right_imp_eq
% 0.30/0.50 thf(fact_1286_gbinomial__Suc__Suc,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [A3: A,K: nat] :
% 0.30/0.50 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.30/0.50 = ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_Suc_Suc
% 0.30/0.50 thf(fact_1287_nat__int__add,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
% 0.30/0.50 = ( plus_plus @ nat @ A3 @ B2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_int_add
% 0.30/0.50 thf(fact_1288_ceiling__add__le,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.50 => ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y3 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ceiling_add_le
% 0.30/0.50 thf(fact_1289_of__bool__conj,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring_1 @ A )
% 0.30/0.50 => ! [P: $o,Q: $o] :
% 0.30/0.50 ( ( zero_neq_one_of_bool @ A
% 0.30/0.50 @ ( P
% 0.30/0.50 & Q ) )
% 0.30/0.50 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_conj
% 0.30/0.50 thf(fact_1290_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.50 & ( K = L ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_semiring(3)
% 0.30/0.50 thf(fact_1291_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( I = J )
% 0.30/0.50 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_semiring(2)
% 0.30/0.50 thf(fact_1292_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.50 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_semiring(1)
% 0.30/0.50 thf(fact_1293_add__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono
% 0.30/0.50 thf(fact_1294_add__left__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_left_mono
% 0.30/0.50 thf(fact_1295_less__eqE,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ~ ! [C3: A] :
% 0.30/0.50 ( B2
% 0.30/0.50 != ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_eqE
% 0.30/0.50 thf(fact_1296_add__right__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_right_mono
% 0.30/0.50 thf(fact_1297_le__iff__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.50 => ( ( ord_less_eq @ A )
% 0.30/0.50 = ( ^ [A5: A,B3: A] :
% 0.30/0.50 ? [C4: A] :
% 0.30/0.50 ( B3
% 0.30/0.50 = ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_iff_add
% 0.30/0.50 thf(fact_1298_add__le__imp__le__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.50 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_imp_le_left
% 0.30/0.50 thf(fact_1299_add__le__imp__le__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_imp_le_right
% 0.30/0.50 thf(fact_1300_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_monoid_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_add_class.add_0
% 0.30/0.50 thf(fact_1301_add_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_monoid_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.comm_neutral
% 0.30/0.50 thf(fact_1302_add_Ogroup__left__neutral,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.group_left_neutral
% 0.30/0.50 thf(fact_1303_verit__sum__simplify,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [A3: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 = A3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % verit_sum_simplify
% 0.30/0.50 thf(fact_1304_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less @ A @ I @ J )
% 0.30/0.50 & ( ord_less @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_field(5)
% 0.30/0.50 thf(fact_1305_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( I = J )
% 0.30/0.50 & ( ord_less @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_field(2)
% 0.30/0.50 thf(fact_1306_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less @ A @ I @ J )
% 0.30/0.50 & ( K = L ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_field(1)
% 0.30/0.50 thf(fact_1307_add__strict__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( strict9044650504122735259up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_strict_mono
% 0.30/0.50 thf(fact_1308_add__strict__left__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_strict_left_mono
% 0.30/0.50 thf(fact_1309_add__strict__right__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_strict_right_mono
% 0.30/0.50 thf(fact_1310_add__less__imp__less__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.30/0.50 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_imp_less_left
% 0.30/0.50 thf(fact_1311_add__less__imp__less__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_imp_less_right
% 0.30/0.50 thf(fact_1312_combine__common__factor,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring @ A )
% 0.30/0.50 => ! [A3: A,E: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % combine_common_factor
% 0.30/0.50 thf(fact_1313_distrib__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_right
% 0.30/0.50 thf(fact_1314_distrib__left,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( semiring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_left
% 0.30/0.50 thf(fact_1315_comm__semiring__class_Odistrib,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_semiring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % comm_semiring_class.distrib
% 0.30/0.50 thf(fact_1316_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ring_class.ring_distribs(1)
% 0.30/0.50 thf(fact_1317_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ring_class.ring_distribs(2)
% 0.30/0.50 thf(fact_1318_group__cancel_Osub1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_group_add @ A )
% 0.30/0.50 => ! [A6: A,K: A,A3: A,B2: A] :
% 0.30/0.50 ( ( A6
% 0.30/0.50 = ( plus_plus @ A @ K @ A3 ) )
% 0.30/0.50 => ( ( minus_minus @ A @ A6 @ B2 )
% 0.30/0.50 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.sub1
% 0.30/0.50 thf(fact_1319_diff__eq__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.30/0.50 = C2 )
% 0.30/0.50 = ( A3
% 0.30/0.50 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_eq
% 0.30/0.50 thf(fact_1320_eq__diff__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( A3
% 0.30/0.50 = ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.50 = C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_eq
% 0.30/0.50 thf(fact_1321_add__diff__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_eq
% 0.30/0.50 thf(fact_1322_diff__diff__eq2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq2
% 0.30/0.50 thf(fact_1323_diff__add__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq
% 0.30/0.50 thf(fact_1324_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq_diff_diff_swap
% 0.30/0.50 thf(fact_1325_add__implies__diff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel1802427076303600483id_add @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ( plus_plus @ A @ C2 @ B2 )
% 0.30/0.50 = A3 )
% 0.30/0.50 => ( C2
% 0.30/0.50 = ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_implies_diff
% 0.30/0.50 thf(fact_1326_diff__diff__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( cancel2418104881723323429up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq
% 0.30/0.50 thf(fact_1327_is__num__normalize_I8_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( neg_numeral @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % is_num_normalize(8)
% 0.30/0.50 thf(fact_1328_group__cancel_Oneg1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ab_group_add @ A )
% 0.30/0.50 => ! [A6: A,K: A,A3: A] :
% 0.30/0.50 ( ( A6
% 0.30/0.50 = ( plus_plus @ A @ K @ A3 ) )
% 0.30/0.50 => ( ( uminus_uminus @ A @ A6 )
% 0.30/0.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.neg1
% 0.30/0.50 thf(fact_1329_add_Oinverse__distrib__swap,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add.inverse_distrib_swap
% 0.30/0.50 thf(fact_1330_dvd__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_semiring_1 @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.50 => ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add
% 0.30/0.50 thf(fact_1331_dvd__add__left__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.50 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_left_iff
% 0.30/0.50 thf(fact_1332_dvd__add__right__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( comm_s4317794764714335236cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.50 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dvd_add_right_iff
% 0.30/0.50 thf(fact_1333_add__divide__distrib,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( division_ring @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_divide_distrib
% 0.30/0.50 thf(fact_1334_nat__arith_Osuc1,axiom,
% 0.30/0.50 ! [A6: nat,K: nat,A3: nat] :
% 0.30/0.50 ( ( A6
% 0.30/0.50 = ( plus_plus @ nat @ K @ A3 ) )
% 0.30/0.50 => ( ( suc @ A6 )
% 0.30/0.50 = ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_arith.suc1
% 0.30/0.50 thf(fact_1335_add__Suc,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 0.30/0.50 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc
% 0.30/0.50 thf(fact_1336_add__Suc__shift,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 0.30/0.50 = ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc_shift
% 0.30/0.50 thf(fact_1337_plus__nat_Oadd__0,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.50 = N ) ).
% 0.30/0.50
% 0.30/0.50 % plus_nat.add_0
% 0.30/0.50 thf(fact_1338_add__eq__self__zero,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ( plus_plus @ nat @ M @ N )
% 0.30/0.50 = M )
% 0.30/0.50 => ( N
% 0.30/0.50 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_eq_self_zero
% 0.30/0.50 thf(fact_1339_mod__add__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_eq
% 0.30/0.50 thf(fact_1340_mod__add__cong,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 0.30/0.50 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 0.30/0.50 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 0.30/0.50 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_cong
% 0.30/0.50 thf(fact_1341_mod__add__left__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_left_eq
% 0.30/0.50 thf(fact_1342_mod__add__right__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.30/0.50 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mod_add_right_eq
% 0.30/0.50 thf(fact_1343_plus__int__code_I2_J,axiom,
% 0.30/0.50 ! [L: int] :
% 0.30/0.50 ( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.50 = L ) ).
% 0.30/0.50
% 0.30/0.50 % plus_int_code(2)
% 0.30/0.50 thf(fact_1344_plus__int__code_I1_J,axiom,
% 0.30/0.50 ! [K: int] :
% 0.30/0.50 ( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.50 = K ) ).
% 0.30/0.50
% 0.30/0.50 % plus_int_code(1)
% 0.30/0.50 thf(fact_1345_add__lessD1,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.30/0.50 => ( ord_less @ nat @ I @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_lessD1
% 0.30/0.50 thf(fact_1346_add__less__mono,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ I @ J )
% 0.30/0.50 => ( ( ord_less @ nat @ K @ L )
% 0.30/0.50 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_mono
% 0.30/0.50 thf(fact_1347_not__add__less1,axiom,
% 0.30/0.50 ! [I: nat,J: nat] :
% 0.30/0.50 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 0.30/0.50
% 0.30/0.50 % not_add_less1
% 0.30/0.50 thf(fact_1348_not__add__less2,axiom,
% 0.30/0.50 ! [J: nat,I: nat] :
% 0.30/0.50 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 0.30/0.50
% 0.30/0.50 % not_add_less2
% 0.30/0.50 thf(fact_1349_add__less__mono1,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ I @ J )
% 0.30/0.50 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_mono1
% 0.30/0.50 thf(fact_1350_trans__less__add1,axiom,
% 0.30/0.50 ! [I: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ I @ J )
% 0.30/0.50 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_less_add1
% 0.30/0.50 thf(fact_1351_trans__less__add2,axiom,
% 0.30/0.50 ! [I: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ I @ J )
% 0.30/0.50 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_less_add2
% 0.30/0.50 thf(fact_1352_less__add__eq__less,axiom,
% 0.30/0.50 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less @ nat @ K @ L )
% 0.30/0.50 => ( ( ( plus_plus @ nat @ M @ L )
% 0.30/0.50 = ( plus_plus @ nat @ K @ N ) )
% 0.30/0.50 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_eq_less
% 0.30/0.50 thf(fact_1353_nat__le__iff__add,axiom,
% 0.30/0.50 ( ( ord_less_eq @ nat )
% 0.30/0.50 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.50 ? [K3: nat] :
% 0.30/0.50 ( N3
% 0.30/0.50 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_iff_add
% 0.30/0.50 thf(fact_1354_trans__le__add2,axiom,
% 0.30/0.50 ! [I: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.50 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_le_add2
% 0.30/0.50 thf(fact_1355_trans__le__add1,axiom,
% 0.30/0.50 ! [I: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.50 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_le_add1
% 0.30/0.50 thf(fact_1356_add__le__mono1,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.50 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_mono1
% 0.30/0.50 thf(fact_1357_add__le__mono,axiom,
% 0.30/0.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.50 => ( ( ord_less_eq @ nat @ K @ L )
% 0.30/0.50 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_mono
% 0.30/0.50 thf(fact_1358_le__Suc__ex,axiom,
% 0.30/0.50 ! [K: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ K @ L )
% 0.30/0.50 => ? [N4: nat] :
% 0.30/0.50 ( L
% 0.30/0.50 = ( plus_plus @ nat @ K @ N4 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_Suc_ex
% 0.30/0.50 thf(fact_1359_add__leD2,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.30/0.50 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_leD2
% 0.30/0.50 thf(fact_1360_add__leD1,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.30/0.50 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_leD1
% 0.30/0.50 thf(fact_1361_le__add2,axiom,
% 0.30/0.50 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add2
% 0.30/0.50 thf(fact_1362_le__add1,axiom,
% 0.30/0.50 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add1
% 0.30/0.50 thf(fact_1363_add__leE,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.30/0.50 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.50 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_leE
% 0.30/0.50 thf(fact_1364_Nat_Odiff__cancel,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.30/0.50 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_cancel
% 0.30/0.50 thf(fact_1365_diff__cancel2,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.30/0.50 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_cancel2
% 0.30/0.50 thf(fact_1366_diff__add__inverse,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse
% 0.30/0.50 thf(fact_1367_diff__add__inverse2,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse2
% 0.30/0.50 thf(fact_1368_left__add__mult__distrib,axiom,
% 0.30/0.50 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_mult_distrib
% 0.30/0.50 thf(fact_1369_add__mult__distrib2,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mult_distrib2
% 0.30/0.50 thf(fact_1370_add__mult__distrib,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mult_distrib
% 0.30/0.50 thf(fact_1371_int__distrib_I2_J,axiom,
% 0.30/0.50 ! [W2: int,Z1: int,Z22: int] :
% 0.30/0.50 ( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
% 0.30/0.50 = ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_distrib(2)
% 0.30/0.50 thf(fact_1372_int__distrib_I1_J,axiom,
% 0.30/0.50 ! [Z1: int,Z22: int,W2: int] :
% 0.30/0.50 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
% 0.30/0.50 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % int_distrib(1)
% 0.30/0.50 thf(fact_1373_div__mod__decomp__int,axiom,
% 0.30/0.50 ! [A6: int,N: int] :
% 0.30/0.50 ( A6
% 0.30/0.50 = ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A6 @ N ) @ N ) @ ( modulo_modulo @ int @ A6 @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mod_decomp_int
% 0.30/0.50 thf(fact_1374_same__sgn__sgn__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [B2: A,A3: A] :
% 0.30/0.50 ( ( ( sgn_sgn @ A @ B2 )
% 0.30/0.50 = ( sgn_sgn @ A @ A3 ) )
% 0.30/0.50 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.50 = ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % same_sgn_sgn_add
% 0.30/0.50 thf(fact_1375_gbinomial__addition__formula,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [A3: A,K: nat] :
% 0.30/0.50 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 0.30/0.50 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_addition_formula
% 0.30/0.50 thf(fact_1376_bezout__lemma__nat,axiom,
% 0.30/0.50 ! [D2: nat,A3: nat,B2: nat,X: nat,Y3: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 0.30/0.50 => ( ( dvd_dvd @ nat @ D2 @ B2 )
% 0.30/0.50 => ( ( ( ( times_times @ nat @ A3 @ X )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) )
% 0.30/0.50 | ( ( times_times @ nat @ B2 @ X )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) )
% 0.30/0.50 => ? [X3: nat,Y5: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 0.30/0.50 & ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A3 @ B2 ) )
% 0.30/0.50 & ( ( ( times_times @ nat @ A3 @ X3 )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y5 ) @ D2 ) )
% 0.30/0.50 | ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X3 )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bezout_lemma_nat
% 0.30/0.50 thf(fact_1377_bezout__add__nat,axiom,
% 0.30/0.50 ! [A3: nat,B2: nat] :
% 0.30/0.50 ? [D6: nat,X3: nat,Y5: nat] :
% 0.30/0.50 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 0.30/0.50 & ( dvd_dvd @ nat @ D6 @ B2 )
% 0.30/0.50 & ( ( ( times_times @ nat @ A3 @ X3 )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) )
% 0.30/0.50 | ( ( times_times @ nat @ B2 @ X3 )
% 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D6 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bezout_add_nat
% 0.30/0.50 thf(fact_1378_gbinomial__mult__1_H,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [A3: A,K: nat] :
% 0.30/0.50 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
% 0.30/0.50 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_mult_1'
% 0.30/0.50 thf(fact_1379_gbinomial__mult__1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [A3: A,K: nat] :
% 0.30/0.50 ( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
% 0.30/0.50 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % gbinomial_mult_1
% 0.30/0.50 thf(fact_1380_nat__add__distrib,axiom,
% 0.30/0.50 ! [Z3: int,Z6: int] :
% 0.30/0.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 0.30/0.50 => ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z6 ) )
% 0.30/0.50 = ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_add_distrib
% 0.30/0.50 thf(fact_1381_Suc__times__gbinomial,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( field_char_0 @ A )
% 0.30/0.50 => ! [K: nat,A3: A] :
% 0.30/0.50 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
% 0.30/0.50 = ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_times_gbinomial
% 0.30/0.50 thf(fact_1382_zero__less__eq__of__bool,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_less_eq_of_bool
% 0.30/0.50 thf(fact_1383_of__bool__less__eq__one,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_less_eq_one
% 0.30/0.50 thf(fact_1384_of__bool__def,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ( ( zero_neq_one_of_bool @ A )
% 0.30/0.50 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % of_bool_def
% 0.30/0.50 thf(fact_1385_split__of__bool,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ! [P: A > $o,P5: $o] :
% 0.30/0.50 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 0.30/0.50 = ( ( P5
% 0.30/0.50 => ( P @ ( one_one @ A ) ) )
% 0.30/0.50 & ( ~ P5
% 0.30/0.50 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % split_of_bool
% 0.30/0.50 thf(fact_1386_split__of__bool__asm,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( zero_neq_one @ A )
% 0.30/0.50 => ! [P: A > $o,P5: $o] :
% 0.30/0.50 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 0.30/0.50 = ( ~ ( ( P5
% 0.30/0.50 & ~ ( P @ ( one_one @ A ) ) )
% 0.30/0.50 | ( ~ P5
% 0.30/0.50 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % split_of_bool_asm
% 0.30/0.50 thf(fact_1387_add__decreasing,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_decreasing
% 0.30/0.50 thf(fact_1388_add__increasing,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.50 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_increasing
% 0.30/0.50 thf(fact_1389_add__decreasing2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_decreasing2
% 0.30/0.50 thf(fact_1390_add__increasing2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.50 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_increasing2
% 0.30/0.50 thf(fact_1391_add__nonneg__nonneg,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_nonneg_nonneg
% 0.30/0.50 thf(fact_1392_add__nonpos__nonpos,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_nonpos_nonpos
% 0.30/0.50 thf(fact_1393_add__nonneg__eq__0__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [X: A,Y3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.50 => ( ( ( plus_plus @ A @ X @ Y3 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( ( X
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 & ( Y3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_nonneg_eq_0_iff
% 0.30/0.50 thf(fact_1394_add__nonpos__eq__0__iff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [X: A,Y3: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ( plus_plus @ A @ X @ Y3 )
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 = ( ( X
% 0.30/0.50 = ( zero_zero @ A ) )
% 0.30/0.50 & ( Y3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_nonpos_eq_0_iff
% 0.30/0.50 thf(fact_1395_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.50 & ( ord_less @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_field(4)
% 0.30/0.50 thf(fact_1396_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.50 ( ( ( ord_less @ A @ I @ J )
% 0.30/0.50 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono_thms_linordered_field(3)
% 0.30/0.50 thf(fact_1397_add__le__less__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_less_mono
% 0.30/0.50 thf(fact_1398_add__less__le__mono,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere580206878836729694up_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_le_mono
% 0.30/0.50 thf(fact_1399_add__neg__neg,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_neg_neg
% 0.30/0.50 thf(fact_1400_add__pos__pos,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_pos_pos
% 0.30/0.50 thf(fact_1401_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ~ ! [C3: A] :
% 0.30/0.50 ( ( B2
% 0.30/0.50 = ( plus_plus @ A @ A3 @ C3 ) )
% 0.30/0.50 => ( C3
% 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % canonically_ordered_monoid_add_class.lessE
% 0.30/0.50 thf(fact_1402_pos__add__strict,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( strict7427464778891057005id_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.50 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.50 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pos_add_strict
% 0.30/0.50 thf(fact_1403_add__less__zeroD,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_idom @ A )
% 0.30/0.50 => ! [X: A,Y3: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( zero_zero @ A ) )
% 0.30/0.50 => ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.50 | ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_zeroD
% 0.30/0.50 thf(fact_1404_add__le__imp__le__diff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [I: A,K: A,N: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.50 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_imp_le_diff
% 0.30/0.50 thf(fact_1405_add__le__add__imp__diff__le,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [I: A,K: A,N: A,J: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.50 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.30/0.50 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.30/0.50 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.30/0.50 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_le_add_imp_diff_le
% 0.30/0.50 thf(fact_1406_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ( minus_minus @ A @ B2 @ A3 )
% 0.30/0.50 = C2 )
% 0.30/0.50 = ( B2
% 0.30/0.50 = ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.50 thf(fact_1407_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.30/0.50 = B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.50 thf(fact_1408_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.50 thf(fact_1409_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.50 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.50 thf(fact_1410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.50 thf(fact_1411_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
% 0.30/0.50 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.50 thf(fact_1412_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.50 thf(fact_1413_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.50 thf(fact_1414_le__add__diff,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_diff
% 0.30/0.50 thf(fact_1415_diff__add,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.50 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
% 0.30/0.50 = B2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add
% 0.30/0.50 thf(fact_1416_le__diff__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordered_ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_eq
% 0.30/0.50 thf(fact_1417_diff__le__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordered_ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_eq
% 0.30/0.50 thf(fact_1418_less__add__one,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_one
% 0.30/0.50 thf(fact_1419_add__mono1,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linord181362715937106298miring @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mono1
% 0.30/0.50 thf(fact_1420_diff__less__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordered_ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.50 ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.50 = ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_eq
% 0.30/0.50 thf(fact_1421_less__diff__eq,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( ordered_ab_group_add @ A )
% 0.30/0.50 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.50 ( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.30/0.50 = ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_eq
% 0.30/0.50 thf(fact_1422_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.50 ! [A: $tType] :
% 0.30/0.50 ( ( linordered_semidom @ A )
% 0.30/0.50 => ! [A3: A,B2: A] :
% 0.30/0.50 ( ~ ( ord_less @ A @ A3 @ B2 )
% 0.30/0.51 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.51 = A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_semidom_class.add_diff_inverse
% 0.30/0.51 thf(fact_1423_add__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( group_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( B2
% 0.30/0.51 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_eq_0_iff
% 0.30/0.51 thf(fact_1424_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ab_group_add @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ab_group_add_class.ab_left_minus
% 0.30/0.51 thf(fact_1425_add_Oinverse__unique,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( group_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( uminus_uminus @ A @ A3 )
% 0.30/0.51 = B2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add.inverse_unique
% 0.30/0.51 thf(fact_1426_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( group_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( uminus_uminus @ A @ B2 ) )
% 0.30/0.51 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_neg_iff_add_eq_0
% 0.30/0.51 thf(fact_1427_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( group_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ( uminus_uminus @ A @ A3 )
% 0.30/0.51 = B2 )
% 0.30/0.51 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % neg_eq_iff_add_eq_0
% 0.30/0.51 thf(fact_1428_square__diff__square__factored,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
% 0.30/0.51 = ( times_times @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1429_eq__add__iff2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( C2
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff2
% 0.30/0.51 thf(fact_1430_eq__add__iff1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 )
% 0.30/0.51 = D2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff1
% 0.30/0.51 thf(fact_1431_pinf_I9_J,axiom,
% 0.30/0.51 ! [B: $tType] :
% 0.30/0.51 ( ( ( plus @ B )
% 0.30/0.51 & ( linorder @ B )
% 0.30/0.51 & ( dvd @ B ) )
% 0.30/0.51 => ! [D2: B,S: B] :
% 0.30/0.51 ? [Z4: B] :
% 0.30/0.51 ! [X4: B] :
% 0.30/0.51 ( ( ord_less @ B @ Z4 @ X4 )
% 0.30/0.51 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
% 0.30/0.51 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pinf(9)
% 0.30/0.51 thf(fact_1432_pinf_I10_J,axiom,
% 0.30/0.51 ! [B: $tType] :
% 0.30/0.51 ( ( ( plus @ B )
% 0.30/0.51 & ( linorder @ B )
% 0.30/0.51 & ( dvd @ B ) )
% 0.30/0.51 => ! [D2: B,S: B] :
% 0.30/0.51 ? [Z4: B] :
% 0.30/0.51 ! [X4: B] :
% 0.30/0.51 ( ( ord_less @ B @ Z4 @ X4 )
% 0.30/0.51 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
% 0.30/0.51 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pinf(10)
% 0.30/0.51 thf(fact_1433_minf_I9_J,axiom,
% 0.30/0.51 ! [B: $tType] :
% 0.30/0.51 ( ( ( plus @ B )
% 0.30/0.51 & ( linorder @ B )
% 0.30/0.51 & ( dvd @ B ) )
% 0.30/0.51 => ! [D2: B,S: B] :
% 0.30/0.51 ? [Z4: B] :
% 0.30/0.51 ! [X4: B] :
% 0.30/0.51 ( ( ord_less @ B @ X4 @ Z4 )
% 0.30/0.51 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
% 0.30/0.51 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minf(9)
% 0.30/0.51 thf(fact_1434_minf_I10_J,axiom,
% 0.30/0.51 ! [B: $tType] :
% 0.30/0.51 ( ( ( plus @ B )
% 0.30/0.51 & ( linorder @ B )
% 0.30/0.51 & ( dvd @ B ) )
% 0.30/0.51 => ! [D2: B,S: B] :
% 0.30/0.51 ? [Z4: B] :
% 0.30/0.51 ! [X4: B] :
% 0.30/0.51 ( ( ord_less @ B @ X4 @ Z4 )
% 0.30/0.51 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
% 0.30/0.51 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minf(10)
% 0.30/0.51 thf(fact_1435_group__cancel_Osub2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ab_group_add @ A )
% 0.30/0.51 => ! [B7: A,K: A,B2: A,A3: A] :
% 0.30/0.51 ( ( B7
% 0.30/0.51 = ( plus_plus @ A @ K @ B2 ) )
% 0.30/0.51 => ( ( minus_minus @ A @ A3 @ B7 )
% 0.30/0.51 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % group_cancel.sub2
% 0.30/0.51 thf(fact_1436_diff__conv__add__uminus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( group_add @ A )
% 0.30/0.51 => ( ( minus_minus @ A )
% 0.30/0.51 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % diff_conv_add_uminus
% 0.30/0.51 thf(fact_1437_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ab_group_add @ A )
% 0.30/0.51 => ( ( minus_minus @ A )
% 0.30/0.51 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.51 thf(fact_1438_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.51 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_plus_div_distrib_dvd_left
% 0.30/0.51 thf(fact_1439_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.51 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_plus_div_distrib_dvd_right
% 0.30/0.51 thf(fact_1440_mod__eqE,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.51 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.51 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.30/0.51 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 0.30/0.51 => ~ ! [D6: A] :
% 0.30/0.51 ( B2
% 0.30/0.51 != ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D6 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_eqE
% 0.30/0.51 thf(fact_1441_div__add1__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.51 = ( 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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_add1_eq
% 0.30/0.51 thf(fact_1442_add__is__1,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( ( plus_plus @ nat @ M @ N )
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 = ( ( ( M
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) )
% 0.30/0.51 | ( ( M
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_is_1
% 0.30/0.51 thf(fact_1443_one__is__add,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.30/0.51 = ( plus_plus @ nat @ M @ N ) )
% 0.30/0.51 = ( ( ( M
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) )
% 0.30/0.51 | ( ( M
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % one_is_add
% 0.30/0.51 thf(fact_1444_less__natE,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ M @ N )
% 0.30/0.51 => ~ ! [Q5: nat] :
% 0.30/0.51 ( N
% 0.30/0.51 != ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_natE
% 0.30/0.51 thf(fact_1445_less__add__Suc1,axiom,
% 0.30/0.51 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_Suc1
% 0.30/0.51 thf(fact_1446_less__add__Suc2,axiom,
% 0.30/0.51 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_Suc2
% 0.30/0.51 thf(fact_1447_less__iff__Suc__add,axiom,
% 0.30/0.51 ( ( ord_less @ nat )
% 0.30/0.51 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.51 ? [K3: nat] :
% 0.30/0.51 ( N3
% 0.30/0.51 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_iff_Suc_add
% 0.30/0.51 thf(fact_1448_less__imp__Suc__add,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ M @ N )
% 0.30/0.51 => ? [K2: nat] :
% 0.30/0.51 ( N
% 0.30/0.51 = ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_imp_Suc_add
% 0.30/0.51 thf(fact_1449_less__imp__add__positive,axiom,
% 0.30/0.51 ! [I: nat,J: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ I @ J )
% 0.30/0.51 => ? [K2: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
% 0.30/0.51 & ( ( plus_plus @ nat @ I @ K2 )
% 0.30/0.51 = J ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_imp_add_positive
% 0.30/0.51 thf(fact_1450_mono__nat__linear__lb,axiom,
% 0.30/0.51 ! [F: nat > nat,M: nat,K: nat] :
% 0.30/0.51 ( ! [M6: nat,N4: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ M6 @ N4 )
% 0.30/0.51 => ( ord_less @ nat @ ( F @ M6 ) @ ( F @ N4 ) ) )
% 0.30/0.51 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mono_nat_linear_lb
% 0.30/0.51 thf(fact_1451_ex__has__greatest__nat__lemma,axiom,
% 0.30/0.51 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 0.30/0.51 ( ( P @ K )
% 0.30/0.51 => ( ! [X3: A] :
% 0.30/0.51 ( ( P @ X3 )
% 0.30/0.51 => ? [Y4: A] :
% 0.30/0.51 ( ( P @ Y4 )
% 0.30/0.51 & ~ ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) )
% 0.30/0.51 => ? [Y5: A] :
% 0.30/0.51 ( ( P @ Y5 )
% 0.30/0.51 & ~ ( ord_less @ nat @ ( F @ Y5 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ex_has_greatest_nat_lemma
% 0.30/0.51 thf(fact_1452_bezout__add__strong__nat,axiom,
% 0.30/0.51 ! [A3: nat,B2: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 => ? [D6: nat,X3: nat,Y5: nat] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 0.30/0.51 & ( dvd_dvd @ nat @ D6 @ B2 )
% 0.30/0.51 & ( ( times_times @ nat @ A3 @ X3 )
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % bezout_add_strong_nat
% 0.30/0.51 thf(fact_1453_diff__add__0,axiom,
% 0.30/0.51 ! [N: nat,M: nat] :
% 0.30/0.51 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 0.30/0.51 = ( zero_zero @ nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % diff_add_0
% 0.30/0.51 thf(fact_1454_odd__nonzero,axiom,
% 0.30/0.51 ! [Z3: int] :
% 0.30/0.51 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
% 0.30/0.51 != ( zero_zero @ int ) ) ).
% 0.30/0.51
% 0.30/0.51 % odd_nonzero
% 0.30/0.51 thf(fact_1455_mult__Suc,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( times_times @ nat @ ( suc @ M ) @ N )
% 0.30/0.51 = ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_Suc
% 0.30/0.51 thf(fact_1456_less__diff__conv,axiom,
% 0.30/0.51 ! [I: nat,J: nat,K: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.30/0.51 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_diff_conv
% 0.30/0.51 thf(fact_1457_add__diff__inverse__nat,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ~ ( ord_less @ nat @ M @ N )
% 0.30/0.51 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.51 = M ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_inverse_nat
% 0.30/0.51 thf(fact_1458_Suc__eq__plus1__left,axiom,
% 0.30/0.51 ( suc
% 0.30/0.51 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_eq_plus1_left
% 0.30/0.51 thf(fact_1459_plus__1__eq__Suc,axiom,
% 0.30/0.51 ( ( plus_plus @ nat @ ( one_one @ nat ) )
% 0.30/0.51 = suc ) ).
% 0.30/0.51
% 0.30/0.51 % plus_1_eq_Suc
% 0.30/0.51 thf(fact_1460_Suc__eq__plus1,axiom,
% 0.30/0.51 ( suc
% 0.30/0.51 = ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_eq_plus1
% 0.30/0.51 thf(fact_1461_Nat_Ole__imp__diff__is__add,axiom,
% 0.30/0.51 ! [I: nat,J: nat,K: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.51 => ( ( ( minus_minus @ nat @ J @ I )
% 0.30/0.51 = K )
% 0.30/0.51 = ( J
% 0.30/0.51 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Nat.le_imp_diff_is_add
% 0.30/0.51 thf(fact_1462_Nat_Odiff__add__assoc2,axiom,
% 0.30/0.51 ! [K: nat,J: nat,I: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.51 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 0.30/0.51 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Nat.diff_add_assoc2
% 0.30/0.51 thf(fact_1463_Nat_Odiff__add__assoc,axiom,
% 0.30/0.51 ! [K: nat,J: nat,I: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.51 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.30/0.51 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Nat.diff_add_assoc
% 0.30/0.51 thf(fact_1464_Nat_Ole__diff__conv2,axiom,
% 0.30/0.51 ! [K: nat,J: nat,I: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Nat.le_diff_conv2
% 0.30/0.51 thf(fact_1465_le__diff__conv,axiom,
% 0.30/0.51 ! [J: nat,K: nat,I: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.30/0.51 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_diff_conv
% 0.30/0.51 thf(fact_1466_int__ge__induct,axiom,
% 0.30/0.51 ! [K: int,I: int,P: int > $o] :
% 0.30/0.51 ( ( ord_less_eq @ int @ K @ I )
% 0.30/0.51 => ( ( P @ K )
% 0.30/0.51 => ( ! [I4: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ K @ I4 )
% 0.30/0.51 => ( ( P @ I4 )
% 0.30/0.51 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.51 => ( P @ I ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_ge_induct
% 0.30/0.51 thf(fact_1467_zless__add1__eq,axiom,
% 0.30/0.51 ! [W2: int,Z3: int] :
% 0.30/0.51 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
% 0.30/0.51 = ( ( ord_less @ int @ W2 @ Z3 )
% 0.30/0.51 | ( W2 = Z3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zless_add1_eq
% 0.30/0.51 thf(fact_1468_int__gr__induct,axiom,
% 0.30/0.51 ! [K: int,I: int,P: int > $o] :
% 0.30/0.51 ( ( ord_less @ int @ K @ I )
% 0.30/0.51 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.30/0.51 => ( ! [I4: int] :
% 0.30/0.51 ( ( ord_less @ int @ K @ I4 )
% 0.30/0.51 => ( ( P @ I4 )
% 0.30/0.51 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.51 => ( P @ I ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_gr_induct
% 0.30/0.51 thf(fact_1469_zle__iff__zadd,axiom,
% 0.30/0.51 ( ( ord_less_eq @ int )
% 0.30/0.51 = ( ^ [W3: int,Z7: int] :
% 0.30/0.51 ? [N3: nat] :
% 0.30/0.51 ( Z7
% 0.30/0.51 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zle_iff_zadd
% 0.30/0.51 thf(fact_1470_zdvd__period,axiom,
% 0.30/0.51 ! [A3: int,D2: int,X: int,T2: int,C2: int] :
% 0.30/0.51 ( ( dvd_dvd @ int @ A3 @ D2 )
% 0.30/0.51 => ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
% 0.30/0.51 = ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdvd_period
% 0.30/0.51 thf(fact_1471_zdvd__reduce,axiom,
% 0.30/0.51 ! [K: int,N: int,M: int] :
% 0.30/0.51 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
% 0.30/0.51 = ( dvd_dvd @ int @ K @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdvd_reduce
% 0.30/0.51 thf(fact_1472_nat__mod__eq__iff,axiom,
% 0.30/0.51 ! [X: nat,N: nat,Y3: nat] :
% 0.30/0.51 ( ( ( modulo_modulo @ nat @ X @ N )
% 0.30/0.51 = ( modulo_modulo @ nat @ Y3 @ N ) )
% 0.30/0.51 = ( ? [Q1: nat,Q22: nat] :
% 0.30/0.51 ( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
% 0.30/0.51 = ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mod_eq_iff
% 0.30/0.51 thf(fact_1473_frac__1__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 0.30/0.51 = ( archimedean_frac @ A @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_1_eq
% 0.30/0.51 thf(fact_1474_gbinomial__factors,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,K: nat] :
% 0.30/0.51 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.30/0.51 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_factors
% 0.30/0.51 thf(fact_1475_gbinomial__rec,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,K: nat] :
% 0.30/0.51 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.30/0.51 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_rec
% 0.30/0.51 thf(fact_1476_dbl__inc__def,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( neg_numeral @ A )
% 0.30/0.51 => ( ( neg_numeral_dbl_inc @ A )
% 0.30/0.51 = ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dbl_inc_def
% 0.30/0.51 thf(fact_1477_gbinomial__reduce__nat,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [K: nat,A3: A] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.51 => ( ( gbinomial @ A @ A3 @ K )
% 0.30/0.51 = ( 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 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_reduce_nat
% 0.30/0.51 thf(fact_1478_field__le__epsilon,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ! [E2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 0.30/0.51 => ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y3 @ E2 ) ) )
% 0.30/0.51 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % field_le_epsilon
% 0.30/0.51 thf(fact_1479_add__strict__increasing2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere8940638589300402666id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.51 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_strict_increasing2
% 0.30/0.51 thf(fact_1480_add__strict__increasing,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere8940638589300402666id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.51 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_strict_increasing
% 0.30/0.51 thf(fact_1481_add__pos__nonneg,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_pos_nonneg
% 0.30/0.51 thf(fact_1482_add__nonpos__neg,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_nonpos_neg
% 0.30/0.51 thf(fact_1483_add__nonneg__pos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_nonneg_pos
% 0.30/0.51 thf(fact_1484_add__neg__nonpos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_neg_nonpos
% 0.30/0.51 thf(fact_1485_sum__squares__ge__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_ring @ A )
% 0.30/0.51 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sum_squares_ge_zero
% 0.30/0.51 thf(fact_1486_discrete,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.51 => ( ( ord_less @ A )
% 0.30/0.51 = ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % discrete
% 0.30/0.51 thf(fact_1487_not__sum__squares__lt__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_ring @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % not_sum_squares_lt_zero
% 0.30/0.51 thf(fact_1488_zero__less__two,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord181362715937106298miring @ A )
% 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_less_two
% 0.30/0.51 thf(fact_1489_le__add__iff2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordered_ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_iff2
% 0.30/0.51 thf(fact_1490_le__add__iff1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordered_ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_iff1
% 0.30/0.51 thf(fact_1491_less__add__iff2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordered_ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff2
% 0.30/0.51 thf(fact_1492_less__add__iff1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordered_ring @ A )
% 0.30/0.51 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 0.30/0.51 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff1
% 0.30/0.51 thf(fact_1493_unity__coeff__ex,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( dvd @ A )
% 0.30/0.51 & ( semiring_0 @ A ) )
% 0.30/0.51 => ! [P: A > $o,L: A] :
% 0.30/0.51 ( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
% 0.30/0.51 = ( ? [X2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
% 0.30/0.51 & ( P @ X2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % unity_coeff_ex
% 0.30/0.51 thf(fact_1494_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.51 ( ( ( Z3
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 0.30/0.51 = B2 ) )
% 0.30/0.51 & ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_eq_if_simps(2)
% 0.30/0.51 thf(fact_1495_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.51 ( ( ( Z3
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 0.30/0.51 = A3 ) )
% 0.30/0.51 & ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_eq_if_simps(1)
% 0.30/0.51 thf(fact_1496_add__frac__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field @ A )
% 0.30/0.51 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 0.30/0.51 ( ( Y3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_frac_eq
% 0.30/0.51 thf(fact_1497_add__frac__num,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field @ A )
% 0.30/0.51 => ! [Y3: A,X: A,Z3: A] :
% 0.30/0.51 ( ( Y3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_frac_num
% 0.30/0.51 thf(fact_1498_add__num__frac,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field @ A )
% 0.30/0.51 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.51 ( ( Y3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_num_frac
% 0.30/0.51 thf(fact_1499_add__divide__eq__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.51 ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_eq_iff
% 0.30/0.51 thf(fact_1500_divide__add__eq__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.51 ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_add_eq_iff
% 0.30/0.51 thf(fact_1501_div__add__self1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_add_self1
% 0.30/0.51 thf(fact_1502_div__add__self2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_add_self2
% 0.30/0.51 thf(fact_1503_square__diff__one__factored,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
% 0.30/0.51 = ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1504_gt__half__sum,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.51 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gt_half_sum
% 0.30/0.51 thf(fact_1505_less__half__sum,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.51 => ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_half_sum
% 0.30/0.51 thf(fact_1506_inf__period_I4_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( comm_ring @ A )
% 0.30/0.51 & ( dvd @ A ) )
% 0.30/0.51 => ! [D2: A,D4: A,T2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ D2 @ D4 )
% 0.30/0.51 => ! [X4: A,K4: A] :
% 0.30/0.51 ( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
% 0.30/0.51 = ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % inf_period(4)
% 0.30/0.51 thf(fact_1507_inf__period_I3_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( comm_ring @ A )
% 0.30/0.51 & ( dvd @ A ) )
% 0.30/0.51 => ! [D2: A,D4: A,T2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ D2 @ D4 )
% 0.30/0.51 => ! [X4: A,K4: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
% 0.30/0.51 = ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % inf_period(3)
% 0.30/0.51 thf(fact_1508_gbinomial__of__nat__symmetric,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [K: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.51 => ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
% 0.30/0.51 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_of_nat_symmetric
% 0.30/0.51 thf(fact_1509_div__mult1__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.51 = ( 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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_mult1_eq
% 0.30/0.51 thf(fact_1510_mult__div__mod__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_modulo @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_div_mod_eq
% 0.30/0.51 thf(fact_1511_mod__mult__div__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_modulo @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_mult_div_eq
% 0.30/0.51 thf(fact_1512_mod__div__mult__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_modulo @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_div_mult_eq
% 0.30/0.51 thf(fact_1513_div__mult__mod__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_modulo @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_mult_mod_eq
% 0.30/0.51 thf(fact_1514_mod__div__decomp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_modulo @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( A3
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_div_decomp
% 0.30/0.51 thf(fact_1515_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semidom_modulo @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_div_mod_rules(1)
% 0.30/0.51 thf(fact_1516_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semidom_modulo @ A )
% 0.30/0.51 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.51 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 0.30/0.51 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_div_mod_rules(2)
% 0.30/0.51 thf(fact_1517_nat__diff__split__asm,axiom,
% 0.30/0.51 ! [P: nat > $o,A3: nat,B2: nat] :
% 0.30/0.51 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.30/0.51 = ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
% 0.30/0.51 & ~ ( P @ ( zero_zero @ nat ) ) )
% 0.30/0.51 | ? [D5: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ nat @ B2 @ D5 ) )
% 0.30/0.51 & ~ ( P @ D5 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_diff_split_asm
% 0.30/0.51 thf(fact_1518_nat__diff__split,axiom,
% 0.30/0.51 ! [P: nat > $o,A3: nat,B2: nat] :
% 0.30/0.51 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.30/0.51 = ( ( ( ord_less @ nat @ A3 @ B2 )
% 0.30/0.51 => ( P @ ( zero_zero @ nat ) ) )
% 0.30/0.51 & ! [D5: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ nat @ B2 @ D5 ) )
% 0.30/0.51 => ( P @ D5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_diff_split
% 0.30/0.51 thf(fact_1519_mod__pos__neg__trivial,axiom,
% 0.30/0.51 ! [K: int,L: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( modulo_modulo @ int @ K @ L )
% 0.30/0.51 = ( plus_plus @ int @ K @ L ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_pos_neg_trivial
% 0.30/0.51 thf(fact_1520_less__diff__conv2,axiom,
% 0.30/0.51 ! [K: nat,J: nat,I: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ J )
% 0.30/0.51 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.30/0.51 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_diff_conv2
% 0.30/0.51 thf(fact_1521_odd__less__0__iff,axiom,
% 0.30/0.51 ! [Z3: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 ) @ ( zero_zero @ int ) )
% 0.30/0.51 = ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % odd_less_0_iff
% 0.30/0.51 thf(fact_1522_zless__iff__Suc__zadd,axiom,
% 0.30/0.51 ( ( ord_less @ int )
% 0.30/0.51 = ( ^ [W3: int,Z7: int] :
% 0.30/0.51 ? [N3: nat] :
% 0.30/0.51 ( Z7
% 0.30/0.51 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zless_iff_Suc_zadd
% 0.30/0.51 thf(fact_1523_int__ops_I4_J,axiom,
% 0.30/0.51 ! [A3: nat] :
% 0.30/0.51 ( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
% 0.30/0.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_ops(4)
% 0.30/0.51 thf(fact_1524_int__Suc,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( semiring_1_of_nat @ int @ ( suc @ N ) )
% 0.30/0.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_Suc
% 0.30/0.51 thf(fact_1525_zmod__zmult2__eq,axiom,
% 0.30/0.51 ! [C2: int,A3: int,B2: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 0.30/0.51 => ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zmod_zmult2_eq
% 0.30/0.51 thf(fact_1526_zless__imp__add1__zle,axiom,
% 0.30/0.51 ! [W2: int,Z3: int] :
% 0.30/0.51 ( ( ord_less @ int @ W2 @ Z3 )
% 0.30/0.51 => ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % zless_imp_add1_zle
% 0.30/0.51 thf(fact_1527_add1__zle__eq,axiom,
% 0.30/0.51 ! [W2: int,Z3: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 )
% 0.30/0.51 = ( ord_less @ int @ W2 @ Z3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % add1_zle_eq
% 0.30/0.51 thf(fact_1528_nat__eq__add__iff1,axiom,
% 0.30/0.51 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.51 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
% 0.30/0.51 = N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_eq_add_iff1
% 0.30/0.51 thf(fact_1529_nat__eq__add__iff2,axiom,
% 0.30/0.51 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.51 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( M
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_eq_add_iff2
% 0.30/0.51 thf(fact_1530_nat__le__add__iff1,axiom,
% 0.30/0.51 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_le_add_iff1
% 0.30/0.51 thf(fact_1531_nat__le__add__iff2,axiom,
% 0.30/0.51 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_le_add_iff2
% 0.30/0.51 thf(fact_1532_nat__diff__add__eq1,axiom,
% 0.30/0.51 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.51 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_diff_add_eq1
% 0.30/0.51 thf(fact_1533_nat__diff__add__eq2,axiom,
% 0.30/0.51 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.51 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_diff_add_eq2
% 0.30/0.51 thf(fact_1534_fact__fact__dvd__fact,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % fact_fact_dvd_fact
% 0.30/0.51 thf(fact_1535_int__induct,axiom,
% 0.30/0.51 ! [P: int > $o,K: int,I: int] :
% 0.30/0.51 ( ( P @ K )
% 0.30/0.51 => ( ! [I4: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ K @ I4 )
% 0.30/0.51 => ( ( P @ I4 )
% 0.30/0.51 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.51 => ( ! [I4: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ I4 @ K )
% 0.30/0.51 => ( ( P @ I4 )
% 0.30/0.51 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.51 => ( P @ I ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_induct
% 0.30/0.51 thf(fact_1536_mod__eq__nat1E,axiom,
% 0.30/0.51 ! [M: nat,Q4: nat,N: nat] :
% 0.30/0.51 ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 0.30/0.51 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.51 => ~ ! [S2: nat] :
% 0.30/0.51 ( M
% 0.30/0.51 != ( plus_plus @ nat @ N @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_eq_nat1E
% 0.30/0.51 thf(fact_1537_mod__eq__nat2E,axiom,
% 0.30/0.51 ! [M: nat,Q4: nat,N: nat] :
% 0.30/0.51 ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 0.30/0.51 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.51 => ~ ! [S2: nat] :
% 0.30/0.51 ( N
% 0.30/0.51 != ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_eq_nat2E
% 0.30/0.51 thf(fact_1538_nat__mod__eq__lemma,axiom,
% 0.30/0.51 ! [X: nat,N: nat,Y3: nat] :
% 0.30/0.51 ( ( ( modulo_modulo @ nat @ X @ N )
% 0.30/0.51 = ( modulo_modulo @ nat @ Y3 @ N ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ Y3 @ X )
% 0.30/0.51 => ? [Q5: nat] :
% 0.30/0.51 ( X
% 0.30/0.51 = ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mod_eq_lemma
% 0.30/0.51 thf(fact_1539_mod__mult2__eq,axiom,
% 0.30/0.51 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.51 ( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_mult2_eq
% 0.30/0.51 thf(fact_1540_div__mod__decomp,axiom,
% 0.30/0.51 ! [A6: nat,N: nat] :
% 0.30/0.51 ( A6
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A6 @ N ) @ N ) @ ( modulo_modulo @ nat @ A6 @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_mod_decomp
% 0.30/0.51 thf(fact_1541_dbl__dec__def,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( neg_numeral @ A )
% 0.30/0.51 => ( ( neg_numeral_dbl_dec @ A )
% 0.30/0.51 = ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dbl_dec_def
% 0.30/0.51 thf(fact_1542_gbinomial__absorb__comp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,K: nat] :
% 0.30/0.51 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
% 0.30/0.51 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_absorb_comp
% 0.30/0.51 thf(fact_1543_convex__bound__le,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord6961819062388156250ring_1 @ A )
% 0.30/0.51 => ! [X: A,A3: A,Y3: A,U: A,V: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ X @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ Y3 @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 0.30/0.51 => ( ( ( plus_plus @ A @ U @ V )
% 0.30/0.51 = ( one_one @ A ) )
% 0.30/0.51 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % convex_bound_le
% 0.30/0.51 thf(fact_1544_divide__int__unfold,axiom,
% 0.30/0.51 ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.51 ( ( ( ( ( sgn_sgn @ int @ L )
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 | ( ( sgn_sgn @ int @ K )
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) )
% 0.30/0.51 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.51 = ( zero_zero @ int ) ) )
% 0.30/0.51 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 | ( ( sgn_sgn @ int @ K )
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) )
% 0.30/0.51 => ( ( ( ( sgn_sgn @ int @ K )
% 0.30/0.51 = ( sgn_sgn @ int @ L ) )
% 0.30/0.51 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.51 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
% 0.30/0.51 & ( ( ( sgn_sgn @ int @ K )
% 0.30/0.51 != ( sgn_sgn @ int @ L ) )
% 0.30/0.51 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.51 = ( uminus_uminus @ int
% 0.30/0.51 @ ( semiring_1_of_nat @ int
% 0.30/0.51 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
% 0.30/0.51 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.51 @ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_int_unfold
% 0.30/0.51 thf(fact_1545_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,A3: A,B2: A] :
% 0.30/0.51 ( ( ( Z3
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 0.30/0.51 = B2 ) )
% 0.30/0.51 & ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_eq_if_simps(3)
% 0.30/0.51 thf(fact_1546_minus__divide__add__eq__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.51 ( ( Z3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
% 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_divide_add_eq_iff
% 0.30/0.51 thf(fact_1547_mod__mult2__eq_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_mult2_eq'
% 0.30/0.51 thf(fact_1548_add__eq__if,axiom,
% 0.30/0.51 ( ( plus_plus @ nat )
% 0.30/0.51 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.51 ( if @ nat
% 0.30/0.51 @ ( M2
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 @ N3
% 0.30/0.51 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_eq_if
% 0.30/0.51 thf(fact_1549_le__imp__0__less,axiom,
% 0.30/0.51 ! [Z3: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.51 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_imp_0_less
% 0.30/0.51 thf(fact_1550_verit__le__mono__div__int,axiom,
% 0.30/0.51 ! [A6: int,B7: int,N: int] :
% 0.30/0.51 ( ( ord_less @ int @ A6 @ B7 )
% 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.51 => ( ord_less_eq @ int
% 0.30/0.51 @ ( plus_plus @ int @ ( divide_divide @ int @ A6 @ N )
% 0.30/0.51 @ ( if @ int
% 0.30/0.51 @ ( ( modulo_modulo @ int @ B7 @ N )
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 @ ( one_one @ int )
% 0.30/0.51 @ ( zero_zero @ int ) ) )
% 0.30/0.51 @ ( divide_divide @ int @ B7 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % verit_le_mono_div_int
% 0.30/0.51 thf(fact_1551_nat__less__add__iff1,axiom,
% 0.30/0.51 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.51 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_less_add_iff1
% 0.30/0.51 thf(fact_1552_nat__less__add__iff2,axiom,
% 0.30/0.51 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.51 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.30/0.51 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_less_add_iff2
% 0.30/0.51 thf(fact_1553_incr__mult__lemma,axiom,
% 0.30/0.51 ! [D2: int,P: int > $o,K: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.51 => ( ! [X3: int] :
% 0.30/0.51 ( ( P @ X3 )
% 0.30/0.51 => ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ! [X4: int] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % incr_mult_lemma
% 0.30/0.51 thf(fact_1554_unique__quotient__lemma__neg,axiom,
% 0.30/0.51 ! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( ord_less @ int @ B2 @ R )
% 0.30/0.51 => ( ( ord_less @ int @ B2 @ R4 )
% 0.30/0.51 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % unique_quotient_lemma_neg
% 0.30/0.51 thf(fact_1555_unique__quotient__lemma,axiom,
% 0.30/0.51 ! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 0.30/0.51 => ( ( ord_less @ int @ R4 @ B2 )
% 0.30/0.51 => ( ( ord_less @ int @ R @ B2 )
% 0.30/0.51 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % unique_quotient_lemma
% 0.30/0.51 thf(fact_1556_zdiv__mono2__neg__lemma,axiom,
% 0.30/0.51 ! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
% 0.30/0.51 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 0.30/0.51 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( ord_less @ int @ R @ B2 )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.30/0.51 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.30/0.51 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdiv_mono2_neg_lemma
% 0.30/0.51 thf(fact_1557_zdiv__mono2__lemma,axiom,
% 0.30/0.51 ! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
% 0.30/0.51 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 0.30/0.51 => ( ( ord_less @ int @ R4 @ B4 )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.30/0.51 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.30/0.51 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdiv_mono2_lemma
% 0.30/0.51 thf(fact_1558_q__pos__lemma,axiom,
% 0.30/0.51 ! [B4: int,Q6: int,R4: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 0.30/0.51 => ( ( ord_less @ int @ R4 @ B4 )
% 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.30/0.51 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % q_pos_lemma
% 0.30/0.51 thf(fact_1559_int__div__pos__eq,axiom,
% 0.30/0.51 ! [A3: int,B2: int,Q4: int,R: int] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.30/0.51 => ( ( ord_less @ int @ R @ B2 )
% 0.30/0.51 => ( ( divide_divide @ int @ A3 @ B2 )
% 0.30/0.51 = Q4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_div_pos_eq
% 0.30/0.51 thf(fact_1560_int__div__neg__eq,axiom,
% 0.30/0.51 ! [A3: int,B2: int,Q4: int,R: int] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( ord_less @ int @ B2 @ R )
% 0.30/0.51 => ( ( divide_divide @ int @ A3 @ B2 )
% 0.30/0.51 = Q4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_div_neg_eq
% 0.30/0.51 thf(fact_1561_split__zdiv,axiom,
% 0.30/0.51 ! [P: int > $o,N: int,K: int] :
% 0.30/0.51 ( ( P @ ( divide_divide @ int @ N @ K ) )
% 0.30/0.51 = ( ( ( K
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 => ( P @ ( zero_zero @ int ) ) )
% 0.30/0.51 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 0.30/0.51 & ( ord_less @ int @ J3 @ K )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ I5 ) ) )
% 0.30/0.51 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.51 => ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less @ int @ K @ J3 )
% 0.30/0.51 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ I5 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_zdiv
% 0.30/0.51 thf(fact_1562_int__mod__pos__eq,axiom,
% 0.30/0.51 ! [A3: int,B2: int,Q4: int,R: int] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.30/0.51 => ( ( ord_less @ int @ R @ B2 )
% 0.30/0.51 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.51 = R ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_mod_pos_eq
% 0.30/0.51 thf(fact_1563_int__mod__neg__eq,axiom,
% 0.30/0.51 ! [A3: int,B2: int,Q4: int,R: int] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( ord_less @ int @ B2 @ R )
% 0.30/0.51 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.30/0.51 = R ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_mod_neg_eq
% 0.30/0.51 thf(fact_1564_split__zmod,axiom,
% 0.30/0.51 ! [P: int > $o,N: int,K: int] :
% 0.30/0.51 ( ( P @ ( modulo_modulo @ int @ N @ K ) )
% 0.30/0.51 = ( ( ( K
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 => ( P @ N ) )
% 0.30/0.51 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 0.30/0.51 & ( ord_less @ int @ J3 @ K )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ J3 ) ) )
% 0.30/0.51 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.51 => ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less @ int @ K @ J3 )
% 0.30/0.51 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ J3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_zmod
% 0.30/0.51 thf(fact_1565_split__pos__lemma,axiom,
% 0.30/0.51 ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 0.30/0.51 = ( ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 0.30/0.51 & ( ord_less @ int @ J3 @ K )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ I5 @ J3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_pos_lemma
% 0.30/0.51 thf(fact_1566_split__neg__lemma,axiom,
% 0.30/0.51 ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.51 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 0.30/0.51 = ( ! [I5: int,J3: int] :
% 0.30/0.51 ( ( ( ord_less @ int @ K @ J3 )
% 0.30/0.51 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 0.30/0.51 & ( N
% 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 0.30/0.51 => ( P @ I5 @ J3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_neg_lemma
% 0.30/0.51 thf(fact_1567_frac__add,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 0.30/0.51 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.51 = ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) ) )
% 0.30/0.51 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 0.30/0.51 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.51 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_add
% 0.30/0.51 thf(fact_1568_mult__eq__if,axiom,
% 0.30/0.51 ( ( times_times @ nat )
% 0.30/0.51 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.51 ( if @ nat
% 0.30/0.51 @ ( M2
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 @ ( zero_zero @ nat )
% 0.30/0.51 @ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_if
% 0.30/0.51 thf(fact_1569_Suc__as__int,axiom,
% 0.30/0.51 ( suc
% 0.30/0.51 = ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_as_int
% 0.30/0.51 thf(fact_1570_split__mod,axiom,
% 0.30/0.51 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.51 ( ( P @ ( modulo_modulo @ nat @ M @ N ) )
% 0.30/0.51 = ( ( ( N
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 => ( P @ M ) )
% 0.30/0.51 & ( ( N
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 => ! [I5: nat,J3: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ J3 @ N )
% 0.30/0.51 => ( ( M
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
% 0.30/0.51 => ( P @ J3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_mod
% 0.30/0.51 thf(fact_1571_dividend__less__times__div,axiom,
% 0.30/0.51 ! [N: nat,M: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dividend_less_times_div
% 0.30/0.51 thf(fact_1572_dividend__less__div__times,axiom,
% 0.30/0.51 ! [N: nat,M: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dividend_less_div_times
% 0.30/0.51 thf(fact_1573_split__div,axiom,
% 0.30/0.51 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.51 ( ( P @ ( divide_divide @ nat @ M @ N ) )
% 0.30/0.51 = ( ( ( N
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 => ( P @ ( zero_zero @ nat ) ) )
% 0.30/0.51 & ( ( N
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 => ! [I5: nat,J3: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ J3 @ N )
% 0.30/0.51 => ( ( M
% 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
% 0.30/0.51 => ( P @ I5 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % split_div
% 0.30/0.51 thf(fact_1574_dvd__minus__add,axiom,
% 0.30/0.51 ! [Q4: nat,N: nat,R: nat,M: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ Q4 @ N )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R @ M ) )
% 0.30/0.51 => ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q4 ) )
% 0.30/0.51 = ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M ) @ Q4 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_minus_add
% 0.30/0.51 thf(fact_1575_convex__bound__lt,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord715952674999750819strict @ A )
% 0.30/0.51 => ! [X: A,A3: A,Y3: A,U: A,V: A] :
% 0.30/0.51 ( ( ord_less @ A @ X @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ Y3 @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 0.30/0.51 => ( ( ( plus_plus @ A @ U @ V )
% 0.30/0.51 = ( one_one @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % convex_bound_lt
% 0.30/0.51 thf(fact_1576_gbinomial__absorption,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [K: nat,A3: A] :
% 0.30/0.51 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
% 0.30/0.51 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_absorption
% 0.30/0.51 thf(fact_1577_scaling__mono,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [U: A,V: A,R: A,S: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ U @ V )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 0.30/0.51 => ( ( ord_less_eq @ A @ R @ S )
% 0.30/0.51 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % scaling_mono
% 0.30/0.51 thf(fact_1578_gbinomial__trinomial__revision,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [K: nat,M: nat,A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.51 => ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
% 0.30/0.51 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_trinomial_revision
% 0.30/0.51 thf(fact_1579_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.51 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.51 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.51 thf(fact_1580_Suc__nat__eq__nat__zadd1,axiom,
% 0.30/0.51 ! [Z3: int] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.51 => ( ( suc @ ( nat2 @ Z3 ) )
% 0.30/0.51 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_nat_eq_nat_zadd1
% 0.30/0.51 thf(fact_1581_div__pos__neg__trivial,axiom,
% 0.30/0.51 ! [K: int,L: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.51 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.51 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_pos_neg_trivial
% 0.30/0.51 thf(fact_1582_div__pos__geq,axiom,
% 0.30/0.51 ! [L: int,K: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.51 => ( ( ord_less_eq @ int @ L @ K )
% 0.30/0.51 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.51 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_pos_geq
% 0.30/0.51 thf(fact_1583_div__eqI,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.51 => ! [B2: A,R: A,Q4: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.30/0.51 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.30/0.51 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.51 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R )
% 0.30/0.51 = A3 )
% 0.30/0.51 => ( ( divide_divide @ A @ A3 @ B2 )
% 0.30/0.51 = Q4 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_eqI
% 0.30/0.51 thf(fact_1584_div__bounded,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.51 => ! [B2: A,R: A,Q4: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.30/0.51 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.30/0.51 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R ) @ B2 )
% 0.30/0.51 = Q4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_bounded
% 0.30/0.51 thf(fact_1585_mod__eqI,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid3128863361964157862miring @ A )
% 0.30/0.51 => ! [B2: A,R: A,Q4: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.30/0.51 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.30/0.51 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.30/0.51 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R )
% 0.30/0.51 = A3 )
% 0.30/0.51 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.30/0.51 = R ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_eqI
% 0.30/0.51 thf(fact_1586_verit__le__mono__div,axiom,
% 0.30/0.51 ! [A6: nat,B7: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ A6 @ B7 )
% 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ord_less_eq @ nat
% 0.30/0.51 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A6 @ N )
% 0.30/0.51 @ ( if @ nat
% 0.30/0.51 @ ( ( modulo_modulo @ nat @ B7 @ N )
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 @ ( one_one @ nat )
% 0.30/0.51 @ ( zero_zero @ nat ) ) )
% 0.30/0.51 @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % verit_le_mono_div
% 0.30/0.51 thf(fact_1587_division__decomp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_gcd @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.51 => ? [B8: A,C5: A] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( times_times @ A @ B8 @ C5 ) )
% 0.30/0.51 & ( dvd_dvd @ A @ B8 @ B2 )
% 0.30/0.51 & ( dvd_dvd @ A @ C5 @ C2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % division_decomp
% 0.30/0.51 thf(fact_1588_dvd__productE,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_gcd @ A )
% 0.30/0.51 => ! [P5: A,A3: A,B2: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.51 => ~ ! [X3: A,Y5: A] :
% 0.30/0.51 ( ( P5
% 0.30/0.51 = ( times_times @ A @ X3 @ Y5 ) )
% 0.30/0.51 => ( ( dvd_dvd @ A @ X3 @ A3 )
% 0.30/0.51 => ~ ( dvd_dvd @ A @ Y5 @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_productE
% 0.30/0.51 thf(fact_1589_gcd__nat_Oextremum__uniqueI,axiom,
% 0.30/0.51 ! [A3: nat] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.51 => ( A3
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gcd_nat.extremum_uniqueI
% 0.30/0.51 thf(fact_1590_gcd__nat_Onot__eq__extremum,axiom,
% 0.30/0.51 ! [A3: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 = ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.51 & ( A3
% 0.30/0.51 != ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gcd_nat.not_eq_extremum
% 0.30/0.51 thf(fact_1591_gcd__nat_Oextremum__unique,axiom,
% 0.30/0.51 ! [A3: nat] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gcd_nat.extremum_unique
% 0.30/0.51 thf(fact_1592_gcd__nat_Oextremum__strict,axiom,
% 0.30/0.51 ! [A3: nat] :
% 0.30/0.51 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.51 & ( ( zero_zero @ nat )
% 0.30/0.51 != A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % gcd_nat.extremum_strict
% 0.30/0.51 thf(fact_1593_sum__squares__gt__zero__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord4710134922213307826strict @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) )
% 0.30/0.51 = ( ( X
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 | ( Y3
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sum_squares_gt_zero_iff
% 0.30/0.51 thf(fact_1594_sum__squares__le__zero__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord4710134922213307826strict @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) )
% 0.30/0.51 = ( ( X
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 & ( Y3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sum_squares_le_zero_iff
% 0.30/0.51 thf(fact_1595_sum__squares__eq__zero__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linord4710134922213307826strict @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( ( X
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 & ( Y3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sum_squares_eq_zero_iff
% 0.30/0.51 thf(fact_1596_add__scale__eq__noteq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri1453513574482234551roduct @ A )
% 0.30/0.51 => ! [R: A,A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.51 ( ( R
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ( A3 = B2 )
% 0.30/0.51 & ( C2 != D2 ) )
% 0.30/0.51 => ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
% 0.30/0.51 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_scale_eq_noteq
% 0.30/0.51 thf(fact_1597_divide__int__def,axiom,
% 0.30/0.51 ( ( divide_divide @ int )
% 0.30/0.51 = ( ^ [K3: int,L3: int] :
% 0.30/0.51 ( if @ int
% 0.30/0.51 @ ( L3
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 @ ( zero_zero @ int )
% 0.30/0.51 @ ( if @ int
% 0.30/0.51 @ ( ( sgn_sgn @ int @ K3 )
% 0.30/0.51 = ( sgn_sgn @ int @ L3 ) )
% 0.30/0.51 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
% 0.30/0.51 @ ( uminus_uminus @ int
% 0.30/0.51 @ ( semiring_1_of_nat @ int
% 0.30/0.51 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
% 0.30/0.51 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.51 @ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_int_def
% 0.30/0.51 thf(fact_1598_gbinomial__pochhammer_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ( ( gbinomial @ A )
% 0.30/0.51 = ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_pochhammer'
% 0.30/0.51 thf(fact_1599_abs__idempotent,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_idempotent
% 0.30/0.51 thf(fact_1600_abs__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_abs
% 0.30/0.51 thf(fact_1601_abs__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_0
% 0.30/0.51 thf(fact_1602_abs__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_zero
% 0.30/0.51 thf(fact_1603_abs__eq__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_eq_0
% 0.30/0.51 thf(fact_1604_abs__0__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ( zero_zero @ A )
% 0.30/0.51 = ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_0_eq
% 0.30/0.51 thf(fact_1605_abs__mult__self__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( times_times @ A @ A3 @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mult_self_eq
% 0.30/0.51 thf(fact_1606_abs__add__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
% 0.30/0.51 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_add_abs
% 0.30/0.51 thf(fact_1607_abs__1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_1
% 0.30/0.51 thf(fact_1608_abs__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.51 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_minus
% 0.30/0.51 thf(fact_1609_abs__minus__cancel,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.51 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_minus_cancel
% 0.30/0.51 thf(fact_1610_dvd__abs__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [M: A,K: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
% 0.30/0.51 = ( dvd_dvd @ A @ M @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_abs_iff
% 0.30/0.51 thf(fact_1611_abs__dvd__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [M: A,K: A] :
% 0.30/0.51 ( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
% 0.30/0.51 = ( dvd_dvd @ A @ M @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_dvd_iff
% 0.30/0.51 thf(fact_1612_abs__divide,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_divide
% 0.30/0.51 thf(fact_1613_abs__of__nat,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.51 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_of_nat
% 0.30/0.51 thf(fact_1614_abs__bool__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [P: $o] :
% 0.30/0.51 ( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
% 0.30/0.51 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_bool_eq
% 0.30/0.51 thf(fact_1615_pochhammer__1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_1
% 0.30/0.51 thf(fact_1616_nat__of__bool,axiom,
% 0.30/0.51 ! [P: $o] :
% 0.30/0.51 ( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
% 0.30/0.51 = ( zero_neq_one_of_bool @ nat @ P ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_of_bool
% 0.30/0.51 thf(fact_1617_abs__le__zero__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_le_zero_iff
% 0.30/0.51 thf(fact_1618_abs__le__self__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
% 0.30/0.51 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_le_self_iff
% 0.30/0.51 thf(fact_1619_abs__of__nonneg,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_of_nonneg
% 0.30/0.51 thf(fact_1620_zero__less__abs__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_less_abs_iff
% 0.30/0.51 thf(fact_1621_abs__neg__one,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_neg_one
% 0.30/0.51 thf(fact_1622_pochhammer__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_0
% 0.30/0.51 thf(fact_1623_pochhammer__Suc0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_Suc0
% 0.30/0.51 thf(fact_1624_zdvd1__eq,axiom,
% 0.30/0.51 ! [X: int] :
% 0.30/0.51 ( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
% 0.30/0.51 = ( ( abs_abs @ int @ X )
% 0.30/0.51 = ( one_one @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdvd1_eq
% 0.30/0.51 thf(fact_1625_abs__division__segment,axiom,
% 0.30/0.51 ! [K: int] :
% 0.30/0.51 ( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
% 0.30/0.51 = ( one_one @ int ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_division_segment
% 0.30/0.51 thf(fact_1626_abs__of__nonpos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_of_nonpos
% 0.30/0.51 thf(fact_1627_zero__le__divide__abs__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
% 0.30/0.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 | ( B2
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_le_divide_abs_iff
% 0.30/0.51 thf(fact_1628_divide__le__0__abs__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
% 0.30/0.51 = ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.51 | ( B2
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_le_0_abs_iff
% 0.30/0.51 thf(fact_1629_abs__sgn__eq__1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_sgn_eq_1
% 0.30/0.51 thf(fact_1630_zabs__less__one__iff,axiom,
% 0.30/0.51 ! [Z3: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( abs_abs @ int @ Z3 ) @ ( one_one @ int ) )
% 0.30/0.51 = ( Z3
% 0.30/0.51 = ( zero_zero @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zabs_less_one_iff
% 0.30/0.51 thf(fact_1631_idom__abs__sgn__class_Oabs__sgn,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = ( zero_neq_one_of_bool @ A
% 0.30/0.51 @ ( A3
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % idom_abs_sgn_class.abs_sgn
% 0.30/0.51 thf(fact_1632_sgn__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 = ( zero_neq_one_of_bool @ A
% 0.30/0.51 @ ( A3
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sgn_abs
% 0.30/0.51 thf(fact_1633_nat__abs__dvd__iff,axiom,
% 0.30/0.51 ! [K: int,N: nat] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 0.30/0.51 = ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_abs_dvd_iff
% 0.30/0.51 thf(fact_1634_dvd__nat__abs__iff,axiom,
% 0.30/0.51 ! [N: nat,K: int] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 0.30/0.51 = ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_nat_abs_iff
% 0.30/0.51 thf(fact_1635_pochhammer__of__nat,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [X: nat,N: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
% 0.30/0.51 = ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_of_nat
% 0.30/0.51 thf(fact_1636_abs__ge__self,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_ge_self
% 0.30/0.51 thf(fact_1637_abs__le__D1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.30/0.51 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_le_D1
% 0.30/0.51 thf(fact_1638_abs__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_eq_0_iff
% 0.30/0.51 thf(fact_1639_abs__mult,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mult
% 0.30/0.51 thf(fact_1640_abs__one,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_one
% 0.30/0.51 thf(fact_1641_abs__minus__commute,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_minus_commute
% 0.30/0.51 thf(fact_1642_abs__eq__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_ring @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ( abs_abs @ A @ X )
% 0.30/0.51 = ( abs_abs @ A @ Y3 ) )
% 0.30/0.51 = ( ( X = Y3 )
% 0.30/0.51 | ( X
% 0.30/0.51 = ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_eq_iff
% 0.30/0.51 thf(fact_1643_dvd__if__abs__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [L: A,K: A] :
% 0.30/0.51 ( ( ( abs_abs @ A @ L )
% 0.30/0.51 = ( abs_abs @ A @ K ) )
% 0.30/0.51 => ( dvd_dvd @ A @ L @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_if_abs_eq
% 0.30/0.51 thf(fact_1644_zdvd__antisym__abs,axiom,
% 0.30/0.51 ! [A3: int,B2: int] :
% 0.30/0.51 ( ( dvd_dvd @ int @ A3 @ B2 )
% 0.30/0.51 => ( ( dvd_dvd @ int @ B2 @ A3 )
% 0.30/0.51 => ( ( abs_abs @ int @ A3 )
% 0.30/0.51 = ( abs_abs @ int @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdvd_antisym_abs
% 0.30/0.51 thf(fact_1645_abs__ge__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_ge_zero
% 0.30/0.51 thf(fact_1646_abs__not__less__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_not_less_zero
% 0.30/0.51 thf(fact_1647_abs__of__pos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_of_pos
% 0.30/0.51 thf(fact_1648_abs__triangle__ineq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_triangle_ineq
% 0.30/0.51 thf(fact_1649_abs__triangle__ineq2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_triangle_ineq2
% 0.30/0.51 thf(fact_1650_abs__triangle__ineq3,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_triangle_ineq3
% 0.30/0.51 thf(fact_1651_abs__triangle__ineq2__sym,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_triangle_ineq2_sym
% 0.30/0.51 thf(fact_1652_abs__mult__less,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
% 0.30/0.51 => ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
% 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mult_less
% 0.30/0.51 thf(fact_1653_abs__ge__minus__self,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_ge_minus_self
% 0.30/0.51 thf(fact_1654_abs__le__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.30/0.51 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.51 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_le_iff
% 0.30/0.51 thf(fact_1655_abs__le__D2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_le_D2
% 0.30/0.51 thf(fact_1656_abs__leI,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_leI
% 0.30/0.51 thf(fact_1657_abs__less__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.30/0.51 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.51 & ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_less_iff
% 0.30/0.51 thf(fact_1658_nonzero__abs__divide,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nonzero_abs_divide
% 0.30/0.51 thf(fact_1659_linordered__idom__class_Oabs__sgn,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ( ( abs_abs @ A )
% 0.30/0.51 = ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_idom_class.abs_sgn
% 0.30/0.51 thf(fact_1660_abs__mult__sgn,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mult_sgn
% 0.30/0.51 thf(fact_1661_sgn__mult__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom_abs_sgn @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % sgn_mult_abs
% 0.30/0.51 thf(fact_1662_mult__sgn__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
% 0.30/0.51 = X ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_sgn_abs
% 0.30/0.51 thf(fact_1663_same__sgn__abs__add,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( ( sgn_sgn @ A @ B2 )
% 0.30/0.51 = ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % same_sgn_abs_add
% 0.30/0.51 thf(fact_1664_abs__zmult__eq__1,axiom,
% 0.30/0.51 ! [M: int,N: int] :
% 0.30/0.51 ( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
% 0.30/0.51 = ( one_one @ int ) )
% 0.30/0.51 => ( ( abs_abs @ int @ M )
% 0.30/0.51 = ( one_one @ int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_zmult_eq_1
% 0.30/0.51 thf(fact_1665_nat__abs__triangle__ineq,axiom,
% 0.30/0.51 ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_abs_triangle_ineq
% 0.30/0.51 thf(fact_1666_pochhammer__pos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [X: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_pos
% 0.30/0.51 thf(fact_1667_pochhammer__neq__0__mono,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_neq_0_mono
% 0.30/0.51 thf(fact_1668_pochhammer__eq__0__mono,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,N: nat,M: nat] :
% 0.30/0.51 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_eq_0_mono
% 0.30/0.51 thf(fact_1669_pochhammer__fact,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( semiring_char_0 @ A )
% 0.30/0.51 & ( comm_semiring_1 @ A ) )
% 0.30/0.51 => ( ( semiring_char_0_fact @ A )
% 0.30/0.51 = ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_fact
% 0.30/0.51 thf(fact_1670_dense__eq0__I,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 & ( dense_linorder @ A ) )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ! [E2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
% 0.30/0.51 => ( X
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dense_eq0_I
% 0.30/0.51 thf(fact_1671_abs__eq__mult,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordered_ring_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 | ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 0.30/0.51 & ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 | ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.51 = ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_eq_mult
% 0.30/0.51 thf(fact_1672_abs__mult__pos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [X: A,Y3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.51 => ( ( times_times @ A @ ( abs_abs @ A @ Y3 ) @ X )
% 0.30/0.51 = ( abs_abs @ A @ ( times_times @ A @ Y3 @ X ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mult_pos
% 0.30/0.51 thf(fact_1673_abs__minus__le__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_minus_le_zero
% 0.30/0.51 thf(fact_1674_eq__abs__iff_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_ring @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 = ( abs_abs @ A @ B2 ) )
% 0.30/0.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 & ( ( B2 = A3 )
% 0.30/0.51 | ( B2
% 0.30/0.51 = ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_abs_iff'
% 0.30/0.51 thf(fact_1675_abs__eq__iff_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_ring @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = B2 )
% 0.30/0.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 & ( ( A3 = B2 )
% 0.30/0.51 | ( A3
% 0.30/0.51 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_eq_iff'
% 0.30/0.51 thf(fact_1676_abs__of__neg,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ A3 )
% 0.30/0.51 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_of_neg
% 0.30/0.51 thf(fact_1677_abs__if__raw,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( abs_if @ A )
% 0.30/0.51 => ( ( abs_abs @ A )
% 0.30/0.51 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_if_raw
% 0.30/0.51 thf(fact_1678_abs__if,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( abs_if @ A )
% 0.30/0.51 => ( ( abs_abs @ A )
% 0.30/0.51 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_if
% 0.30/0.51 thf(fact_1679_abs__diff__le__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [X: A,A3: A,R: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 0.30/0.51 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 0.30/0.51 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_diff_le_iff
% 0.30/0.51 thf(fact_1680_abs__diff__triangle__ineq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_diff_triangle_ineq
% 0.30/0.51 thf(fact_1681_abs__triangle__ineq4,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.51 => ! [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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_triangle_ineq4
% 0.30/0.51 thf(fact_1682_abs__div__pos,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [Y3: A,X: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.51 => ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y3 )
% 0.30/0.51 = ( abs_abs @ A @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_div_pos
% 0.30/0.51 thf(fact_1683_abs__diff__less__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [X: A,A3: A,R: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 0.30/0.51 = ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 0.30/0.51 & ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_diff_less_iff
% 0.30/0.51 thf(fact_1684_div__noneq__sgn__abs,axiom,
% 0.30/0.51 ! [L: int,K: int] :
% 0.30/0.51 ( ( L
% 0.30/0.51 != ( zero_zero @ int ) )
% 0.30/0.51 => ( ( ( sgn_sgn @ int @ K )
% 0.30/0.51 != ( sgn_sgn @ int @ L ) )
% 0.30/0.51 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.51 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
% 0.30/0.51 @ ( zero_neq_one_of_bool @ int
% 0.30/0.51 @ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_noneq_sgn_abs
% 0.30/0.51 thf(fact_1685_abs__sgn__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( ( A3
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 = ( zero_zero @ A ) ) )
% 0.30/0.51 & ( ( A3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_sgn_eq
% 0.30/0.51 thf(fact_1686_zabs__def,axiom,
% 0.30/0.51 ( ( abs_abs @ int )
% 0.30/0.51 = ( ^ [I5: int] : ( if @ int @ ( ord_less @ int @ I5 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I5 ) @ I5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zabs_def
% 0.30/0.51 thf(fact_1687_dvd__imp__le__int,axiom,
% 0.30/0.51 ! [I: int,D2: int] :
% 0.30/0.51 ( ( I
% 0.30/0.51 != ( zero_zero @ int ) )
% 0.30/0.51 => ( ( dvd_dvd @ int @ D2 @ I )
% 0.30/0.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_imp_le_int
% 0.30/0.51 thf(fact_1688_nat__abs__mult__distrib,axiom,
% 0.30/0.51 ! [W2: int,Z3: int] :
% 0.30/0.51 ( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z3 ) ) )
% 0.30/0.51 = ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_abs_mult_distrib
% 0.30/0.51 thf(fact_1689_abs__mod__less,axiom,
% 0.30/0.51 ! [L: int,K: int] :
% 0.30/0.51 ( ( L
% 0.30/0.51 != ( zero_zero @ int ) )
% 0.30/0.51 => ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_mod_less
% 0.30/0.51 thf(fact_1690_pochhammer__nonneg,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [X: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_nonneg
% 0.30/0.51 thf(fact_1691_pochhammer__0__left,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( ( N
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.51 = ( one_one @ A ) ) )
% 0.30/0.51 & ( ( N
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_0_left
% 0.30/0.51 thf(fact_1692_pochhammer__product_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [Z3: A,N: nat,M: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N @ M ) )
% 0.30/0.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_product'
% 0.30/0.51 thf(fact_1693_abs__add__one__gt__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_add_one_gt_zero
% 0.30/0.51 thf(fact_1694_zdvd__mult__cancel1,axiom,
% 0.30/0.51 ! [M: int,N: int] :
% 0.30/0.51 ( ( M
% 0.30/0.51 != ( zero_zero @ int ) )
% 0.30/0.51 => ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
% 0.30/0.51 = ( ( abs_abs @ int @ N )
% 0.30/0.51 = ( one_one @ int ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdvd_mult_cancel1
% 0.30/0.51 thf(fact_1695_div__sgn__abs__cancel,axiom,
% 0.30/0.51 ! [V: int,K: int,L: int] :
% 0.30/0.51 ( ( V
% 0.30/0.51 != ( zero_zero @ int ) )
% 0.30/0.51 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ L ) ) )
% 0.30/0.51 = ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_sgn_abs_cancel
% 0.30/0.51 thf(fact_1696_div__abs__eq__div__nat,axiom,
% 0.30/0.51 ! [K: int,L: int] :
% 0.30/0.51 ( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
% 0.30/0.51 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_abs_eq_div_nat
% 0.30/0.51 thf(fact_1697_mod__abs__eq__div__nat,axiom,
% 0.30/0.51 ! [K: int,L: int] :
% 0.30/0.51 ( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
% 0.30/0.51 = ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mod_abs_eq_div_nat
% 0.30/0.51 thf(fact_1698_pochhammer__rec,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.30/0.51 = ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_rec
% 0.30/0.51 thf(fact_1699_pochhammer__Suc,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.30/0.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_Suc
% 0.30/0.51 thf(fact_1700_pochhammer__rec_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [Z3: A,N: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) )
% 0.30/0.51 = ( times_times @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_rec'
% 0.30/0.51 thf(fact_1701_div__dvd__sgn__abs,axiom,
% 0.30/0.51 ! [L: int,K: int] :
% 0.30/0.51 ( ( dvd_dvd @ int @ L @ K )
% 0.30/0.51 => ( ( divide_divide @ int @ K @ L )
% 0.30/0.51 = ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_dvd_sgn_abs
% 0.30/0.51 thf(fact_1702_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( idom @ A )
% 0.30/0.51 => ! [N: nat,K: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ N @ K )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.51 thf(fact_1703_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( ring_char_0 @ A )
% 0.30/0.51 & ( idom @ A ) )
% 0.30/0.51 => ! [N: nat,K: nat] :
% 0.30/0.51 ( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( ord_less @ nat @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_of_nat_eq_0_iff
% 0.30/0.51 thf(fact_1704_pochhammer__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( ? [K3: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ K3 @ N )
% 0.30/0.51 & ( A3
% 0.30/0.51 = ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_eq_0_iff
% 0.30/0.51 thf(fact_1705_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( ring_char_0 @ A )
% 0.30/0.51 & ( idom @ A ) )
% 0.30/0.51 => ! [K: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.51 thf(fact_1706_modulo__int__def,axiom,
% 0.30/0.51 ( ( modulo_modulo @ int )
% 0.30/0.51 = ( ^ [K3: int,L3: int] :
% 0.30/0.51 ( if @ int
% 0.30/0.51 @ ( L3
% 0.30/0.51 = ( zero_zero @ int ) )
% 0.30/0.51 @ K3
% 0.30/0.51 @ ( if @ int
% 0.30/0.51 @ ( ( sgn_sgn @ int @ K3 )
% 0.30/0.51 = ( sgn_sgn @ int @ L3 ) )
% 0.30/0.51 @ ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) )
% 0.30/0.51 @ ( times_times @ int @ ( sgn_sgn @ int @ L3 )
% 0.30/0.51 @ ( minus_minus @ int
% 0.30/0.51 @ ( times_times @ int @ ( abs_abs @ int @ L3 )
% 0.30/0.51 @ ( zero_neq_one_of_bool @ int
% 0.30/0.51 @ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) )
% 0.30/0.51 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % modulo_int_def
% 0.30/0.51 thf(fact_1707_incr__lemma,axiom,
% 0.30/0.51 ! [D2: int,Z3: int,X: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.51 => ( ord_less @ int @ Z3 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % incr_lemma
% 0.30/0.51 thf(fact_1708_decr__lemma,axiom,
% 0.30/0.51 ! [D2: int,X: int,Z3: int] :
% 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.51 => ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % decr_lemma
% 0.30/0.51 thf(fact_1709_nat__abs__int__diff,axiom,
% 0.30/0.51 ! [A3: nat,B2: nat] :
% 0.30/0.51 ( ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.51 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 0.30/0.51 = ( minus_minus @ nat @ B2 @ A3 ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.51 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 0.30/0.51 = ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_abs_int_diff
% 0.30/0.51 thf(fact_1710_pochhammer__product,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [M: nat,N: nat,Z3: A] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.51 => ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N )
% 0.30/0.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_product
% 0.30/0.51 thf(fact_1711_nat__intermed__int__val,axiom,
% 0.30/0.51 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.30/0.51 ( ! [I4: nat] :
% 0.30/0.51 ( ( ( ord_less_eq @ nat @ M @ I4 )
% 0.30/0.51 & ( ord_less @ nat @ I4 @ N ) )
% 0.30/0.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( F @ M ) @ K )
% 0.30/0.51 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 0.30/0.51 => ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ M @ I4 )
% 0.30/0.51 & ( ord_less_eq @ nat @ I4 @ N )
% 0.30/0.51 & ( ( F @ I4 )
% 0.30/0.51 = K ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_intermed_int_val
% 0.30/0.51 thf(fact_1712_pochhammer__absorb__comp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring_1 @ A )
% 0.30/0.51 => ! [R: A,K: nat] :
% 0.30/0.51 ( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
% 0.30/0.51 = ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_absorb_comp
% 0.30/0.51 thf(fact_1713_add__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri1453513574482234551roduct @ A )
% 0.30/0.51 => ! [B2: A,A3: A] :
% 0.30/0.51 ( ( B2
% 0.30/0.51 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_0_iff
% 0.30/0.51 thf(fact_1714_crossproduct__noteq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri1453513574482234551roduct @ A )
% 0.30/0.51 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.51 ( ( ( A3 != B2 )
% 0.30/0.51 & ( C2 != D2 ) )
% 0.30/0.51 = ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
% 0.30/0.51 != ( plus_plus @ A @ ( times_times @ A @ A3 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % crossproduct_noteq
% 0.30/0.51 thf(fact_1715_crossproduct__eq,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri1453513574482234551roduct @ A )
% 0.30/0.51 => ! [W2: A,Y3: A,X: A,Z3: A] :
% 0.30/0.51 ( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y3 ) @ ( times_times @ A @ X @ Z3 ) )
% 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ W2 @ Z3 ) @ ( times_times @ A @ X @ Y3 ) ) )
% 0.30/0.51 = ( ( W2 = X )
% 0.30/0.51 | ( Y3 = Z3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % crossproduct_eq
% 0.30/0.51 thf(fact_1716_nat__ivt__aux,axiom,
% 0.30/0.51 ! [N: nat,F: nat > int,K: int] :
% 0.30/0.51 ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ I4 @ N )
% 0.30/0.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 0.30/0.51 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 0.30/0.51 => ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I4 @ N )
% 0.30/0.51 & ( ( F @ I4 )
% 0.30/0.51 = K ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_ivt_aux
% 0.30/0.51 thf(fact_1717_nat0__intermed__int__val,axiom,
% 0.30/0.51 ! [N: nat,F: nat > int,K: int] :
% 0.30/0.51 ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ I4 @ N )
% 0.30/0.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 0.30/0.51 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 0.30/0.51 => ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ I4 @ N )
% 0.30/0.51 & ( ( F @ I4 )
% 0.30/0.51 = K ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat0_intermed_int_val
% 0.30/0.51 thf(fact_1718_gbinomial__pochhammer,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ( ( gbinomial @ A )
% 0.30/0.51 = ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_pochhammer
% 0.30/0.51 thf(fact_1719_int__power__div__base,axiom,
% 0.30/0.51 ! [M: nat,K: int] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.51 => ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K )
% 0.30/0.51 = ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % int_power_div_base
% 0.30/0.51 thf(fact_1720_pochhammer__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring_1 @ A )
% 0.30/0.51 => ! [B2: A,K: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
% 0.30/0.51 = ( 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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_minus
% 0.30/0.51 thf(fact_1721_pochhammer__minus_H,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring_1 @ A )
% 0.30/0.51 => ! [B2: A,K: nat] :
% 0.30/0.51 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
% 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pochhammer_minus'
% 0.30/0.51 thf(fact_1722_gbinomial__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( field_char_0 @ A )
% 0.30/0.51 => ! [A3: A,K: nat] :
% 0.30/0.51 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
% 0.30/0.51 = ( 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 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_minus
% 0.30/0.51 thf(fact_1723_triangle__Suc,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( nat_triangle @ ( suc @ N ) )
% 0.30/0.51 = ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % triangle_Suc
% 0.30/0.51 thf(fact_1724_frac__unique__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A,A3: A] :
% 0.30/0.51 ( ( ( archimedean_frac @ A @ X )
% 0.30/0.51 = A3 )
% 0.30/0.51 = ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
% 0.30/0.51 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 & ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_unique_iff
% 0.30/0.51 thf(fact_1725_power__Suc__0,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc_0
% 0.30/0.51 thf(fact_1726_nat__power__eq__Suc__0__iff,axiom,
% 0.30/0.51 ! [X: nat,M: nat] :
% 0.30/0.51 ( ( ( power_power @ nat @ X @ M )
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 = ( ( M
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 | ( X
% 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_power_eq_Suc_0_iff
% 0.30/0.51 thf(fact_1727_power__one,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_one
% 0.30/0.51 thf(fact_1728_nat__zero__less__power__iff,axiom,
% 0.30/0.51 ! [X: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.30/0.51 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_zero_less_power_iff
% 0.30/0.51 thf(fact_1729_of__nat__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_1 @ A )
% 0.30/0.51 => ! [M: nat,N: nat] :
% 0.30/0.51 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 0.30/0.51 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_power
% 0.30/0.51 thf(fact_1730_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_char_0 @ A )
% 0.30/0.51 => ! [B2: nat,W2: nat,X: nat] :
% 0.30/0.51 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 )
% 0.30/0.51 = ( semiring_1_of_nat @ A @ X ) )
% 0.30/0.51 = ( ( power_power @ nat @ B2 @ W2 )
% 0.30/0.51 = X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.51 thf(fact_1731_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_char_0 @ A )
% 0.30/0.51 => ! [X: nat,B2: nat,W2: nat] :
% 0.30/0.51 ( ( ( semiring_1_of_nat @ A @ X )
% 0.30/0.51 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 0.30/0.51 = ( X
% 0.30/0.51 = ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.51 thf(fact_1732_power__one__right,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( one_one @ nat ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_one_right
% 0.30/0.51 thf(fact_1733_frac__in__Ints__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
% 0.30/0.51 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_in_Ints_iff
% 0.30/0.51 thf(fact_1734_triangle__0,axiom,
% 0.30/0.51 ( ( nat_triangle @ ( zero_zero @ nat ) )
% 0.30/0.51 = ( zero_zero @ nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % triangle_0
% 0.30/0.51 thf(fact_1735_power__inject__exp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ( ( power_power @ A @ A3 @ M )
% 0.30/0.51 = ( power_power @ A @ A3 @ N ) )
% 0.30/0.51 = ( M = N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_inject_exp
% 0.30/0.51 thf(fact_1736_power__0__Suc,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_1 @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_0_Suc
% 0.30/0.51 thf(fact_1737_power__Suc0__right,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc0_right
% 0.30/0.51 thf(fact_1738_abs__power__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 0.30/0.51 = ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % abs_power_minus
% 0.30/0.51 thf(fact_1739_frac__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( ( archimedean_frac @ A @ X )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_eq_0_iff
% 0.30/0.51 thf(fact_1740_minus__one__mult__self,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring_1 @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_one_mult_self
% 0.30/0.51 thf(fact_1741_left__minus__one__mult__self,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_ring_1 @ A )
% 0.30/0.51 => ! [N: nat,A3: A] :
% 0.30/0.51 ( ( 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 ) )
% 0.30/0.51 = A3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_minus_one_mult_self
% 0.30/0.51 thf(fact_1742_power__strict__increasing__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: A,X: nat,Y3: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
% 0.30/0.51 = ( ord_less @ nat @ X @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_strict_increasing_iff
% 0.30/0.51 thf(fact_1743_power__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri2026040879449505780visors @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ( power_power @ A @ A3 @ N )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( ( A3
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_eq_0_iff
% 0.30/0.51 thf(fact_1744_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [X: nat,B2: nat,W2: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 0.30/0.51 = ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.51 thf(fact_1745_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: nat,W2: nat,X: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.30/0.51 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.51 thf(fact_1746_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [X: nat,B2: nat,W2: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.51 thf(fact_1747_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: nat,W2: nat,X: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.51 thf(fact_1748_pow__divides__pow__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_gcd @ A )
% 0.30/0.51 => ! [N: nat,A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pow_divides_pow_iff
% 0.30/0.51 thf(fact_1749_frac__gt__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
% 0.30/0.51 = ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % frac_gt_0_iff
% 0.30/0.51 thf(fact_1750_power__strict__decreasing__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.51 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 = ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_strict_decreasing_iff
% 0.30/0.51 thf(fact_1751_power__increasing__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: A,X: nat,Y3: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ X @ Y3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_increasing_iff
% 0.30/0.51 thf(fact_1752_power__mono__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mono_iff
% 0.30/0.51 thf(fact_1753_zero__less__power__abs__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
% 0.30/0.51 = ( ( A3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_less_power_abs_iff
% 0.30/0.51 thf(fact_1754_power__decreasing__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [B2: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 = ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_decreasing_iff
% 0.30/0.51 thf(fact_1755_of__nat__zero__less__power__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [X: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 0.30/0.51 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % of_nat_zero_less_power_iff
% 0.30/0.51 thf(fact_1756_Ints__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_power
% 0.30/0.51 thf(fact_1757_power__not__zero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiri2026040879449505780visors @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( power_power @ A @ A3 @ N )
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_not_zero
% 0.30/0.51 thf(fact_1758_power__commutes,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
% 0.30/0.51 = ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_commutes
% 0.30/0.51 thf(fact_1759_power__mult__distrib,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_monoid_mult @ A )
% 0.30/0.51 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
% 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult_distrib
% 0.30/0.51 thf(fact_1760_power__commuting__commutes,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [X: A,Y3: A,N: nat] :
% 0.30/0.51 ( ( ( times_times @ A @ X @ Y3 )
% 0.30/0.51 = ( times_times @ A @ Y3 @ X ) )
% 0.30/0.51 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y3 )
% 0.30/0.51 = ( times_times @ A @ Y3 @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_commuting_commutes
% 0.30/0.51 thf(fact_1761_power__mod,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.51 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.51 ( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ N ) @ B2 )
% 0.30/0.51 = ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mod
% 0.30/0.51 thf(fact_1762_power__mult,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( times_times @ nat @ M @ N ) )
% 0.30/0.51 = ( power_power @ A @ ( power_power @ A @ A3 @ M ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult
% 0.30/0.51 thf(fact_1763_Ints__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_0
% 0.30/0.51 thf(fact_1764_Ints__mult,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_mult
% 0.30/0.51 thf(fact_1765_Ints__add,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_add
% 0.30/0.51 thf(fact_1766_Ints__1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_1
% 0.30/0.51 thf(fact_1767_Ints__diff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A,B2: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_diff
% 0.30/0.51 thf(fact_1768_minus__in__Ints__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [X: A] :
% 0.30/0.51 ( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
% 0.30/0.51 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_in_Ints_iff
% 0.30/0.51 thf(fact_1769_Ints__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_minus
% 0.30/0.51 thf(fact_1770_Ints__of__nat,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_of_nat
% 0.30/0.51 thf(fact_1771_Ints__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( member @ A @ ( abs_abs @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_abs
% 0.30/0.51 thf(fact_1772_fact__in__Ints,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ( semiring_char_0 @ A )
% 0.30/0.51 & ( ring_1 @ A ) )
% 0.30/0.51 => ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % fact_in_Ints
% 0.30/0.51 thf(fact_1773_nat__power__eq,axiom,
% 0.30/0.51 ! [Z3: int,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.51 => ( ( nat2 @ ( power_power @ int @ Z3 @ N ) )
% 0.30/0.51 = ( power_power @ nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_power_eq
% 0.30/0.51 thf(fact_1774_power__mono,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mono
% 0.30/0.51 thf(fact_1775_zero__le__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_le_power
% 0.30/0.51 thf(fact_1776_zero__less__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_less_power
% 0.30/0.51 thf(fact_1777_one__le__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % one_le_power
% 0.30/0.51 thf(fact_1778_left__right__inverse__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [X: A,Y3: A,N: nat] :
% 0.30/0.51 ( ( ( times_times @ A @ X @ Y3 )
% 0.30/0.51 = ( one_one @ A ) )
% 0.30/0.51 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_right_inverse_power
% 0.30/0.51 thf(fact_1779_power__Suc2,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc2
% 0.30/0.51 thf(fact_1780_power__Suc,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( power @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 0.30/0.51 = ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc
% 0.30/0.51 thf(fact_1781_power__0,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( power @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.51 = ( one_one @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_0
% 0.30/0.51 thf(fact_1782_power__one__over,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( division_ring @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
% 0.30/0.51 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_one_over
% 0.30/0.51 thf(fact_1783_power__add,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( monoid_mult @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_add
% 0.30/0.51 thf(fact_1784_le__imp__power__dvd,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [M: nat,N: nat,A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.51 => ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_imp_power_dvd
% 0.30/0.51 thf(fact_1785_power__le__dvd,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [A3: A,N: nat,B2: A,M: nat] :
% 0.30/0.51 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.51 => ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_le_dvd
% 0.30/0.51 thf(fact_1786_dvd__power__le,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [X: A,Y3: A,N: nat,M: nat] :
% 0.30/0.51 ( ( dvd_dvd @ A @ X @ Y3 )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.51 => ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ M ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_power_le
% 0.30/0.51 thf(fact_1787_nat__power__less__imp__less,axiom,
% 0.30/0.51 ! [I: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 0.30/0.51 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 0.30/0.51 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_power_less_imp_less
% 0.30/0.51 thf(fact_1788_Ints__double__eq__0__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_char_0 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.30/0.51 = ( zero_zero @ A ) )
% 0.30/0.51 = ( A3
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_double_eq_0_iff
% 0.30/0.51 thf(fact_1789_power__less__imp__less__base,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat,B2: A] :
% 0.30/0.51 ( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_less_imp_less_base
% 0.30/0.51 thf(fact_1790_power__le__one,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_le_one
% 0.30/0.51 thf(fact_1791_power__inject__base,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat,B2: A] :
% 0.30/0.51 ( ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 0.30/0.51 = ( power_power @ A @ B2 @ ( suc @ N ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( A3 = B2 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_inject_base
% 0.30/0.51 thf(fact_1792_power__le__imp__le__base,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat,B2: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_le_imp_le_base
% 0.30/0.51 thf(fact_1793_power__gt1__lemma,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_gt1_lemma
% 0.30/0.51 thf(fact_1794_power__less__power__Suc,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_less_power_Suc
% 0.30/0.51 thf(fact_1795_power__0__left,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_1 @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( ( N
% 0.30/0.51 = ( zero_zero @ nat ) )
% 0.30/0.51 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.51 = ( one_one @ A ) ) )
% 0.30/0.51 & ( ( N
% 0.30/0.51 != ( zero_zero @ nat ) )
% 0.30/0.51 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_0_left
% 0.30/0.51 thf(fact_1796_power__gt1,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_gt1
% 0.30/0.51 thf(fact_1797_power__minus,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_1 @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_minus
% 0.30/0.51 thf(fact_1798_power__less__imp__less__exp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.51 => ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_less_imp_less_exp
% 0.30/0.51 thf(fact_1799_power__strict__increasing,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat,N6: nat,A3: A] :
% 0.30/0.51 ( ( ord_less @ nat @ N @ N6 )
% 0.30/0.51 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_strict_increasing
% 0.30/0.51 thf(fact_1800_power__increasing,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat,N6: nat,A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ N @ N6 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_increasing
% 0.30/0.51 thf(fact_1801_zero__le__power__abs,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_idom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_le_power_abs
% 0.30/0.51 thf(fact_1802_zero__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semiring_1 @ A )
% 0.30/0.51 => ! [N: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.30/0.51 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_power
% 0.30/0.51 thf(fact_1803_is__unit__power__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( algebraic_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
% 0.30/0.51 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 | ( N
% 0.30/0.51 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % is_unit_power_iff
% 0.30/0.51 thf(fact_1804_power__gt__expt,axiom,
% 0.30/0.51 ! [N: nat,K: nat] :
% 0.30/0.51 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.51 => ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_gt_expt
% 0.30/0.51 thf(fact_1805_nat__one__le__power,axiom,
% 0.30/0.51 ! [I: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 0.30/0.51 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_one_le_power
% 0.30/0.51 thf(fact_1806_fact__le__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % fact_le_power
% 0.30/0.51 thf(fact_1807_Ints__odd__nonzero,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( ring_char_0 @ A )
% 0.30/0.51 => ! [A3: A] :
% 0.30/0.51 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.51 => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
% 0.30/0.51 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Ints_odd_nonzero
% 0.30/0.51 thf(fact_1808_bounded__Max__nat,axiom,
% 0.30/0.51 ! [P: nat > $o,X: nat,M7: nat] :
% 0.30/0.51 ( ( P @ X )
% 0.30/0.51 => ( ! [X3: nat] :
% 0.30/0.51 ( ( P @ X3 )
% 0.30/0.51 => ( ord_less_eq @ nat @ X3 @ M7 ) )
% 0.30/0.51 => ~ ! [M6: nat] :
% 0.30/0.51 ( ( P @ M6 )
% 0.30/0.51 => ~ ! [X4: nat] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( ord_less_eq @ nat @ X4 @ M6 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % bounded_Max_nat
% 0.30/0.51 thf(fact_1809_power__Suc__less,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc_less
% 0.30/0.51 thf(fact_1810_power__Suc__le__self,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc_le_self
% 0.30/0.51 thf(fact_1811_power__Suc__less__one,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_Suc_less_one
% 0.30/0.51 thf(fact_1812_power__strict__decreasing,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat,N6: nat,A3: A] :
% 0.30/0.51 ( ( ord_less @ nat @ N @ N6 )
% 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_strict_decreasing
% 0.30/0.51 thf(fact_1813_power__decreasing,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat,N6: nat,A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ nat @ N @ N6 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_decreasing
% 0.30/0.51 thf(fact_1814_power__le__imp__le__exp,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.51 => ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_le_imp_le_exp
% 0.30/0.51 thf(fact_1815_power__eq__imp__eq__base,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat,B2: A] :
% 0.30/0.51 ( ( ( power_power @ A @ A3 @ N )
% 0.30/0.51 = ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( A3 = B2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_eq_imp_eq_base
% 0.30/0.51 thf(fact_1816_power__eq__iff__eq__base,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [N: nat,A3: A,B2: A] :
% 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.51 => ( ( ( power_power @ A @ A3 @ N )
% 0.30/0.51 = ( power_power @ A @ B2 @ N ) )
% 0.30/0.51 = ( A3 = B2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_eq_iff_eq_base
% 0.30/0.51 thf(fact_1817_self__le__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % self_le_power
% 0.30/0.51 thf(fact_1818_one__less__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_semidom @ A )
% 0.30/0.51 => ! [A3: A,N: nat] :
% 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % one_less_power
% 0.30/0.51 thf(fact_1819_dvd__power__iff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( algebraic_semidom @ A )
% 0.30/0.51 => ! [X: A,M: nat,N: nat] :
% 0.30/0.51 ( ( X
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
% 0.30/0.51 = ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
% 0.30/0.51 | ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_power_iff
% 0.30/0.51 thf(fact_1820_dvd__power,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( comm_semiring_1 @ A )
% 0.30/0.51 => ! [N: nat,X: A] :
% 0.30/0.51 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.51 | ( X
% 0.30/0.51 = ( one_one @ A ) ) )
% 0.30/0.51 => ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % dvd_power
% 0.30/0.51 thf(fact_1821_power__diff,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( semidom_divide @ A )
% 0.30/0.51 => ! [A3: A,N: nat,M: nat] :
% 0.30/0.51 ( ( A3
% 0.30/0.51 != ( zero_zero @ A ) )
% 0.30/0.51 => ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.51 => ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.51 = ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_diff
% 0.30/0.51 thf(fact_1822_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.51 ! [A: $tType] :
% 0.30/0.51 ( ( linordered_field @ A )
% 0.30/0.51 => ! [K: nat,A3: A] :
% 0.30/0.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
% 0.30/0.51 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % gbinomial_ge_n_over_k_pow_k
% 0.30/0.51 thf(fact_1823_power__dvd__imp__le,axiom,
% 0.30/0.51 ! [I: nat,M: nat,N: nat] :
% 0.30/0.51 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 0.30/0.52 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
% 0.30/0.52 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_dvd_imp_le
% 0.30/0.52 thf(fact_1824_Ints__odd__less__0,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.52 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_odd_less_0
% 0.30/0.52 thf(fact_1825_fact__div__fact__le__pow,axiom,
% 0.30/0.52 ! [R: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ nat @ R @ N )
% 0.30/0.52 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R ) ) ) @ ( power_power @ nat @ N @ R ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % fact_div_fact_le_pow
% 0.30/0.52 thf(fact_1826_Ints__nonzero__abs__ge1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( X
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_nonzero_abs_ge1
% 0.30/0.52 thf(fact_1827_Ints__nonzero__abs__less1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 0.30/0.52 => ( X
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_nonzero_abs_less1
% 0.30/0.52 thf(fact_1828_Ints__eq__abs__less1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: A,Y3: A] :
% 0.30/0.52 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( member @ A @ Y3 @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( X = Y3 )
% 0.30/0.52 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_eq_abs_less1
% 0.30/0.52 thf(fact_1829_power__strict__mono,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_semidom @ A )
% 0.30/0.52 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.52 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.52 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_strict_mono
% 0.30/0.52 thf(fact_1830_power__diff__power__eq,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.52 => ! [A3: A,N: nat,M: nat] :
% 0.30/0.52 ( ( A3
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.52 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.52 = ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
% 0.30/0.52 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 0.30/0.52 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.52 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_diff_power_eq
% 0.30/0.52 thf(fact_1831_power__eq__if,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( power @ A )
% 0.30/0.52 => ( ( power_power @ A )
% 0.30/0.52 = ( ^ [P6: A,M2: nat] :
% 0.30/0.52 ( if @ A
% 0.30/0.52 @ ( M2
% 0.30/0.52 = ( zero_zero @ nat ) )
% 0.30/0.52 @ ( one_one @ A )
% 0.30/0.52 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_if
% 0.30/0.52 thf(fact_1832_power__minus__mult,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( monoid_mult @ A )
% 0.30/0.52 => ! [N: nat,A3: A] :
% 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.52 => ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
% 0.30/0.52 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_minus_mult
% 0.30/0.52 thf(fact_1833_frac__neg,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.52 = ( zero_zero @ A ) ) )
% 0.30/0.52 & ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.52 = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % frac_neg
% 0.30/0.52 thf(fact_1834_gbinomial__negated__upper,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( field_char_0 @ A )
% 0.30/0.52 => ( ( gbinomial @ A )
% 0.30/0.52 = ( ^ [A5: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A5 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % gbinomial_negated_upper
% 0.30/0.52 thf(fact_1835_gbinomial__index__swap,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( field_char_0 @ A )
% 0.30/0.52 => ! [K: nat,N: nat] :
% 0.30/0.52 ( ( 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 ) )
% 0.30/0.52 = ( 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 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % gbinomial_index_swap
% 0.30/0.52 thf(fact_1836_pochhammer__same,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( semiring_char_0 @ A )
% 0.30/0.52 & ( comm_ring_1 @ A )
% 0.30/0.52 & ( semiri3467727345109120633visors @ A ) )
% 0.30/0.52 => ! [N: nat] :
% 0.30/0.52 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
% 0.30/0.52 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % pochhammer_same
% 0.30/0.52 thf(fact_1837_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [K: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.52 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
% 0.30/0.52 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.52 thf(fact_1838_power__minus_H,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [X: A,N: nat] :
% 0.30/0.52 ( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
% 0.30/0.52 => ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
% 0.30/0.52 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_minus'
% 0.30/0.52 thf(fact_1839_ceiling__divide__lower,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Q4: A,P5: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 0.30/0.52 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_divide_lower
% 0.30/0.52 thf(fact_1840_power_Opower__eq__if,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( power2 @ A )
% 0.30/0.52 = ( ^ [One: A,Times: A > A > A,P6: A,M2: nat] :
% 0.30/0.52 ( if @ A
% 0.30/0.52 @ ( M2
% 0.30/0.52 = ( zero_zero @ nat ) )
% 0.30/0.52 @ One
% 0.30/0.52 @ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power.power_eq_if
% 0.30/0.52 thf(fact_1841_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_numeral(2)
% 0.30/0.52 thf(fact_1842_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_numeral(2)
% 0.30/0.52 thf(fact_1843_mult__ceiling__le__Ints,axiom,
% 0.30/0.52 ! [A: $tType,B: $tType] :
% 0.30/0.52 ( ( ( archim2362893244070406136eiling @ B )
% 0.30/0.52 & ( linordered_idom @ A ) )
% 0.30/0.52 => ! [A3: B,B2: B] :
% 0.30/0.52 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
% 0.30/0.52 => ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_ceiling_le_Ints
% 0.30/0.52 thf(fact_1844_numeral__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ( numeral_numeral @ A @ M )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( M = N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_eq_iff
% 0.30/0.52 thf(fact_1845_int__eq__iff__numeral,axiom,
% 0.30/0.52 ! [M: nat,V: num] :
% 0.30/0.52 ( ( ( semiring_1_of_nat @ int @ M )
% 0.30/0.52 = ( numeral_numeral @ int @ V ) )
% 0.30/0.52 = ( M
% 0.30/0.52 = ( numeral_numeral @ nat @ V ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % int_eq_iff_numeral
% 0.30/0.52 thf(fact_1846_of__int__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ W2 )
% 0.30/0.52 = ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( W2 = Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_iff
% 0.30/0.52 thf(fact_1847_nat__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 0.30/0.52 = ( numeral_numeral @ nat @ K ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_numeral
% 0.30/0.52 thf(fact_1848_numeral__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_le_iff
% 0.30/0.52 thf(fact_1849_numeral__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( ord_less @ num @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_less_iff
% 0.30/0.52 thf(fact_1850_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_numeral @ A )
% 0.30/0.52 => ! [V: num,W2: num,Z3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
% 0.30/0.52 = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_numeral_left_semiring_numeral
% 0.30/0.52 thf(fact_1851_numeral__times__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_numeral @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_times_numeral
% 0.30/0.52 thf(fact_1852_add__numeral__left,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [V: num,W2: num,Z3: A] :
% 0.30/0.52 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
% 0.30/0.52 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_numeral_left
% 0.30/0.52 thf(fact_1853_numeral__plus__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_plus_numeral
% 0.30/0.52 thf(fact_1854_neg__numeral__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( M = N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_eq_iff
% 0.30/0.52 thf(fact_1855_power__zero__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_1 @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1856_power__add__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( monoid_mult @ A )
% 0.30/0.52 => ! [A3: A,M: num,N: num] :
% 0.30/0.52 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
% 0.30/0.52 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_add_numeral
% 0.30/0.52 thf(fact_1857_power__add__numeral2,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( monoid_mult @ A )
% 0.30/0.52 => ! [A3: A,M: num,N: num,B2: A] :
% 0.30/0.52 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
% 0.30/0.52 = ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_add_numeral2
% 0.30/0.52 thf(fact_1858_of__nat__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_1 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_numeral
% 0.30/0.52 thf(fact_1859_abs__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % abs_numeral
% 0.30/0.52 thf(fact_1860_of__int__eq__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Z3: int,N: num] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ Z3 )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( Z3
% 0.30/0.52 = ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_numeral_iff
% 0.30/0.52 thf(fact_1861_of__int__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
% 0.30/0.52 = ( numeral_numeral @ A @ K ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_numeral
% 0.30/0.52 thf(fact_1862_ceiling__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
% 0.30/0.52 = ( numeral_numeral @ int @ V ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_numeral
% 0.30/0.52 thf(fact_1863_euclidean__size__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( euclid6346220572633701492n_size @ A @ ( numeral_numeral @ A @ K ) )
% 0.30/0.52 = ( numeral_numeral @ nat @ K ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % euclidean_size_numeral
% 0.30/0.52 thf(fact_1864_of__int__fact,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( semiring_char_0 @ A )
% 0.30/0.52 & ( ring_1 @ A ) )
% 0.30/0.52 => ! [N: nat] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( semiring_char_0_fact @ int @ N ) )
% 0.30/0.52 = ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_fact
% 0.30/0.52 thf(fact_1865_ceiling__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = Z3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_of_int
% 0.30/0.52 thf(fact_1866_round__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( numeral_numeral @ int @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % round_numeral
% 0.30/0.52 thf(fact_1867_round__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [N: int] :
% 0.30/0.52 ( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
% 0.30/0.52 = N ) ) ).
% 0.30/0.52
% 0.30/0.52 % round_of_int
% 0.30/0.52 thf(fact_1868_neg__numeral__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_iff
% 0.30/0.52 thf(fact_1869_distrib__left__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( numeral @ A )
% 0.30/0.52 & ( semiring @ A ) )
% 0.30/0.52 => ! [V: num,B2: A,C2: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % distrib_left_numeral
% 0.30/0.52 thf(fact_1870_distrib__right__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( numeral @ A )
% 0.30/0.52 & ( semiring @ A ) )
% 0.30/0.52 => ! [A3: A,B2: A,V: num] :
% 0.30/0.52 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % distrib_right_numeral
% 0.30/0.52 thf(fact_1871_neg__numeral__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( ord_less @ num @ N @ M ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_iff
% 0.30/0.52 thf(fact_1872_right__diff__distrib__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( numeral @ A )
% 0.30/0.52 & ( ring @ A ) )
% 0.30/0.52 => ! [V: num,B2: A,C2: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % right_diff_distrib_numeral
% 0.30/0.52 thf(fact_1873_left__diff__distrib__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ( numeral @ A )
% 0.30/0.52 & ( ring @ A ) )
% 0.30/0.52 => ! [A3: A,B2: A,V: num] :
% 0.30/0.52 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % left_diff_distrib_numeral
% 0.30/0.52 thf(fact_1874_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_neg_numeral_simps(1)
% 0.30/0.52 thf(fact_1875_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_neg_numeral_simps(2)
% 0.30/0.52 thf(fact_1876_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_neg_numeral_simps(3)
% 0.30/0.52 thf(fact_1877_semiring__norm_I172_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 0.30/0.52 = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(172)
% 0.30/0.52 thf(fact_1878_semiring__norm_I171_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 0.30/0.52 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(171)
% 0.30/0.52 thf(fact_1879_semiring__norm_I170_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
% 0.30/0.52 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(170)
% 0.30/0.52 thf(fact_1880_semiring__norm_I168_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.52 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 0.30/0.52 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(168)
% 0.30/0.52 thf(fact_1881_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_neg_numeral_simps(3)
% 0.30/0.52 thf(fact_1882_diff__numeral__simps_I3_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_simps(3)
% 0.30/0.52 thf(fact_1883_diff__numeral__simps_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_simps(2)
% 0.30/0.52 thf(fact_1884_abs__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % abs_neg_numeral
% 0.30/0.52 thf(fact_1885_of__int__0,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_0
% 0.30/0.52 thf(fact_1886_of__int__0__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ( zero_zero @ A )
% 0.30/0.52 = ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( Z3
% 0.30/0.52 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_0_eq_iff
% 0.30/0.52 thf(fact_1887_of__int__eq__0__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ Z3 )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 = ( Z3
% 0.30/0.52 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_0_iff
% 0.30/0.52 thf(fact_1888_of__int__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_iff
% 0.30/0.52 thf(fact_1889_of__int__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_iff
% 0.30/0.52 thf(fact_1890_of__int__eq__1__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ Z3 )
% 0.30/0.52 = ( one_one @ A ) )
% 0.30/0.52 = ( Z3
% 0.30/0.52 = ( one_one @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_1_iff
% 0.30/0.52 thf(fact_1891_of__int__1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 0.30/0.52 = ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_1
% 0.30/0.52 thf(fact_1892_of__int__mult,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z3 ) )
% 0.30/0.52 = ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_mult
% 0.30/0.52 thf(fact_1893_of__int__add,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z3 ) )
% 0.30/0.52 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_add
% 0.30/0.52 thf(fact_1894_of__int__minus,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z3 ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_minus
% 0.30/0.52 thf(fact_1895_nat__neg__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
% 0.30/0.52 = ( zero_zero @ nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_neg_numeral
% 0.30/0.52 thf(fact_1896_of__int__diff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [W2: int,Z3: int] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z3 ) )
% 0.30/0.52 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_diff
% 0.30/0.52 thf(fact_1897_division__segment__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
% 0.30/0.52 = ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % division_segment_numeral
% 0.30/0.52 thf(fact_1898_of__int__of__nat__eq,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [N: nat] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.52 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_of_nat_eq
% 0.30/0.52 thf(fact_1899_diff__nat__numeral,axiom,
% 0.30/0.52 ! [V: num,V2: num] :
% 0.30/0.52 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V2 ) )
% 0.30/0.52 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_nat_numeral
% 0.30/0.52 thf(fact_1900_of__int__abs,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: int] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
% 0.30/0.52 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_abs
% 0.30/0.52 thf(fact_1901_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [X: int,B2: int,W2: nat] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ X )
% 0.30/0.52 = ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 0.30/0.52 = ( X
% 0.30/0.52 = ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.52 thf(fact_1902_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [B2: int,W2: nat,X: int] :
% 0.30/0.52 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
% 0.30/0.52 = ( ring_1_of_int @ A @ X ) )
% 0.30/0.52 = ( ( power_power @ int @ B2 @ W2 )
% 0.30/0.52 = X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.52 thf(fact_1903_of__int__power,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Z3: int,N: nat] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z3 @ N ) )
% 0.30/0.52 = ( power_power @ A @ ( ring_1_of_int @ A @ Z3 ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_power
% 0.30/0.52 thf(fact_1904_frac__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % frac_of_int
% 0.30/0.52 thf(fact_1905_nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [Y3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ( nat2 @ Y3 )
% 0.30/0.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 0.30/0.52 = ( Y3
% 0.30/0.52 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_eq_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1906_numeral__power__eq__nat__cancel__iff,axiom,
% 0.30/0.52 ! [X: num,N: nat,Y3: int] :
% 0.30/0.52 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 0.30/0.52 = ( nat2 @ Y3 ) )
% 0.30/0.52 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 0.30/0.52 = Y3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_eq_nat_cancel_iff
% 0.30/0.52 thf(fact_1907_of__int__of__bool,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [P: $o] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
% 0.30/0.52 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_of_bool
% 0.30/0.52 thf(fact_1908_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1909_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
% 0.30/0.52 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_numeral1(1)
% 0.30/0.52 thf(fact_1910_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = A3 )
% 0.30/0.52 = ( ( ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( B2
% 0.30/0.52 = ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
% 0.30/0.52 & ( ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( A3
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral1(1)
% 0.30/0.52 thf(fact_1911_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( A3
% 0.30/0.52 = ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ( ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = B2 ) )
% 0.30/0.52 & ( ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( A3
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1912_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1913_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
% 0.30/0.52 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_numeral1(1)
% 0.30/0.52 thf(fact_1914_of__int__le__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int,N: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( ord_less_eq @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_numeral_iff
% 0.30/0.52 thf(fact_1915_of__int__numeral__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num,Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_numeral_le_iff
% 0.30/0.52 thf(fact_1916_of__int__numeral__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num,Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_numeral_less_iff
% 0.30/0.52 thf(fact_1917_of__int__less__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int,N: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( ord_less @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_numeral_iff
% 0.30/0.52 thf(fact_1918_ceiling__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_le_numeral
% 0.30/0.52 thf(fact_1919_numeral__less__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_less_ceiling
% 0.30/0.52 thf(fact_1920_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [Y3: nat,X: num,N: nat] :
% 0.30/0.52 ( ( ( semiring_1_of_nat @ A @ Y3 )
% 0.30/0.52 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.30/0.52 = ( Y3
% 0.30/0.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1921_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [X: num,N: nat,Y3: nat] :
% 0.30/0.52 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 0.30/0.52 = ( semiring_1_of_nat @ A @ Y3 ) )
% 0.30/0.52 = ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 0.30/0.52 = Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.52 thf(fact_1922_ceiling__add__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.52 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_add_numeral
% 0.30/0.52 thf(fact_1923_ceiling__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.52 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_neg_numeral
% 0.30/0.52 thf(fact_1924_ceiling__diff__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.52 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_diff_numeral
% 0.30/0.52 thf(fact_1925_Suc__times__numeral__mod__eq,axiom,
% 0.30/0.52 ! [K: num,N: nat] :
% 0.30/0.52 ( ( ( numeral_numeral @ nat @ K )
% 0.30/0.52 != ( one_one @ nat ) )
% 0.30/0.52 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.52 = ( one_one @ nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_times_numeral_mod_eq
% 0.30/0.52 thf(fact_1926_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [X: num,N: nat,Y3: int] :
% 0.30/0.52 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 0.30/0.52 = ( ring_1_of_int @ A @ Y3 ) )
% 0.30/0.52 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 0.30/0.52 = Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.52 thf(fact_1927_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Y3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ Y3 )
% 0.30/0.52 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.30/0.52 = ( Y3
% 0.30/0.52 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1928_ceiling__add__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 0.30/0.52 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_add_of_int
% 0.30/0.52 thf(fact_1929_ceiling__diff__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 0.30/0.52 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_diff_of_int
% 0.30/0.52 thf(fact_1930_round__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % round_neg_numeral
% 0.30/0.52 thf(fact_1931_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 0.30/0.52 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_numeral1(2)
% 0.30/0.52 thf(fact_1932_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
% 0.30/0.52 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_numeral1(2)
% 0.30/0.52 thf(fact_1933_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = A3 )
% 0.30/0.52 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( B2
% 0.30/0.52 = ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
% 0.30/0.52 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( A3
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral1(2)
% 0.30/0.52 thf(fact_1934_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( A3
% 0.30/0.52 = ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 0.30/0.52 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = B2 ) )
% 0.30/0.52 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( A3
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral1(2)
% 0.30/0.52 thf(fact_1935_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [A3: A,B2: A,W2: num] :
% 0.30/0.52 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 0.30/0.52 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_numeral1(2)
% 0.30/0.52 thf(fact_1936_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,W2: num,A3: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
% 0.30/0.52 = ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_numeral1(2)
% 0.30/0.52 thf(fact_1937_dbl__dec__simps_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % dbl_dec_simps(1)
% 0.30/0.52 thf(fact_1938_dbl__inc__simps_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % dbl_inc_simps(1)
% 0.30/0.52 thf(fact_1939_of__int__le__0__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
% 0.30/0.52 = ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_0_iff
% 0.30/0.52 thf(fact_1940_of__int__0__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_0_le_iff
% 0.30/0.52 thf(fact_1941_of__int__less__0__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
% 0.30/0.52 = ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_0_iff
% 0.30/0.52 thf(fact_1942_of__int__0__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_0_less_iff
% 0.30/0.52 thf(fact_1943_of__int__1__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_1_le_iff
% 0.30/0.52 thf(fact_1944_of__int__le__1__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
% 0.30/0.52 = ( ord_less_eq @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_1_iff
% 0.30/0.52 thf(fact_1945_of__int__1__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_1_less_iff
% 0.30/0.52 thf(fact_1946_of__int__less__1__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
% 0.30/0.52 = ( ord_less @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_1_iff
% 0.30/0.52 thf(fact_1947_of__nat__nat,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.52 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z3 ) )
% 0.30/0.52 = ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_nat
% 0.30/0.52 thf(fact_1948_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [B2: int,W2: nat,X: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_of_int_power_cancel_iff
% 0.30/0.52 thf(fact_1949_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: int,B2: int,W2: nat] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_power_le_of_int_cancel_iff
% 0.30/0.52 thf(fact_1950_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: int,B2: int,W2: nat] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 0.30/0.52 = ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_power_less_of_int_cancel_iff
% 0.30/0.52 thf(fact_1951_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [B2: int,W2: nat,X: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
% 0.30/0.52 = ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_of_int_power_cancel_iff
% 0.30/0.52 thf(fact_1952_nat__numeral__diff__1,axiom,
% 0.30/0.52 ! [V: num] :
% 0.30/0.52 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
% 0.30/0.52 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_numeral_diff_1
% 0.30/0.52 thf(fact_1953_nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 0.30/0.52 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_less_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1954_numeral__power__less__nat__cancel__iff,axiom,
% 0.30/0.52 ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_less_nat_cancel_iff
% 0.30/0.52 thf(fact_1955_numeral__power__le__nat__cancel__iff,axiom,
% 0.30/0.52 ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_le_nat_cancel_iff
% 0.30/0.52 thf(fact_1956_nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 0.30/0.52 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_le_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1957_ceiling__less__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_less_numeral
% 0.30/0.52 thf(fact_1958_numeral__le__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_le_ceiling
% 0.30/0.52 thf(fact_1959_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_semidom @ A )
% 0.30/0.52 => ! [I: num,N: nat,X: nat] :
% 0.30/0.52 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.30/0.52 = ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_less_of_nat_cancel_iff
% 0.30/0.52 thf(fact_1960_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_semidom @ A )
% 0.30/0.52 => ! [X: nat,I: num,N: nat] :
% 0.30/0.52 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
% 0.30/0.52 = ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_less_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1961_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_semidom @ A )
% 0.30/0.52 => ! [X: nat,I: num,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
% 0.30/0.52 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1962_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_semidom @ A )
% 0.30/0.52 => ! [I: num,N: nat,X: nat] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.52 thf(fact_1963_ceiling__le__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_le_neg_numeral
% 0.30/0.52 thf(fact_1964_neg__numeral__less__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_ceiling
% 0.30/0.52 thf(fact_1965_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.30/0.52 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1966_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_le_of_int_cancel_iff
% 0.30/0.52 thf(fact_1967_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_power_less_of_int_cancel_iff
% 0.30/0.52 thf(fact_1968_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 0.30/0.52 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1969_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [Y3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ( ring_1_of_int @ A @ Y3 )
% 0.30/0.52 = ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 0.30/0.52 = ( Y3
% 0.30/0.52 = ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1970_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [X: num,N: nat,Y3: int] :
% 0.30/0.52 ( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
% 0.30/0.52 = ( ring_1_of_int @ A @ Y3 ) )
% 0.30/0.52 = ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
% 0.30/0.52 = Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.52 thf(fact_1971_ceiling__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_less_neg_numeral
% 0.30/0.52 thf(fact_1972_neg__numeral__le__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_ceiling
% 0.30/0.52 thf(fact_1973_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 0.30/0.52 = ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.52 thf(fact_1974_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 0.30/0.52 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1975_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [X: num,N: nat,A3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 0.30/0.52 = ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.52 thf(fact_1976_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [A3: int,X: num,N: nat] :
% 0.30/0.52 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 0.30/0.52 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.52 thf(fact_1977_of__int__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [K: num] :
% 0.30/0.52 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_neg_numeral
% 0.30/0.52 thf(fact_1978_nat__numeral__as__int,axiom,
% 0.30/0.52 ( ( numeral_numeral @ nat )
% 0.30/0.52 = ( ^ [I5: num] : ( nat2 @ ( numeral_numeral @ int @ I5 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_numeral_as_int
% 0.30/0.52 thf(fact_1979_int__ops_I3_J,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.52 = ( numeral_numeral @ int @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % int_ops(3)
% 0.30/0.52 thf(fact_1980_ex__le__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim462609752435547400_field @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ? [Z4: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ex_le_of_int
% 0.30/0.52 thf(fact_1981_ex__of__int__less,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim462609752435547400_field @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ? [Z4: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z4 ) @ X ) ) ).
% 0.30/0.52
% 0.30/0.52 % ex_of_int_less
% 0.30/0.52 thf(fact_1982_ex__less__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim462609752435547400_field @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ? [Z4: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ex_less_of_int
% 0.30/0.52 thf(fact_1983_mult__of__int__commute,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [X: int,Y3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y3 )
% 0.30/0.52 = ( times_times @ A @ Y3 @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_of_int_commute
% 0.30/0.52 thf(fact_1984_zero__neq__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( zero_zero @ A )
% 0.30/0.52 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1985_numeral__neq__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( numeral_numeral @ A @ M )
% 0.30/0.52 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_neq_neg_numeral
% 0.30/0.52 thf(fact_1986_neg__numeral__neq__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 0.30/0.52 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_neq_numeral
% 0.30/0.52 thf(fact_1987_div__mult2__numeral__eq,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.52 => ! [A3: A,K: num,L: num] :
% 0.30/0.52 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
% 0.30/0.52 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult2_numeral_eq
% 0.30/0.52 thf(fact_1988_pochhammer__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( comm_ring_1 @ A )
% 0.30/0.52 => ! [X: int,N: nat] :
% 0.30/0.52 ( ( comm_s3205402744901411588hammer @ A @ ( ring_1_of_int @ A @ X ) @ N )
% 0.30/0.52 = ( ring_1_of_int @ A @ ( comm_s3205402744901411588hammer @ int @ X @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % pochhammer_of_int
% 0.30/0.52 thf(fact_1989_Ints__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Z3: int] : ( member @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_of_int
% 0.30/0.52 thf(fact_1990_Ints__induct,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Q4: A,P: A > $o] :
% 0.30/0.52 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ( ! [Z4: int] : ( P @ ( ring_1_of_int @ A @ Z4 ) )
% 0.30/0.52 => ( P @ Q4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_induct
% 0.30/0.52 thf(fact_1991_Ints__cases,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [Q4: A] :
% 0.30/0.52 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
% 0.30/0.52 => ~ ! [Z4: int] :
% 0.30/0.52 ( Q4
% 0.30/0.52 != ( ring_1_of_int @ A @ Z4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_cases
% 0.30/0.52 thf(fact_1992_Ints__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Ints_numeral
% 0.30/0.52 thf(fact_1993_not__numeral__le__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_le_zero
% 0.30/0.52 thf(fact_1994_zero__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_numeral
% 0.30/0.52 thf(fact_1995_not__numeral__less__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_less_zero
% 0.30/0.52 thf(fact_1996_zero__less__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_numeral
% 0.30/0.52 thf(fact_1997_le__of__int__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_of_int_ceiling
% 0.30/0.52 thf(fact_1998_one__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_le_numeral
% 0.30/0.52 thf(fact_1999_not__numeral__less__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_less_one
% 0.30/0.52 thf(fact_2000_not__numeral__le__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_le_neg_numeral
% 0.30/0.52 thf(fact_2001_neg__numeral__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_numeral
% 0.30/0.52 thf(fact_2002_zero__neq__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( zero_zero @ A )
% 0.30/0.52 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_neg_numeral
% 0.30/0.52 thf(fact_2003_not__numeral__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_less_neg_numeral
% 0.30/0.52 thf(fact_2004_neg__numeral__less__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_numeral
% 0.30/0.52 thf(fact_2005_one__plus__numeral__commute,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [X: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.30/0.52 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_plus_numeral_commute
% 0.30/0.52 thf(fact_2006_numeral__times__minus__swap,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( comm_ring_1 @ A )
% 0.30/0.52 => ! [W2: num,X: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
% 0.30/0.52 = ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_times_minus_swap
% 0.30/0.52 thf(fact_2007_numeral__neq__neg__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( numeral_numeral @ A @ N )
% 0.30/0.52 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_neq_neg_one
% 0.30/0.52 thf(fact_2008_one__neq__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( one_one @ A )
% 0.30/0.52 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_neq_neg_numeral
% 0.30/0.52 thf(fact_2009_power_Opower_Opower__Suc,axiom,
% 0.30/0.52 ! [A: $tType,One2: A,Times2: A > A > A,A3: A,N: nat] :
% 0.30/0.52 ( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N ) )
% 0.30/0.52 = ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power.power.power_Suc
% 0.30/0.52 thf(fact_2010_power_Opower_Opower__0,axiom,
% 0.30/0.52 ! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
% 0.30/0.52 ( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.52 = One2 ) ).
% 0.30/0.52
% 0.30/0.52 % power.power.power_0
% 0.30/0.52 thf(fact_2011_ceiling__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_le_iff
% 0.30/0.52 thf(fact_2012_less__ceiling__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_ceiling_iff
% 0.30/0.52 thf(fact_2013_neg__numeral__le__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_zero
% 0.30/0.52 thf(fact_2014_not__zero__le__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_zero_le_neg_numeral
% 0.30/0.52 thf(fact_2015_neg__numeral__less__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_zero
% 0.30/0.52 thf(fact_2016_not__zero__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_zero_less_neg_numeral
% 0.30/0.52 thf(fact_2017_not__one__le__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_one_le_neg_numeral
% 0.30/0.52 thf(fact_2018_not__numeral__le__neg__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_le_neg_one
% 0.30/0.52 thf(fact_2019_neg__numeral__le__neg__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_neg_one
% 0.30/0.52 thf(fact_2020_neg__one__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_one_le_numeral
% 0.30/0.52 thf(fact_2021_neg__numeral__le__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_one
% 0.30/0.52 thf(fact_2022_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ( divide_divide @ A @ B2 @ C2 )
% 0.30/0.52 = ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( ( ( C2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( B2
% 0.30/0.52 = ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 0.30/0.52 & ( ( C2
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral(1)
% 0.30/0.52 thf(fact_2023_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( C2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
% 0.30/0.52 = B2 ) )
% 0.30/0.52 & ( ( C2
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( ( numeral_numeral @ A @ W2 )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral(1)
% 0.30/0.52 thf(fact_2024_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_neg_one_less_neg_numeral
% 0.30/0.52 thf(fact_2025_not__one__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_one_less_neg_numeral
% 0.30/0.52 thf(fact_2026_not__numeral__less__neg__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_numeral_less_neg_one
% 0.30/0.52 thf(fact_2027_neg__one__less__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_one_less_numeral
% 0.30/0.52 thf(fact_2028_neg__numeral__less__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_one
% 0.30/0.52 thf(fact_2029_of__int__divide__in__Ints,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( idom_divide @ A )
% 0.30/0.52 => ! [B2: int,A3: int] :
% 0.30/0.52 ( ( dvd_dvd @ int @ B2 @ A3 )
% 0.30/0.52 => ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_divide_in_Ints
% 0.30/0.52 thf(fact_2030_of__int__nonneg,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_nonneg
% 0.30/0.52 thf(fact_2031_of__int__pos,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 0.30/0.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_pos
% 0.30/0.52 thf(fact_2032_of__int__leD,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: int,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.30/0.52 => ( ( N
% 0.30/0.52 = ( zero_zero @ int ) )
% 0.30/0.52 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_leD
% 0.30/0.52 thf(fact_2033_of__int__lessD,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: int,X: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.30/0.52 => ( ( N
% 0.30/0.52 = ( zero_zero @ int ) )
% 0.30/0.52 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_lessD
% 0.30/0.52 thf(fact_2034_floor__exists1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim462609752435547400_field @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ? [X3: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
% 0.30/0.52 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
% 0.30/0.52 & ! [Y4: int] :
% 0.30/0.52 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
% 0.30/0.52 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
% 0.30/0.52 => ( Y4 = X3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_exists1
% 0.30/0.52 thf(fact_2035_floor__exists,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim462609752435547400_field @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ? [Z4: int] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z4 ) @ X )
% 0.30/0.52 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_exists
% 0.30/0.52 thf(fact_2036_of__nat__less__of__int__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [N: nat,X: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
% 0.30/0.52 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_less_of_int_iff
% 0.30/0.52 thf(fact_2037_divide__less__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_numeral(1)
% 0.30/0.52 thf(fact_2038_less__divide__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_numeral(1)
% 0.30/0.52 thf(fact_2039_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ( divide_divide @ A @ B2 @ C2 )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ( ( C2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( B2
% 0.30/0.52 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 0.30/0.52 & ( ( C2
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral(2)
% 0.30/0.52 thf(fact_2040_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( division_ring @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( C2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
% 0.30/0.52 = B2 ) )
% 0.30/0.52 & ( ( C2
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral(2)
% 0.30/0.52 thf(fact_2041_round__diff__minimal,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z3 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % round_diff_minimal
% 0.30/0.52 thf(fact_2042_ceiling__correct,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
% 0.30/0.52 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_correct
% 0.30/0.52 thf(fact_2043_ceiling__unique,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X )
% 0.30/0.52 => ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 => ( ( archimedean_ceiling @ A @ X )
% 0.30/0.52 = Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_unique
% 0.30/0.52 thf(fact_2044_ceiling__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,A3: int] :
% 0.30/0.52 ( ( ( archimedean_ceiling @ A @ X )
% 0.30/0.52 = A3 )
% 0.30/0.52 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
% 0.30/0.52 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_eq_iff
% 0.30/0.52 thf(fact_2045_ceiling__split,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [P: int > $o,T2: A] :
% 0.30/0.52 ( ( P @ ( archimedean_ceiling @ A @ T2 ) )
% 0.30/0.52 = ( ! [I5: int] :
% 0.30/0.52 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) @ T2 )
% 0.30/0.52 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I5 ) ) )
% 0.30/0.52 => ( P @ I5 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_split
% 0.30/0.52 thf(fact_2046_ceiling__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
% 0.30/0.52 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_less_iff
% 0.30/0.52 thf(fact_2047_le__ceiling__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
% 0.30/0.52 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_ceiling_iff
% 0.30/0.52 thf(fact_2048_divide__le__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_numeral(1)
% 0.30/0.52 thf(fact_2049_le__divide__eq__numeral_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_numeral(1)
% 0.30/0.52 thf(fact_2050_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [B2: A,C2: A,W2: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_numeral(2)
% 0.30/0.52 thf(fact_2051_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_field @ A )
% 0.30/0.52 => ! [W2: num,B2: A,C2: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.52 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.52 => ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_numeral(2)
% 0.30/0.52 thf(fact_2052_distrib__left__NO__MATCH,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( semiring @ A )
% 0.30/0.52 => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
% 0.30/0.52 => ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % distrib_left_NO_MATCH
% 0.30/0.52 thf(fact_2053_distrib__right__NO__MATCH,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( semiring @ A )
% 0.30/0.52 => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
% 0.30/0.52 => ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % distrib_right_NO_MATCH
% 0.30/0.52 thf(fact_2054_left__diff__distrib__NO__MATCH,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( ring @ A )
% 0.30/0.52 => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
% 0.30/0.52 => ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % left_diff_distrib_NO_MATCH
% 0.30/0.52 thf(fact_2055_right__diff__distrib__NO__MATCH,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( ring @ A )
% 0.30/0.52 => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
% 0.30/0.52 => ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % right_diff_distrib_NO_MATCH
% 0.30/0.52 thf(fact_2056_ceiling__divide__upper,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Q4: A,P5: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 0.30/0.52 => ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_divide_upper
% 0.30/0.52 thf(fact_2057_of__int__of__nat,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ( ( ring_1_of_int @ A )
% 0.30/0.52 = ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_of_nat
% 0.30/0.52 thf(fact_2058_div__add__self2__no__field,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.52 & ( field @ B ) )
% 0.30/0.52 => ! [X: B,B2: A,A3: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 0.30/0.52 => ( ( B2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.52 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_add_self2_no_field
% 0.30/0.52 thf(fact_2059_div__add__self1__no__field,axiom,
% 0.30/0.52 ! [B: $tType,A: $tType] :
% 0.30/0.52 ( ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.52 & ( field @ B ) )
% 0.30/0.52 => ! [X: B,B2: A,A3: A] :
% 0.30/0.52 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 0.30/0.52 => ( ( B2
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.30/0.52 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_add_self1_no_field
% 0.30/0.52 thf(fact_2060_floor__le__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_neg_numeral
% 0.30/0.52 thf(fact_2061_neg__numeral__less__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_floor
% 0.30/0.52 thf(fact_2062_diff__numeral__special_I5_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(5)
% 0.30/0.52 thf(fact_2063_diff__numeral__special_I6_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(6)
% 0.30/0.52 thf(fact_2064_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_neg_numeral_special(5)
% 0.30/0.52 thf(fact_2065_floor__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 0.30/0.52 = Z3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_of_int
% 0.30/0.52 thf(fact_2066_floor__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
% 0.30/0.52 = ( zero_zero @ int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_zero
% 0.30/0.52 thf(fact_2067_floor__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
% 0.30/0.52 = ( numeral_numeral @ int @ V ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_numeral
% 0.30/0.52 thf(fact_2068_floor__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
% 0.30/0.52 = ( one_one @ int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_one
% 0.30/0.52 thf(fact_2069_floor__of__nat,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [N: nat] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.52 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_of_nat
% 0.30/0.52 thf(fact_2070_floor__uminus__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) )
% 0.30/0.52 = ( uminus_uminus @ int @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_uminus_of_int
% 0.30/0.52 thf(fact_2071_floor__diff__of__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 0.30/0.52 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_diff_of_int
% 0.30/0.52 thf(fact_2072_floor__add2,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: A] :
% 0.30/0.52 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.30/0.52 | ( member @ A @ Y3 @ ( ring_1_Ints @ A ) ) )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.52 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_add2
% 0.30/0.52 thf(fact_2073_zero__le__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_floor
% 0.30/0.52 thf(fact_2074_floor__less__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_zero
% 0.30/0.52 thf(fact_2075_numeral__le__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_le_floor
% 0.30/0.52 thf(fact_2076_zero__less__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_floor
% 0.30/0.52 thf(fact_2077_floor__le__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_zero
% 0.30/0.52 thf(fact_2078_floor__less__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_numeral
% 0.30/0.52 thf(fact_2079_one__le__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_le_floor
% 0.30/0.52 thf(fact_2080_floor__less__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_one
% 0.30/0.52 thf(fact_2081_floor__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.52 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_neg_numeral
% 0.30/0.52 thf(fact_2082_floor__diff__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.52 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_diff_numeral
% 0.30/0.52 thf(fact_2083_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_neg_numeral_special(6)
% 0.30/0.52 thf(fact_2084_floor__diff__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 0.30/0.52 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_diff_one
% 0.30/0.52 thf(fact_2085_numeral__less__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_less_floor
% 0.30/0.52 thf(fact_2086_floor__le__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_numeral
% 0.30/0.52 thf(fact_2087_neg__numeral__le__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [V: num,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_le_floor
% 0.30/0.52 thf(fact_2088_floor__less__neg__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,V: num] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.30/0.52 = ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_neg_numeral
% 0.30/0.52 thf(fact_2089_add__inc,axiom,
% 0.30/0.52 ! [X: num,Y3: num] :
% 0.30/0.52 ( ( plus_plus @ num @ X @ ( inc @ Y3 ) )
% 0.30/0.52 = ( inc @ ( plus_plus @ num @ X @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_inc
% 0.30/0.52 thf(fact_2090_mult__inc,axiom,
% 0.30/0.52 ! [X: num,Y3: num] :
% 0.30/0.52 ( ( times_times @ num @ X @ ( inc @ Y3 ) )
% 0.30/0.52 = ( plus_plus @ num @ ( times_times @ num @ X @ Y3 ) @ X ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_inc
% 0.30/0.52 thf(fact_2091_floor__mono,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.52 => ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_mono
% 0.30/0.52 thf(fact_2092_floor__less__cancel,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: A] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) )
% 0.30/0.52 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_cancel
% 0.30/0.52 thf(fact_2093_of__int__floor__le,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_int_floor_le
% 0.30/0.52 thf(fact_2094_floor__le__ceiling,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_ceiling
% 0.30/0.52 thf(fact_2095_floor__le__round,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_round
% 0.30/0.52 thf(fact_2096_le__floor__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_floor_iff
% 0.30/0.52 thf(fact_2097_floor__less__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 0.30/0.52 = ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_less_iff
% 0.30/0.52 thf(fact_2098_le__floor__add,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_floor_add
% 0.30/0.52 thf(fact_2099_int__add__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( plus_plus @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % int_add_floor
% 0.30/0.52 thf(fact_2100_floor__add__int,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 0.30/0.52 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_add_int
% 0.30/0.52 thf(fact_2101_floor__divide__of__int__eq,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [K: int,L: int] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
% 0.30/0.52 = ( divide_divide @ int @ K @ L ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_divide_of_int_eq
% 0.30/0.52 thf(fact_2102_ceiling__def,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ( ( archimedean_ceiling @ A )
% 0.30/0.52 = ( ^ [X2: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_def
% 0.30/0.52 thf(fact_2103_floor__minus,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.52 = ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_minus
% 0.30/0.52 thf(fact_2104_ceiling__minus,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.52 = ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_minus
% 0.30/0.52 thf(fact_2105_frac__def,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ( ( archimedean_frac @ A )
% 0.30/0.52 = ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % frac_def
% 0.30/0.52 thf(fact_2106_of__nat__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [R: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 0.30/0.52 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R ) ) ) @ R ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % of_nat_floor
% 0.30/0.52 thf(fact_2107_one__add__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 0.30/0.52 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_add_floor
% 0.30/0.52 thf(fact_2108_numeral__inc,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [X: num] :
% 0.30/0.52 ( ( numeral_numeral @ A @ ( inc @ X ) )
% 0.30/0.52 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_inc
% 0.30/0.52 thf(fact_2109_floor__divide__of__nat__eq,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [M: nat,N: nat] :
% 0.30/0.52 ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 0.30/0.52 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_divide_of_nat_eq
% 0.30/0.52 thf(fact_2110_ceiling__diff__floor__le__1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_diff_floor_le_1
% 0.30/0.52 thf(fact_2111_ceiling__altdef,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ( ( archimedean_ceiling @ A )
% 0.30/0.52 = ( ^ [X2: A] :
% 0.30/0.52 ( if @ int
% 0.30/0.52 @ ( X2
% 0.30/0.52 = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
% 0.30/0.52 @ ( archim6421214686448440834_floor @ A @ X2 )
% 0.30/0.52 @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ceiling_altdef
% 0.30/0.52 thf(fact_2112_floor__split,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [P: int > $o,T2: A] :
% 0.30/0.52 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
% 0.30/0.52 = ( ! [I5: int] :
% 0.30/0.52 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I5 ) @ T2 )
% 0.30/0.52 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) ) )
% 0.30/0.52 => ( P @ I5 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_split
% 0.30/0.52 thf(fact_2113_floor__eq__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,A3: int] :
% 0.30/0.52 ( ( ( archim6421214686448440834_floor @ A @ X )
% 0.30/0.52 = A3 )
% 0.30/0.52 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
% 0.30/0.52 & ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_eq_iff
% 0.30/0.52 thf(fact_2114_floor__unique,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
% 0.30/0.52 => ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ X )
% 0.30/0.52 = Z3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_unique
% 0.30/0.52 thf(fact_2115_le__mult__floor,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [A3: A,B2: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.52 => ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_mult_floor
% 0.30/0.52 thf(fact_2116_less__floor__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Z3: int,X: A] :
% 0.30/0.52 ( ( ord_less @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_floor_iff
% 0.30/0.52 thf(fact_2117_floor__le__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Z3: int] :
% 0.30/0.52 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 0.30/0.52 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_le_iff
% 0.30/0.52 thf(fact_2118_floor__correct,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
% 0.30/0.52 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_correct
% 0.30/0.52 thf(fact_2119_floor__divide__lower,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Q4: A,P5: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 0.30/0.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_divide_lower
% 0.30/0.52 thf(fact_2120_le__mult__floor__Ints,axiom,
% 0.30/0.52 ! [A: $tType,B: $tType] :
% 0.30/0.52 ( ( ( archim2362893244070406136eiling @ B )
% 0.30/0.52 & ( linordered_idom @ A ) )
% 0.30/0.52 => ! [A3: B,B2: B] :
% 0.30/0.52 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
% 0.30/0.52 => ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
% 0.30/0.52 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_mult_floor_Ints
% 0.30/0.52 thf(fact_2121_floor__add,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: A] :
% 0.30/0.52 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.52 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) )
% 0.30/0.52 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 0.30/0.52 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 0.30/0.52 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_add
% 0.30/0.52 thf(fact_2122_floor__divide__upper,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [Q4: A,P5: A] :
% 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 0.30/0.52 => ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % floor_divide_upper
% 0.30/0.52 thf(fact_2123_diff__numeral__special_I4_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(4)
% 0.30/0.52 thf(fact_2124_diff__numeral__special_I3_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(3)
% 0.30/0.52 thf(fact_2125_cpmi,axiom,
% 0.30/0.52 ! [D4: int,P: int > $o,P2: int > $o,B7: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ? [Z5: int] :
% 0.30/0.52 ! [X3: int] :
% 0.30/0.52 ( ( ord_less @ int @ X3 @ Z5 )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 = ( P2 @ X3 ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ B7 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int,K2: int] :
% 0.30/0.52 ( ( P2 @ X3 )
% 0.30/0.52 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
% 0.30/0.52 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.52 = ( ? [X2: int] :
% 0.30/0.52 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 & ( P2 @ X2 ) )
% 0.30/0.52 | ? [X2: int] :
% 0.30/0.52 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 & ? [Y6: int] :
% 0.30/0.52 ( ( member @ int @ Y6 @ B7 )
% 0.30/0.52 & ( P @ ( plus_plus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % cpmi
% 0.30/0.52 thf(fact_2126_cppi,axiom,
% 0.30/0.52 ! [D4: int,P: int > $o,P2: int > $o,A6: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ? [Z5: int] :
% 0.30/0.52 ! [X3: int] :
% 0.30/0.52 ( ( ord_less @ int @ Z5 @ X3 )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 = ( P2 @ X3 ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ A6 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int,K2: int] :
% 0.30/0.52 ( ( P2 @ X3 )
% 0.30/0.52 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
% 0.30/0.52 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.52 = ( ? [X2: int] :
% 0.30/0.52 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 & ( P2 @ X2 ) )
% 0.30/0.52 | ? [X2: int] :
% 0.30/0.52 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 & ? [Y6: int] :
% 0.30/0.52 ( ( member @ int @ Y6 @ A6 )
% 0.30/0.52 & ( P @ ( minus_minus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % cppi
% 0.30/0.52 thf(fact_2127_aset_I8_J,axiom,
% 0.30/0.52 ! [D4: int,A6: set @ int,T2: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less_eq @ int @ T2 @ X4 )
% 0.30/0.52 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(8)
% 0.30/0.52 thf(fact_2128_aset_I6_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,A6: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less_eq @ int @ X4 @ T2 )
% 0.30/0.52 => ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(6)
% 0.30/0.52 thf(fact_2129_bset_I8_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,B7: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less_eq @ int @ T2 @ X4 )
% 0.30/0.52 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(8)
% 0.30/0.52 thf(fact_2130_atLeastatMost__subset__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( preorder @ A )
% 0.30/0.52 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 0.30/0.52 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.52 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.52 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % atLeastatMost_subset_iff
% 0.30/0.52 thf(fact_2131_atLeastAtMost__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ord @ A )
% 0.30/0.52 => ! [I: A,L: A,U: A] :
% 0.30/0.52 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 0.30/0.52 = ( ( ord_less_eq @ A @ L @ I )
% 0.30/0.52 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % atLeastAtMost_iff
% 0.30/0.52 thf(fact_2132_Icc__eq__Icc,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( order @ A )
% 0.30/0.52 => ! [L: A,H2: A,L4: A,H: A] :
% 0.30/0.52 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
% 0.30/0.52 = ( set_or1337092689740270186AtMost @ A @ L4 @ H ) )
% 0.30/0.52 = ( ( ( L = L4 )
% 0.30/0.52 & ( H2 = H ) )
% 0.30/0.52 | ( ~ ( ord_less_eq @ A @ L @ H2 )
% 0.30/0.52 & ~ ( ord_less_eq @ A @ L4 @ H ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Icc_eq_Icc
% 0.30/0.52 thf(fact_2133_semiring__norm_I12_J,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( ( times_times @ num @ one2 @ N )
% 0.30/0.52 = N ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(12)
% 0.30/0.52 thf(fact_2134_semiring__norm_I11_J,axiom,
% 0.30/0.52 ! [M: num] :
% 0.30/0.52 ( ( times_times @ num @ M @ one2 )
% 0.30/0.52 = M ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(11)
% 0.30/0.52 thf(fact_2135_semiring__norm_I68_J,axiom,
% 0.30/0.52 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(68)
% 0.30/0.52 thf(fact_2136_semiring__norm_I75_J,axiom,
% 0.30/0.52 ! [M: num] :
% 0.30/0.52 ~ ( ord_less @ num @ M @ one2 ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(75)
% 0.30/0.52 thf(fact_2137_one__eq__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ( one_one @ A )
% 0.30/0.52 = ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( one2 = N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_eq_numeral_iff
% 0.30/0.52 thf(fact_2138_numeral__eq__one__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ( numeral_numeral @ A @ N )
% 0.30/0.52 = ( one_one @ A ) )
% 0.30/0.52 = ( N = one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_eq_one_iff
% 0.30/0.52 thf(fact_2139_numeral__eq__neg__one__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( N = one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_eq_neg_one_iff
% 0.30/0.52 thf(fact_2140_neg__one__eq__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_char_0 @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ( uminus_uminus @ A @ ( one_one @ A ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.52 = ( N = one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_one_eq_numeral_iff
% 0.30/0.52 thf(fact_2141_Suc__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( ( suc @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.52 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_numeral
% 0.30/0.52 thf(fact_2142_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
% 0.30/0.52 = ( M != one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_neg_one_le_neg_numeral_iff
% 0.30/0.52 thf(fact_2143_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linordered_idom @ A )
% 0.30/0.52 => ! [M: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( M != one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % neg_numeral_less_neg_one_iff
% 0.30/0.52 thf(fact_2144_one__plus__numeral,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_plus_numeral
% 0.30/0.52 thf(fact_2145_numeral__plus__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_plus_one
% 0.30/0.52 thf(fact_2146_numeral__le__one__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.30/0.52 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_le_one_iff
% 0.30/0.52 thf(fact_2147_one__less__numeral__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( linord181362715937106298miring @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.52 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_less_numeral_iff
% 0.30/0.52 thf(fact_2148_add__One__commute,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( ( plus_plus @ num @ one2 @ N )
% 0.30/0.52 = ( plus_plus @ num @ N @ one2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_One_commute
% 0.30/0.52 thf(fact_2149_num__induct,axiom,
% 0.30/0.52 ! [P: num > $o,X: num] :
% 0.30/0.52 ( ( P @ one2 )
% 0.30/0.52 => ( ! [X3: num] :
% 0.30/0.52 ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( inc @ X3 ) ) )
% 0.30/0.52 => ( P @ X ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % num_induct
% 0.30/0.52 thf(fact_2150_le__num__One__iff,axiom,
% 0.30/0.52 ! [X: num] :
% 0.30/0.52 ( ( ord_less_eq @ num @ X @ one2 )
% 0.30/0.52 = ( X = one2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_num_One_iff
% 0.30/0.52 thf(fact_2151_atLeastatMost__psubset__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( preorder @ A )
% 0.30/0.52 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.52 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 0.30/0.52 = ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.52 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.52 & ( ord_less_eq @ A @ B2 @ D2 )
% 0.30/0.52 & ( ( ord_less @ A @ C2 @ A3 )
% 0.30/0.52 | ( ord_less @ A @ B2 @ D2 ) ) ) )
% 0.30/0.52 & ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % atLeastatMost_psubset_iff
% 0.30/0.52 thf(fact_2152_mult__numeral__1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_numeral @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
% 0.30/0.52 = A3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_numeral_1
% 0.30/0.52 thf(fact_2153_mult__numeral__1__right,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_numeral @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
% 0.30/0.52 = A3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_numeral_1_right
% 0.30/0.52 thf(fact_2154_numeral__One,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ( ( numeral_numeral @ A @ one2 )
% 0.30/0.52 = ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_One
% 0.30/0.52 thf(fact_2155_divide__numeral__1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( field @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
% 0.30/0.52 = A3 ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_numeral_1
% 0.30/0.52 thf(fact_2156_numerals_I1_J,axiom,
% 0.30/0.52 ( ( numeral_numeral @ nat @ one2 )
% 0.30/0.52 = ( one_one @ nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % numerals(1)
% 0.30/0.52 thf(fact_2157_add__One,axiom,
% 0.30/0.52 ! [X: num] :
% 0.30/0.52 ( ( plus_plus @ num @ X @ one2 )
% 0.30/0.52 = ( inc @ X ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_One
% 0.30/0.52 thf(fact_2158_bset_I1_J,axiom,
% 0.30/0.52 ! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
% 0.30/0.52 ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ B7 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ B7 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( Q @ X3 )
% 0.30/0.52 => ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ( P @ X4 )
% 0.30/0.52 & ( Q @ X4 ) )
% 0.30/0.52 => ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
% 0.30/0.52 & ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(1)
% 0.30/0.52 thf(fact_2159_bset_I2_J,axiom,
% 0.30/0.52 ! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
% 0.30/0.52 ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ B7 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ B7 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( Q @ X3 )
% 0.30/0.52 => ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ( P @ X4 )
% 0.30/0.52 | ( Q @ X4 ) )
% 0.30/0.52 => ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
% 0.30/0.52 | ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(2)
% 0.30/0.52 thf(fact_2160_aset_I1_J,axiom,
% 0.30/0.52 ! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
% 0.30/0.52 ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ A6 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ A6 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( Q @ X3 )
% 0.30/0.52 => ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ( P @ X4 )
% 0.30/0.52 & ( Q @ X4 ) )
% 0.30/0.52 => ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
% 0.30/0.52 & ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(1)
% 0.30/0.52 thf(fact_2161_aset_I2_J,axiom,
% 0.30/0.52 ! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
% 0.30/0.52 ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ A6 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( P @ X3 )
% 0.30/0.52 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ( ! [X3: int] :
% 0.30/0.52 ( ! [Xa: int] :
% 0.30/0.52 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb: int] :
% 0.30/0.52 ( ( member @ int @ Xb @ A6 )
% 0.30/0.52 => ( X3
% 0.30/0.52 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 0.30/0.52 => ( ( Q @ X3 )
% 0.30/0.52 => ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ( P @ X4 )
% 0.30/0.52 | ( Q @ X4 ) )
% 0.30/0.52 => ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
% 0.30/0.52 | ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(2)
% 0.30/0.52 thf(fact_2162_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [B2: A] :
% 0.30/0.52 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
% 0.30/0.52 = ( uminus_uminus @ A @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1s_ring_1(1)
% 0.30/0.52 thf(fact_2163_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [B2: A] :
% 0.30/0.52 ( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1s_ring_1(2)
% 0.30/0.52 thf(fact_2164_uminus__numeral__One,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % uminus_numeral_One
% 0.30/0.52 thf(fact_2165_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.52 => ! [N: num] :
% 0.30/0.52 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % cong_exp_iff_simps(1)
% 0.30/0.52 thf(fact_2166_numeral__1__eq__Suc__0,axiom,
% 0.30/0.52 ( ( numeral_numeral @ nat @ one2 )
% 0.30/0.52 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % numeral_1_eq_Suc_0
% 0.30/0.52 thf(fact_2167_bset_I9_J,axiom,
% 0.30/0.52 ! [D2: int,D4: int,B7: set @ int,T2: int] :
% 0.30/0.52 ( ( dvd_dvd @ int @ D2 @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 0.30/0.52 => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(9)
% 0.30/0.52 thf(fact_2168_bset_I10_J,axiom,
% 0.30/0.52 ! [D2: int,D4: int,B7: set @ int,T2: int] :
% 0.30/0.52 ( ( dvd_dvd @ int @ D2 @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 0.30/0.52 => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(10)
% 0.30/0.52 thf(fact_2169_aset_I9_J,axiom,
% 0.30/0.52 ! [D2: int,D4: int,A6: set @ int,T2: int] :
% 0.30/0.52 ( ( dvd_dvd @ int @ D2 @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 0.30/0.52 => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(9)
% 0.30/0.52 thf(fact_2170_aset_I10_J,axiom,
% 0.30/0.52 ! [D2: int,D4: int,A6: set @ int,T2: int] :
% 0.30/0.52 ( ( dvd_dvd @ int @ D2 @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 0.30/0.52 => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(10)
% 0.30/0.52 thf(fact_2171_Suc__nat__number__of__add,axiom,
% 0.30/0.52 ! [V: num,N: nat] :
% 0.30/0.52 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.30/0.52 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_nat_number_of_add
% 0.30/0.52 thf(fact_2172_periodic__finite__ex,axiom,
% 0.30/0.52 ! [D2: int,P: int > $o] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.52 => ( ! [X3: int,K2: int] :
% 0.30/0.52 ( ( P @ X3 )
% 0.30/0.52 = ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 0.30/0.52 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.52 = ( ? [X2: int] :
% 0.30/0.52 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
% 0.30/0.52 & ( P @ X2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % periodic_finite_ex
% 0.30/0.52 thf(fact_2173_aset_I7_J,axiom,
% 0.30/0.52 ! [D4: int,A6: set @ int,T2: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less @ int @ T2 @ X4 )
% 0.30/0.52 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(7)
% 0.30/0.52 thf(fact_2174_aset_I5_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,A6: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ T2 @ A6 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less @ int @ X4 @ T2 )
% 0.30/0.52 => ( ord_less @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(5)
% 0.30/0.52 thf(fact_2175_aset_I4_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,A6: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ T2 @ A6 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( X4 != T2 )
% 0.30/0.52 => ( ( plus_plus @ int @ X4 @ D4 )
% 0.30/0.52 != T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(4)
% 0.30/0.52 thf(fact_2176_aset_I3_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,A6: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ A6 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( X4 = T2 )
% 0.30/0.52 => ( ( plus_plus @ int @ X4 @ D4 )
% 0.30/0.52 = T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % aset(3)
% 0.30/0.52 thf(fact_2177_bset_I7_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,B7: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ T2 @ B7 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less @ int @ T2 @ X4 )
% 0.30/0.52 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(7)
% 0.30/0.52 thf(fact_2178_bset_I5_J,axiom,
% 0.30/0.52 ! [D4: int,B7: set @ int,T2: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less @ int @ X4 @ T2 )
% 0.30/0.52 => ( ord_less @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(5)
% 0.30/0.52 thf(fact_2179_bset_I4_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,B7: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ T2 @ B7 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( X4 != T2 )
% 0.30/0.52 => ( ( minus_minus @ int @ X4 @ D4 )
% 0.30/0.52 != T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(4)
% 0.30/0.52 thf(fact_2180_bset_I3_J,axiom,
% 0.30/0.52 ! [D4: int,T2: int,B7: set @ int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( X4 = T2 )
% 0.30/0.52 => ( ( minus_minus @ int @ X4 @ D4 )
% 0.30/0.52 = T2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(3)
% 0.30/0.52 thf(fact_2181_bset_I6_J,axiom,
% 0.30/0.52 ! [D4: int,B7: set @ int,T2: int] :
% 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.30/0.52 => ! [X4: int] :
% 0.30/0.52 ( ! [Xa2: int] :
% 0.30/0.52 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.30/0.52 => ! [Xb2: int] :
% 0.30/0.52 ( ( member @ int @ Xb2 @ B7 )
% 0.30/0.52 => ( X4
% 0.30/0.52 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.30/0.52 => ( ( ord_less_eq @ int @ X4 @ T2 )
% 0.30/0.52 => ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bset(6)
% 0.30/0.52 thf(fact_2182_pochhammer__double,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( field_char_0 @ A )
% 0.30/0.52 => ! [Z3: A,N: nat] :
% 0.30/0.52 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.52 = ( 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 @ Z3 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % pochhammer_double
% 0.30/0.52 thf(fact_2183_even__succ__mod__exp,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( bit_semiring_bits @ A )
% 0.30/0.52 => ! [A3: A,N: nat] :
% 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.52 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.52 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_succ_mod_exp
% 0.30/0.52 thf(fact_2184_even__succ__div__exp,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( bit_semiring_bits @ A )
% 0.30/0.52 => ! [A3: A,N: nat] :
% 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.52 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.52 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_succ_div_exp
% 0.30/0.52 thf(fact_2185_fact__double,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( field_char_0 @ A )
% 0.30/0.52 => ! [N: nat] :
% 0.30/0.52 ( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.52 = ( 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 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % fact_double
% 0.30/0.52 thf(fact_2186_round__unique,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.52 => ! [X: A,Y3: int] :
% 0.30/0.52 ( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y3 ) )
% 0.30/0.52 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
% 0.30/0.52 => ( ( archimedean_round @ A @ X )
% 0.30/0.52 = Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % round_unique
% 0.30/0.52 thf(fact_2187_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( bit_semiring_bits @ A )
% 0.30/0.52 => ! [A3: A,M: nat,N: nat] :
% 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.52 = ( ( ord_less @ nat @ N @ M )
% 0.30/0.52 | ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 | ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.52 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_mult_exp_div_exp_iff
% 0.30/0.52 thf(fact_2188_divmod__digit__1_I1_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.52 => ! [A3: A,B2: A] :
% 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.52 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 0.30/0.52 => ( ( 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 ) )
% 0.30/0.52 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divmod_digit_1(1)
% 0.30/0.52 thf(fact_2189_verit__eq__simplify_I8_J,axiom,
% 0.30/0.52 ! [X22: num,Y2: num] :
% 0.30/0.52 ( ( ( bit0 @ X22 )
% 0.30/0.52 = ( bit0 @ Y2 ) )
% 0.30/0.52 = ( X22 = Y2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % verit_eq_simplify(8)
% 0.30/0.52 thf(fact_2190_semiring__norm_I87_J,axiom,
% 0.30/0.52 ! [M: num,N: num] :
% 0.30/0.52 ( ( ( bit0 @ M )
% 0.30/0.52 = ( bit0 @ N ) )
% 0.30/0.52 = ( M = N ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(87)
% 0.30/0.52 thf(fact_2191_semiring__norm_I85_J,axiom,
% 0.30/0.52 ! [M: num] :
% 0.30/0.52 ( ( bit0 @ M )
% 0.30/0.52 != one2 ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(85)
% 0.30/0.52 thf(fact_2192_semiring__norm_I83_J,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( one2
% 0.30/0.52 != ( bit0 @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(83)
% 0.30/0.52 thf(fact_2193_semiring__norm_I6_J,axiom,
% 0.30/0.52 ! [M: num,N: num] :
% 0.30/0.52 ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.52 = ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(6)
% 0.30/0.52 thf(fact_2194_semiring__norm_I13_J,axiom,
% 0.30/0.52 ! [M: num,N: num] :
% 0.30/0.52 ( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.52 = ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(13)
% 0.30/0.52 thf(fact_2195_semiring__norm_I78_J,axiom,
% 0.30/0.52 ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.52 = ( ord_less @ num @ M @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(78)
% 0.30/0.52 thf(fact_2196_semiring__norm_I71_J,axiom,
% 0.30/0.52 ! [M: num,N: num] :
% 0.30/0.52 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.52 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(71)
% 0.30/0.52 thf(fact_2197_semiring__norm_I2_J,axiom,
% 0.30/0.52 ( ( plus_plus @ num @ one2 @ one2 )
% 0.30/0.52 = ( bit0 @ one2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(2)
% 0.30/0.52 thf(fact_2198_num__double,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( ( times_times @ num @ ( bit0 @ one2 ) @ N )
% 0.30/0.52 = ( bit0 @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % num_double
% 0.30/0.52 thf(fact_2199_semiring__norm_I69_J,axiom,
% 0.30/0.52 ! [M: num] :
% 0.30/0.52 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(69)
% 0.30/0.52 thf(fact_2200_semiring__norm_I76_J,axiom,
% 0.30/0.52 ! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % semiring_norm(76)
% 0.30/0.52 thf(fact_2201_one__add__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( numeral @ A )
% 0.30/0.52 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_add_one
% 0.30/0.52 thf(fact_2202_even__mult__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_parity @ A )
% 0.30/0.52 => ! [A3: A,B2: A] :
% 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.52 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.52 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_mult_iff
% 0.30/0.52 thf(fact_2203_bits__one__mod__two__eq__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( bit_semiring_bits @ A )
% 0.30/0.52 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bits_one_mod_two_eq_one
% 0.30/0.52 thf(fact_2204_one__mod__two__eq__one,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.52 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( one_one @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_mod_two_eq_one
% 0.30/0.52 thf(fact_2205_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiri2026040879449505780visors @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( zero_zero @ A ) )
% 0.30/0.52 = ( A3
% 0.30/0.52 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_2206_power2__minus,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( ring_1 @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_minus
% 0.30/0.52 thf(fact_2207_add__2__eq__Suc_H,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_2_eq_Suc'
% 0.30/0.52 thf(fact_2208_add__2__eq__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.52 = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_2_eq_Suc
% 0.30/0.52 thf(fact_2209_Suc__1,axiom,
% 0.30/0.52 ( ( suc @ ( one_one @ nat ) )
% 0.30/0.52 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_1
% 0.30/0.52 thf(fact_2210_even__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) )
% 0.30/0.52 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_Suc
% 0.30/0.52 thf(fact_2211_even__Suc__Suc__iff,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N ) ) )
% 0.30/0.52 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_Suc_Suc_iff
% 0.30/0.52 thf(fact_2212_mod2__Suc__Suc,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mod2_Suc_Suc
% 0.30/0.52 thf(fact_2213_div2__Suc__Suc,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div2_Suc_Suc
% 0.30/0.52 thf(fact_2214_fact__2,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_char_0 @ A )
% 0.30/0.52 => ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % fact_2
% 0.30/0.52 thf(fact_2215_add__self__div__2,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = M ) ).
% 0.30/0.52
% 0.30/0.52 % add_self_div_2
% 0.30/0.52 thf(fact_2216_zmod__numeral__Bit0,axiom,
% 0.30/0.52 ! [V: num,W2: num] :
% 0.30/0.52 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
% 0.30/0.52 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zmod_numeral_Bit0
% 0.30/0.52 thf(fact_2217_one__div__two__eq__zero,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.52 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_div_two_eq_zero
% 0.30/0.52 thf(fact_2218_bits__1__div__2,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( bit_semiring_bits @ A )
% 0.30/0.52 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( zero_zero @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % bits_1_div_2
% 0.30/0.52 thf(fact_2219_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_neg_numeral_special(9)
% 0.30/0.52 thf(fact_2220_diff__numeral__special_I11_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.52 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(11)
% 0.30/0.52 thf(fact_2221_diff__numeral__special_I10_J,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( neg_numeral @ A )
% 0.30/0.52 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 0.30/0.52 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(10)
% 0.30/0.52 thf(fact_2222_even__plus__one__iff,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_parity @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
% 0.30/0.52 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % even_plus_one_iff
% 0.30/0.52 thf(fact_2223_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_parity @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 != ( zero_zero @ A ) )
% 0.30/0.52 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 = ( one_one @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_mod_2_eq_0_eq_1
% 0.30/0.52 thf(fact_2224_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.52 ! [A: $tType] :
% 0.30/0.52 ( ( semiring_parity @ A )
% 0.30/0.52 => ! [A3: A] :
% 0.30/0.52 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.52 != ( one_one @ A ) )
% 0.30/0.53 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_1_eq_0
% 0.30/0.53 thf(fact_2225_minus__1__div__2__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid8789492081693882211th_nat @ A )
% 0.30/0.53 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_1_div_2_eq
% 0.30/0.53 thf(fact_2226_power2__less__eq__zero__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( A3
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_eq_zero_iff
% 0.30/0.53 thf(fact_2227_power2__eq__iff__nonneg,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_semidom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.53 => ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( X = Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff_nonneg
% 0.30/0.53 thf(fact_2228_zero__less__power2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( A3
% 0.30/0.53 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power2
% 0.30/0.53 thf(fact_2229_bits__minus__1__mod__2__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bits_minus_1_mod_2_eq
% 0.30/0.53 thf(fact_2230_minus__1__mod__2__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid8789492081693882211th_nat @ A )
% 0.30/0.53 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_1_mod_2_eq
% 0.30/0.53 thf(fact_2231_of__bool__half__eq__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [B2: $o] :
% 0.30/0.53 ( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_bool_half_eq_0
% 0.30/0.53 thf(fact_2232_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Power.ring_1_class.power_minus_even
% 0.30/0.53 thf(fact_2233_power__minus__odd,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.53 = ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_minus_odd
% 0.30/0.53 thf(fact_2234_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.53 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Parity.ring_1_class.power_minus_even
% 0.30/0.53 thf(fact_2235_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 != ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod2_eq_Suc_0_eq_0
% 0.30/0.53 thf(fact_2236_add__self__mod__2,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_self_mod_2
% 0.30/0.53 thf(fact_2237_half__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % half_nonnegative_int_iff
% 0.30/0.53 thf(fact_2238_half__negative__int__iff,axiom,
% 0.30/0.53 ! [K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % half_negative_int_iff
% 0.30/0.53 thf(fact_2239_even__Suc__div__two,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_Suc_div_two
% 0.30/0.53 thf(fact_2240_odd__Suc__div__two,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_Suc_div_two
% 0.30/0.53 thf(fact_2241_even__succ__div__2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_succ_div_2
% 0.30/0.53 thf(fact_2242_even__succ__div__two,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_succ_div_two
% 0.30/0.53 thf(fact_2243_odd__succ__div__two,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_succ_div_two
% 0.30/0.53 thf(fact_2244_even__power,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.53 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_power
% 0.30/0.53 thf(fact_2245_zero__le__power__eq__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,W2: num] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
% 0.30/0.53 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power_eq_numeral
% 0.30/0.53 thf(fact_2246_power__less__zero__eq__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,W2: num] :
% 0.30/0.53 ( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_less_zero_eq_numeral
% 0.30/0.53 thf(fact_2247_power__less__zero__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_less_zero_eq
% 0.30/0.53 thf(fact_2248_power__minus1__even,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_minus1_even
% 0.30/0.53 thf(fact_2249_neg__one__even__power,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.30/0.53 = ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % neg_one_even_power
% 0.30/0.53 thf(fact_2250_neg__one__odd__power,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % neg_one_odd_power
% 0.30/0.53 thf(fact_2251_even__of__nat,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.53 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_of_nat
% 0.30/0.53 thf(fact_2252_odd__Suc__minus__one,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.53 = N ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_Suc_minus_one
% 0.30/0.53 thf(fact_2253_one__less__floor,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.30/0.53 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_less_floor
% 0.30/0.53 thf(fact_2254_floor__le__one,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 0.30/0.53 = ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % floor_le_one
% 0.30/0.53 thf(fact_2255_even__diff__nat,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
% 0.30/0.53 = ( ( ord_less @ nat @ M @ N )
% 0.30/0.53 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_diff_nat
% 0.30/0.53 thf(fact_2256_mod2__gr__0,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod2_gr_0
% 0.30/0.53 thf(fact_2257_odd__two__times__div__two__succ,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
% 0.30/0.53 = A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_two_times_div_two_succ
% 0.30/0.53 thf(fact_2258_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
% 0.30/0.53 = ( N
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_parity_class.even_mask_iff
% 0.30/0.53 thf(fact_2259_zero__less__power__eq__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,W2: num] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
% 0.30/0.53 = ( ( ( numeral_numeral @ nat @ W2 )
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( A3
% 0.30/0.53 != ( zero_zero @ A ) ) )
% 0.30/0.53 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power_eq_numeral
% 0.30/0.53 thf(fact_2260_bits__1__div__exp,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( zero_neq_one_of_bool @ A
% 0.30/0.53 @ ( N
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bits_1_div_exp
% 0.30/0.53 thf(fact_2261_one__div__2__pow__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( zero_neq_one_of_bool @ A
% 0.30/0.53 @ ( N
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_div_2_pow_eq
% 0.30/0.53 thf(fact_2262_odd__two__times__div__two__nat,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_two_times_div_two_nat
% 0.30/0.53 thf(fact_2263_power__le__zero__eq__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,W2: num] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 0.30/0.53 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 0.30/0.53 & ( A3
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_le_zero_eq_numeral
% 0.30/0.53 thf(fact_2264_one__mod__2__pow__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_mod_2_pow_eq
% 0.30/0.53 thf(fact_2265_verit__eq__simplify_I10_J,axiom,
% 0.30/0.53 ! [X22: num] :
% 0.30/0.53 ( one2
% 0.30/0.53 != ( bit0 @ X22 ) ) ).
% 0.30/0.53
% 0.30/0.53 % verit_eq_simplify(10)
% 0.30/0.53 thf(fact_2266_ex__nat__less,axiom,
% 0.30/0.53 ! [N: nat,P: nat > $o] :
% 0.30/0.53 ( ( ? [M2: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.30/0.53 & ( P @ M2 ) ) )
% 0.30/0.53 = ( ? [X2: nat] :
% 0.30/0.53 ( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.53 & ( P @ X2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ex_nat_less
% 0.30/0.53 thf(fact_2267_all__nat__less,axiom,
% 0.30/0.53 ! [N: nat,P: nat > $o] :
% 0.30/0.53 ( ( ! [M2: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.30/0.53 => ( P @ M2 ) ) )
% 0.30/0.53 = ( ! [X2: nat] :
% 0.30/0.53 ( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.53 => ( P @ X2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % all_nat_less
% 0.30/0.53 thf(fact_2268_numeral__Bit0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( numeral @ A )
% 0.30/0.53 => ! [N: num] :
% 0.30/0.53 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 0.30/0.53 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_Bit0
% 0.30/0.53 thf(fact_2269_inc_Osimps_I1_J,axiom,
% 0.30/0.53 ( ( inc @ one2 )
% 0.30/0.53 = ( bit0 @ one2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % inc.simps(1)
% 0.30/0.53 thf(fact_2270_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [N: num,Q4: num] :
% 0.30/0.53 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(2)
% 0.30/0.53 thf(fact_2271_numeral__Bit0__div__2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [N: num] :
% 0.30/0.53 ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_Bit0_div_2
% 0.30/0.53 thf(fact_2272_power__minus__Bit0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [X: A,K: num] :
% 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
% 0.30/0.53 = ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_minus_Bit0
% 0.30/0.53 thf(fact_2273_mult__2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_numeral @ A )
% 0.30/0.53 => ! [Z3: A] :
% 0.30/0.53 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 )
% 0.30/0.53 = ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_2
% 0.30/0.53 thf(fact_2274_mult__2__right,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_numeral @ A )
% 0.30/0.53 => ! [Z3: A] :
% 0.30/0.53 ( ( times_times @ A @ Z3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_2_right
% 0.30/0.53 thf(fact_2275_left__add__twice,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_numeral @ A )
% 0.30/0.53 => ! [A3: A,B2: A] :
% 0.30/0.53 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.30/0.53 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % left_add_twice
% 0.30/0.53 thf(fact_2276_even__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_zero
% 0.30/0.53 thf(fact_2277_evenE,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ~ ! [B5: A] :
% 0.30/0.53 ( A3
% 0.30/0.53 != ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % evenE
% 0.30/0.53 thf(fact_2278_odd__one,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_one
% 0.30/0.53 thf(fact_2279_even__minus,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.53 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_minus
% 0.30/0.53 thf(fact_2280_zero__power2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_1 @ A )
% 0.30/0.53 => ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power2
% 0.30/0.53 thf(fact_2281_power2__eq__square,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( monoid_mult @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( times_times @ A @ A3 @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_square
% 0.30/0.53 thf(fact_2282_power4__eq__xxxx,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( monoid_mult @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power4_eq_xxxx
% 0.30/0.53 thf(fact_2283_numeral__2__eq__2,axiom,
% 0.30/0.53 ( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
% 0.30/0.53 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_2_eq_2
% 0.30/0.53 thf(fact_2284_one__power2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_1 @ A )
% 0.30/0.53 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_power2
% 0.30/0.53 thf(fact_2285_power2__eq__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ( X = Y3 )
% 0.30/0.53 | ( X
% 0.30/0.53 = ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff
% 0.30/0.53 thf(fact_2286_double__not__eq__Suc__double,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 0.30/0.53 != ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_not_eq_Suc_double
% 0.30/0.53 thf(fact_2287_Suc__double__not__eq__double,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
% 0.30/0.53 != ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_double_not_eq_double
% 0.30/0.53 thf(fact_2288_power__even__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( monoid_mult @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_even_eq
% 0.30/0.53 thf(fact_2289_nat__1__add__1,axiom,
% 0.30/0.53 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_1_add_1
% 0.30/0.53 thf(fact_2290_less__exp,axiom,
% 0.30/0.53 ! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_exp
% 0.30/0.53 thf(fact_2291_self__le__ge2__pow,axiom,
% 0.30/0.53 ! [K: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.53 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % self_le_ge2_pow
% 0.30/0.53 thf(fact_2292_power2__nat__le__eq__le,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_nat_le_eq_le
% 0.30/0.53 thf(fact_2293_power2__nat__le__imp__le,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
% 0.30/0.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_nat_le_imp_le
% 0.30/0.53 thf(fact_2294_even__even__mod__4__iff,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_even_mod_4_iff
% 0.30/0.53 thf(fact_2295_div2__even__ext__nat,axiom,
% 0.30/0.53 ! [X: nat,Y3: nat] :
% 0.30/0.53 ( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( divide_divide @ nat @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
% 0.30/0.53 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y3 ) )
% 0.30/0.53 => ( X = Y3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div2_even_ext_nat
% 0.30/0.53 thf(fact_2296_half__gt__zero__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_field @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % half_gt_zero_iff
% 0.30/0.53 thf(fact_2297_half__gt__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_field @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % half_gt_zero
% 0.30/0.53 thf(fact_2298_even__two__times__div__two,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_two_times_div_two
% 0.30/0.53 thf(fact_2299_zero__le__power2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power2
% 0.30/0.53 thf(fact_2300_power2__eq__imp__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_semidom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.53 => ( X = Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_imp_eq
% 0.30/0.53 thf(fact_2301_power2__le__imp__le,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_semidom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.53 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_le_imp_le
% 0.30/0.53 thf(fact_2302_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_iff_mod_2_eq_zero
% 0.30/0.53 thf(fact_2303_power2__less__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_0
% 0.30/0.53 thf(fact_2304_exp__add__not__zero__imp__left,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.53 != ( zero_zero @ A ) )
% 0.30/0.53 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
% 0.30/0.53 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % exp_add_not_zero_imp_left
% 0.30/0.53 thf(fact_2305_exp__add__not__zero__imp__right,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.53 != ( zero_zero @ A ) )
% 0.30/0.53 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % exp_add_not_zero_imp_right
% 0.30/0.53 thf(fact_2306_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.30/0.53 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_iff_mod_2_eq_one
% 0.30/0.53 thf(fact_2307_sum__power2__eq__zero__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 = ( ( X
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 & ( Y3
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_eq_zero_iff
% 0.30/0.53 thf(fact_2308_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [N: nat,M: nat] :
% 0.30/0.53 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 != ( zero_zero @ A ) )
% 0.30/0.53 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.53 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.53 thf(fact_2309_power2__eq__1__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_15535105094025558882visors @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) )
% 0.30/0.53 = ( ( A3
% 0.30/0.53 = ( one_one @ A ) )
% 0.30/0.53 | ( A3
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_1_iff
% 0.30/0.53 thf(fact_2310_less__2__cases__iff,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( ( N
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( N
% 0.30/0.53 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_2_cases_iff
% 0.30/0.53 thf(fact_2311_less__2__cases,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 => ( ( N
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( N
% 0.30/0.53 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_2_cases
% 0.30/0.53 thf(fact_2312_abs__le__square__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y3 ) )
% 0.30/0.53 = ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % abs_le_square_iff
% 0.30/0.53 thf(fact_2313_power__mono__odd,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [N: nat,A3: A,B2: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.53 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_odd
% 0.30/0.53 thf(fact_2314_abs__square__eq__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) )
% 0.30/0.53 = ( ( abs_abs @ A @ X )
% 0.30/0.53 = ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % abs_square_eq_1
% 0.30/0.53 thf(fact_2315_odd__pos,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_pos
% 0.30/0.53 thf(fact_2316_uminus__power__if,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.53 = ( power_power @ A @ A3 @ N ) ) )
% 0.30/0.53 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.53 = ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % uminus_power_if
% 0.30/0.53 thf(fact_2317_nat__2,axiom,
% 0.30/0.53 ( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_2
% 0.30/0.53 thf(fact_2318_nat__induct2,axiom,
% 0.30/0.53 ! [P: nat > $o,N: nat] :
% 0.30/0.53 ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.53 => ( ( P @ ( one_one @ nat ) )
% 0.30/0.53 => ( ! [N4: nat] :
% 0.30/0.53 ( ( P @ N4 )
% 0.30/0.53 => ( P @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 0.30/0.53 => ( P @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_induct2
% 0.30/0.53 thf(fact_2319_not__exp__less__eq__0__int,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_exp_less_eq_0_int
% 0.30/0.53 thf(fact_2320_diff__le__diff__pow,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.53 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_le_diff_pow
% 0.30/0.53 thf(fact_2321_dvd__power__iff__le,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.53 => ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dvd_power_iff_le
% 0.30/0.53 thf(fact_2322_divmod__digit__0_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [B2: A,A3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.53 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 0.30/0.53 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
% 0.30/0.53 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divmod_digit_0(2)
% 0.30/0.53 thf(fact_2323_oddE,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ~ ! [B5: A] :
% 0.30/0.53 ( A3
% 0.30/0.53 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % oddE
% 0.30/0.53 thf(fact_2324_power2__less__imp__less,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_semidom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.53 => ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_imp_less
% 0.30/0.53 thf(fact_2325_sum__power2__le__zero__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( ( X
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 & ( Y3
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_le_zero_iff
% 0.30/0.53 thf(fact_2326_sum__power2__ge__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_ge_zero
% 0.30/0.53 thf(fact_2327_mod2__eq__if,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) )
% 0.30/0.53 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod2_eq_if
% 0.30/0.53 thf(fact_2328_parity__cases,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_parity @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 != ( zero_zero @ A ) ) )
% 0.30/0.53 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 != ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % parity_cases
% 0.30/0.53 thf(fact_2329_bits__stable__imp__add__self,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = A3 )
% 0.30/0.53 => ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bits_stable_imp_add_self
% 0.30/0.53 thf(fact_2330_not__sum__power2__lt__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_sum_power2_lt_zero
% 0.30/0.53 thf(fact_2331_sum__power2__gt__zero__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 0.30/0.53 = ( ( X
% 0.30/0.53 != ( zero_zero @ A ) )
% 0.30/0.53 | ( Y3
% 0.30/0.53 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_gt_zero_iff
% 0.30/0.53 thf(fact_2332_power2__sum,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( comm_semiring_1 @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_sum
% 0.30/0.53 thf(fact_2333_square__le__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.30/0.53 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 0.30/0.53 => ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % square_le_1
% 0.30/0.53 thf(fact_2334_zero__le__even__power_H,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_even_power'
% 0.30/0.53 thf(fact_2335_power2__le__iff__abs__le,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [Y3: A,X: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_le_iff_abs_le
% 0.30/0.53 thf(fact_2336_zero__le__power__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.53 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power_eq
% 0.30/0.53 thf(fact_2337_zero__le__odd__power,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.53 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_odd_power
% 0.30/0.53 thf(fact_2338_zero__le__even__power,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_even_power
% 0.30/0.53 thf(fact_2339_abs__square__le__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % abs_square_le_1
% 0.30/0.53 thf(fact_2340_abs__square__less__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % abs_square_less_1
% 0.30/0.53 thf(fact_2341_bits__induct,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [P: A > $o,A3: A] :
% 0.30/0.53 ( ! [A8: A] :
% 0.30/0.53 ( ( ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = A8 )
% 0.30/0.53 => ( P @ A8 ) )
% 0.30/0.53 => ( ! [A8: A,B5: $o] :
% 0.30/0.53 ( ( P @ A8 )
% 0.30/0.53 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = A8 )
% 0.30/0.53 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) ) ) )
% 0.30/0.53 => ( P @ A3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bits_induct
% 0.30/0.53 thf(fact_2342_minus__power__mult__self,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( comm_ring_1 @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 0.30/0.53 = ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_power_mult_self
% 0.30/0.53 thf(fact_2343_power__odd__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( monoid_mult @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 = ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_odd_eq
% 0.30/0.53 thf(fact_2344_nat__bit__induct,axiom,
% 0.30/0.53 ! [P: nat > $o,N: nat] :
% 0.30/0.53 ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.53 => ( ! [N4: nat] :
% 0.30/0.53 ( ( P @ N4 )
% 0.30/0.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.53 => ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
% 0.30/0.53 => ( ! [N4: nat] :
% 0.30/0.53 ( ( P @ N4 )
% 0.30/0.53 => ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
% 0.30/0.53 => ( P @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_bit_induct
% 0.30/0.53 thf(fact_2345_minus__one__power__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.30/0.53 = ( one_one @ A ) ) )
% 0.30/0.53 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_one_power_iff
% 0.30/0.53 thf(fact_2346_power__mono__even,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [N: nat,A3: A,B2: A] :
% 0.30/0.53 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) )
% 0.30/0.53 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_even
% 0.30/0.53 thf(fact_2347_Suc__n__div__2__gt__zero,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_n_div_2_gt_zero
% 0.30/0.53 thf(fact_2348_div__2__gt__zero,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_2_gt_zero
% 0.30/0.53 thf(fact_2349_minus__1__div__exp__eq__int,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_1_div_exp_eq_int
% 0.30/0.53 thf(fact_2350_triangle__def,axiom,
% 0.30/0.53 ( nat_triangle
% 0.30/0.53 = ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % triangle_def
% 0.30/0.53 thf(fact_2351_divmod__digit__0_I1_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [B2: A,A3: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.53 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 0.30/0.53 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 0.30/0.53 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divmod_digit_0(1)
% 0.30/0.53 thf(fact_2352_power2__diff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( comm_ring_1 @ A )
% 0.30/0.53 => ! [X: A,Y3: A] :
% 0.30/0.53 ( ( power_power @ A @ ( minus_minus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_diff
% 0.30/0.53 thf(fact_2353_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_0_le_power_imp_0_le
% 0.30/0.53 thf(fact_2354_odd__power__less__zero,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.53 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_power_less_zero
% 0.30/0.53 thf(fact_2355_zero__less__power__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 0.30/0.53 = ( ( N
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( A3
% 0.30/0.53 != ( zero_zero @ A ) ) )
% 0.30/0.53 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power_eq
% 0.30/0.53 thf(fact_2356_mult__exp__mod__exp__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [M: nat,N: nat,A3: A] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.53 => ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_exp_mod_exp_eq
% 0.30/0.53 thf(fact_2357_power__minus1__odd,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_minus1_odd
% 0.30/0.53 thf(fact_2358_exp__mod__exp,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % exp_mod_exp
% 0.30/0.53 thf(fact_2359_int__bit__induct,axiom,
% 0.30/0.53 ! [P: int > $o,K: int] :
% 0.30/0.53 ( ( P @ ( zero_zero @ int ) )
% 0.30/0.53 => ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 => ( ! [K2: int] :
% 0.30/0.53 ( ( P @ K2 )
% 0.30/0.53 => ( ( K2
% 0.30/0.53 != ( zero_zero @ int ) )
% 0.30/0.53 => ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
% 0.30/0.53 => ( ! [K2: int] :
% 0.30/0.53 ( ( P @ K2 )
% 0.30/0.53 => ( ( K2
% 0.30/0.53 != ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 => ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
% 0.30/0.53 => ( P @ K ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % int_bit_induct
% 0.30/0.53 thf(fact_2360_ex__power__ivl1,axiom,
% 0.30/0.53 ! [B2: nat,K: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 0.30/0.53 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
% 0.30/0.53 => ? [N4: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
% 0.30/0.53 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ex_power_ivl1
% 0.30/0.53 thf(fact_2361_ex__power__ivl2,axiom,
% 0.30/0.53 ! [B2: nat,K: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 0.30/0.53 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.53 => ? [N4: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
% 0.30/0.53 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ex_power_ivl2
% 0.30/0.53 thf(fact_2362_round__def,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ( ( archimedean_round @ A )
% 0.30/0.53 = ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % round_def
% 0.30/0.53 thf(fact_2363_mod__double__modulus,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [M: A,X: A] :
% 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.53 => ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
% 0.30/0.53 = ( modulo_modulo @ A @ X @ M ) )
% 0.30/0.53 | ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
% 0.30/0.53 = ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_double_modulus
% 0.30/0.53 thf(fact_2364_divmod__digit__1_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [A3: A,B2: A] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.53 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 0.30/0.53 => ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 0.30/0.53 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divmod_digit_1(2)
% 0.30/0.53 thf(fact_2365_even__mask__div__iff_H,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_mask_div_iff'
% 0.30/0.53 thf(fact_2366_power__le__zero__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_idom @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
% 0.30/0.53 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.53 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 0.30/0.53 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 & ( A3
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_le_zero_eq
% 0.30/0.53 thf(fact_2367_of__int__round__le,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [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 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_int_round_le
% 0.30/0.53 thf(fact_2368_of__int__round__ge,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [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 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_int_round_ge
% 0.30/0.53 thf(fact_2369_of__int__round__gt,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [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 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_int_round_gt
% 0.30/0.53 thf(fact_2370_even__mod__4__div__2,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 => ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_mod_4_div_2
% 0.30/0.53 thf(fact_2371_neg__zdiv__mult__2,axiom,
% 0.30/0.53 ! [A3: int,B2: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 0.30/0.53 => ( ( 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 ) )
% 0.30/0.53 = ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % neg_zdiv_mult_2
% 0.30/0.53 thf(fact_2372_pos__zdiv__mult__2,axiom,
% 0.30/0.53 ! [A3: int,B2: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.53 => ( ( 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 ) )
% 0.30/0.53 = ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_zdiv_mult_2
% 0.30/0.53 thf(fact_2373_even__nat__iff,axiom,
% 0.30/0.53 ! [K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
% 0.30/0.53 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_nat_iff
% 0.30/0.53 thf(fact_2374_pos__zmod__mult__2,axiom,
% 0.30/0.53 ! [A3: int,B2: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.53 => ( ( 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 ) )
% 0.30/0.53 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_zmod_mult_2
% 0.30/0.53 thf(fact_2375_even__mask__div__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 | ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_mask_div_iff
% 0.30/0.53 thf(fact_2376_exp__div__exp__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_semiring_bits @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( times_times @ A
% 0.30/0.53 @ ( zero_neq_one_of_bool @ A
% 0.30/0.53 @ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
% 0.30/0.53 != ( zero_zero @ A ) )
% 0.30/0.53 & ( ord_less_eq @ nat @ N @ M ) ) )
% 0.30/0.53 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % exp_div_exp_eq
% 0.30/0.53 thf(fact_2377_of__int__round__abs__le,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [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 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_int_round_abs_le
% 0.30/0.53 thf(fact_2378_round__unique_H,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ! [X: A,N: int] :
% 0.30/0.53 ( ( 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 ) ) ) )
% 0.30/0.53 => ( ( archimedean_round @ A @ X )
% 0.30/0.53 = N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % round_unique'
% 0.30/0.53 thf(fact_2379_neg__zmod__mult__2,axiom,
% 0.30/0.53 ! [A3: int,B2: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 0.30/0.53 => ( ( 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 ) )
% 0.30/0.53 = ( 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 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % neg_zmod_mult_2
% 0.30/0.53 thf(fact_2380_round__altdef,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.53 => ( ( archimedean_round @ A )
% 0.30/0.53 = ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % round_altdef
% 0.30/0.53 thf(fact_2381_flip__bit__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % flip_bit_0
% 0.30/0.53 thf(fact_2382_set__bit__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.53 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_bit_0
% 0.30/0.53 thf(fact_2383_psubsetI,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ( A6 != B7 )
% 0.30/0.53 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubsetI
% 0.30/0.53 thf(fact_2384_unset__bit__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_0
% 0.30/0.53 thf(fact_2385_unset__bit__Suc,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_Suc
% 0.30/0.53 thf(fact_2386_flip__bit__Suc,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % flip_bit_Suc
% 0.30/0.53 thf(fact_2387_set__bit__Suc,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_bit_Suc
% 0.30/0.53 thf(fact_2388_Compl__subset__Compl__iff,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 0.30/0.53 = ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Compl_subset_Compl_iff
% 0.30/0.53 thf(fact_2389_Compl__anti__mono,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Compl_anti_mono
% 0.30/0.53 thf(fact_2390_unset__bit__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_nonnegative_int_iff
% 0.30/0.53 thf(fact_2391_set__bit__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_bit_nonnegative_int_iff
% 0.30/0.53 thf(fact_2392_flip__bit__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % flip_bit_nonnegative_int_iff
% 0.30/0.53 thf(fact_2393_unset__bit__negative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_negative_int_iff
% 0.30/0.53 thf(fact_2394_set__bit__negative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_bit_negative_int_iff
% 0.30/0.53 thf(fact_2395_flip__bit__negative__int__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % flip_bit_negative_int_iff
% 0.30/0.53 thf(fact_2396_unset__bit__nat__def,axiom,
% 0.30/0.53 ( ( bit_se2638667681897837118et_bit @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M2 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_nat_def
% 0.30/0.53 thf(fact_2397_unset__bit__less__eq,axiom,
% 0.30/0.53 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% 0.30/0.53
% 0.30/0.53 % unset_bit_less_eq
% 0.30/0.53 thf(fact_2398_set__bit__greater__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_bit_greater_eq
% 0.30/0.53 thf(fact_2399_psubset__imp__ex__mem,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubset_imp_ex_mem
% 0.30/0.53 thf(fact_2400_psubset__trans,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
% 0.30/0.53 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubset_trans
% 0.30/0.53 thf(fact_2401_psubsetD,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A,C2: A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ( member @ A @ C2 @ A6 )
% 0.30/0.53 => ( member @ A @ C2 @ B7 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubsetD
% 0.30/0.53 thf(fact_2402_psubsetE,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubsetE
% 0.30/0.53 thf(fact_2403_psubset__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) )
% 0.30/0.53 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.30/0.53 & ( A7 != B6 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubset_eq
% 0.30/0.53 thf(fact_2404_psubset__imp__subset,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubset_imp_subset
% 0.30/0.53 thf(fact_2405_psubset__subset__trans,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C6 )
% 0.30/0.53 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % psubset_subset_trans
% 0.30/0.53 thf(fact_2406_subset__not__subset__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) )
% 0.30/0.53 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 0.30/0.53 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % subset_not_subset_eq
% 0.30/0.53 thf(fact_2407_subset__psubset__trans,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.53 => ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
% 0.30/0.53 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % subset_psubset_trans
% 0.30/0.53 thf(fact_2408_subset__iff__psubset__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.53 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.30/0.53 | ( A7 = B6 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % subset_iff_psubset_eq
% 0.30/0.53 thf(fact_2409_even__set__bit__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [M: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 & ( M
% 0.30/0.53 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_set_bit_iff
% 0.30/0.53 thf(fact_2410_even__flip__bit__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [M: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 != ( M
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_flip_bit_iff
% 0.30/0.53 thf(fact_2411_even__unset__bit__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [M: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 | ( M
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_unset_bit_iff
% 0.30/0.53 thf(fact_2412_signed__take__bit__rec,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 0.30/0.53 = ( ^ [N3: nat,A5: A] :
% 0.30/0.53 ( if @ A
% 0.30/0.53 @ ( N3
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_rec
% 0.30/0.53 thf(fact_2413_dbl__simps_I4_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_simps(4)
% 0.30/0.53 thf(fact_2414_dbl__simps_I3_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_simps(3)
% 0.30/0.53 thf(fact_2415_concat__bit__Suc,axiom,
% 0.30/0.53 ! [N: nat,K: int,L: int] :
% 0.30/0.53 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.30/0.53 = ( 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 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % concat_bit_Suc
% 0.30/0.53 thf(fact_2416_set__decode__Suc,axiom,
% 0.30/0.53 ! [N: nat,X: nat] :
% 0.30/0.53 ( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 0.30/0.53 = ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_decode_Suc
% 0.30/0.53 thf(fact_2417_signed__take__bit__Suc,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc
% 0.30/0.53 thf(fact_2418_Compl__eq__Compl__iff,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.53 ( ( ( uminus_uminus @ ( set @ A ) @ A6 )
% 0.30/0.53 = ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 0.30/0.53 = ( A6 = B7 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Compl_eq_Compl_iff
% 0.30/0.53 thf(fact_2419_Compl__iff,axiom,
% 0.30/0.53 ! [A: $tType,C2: A,A6: set @ A] :
% 0.30/0.53 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.53 = ( ~ ( member @ A @ C2 @ A6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Compl_iff
% 0.30/0.53 thf(fact_2420_ComplI,axiom,
% 0.30/0.53 ! [A: $tType,C2: A,A6: set @ A] :
% 0.30/0.53 ( ~ ( member @ A @ C2 @ A6 )
% 0.30/0.53 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ComplI
% 0.30/0.53 thf(fact_2421_signed__take__bit__of__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_of_0
% 0.30/0.53 thf(fact_2422_concat__bit__0,axiom,
% 0.30/0.53 ! [K: int,L: int] :
% 0.30/0.53 ( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
% 0.30/0.53 = L ) ).
% 0.30/0.53
% 0.30/0.53 % concat_bit_0
% 0.30/0.53 thf(fact_2423_dbl__simps_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_simps(2)
% 0.30/0.53 thf(fact_2424_signed__take__bit__of__minus__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_of_minus_1
% 0.30/0.53 thf(fact_2425_signed__take__bit__Suc__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc_1
% 0.30/0.53 thf(fact_2426_signed__take__bit__numeral__of__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_numeral_of_1
% 0.30/0.53 thf(fact_2427_concat__bit__nonnegative__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int,L: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% 0.30/0.53
% 0.30/0.53 % concat_bit_nonnegative_iff
% 0.30/0.53 thf(fact_2428_concat__bit__negative__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int,L: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % concat_bit_negative_iff
% 0.30/0.53 thf(fact_2429_dbl__simps_I5_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_simps(5)
% 0.30/0.53 thf(fact_2430_dbl__simps_I1_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_simps(1)
% 0.30/0.53 thf(fact_2431_signed__take__bit__Suc__bit0,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc_bit0
% 0.30/0.53 thf(fact_2432_signed__take__bit__Suc__minus__bit0,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc_minus_bit0
% 0.30/0.53 thf(fact_2433_set__decode__0,axiom,
% 0.30/0.53 ! [X: nat] :
% 0.30/0.53 ( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
% 0.30/0.53 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % set_decode_0
% 0.30/0.53 thf(fact_2434_signed__take__bit__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.53 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_0
% 0.30/0.53 thf(fact_2435_double__complement,axiom,
% 0.30/0.53 ! [A: $tType,A6: set @ A] :
% 0.30/0.53 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.53 = A6 ) ).
% 0.30/0.53
% 0.30/0.53 % double_complement
% 0.30/0.53 thf(fact_2436_ComplD,axiom,
% 0.30/0.53 ! [A: $tType,C2: A,A6: set @ A] :
% 0.30/0.53 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.53 => ~ ( member @ A @ C2 @ A6 ) ) ).
% 0.30/0.53
% 0.30/0.53 % ComplD
% 0.30/0.53 thf(fact_2437_signed__take__bit__mult,axiom,
% 0.30/0.53 ! [N: nat,K: int,L: int] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
% 0.30/0.53 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_mult
% 0.30/0.53 thf(fact_2438_signed__take__bit__minus,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
% 0.30/0.53 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_minus
% 0.30/0.53 thf(fact_2439_subset__decode__imp__le,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.30/0.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % subset_decode_imp_le
% 0.30/0.53 thf(fact_2440_dbl__def,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl @ A )
% 0.30/0.53 = ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_def
% 0.30/0.53 thf(fact_2441_signed__take__bit__int__less__exp,axiom,
% 0.30/0.53 ! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_less_exp
% 0.30/0.53 thf(fact_2442_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_greater_eq_self_iff
% 0.30/0.53 thf(fact_2443_signed__take__bit__int__less__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
% 0.30/0.53 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_less_self_iff
% 0.30/0.53 thf(fact_2444_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.30/0.53 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_greater_eq_minus_exp
% 0.30/0.53 thf(fact_2445_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
% 0.30/0.53 = ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_less_eq_self_iff
% 0.30/0.53 thf(fact_2446_signed__take__bit__int__greater__self__iff,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_greater_self_iff
% 0.30/0.53 thf(fact_2447_signed__take__bit__int__less__eq,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_less_eq
% 0.30/0.53 thf(fact_2448_signed__take__bit__int__eq__self,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
% 0.30/0.53 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
% 0.30/0.53 = K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_eq_self
% 0.30/0.53 thf(fact_2449_signed__take__bit__int__eq__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
% 0.30/0.53 = K )
% 0.30/0.53 = ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
% 0.30/0.53 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_eq_self_iff
% 0.30/0.53 thf(fact_2450_signed__take__bit__int__greater__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_int_greater_eq
% 0.30/0.53 thf(fact_2451_signed__take__bit__Suc__minus__bit1,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc_minus_bit1
% 0.30/0.53 thf(fact_2452_fact__code,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_char_0 @ A )
% 0.30/0.53 => ( ( semiring_char_0_fact @ A )
% 0.30/0.53 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % fact_code
% 0.30/0.53 thf(fact_2453_take__bit__rec,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A )
% 0.30/0.53 = ( ^ [N3: nat,A5: A] :
% 0.30/0.53 ( if @ A
% 0.30/0.53 @ ( N3
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 @ ( zero_zero @ A )
% 0.30/0.53 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_rec
% 0.30/0.53 thf(fact_2454_num_Osize__gen_I2_J,axiom,
% 0.30/0.53 ! [X22: num] :
% 0.30/0.53 ( ( size_num @ ( bit0 @ X22 ) )
% 0.30/0.53 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % num.size_gen(2)
% 0.30/0.53 thf(fact_2455_and__int__unfold,axiom,
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int )
% 0.30/0.53 = ( ^ [K3: int,L3: int] :
% 0.30/0.53 ( if @ int
% 0.30/0.53 @ ( ( K3
% 0.30/0.53 = ( zero_zero @ int ) )
% 0.30/0.53 | ( L3
% 0.30/0.53 = ( zero_zero @ int ) ) )
% 0.30/0.53 @ ( zero_zero @ int )
% 0.30/0.53 @ ( if @ int
% 0.30/0.53 @ ( K3
% 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 @ L3
% 0.30/0.53 @ ( if @ int
% 0.30/0.53 @ ( L3
% 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 @ K3
% 0.30/0.53 @ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_int_unfold
% 0.30/0.53 thf(fact_2456_xor__Suc__0__eq,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.53 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_Suc_0_eq
% 0.30/0.53 thf(fact_2457_semiring__norm_I90_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ( bit1 @ M )
% 0.30/0.53 = ( bit1 @ N ) )
% 0.30/0.53 = ( M = N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(90)
% 0.30/0.53 thf(fact_2458_verit__eq__simplify_I9_J,axiom,
% 0.30/0.53 ! [X32: num,Y32: num] :
% 0.30/0.53 ( ( ( bit1 @ X32 )
% 0.30/0.53 = ( bit1 @ Y32 ) )
% 0.30/0.53 = ( X32 = Y32 ) ) ).
% 0.30/0.53
% 0.30/0.53 % verit_eq_simplify(9)
% 0.30/0.53 thf(fact_2459_take__bit__of__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_0
% 0.30/0.53 thf(fact_2460_and__zero__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_zero_eq
% 0.30/0.53 thf(fact_2461_zero__and__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_and_eq
% 0.30/0.53 thf(fact_2462_bit_Oconj__zero__left,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bit.conj_zero_left
% 0.30/0.53 thf(fact_2463_bit_Oconj__zero__right,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bit.conj_zero_right
% 0.30/0.53 thf(fact_2464_semiring__norm_I88_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( bit0 @ M )
% 0.30/0.53 != ( bit1 @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(88)
% 0.30/0.53 thf(fact_2465_semiring__norm_I89_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( bit1 @ M )
% 0.30/0.53 != ( bit0 @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(89)
% 0.30/0.53 thf(fact_2466_semiring__norm_I84_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( one2
% 0.30/0.53 != ( bit1 @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(84)
% 0.30/0.53 thf(fact_2467_semiring__norm_I86_J,axiom,
% 0.30/0.53 ! [M: num] :
% 0.30/0.53 ( ( bit1 @ M )
% 0.30/0.53 != one2 ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(86)
% 0.30/0.53 thf(fact_2468_bit_Oxor__self,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % bit.xor_self
% 0.30/0.53 thf(fact_2469_xor__self__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_self_eq
% 0.30/0.53 thf(fact_2470_xor_Oleft__neutral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.53 = A3 ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor.left_neutral
% 0.30/0.53 thf(fact_2471_xor_Oright__neutral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.53 = A3 ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor.right_neutral
% 0.30/0.53 thf(fact_2472_concat__bit__of__zero__2,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % concat_bit_of_zero_2
% 0.30/0.53 thf(fact_2473_semiring__norm_I73_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(73)
% 0.30/0.53 thf(fact_2474_semiring__norm_I80_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( ord_less @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(80)
% 0.30/0.53 thf(fact_2475_take__bit__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_0
% 0.30/0.53 thf(fact_2476_take__bit__Suc__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_1
% 0.30/0.53 thf(fact_2477_bit_Oconj__one__right,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [X: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = X ) ) ).
% 0.30/0.53
% 0.30/0.53 % bit.conj_one_right
% 0.30/0.53 thf(fact_2478_and_Oright__neutral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = A3 ) ) ).
% 0.30/0.53
% 0.30/0.53 % and.right_neutral
% 0.30/0.53 thf(fact_2479_and_Oleft__neutral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
% 0.30/0.53 = A3 ) ) ).
% 0.30/0.53
% 0.30/0.53 % and.left_neutral
% 0.30/0.53 thf(fact_2480_take__bit__numeral__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [L: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_1
% 0.30/0.53 thf(fact_2481_of__nat__nat__take__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [N: nat,K: int] :
% 0.30/0.53 ( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
% 0.30/0.53 = ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_nat_nat_take_bit_eq
% 0.30/0.53 thf(fact_2482_and__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [K: int,L: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
% 0.30/0.53 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nonnegative_int_iff
% 0.30/0.53 thf(fact_2483_and__negative__int__iff,axiom,
% 0.30/0.53 ! [K: int,L: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.53 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_negative_int_iff
% 0.30/0.53 thf(fact_2484_semiring__norm_I9_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.53 = ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(9)
% 0.30/0.53 thf(fact_2485_semiring__norm_I7_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(7)
% 0.30/0.53 thf(fact_2486_semiring__norm_I14_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(14)
% 0.30/0.53 thf(fact_2487_semiring__norm_I15_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.53 = ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(15)
% 0.30/0.53 thf(fact_2488_semiring__norm_I81_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.53 = ( ord_less @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(81)
% 0.30/0.53 thf(fact_2489_semiring__norm_I72_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(72)
% 0.30/0.53 thf(fact_2490_semiring__norm_I77_J,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(77)
% 0.30/0.53 thf(fact_2491_semiring__norm_I70_J,axiom,
% 0.30/0.53 ! [M: num] :
% 0.30/0.53 ~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(70)
% 0.30/0.53 thf(fact_2492_dbl__inc__simps_I5_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_inc_simps(5)
% 0.30/0.53 thf(fact_2493_take__bit__of__1__eq__0__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 = ( N
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_1_eq_0_iff
% 0.30/0.53 thf(fact_2494_and__numerals_I8_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(8)
% 0.30/0.53 thf(fact_2495_and__numerals_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( one_one @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(2)
% 0.30/0.53 thf(fact_2496_semiring__norm_I10_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(10)
% 0.30/0.53 thf(fact_2497_semiring__norm_I8_J,axiom,
% 0.30/0.53 ! [M: num] :
% 0.30/0.53 ( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
% 0.30/0.53 = ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(8)
% 0.30/0.53 thf(fact_2498_semiring__norm_I5_J,axiom,
% 0.30/0.53 ! [M: num] :
% 0.30/0.53 ( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
% 0.30/0.53 = ( bit1 @ M ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(5)
% 0.30/0.53 thf(fact_2499_semiring__norm_I4_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
% 0.30/0.53 = ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(4)
% 0.30/0.53 thf(fact_2500_semiring__norm_I3_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
% 0.30/0.53 = ( bit1 @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(3)
% 0.30/0.53 thf(fact_2501_take__bit__of__Suc__0,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_Suc_0
% 0.30/0.53 thf(fact_2502_semiring__norm_I16_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(16)
% 0.30/0.53 thf(fact_2503_semiring__norm_I74_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.53 = ( ord_less @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(74)
% 0.30/0.53 thf(fact_2504_semiring__norm_I79_J,axiom,
% 0.30/0.53 ! [M: num,N: num] :
% 0.30/0.53 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.53 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % semiring_norm(79)
% 0.30/0.53 thf(fact_2505_and__numerals_I1_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(1)
% 0.30/0.53 thf(fact_2506_and__numerals_I5_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(5)
% 0.30/0.53 thf(fact_2507_and__numerals_I3_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(3)
% 0.30/0.53 thf(fact_2508_xor__numerals_I3_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(3)
% 0.30/0.53 thf(fact_2509_xor__numerals_I8_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(8)
% 0.30/0.53 thf(fact_2510_xor__numerals_I5_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(5)
% 0.30/0.53 thf(fact_2511_xor__numerals_I2_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(2)
% 0.30/0.53 thf(fact_2512_xor__numerals_I1_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(1)
% 0.30/0.53 thf(fact_2513_take__bit__of__1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 0.30/0.53 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_1
% 0.30/0.53 thf(fact_2514_and__minus__numerals_I2_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.53 = ( one_one @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_minus_numerals(2)
% 0.30/0.53 thf(fact_2515_and__minus__numerals_I6_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
% 0.30/0.53 = ( one_one @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_minus_numerals(6)
% 0.30/0.53 thf(fact_2516_dbl__inc__simps_I3_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
% 0.30/0.53 = ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_inc_simps(3)
% 0.30/0.53 thf(fact_2517_even__take__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 0.30/0.53 = ( ( N
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % even_take_bit_eq
% 0.30/0.53 thf(fact_2518_and__numerals_I4_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(4)
% 0.30/0.53 thf(fact_2519_and__numerals_I6_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(6)
% 0.30/0.53 thf(fact_2520_xor__numerals_I7_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(7)
% 0.30/0.53 thf(fact_2521_mod__Suc__eq__mod__add3,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.53 = ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_Suc_eq_mod_add3
% 0.30/0.53 thf(fact_2522_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.30/0.53 ! [M: nat,V: num] :
% 0.30/0.53 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 0.30/0.53 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_eq_add3_mod_numeral
% 0.30/0.53 thf(fact_2523_div__Suc__eq__div__add3,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.53 = ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_Suc_eq_div_add3
% 0.30/0.53 thf(fact_2524_Suc__div__eq__add3__div__numeral,axiom,
% 0.30/0.53 ! [M: nat,V: num] :
% 0.30/0.53 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 0.30/0.53 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_div_eq_add3_div_numeral
% 0.30/0.53 thf(fact_2525_and__minus__numerals_I5_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
% 0.30/0.53 = ( zero_zero @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_minus_numerals(5)
% 0.30/0.53 thf(fact_2526_and__minus__numerals_I1_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.53 = ( zero_zero @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_minus_numerals(1)
% 0.30/0.53 thf(fact_2527_xor__nat__numerals_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_numerals(4)
% 0.30/0.53 thf(fact_2528_xor__nat__numerals_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_numerals(3)
% 0.30/0.53 thf(fact_2529_xor__nat__numerals_I2_J,axiom,
% 0.30/0.53 ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_numerals(2)
% 0.30/0.53 thf(fact_2530_xor__nat__numerals_I1_J,axiom,
% 0.30/0.53 ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_numerals(1)
% 0.30/0.53 thf(fact_2531_dbl__dec__simps_I4_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( neg_numeral @ A )
% 0.30/0.53 => ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % dbl_dec_simps(4)
% 0.30/0.53 thf(fact_2532_take__bit__Suc__0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
% 0.30/0.53 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_0
% 0.30/0.53 thf(fact_2533_and__numerals_I7_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_numerals(7)
% 0.30/0.53 thf(fact_2534_xor__numerals_I6_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(6)
% 0.30/0.53 thf(fact_2535_xor__numerals_I4_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [X: num,Y3: num] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_numerals(4)
% 0.30/0.53 thf(fact_2536_take__bit__of__exp,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_exp
% 0.30/0.53 thf(fact_2537_take__bit__of__2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_2
% 0.30/0.53 thf(fact_2538_zmod__numeral__Bit1,axiom,
% 0.30/0.53 ! [V: num,W2: num] :
% 0.30/0.53 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
% 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zmod_numeral_Bit1
% 0.30/0.53 thf(fact_2539_signed__take__bit__Suc__bit1,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_Suc_bit1
% 0.30/0.53 thf(fact_2540_take__bit__of__nat,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,M: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 0.30/0.53 = ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_of_nat
% 0.30/0.53 thf(fact_2541_of__nat__xor__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
% 0.30/0.53 = ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_nat_xor_eq
% 0.30/0.53 thf(fact_2542_of__nat__and__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [M: nat,N: nat] :
% 0.30/0.53 ( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
% 0.30/0.53 = ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % of_nat_and_eq
% 0.30/0.53 thf(fact_2543_take__bit__tightened,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A,B2: A,M: nat] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
% 0.30/0.53 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_tightened
% 0.30/0.53 thf(fact_2544_take__bit__tightened__less__eq__nat,axiom,
% 0.30/0.53 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.53 => ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q4 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_tightened_less_eq_nat
% 0.30/0.53 thf(fact_2545_take__bit__nat__less__eq__self,axiom,
% 0.30/0.53 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_less_eq_self
% 0.30/0.53 thf(fact_2546_verit__eq__simplify_I14_J,axiom,
% 0.30/0.53 ! [X22: num,X32: num] :
% 0.30/0.53 ( ( bit0 @ X22 )
% 0.30/0.53 != ( bit1 @ X32 ) ) ).
% 0.30/0.53
% 0.30/0.53 % verit_eq_simplify(14)
% 0.30/0.53 thf(fact_2547_verit__eq__simplify_I12_J,axiom,
% 0.30/0.53 ! [X32: num] :
% 0.30/0.53 ( one2
% 0.30/0.53 != ( bit1 @ X32 ) ) ).
% 0.30/0.53
% 0.30/0.53 % verit_eq_simplify(12)
% 0.30/0.53 thf(fact_2548_take__bit__mult,axiom,
% 0.30/0.53 ! [N: nat,K: int,L: int] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_mult
% 0.30/0.53 thf(fact_2549_take__bit__minus,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_minus
% 0.30/0.53 thf(fact_2550_take__bit__nat__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
% 0.30/0.53 = ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_eq
% 0.30/0.53 thf(fact_2551_nat__take__bit__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 => ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_take_bit_eq
% 0.30/0.53 thf(fact_2552_and__eq__minus__1__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [A3: A,B2: A] :
% 0.30/0.53 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( ( A3
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 & ( B2
% 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_eq_minus_1_iff
% 0.30/0.53 thf(fact_2553_num_Oexhaust,axiom,
% 0.30/0.53 ! [Y3: num] :
% 0.30/0.53 ( ( Y3 != one2 )
% 0.30/0.53 => ( ! [X23: num] :
% 0.30/0.53 ( Y3
% 0.30/0.53 != ( bit0 @ X23 ) )
% 0.30/0.53 => ~ ! [X33: num] :
% 0.30/0.53 ( Y3
% 0.30/0.53 != ( bit1 @ X33 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % num.exhaust
% 0.30/0.53 thf(fact_2554_AND__upper2_H,axiom,
% 0.30/0.53 ! [Y3: int,Z3: int,X: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ( ord_less_eq @ int @ Y3 @ Z3 )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper2'
% 0.30/0.53 thf(fact_2555_AND__upper1_H,axiom,
% 0.30/0.53 ! [Y3: int,Z3: int,Ya: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ( ord_less_eq @ int @ Y3 @ Z3 )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper1'
% 0.30/0.53 thf(fact_2556_AND__upper2,axiom,
% 0.30/0.53 ! [Y3: int,X: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Y3 ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper2
% 0.30/0.53 thf(fact_2557_AND__upper1,axiom,
% 0.30/0.53 ! [X: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper1
% 0.30/0.53 thf(fact_2558_AND__lower,axiom,
% 0.30/0.53 ! [X: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.53 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_lower
% 0.30/0.53 thf(fact_2559_take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_less_eq_self_iff
% 0.30/0.53 thf(fact_2560_take__bit__nonnegative,axiom,
% 0.30/0.53 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nonnegative
% 0.30/0.53 thf(fact_2561_not__take__bit__negative,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_take_bit_negative
% 0.30/0.53 thf(fact_2562_take__bit__int__greater__self__iff,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_greater_self_iff
% 0.30/0.53 thf(fact_2563_take__bit__tightened__less__eq__int,axiom,
% 0.30/0.53 ! [M: nat,N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_tightened_less_eq_int
% 0.30/0.53 thf(fact_2564_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A,B2: A] :
% 0.30/0.53 ( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
% 0.30/0.53 = ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
% 0.30/0.53 = ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_eq_iff_take_bit_eq
% 0.30/0.53 thf(fact_2565_signed__take__bit__take__bit,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [M: nat,N: nat,A3: A] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 0.30/0.53 = ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_take_bit
% 0.30/0.53 thf(fact_2566_num_Osize__gen_I3_J,axiom,
% 0.30/0.53 ! [X32: num] :
% 0.30/0.53 ( ( size_num @ ( bit1 @ X32 ) )
% 0.30/0.53 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % num.size_gen(3)
% 0.30/0.53 thf(fact_2567_take__bit__unset__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,M: nat,A3: A] :
% 0.30/0.53 ( ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_unset_bit_eq
% 0.30/0.53 thf(fact_2568_take__bit__set__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,M: nat,A3: A] :
% 0.30/0.53 ( ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_set_bit_eq
% 0.30/0.53 thf(fact_2569_take__bit__flip__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,M: nat,A3: A] :
% 0.30/0.53 ( ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 0.30/0.53 = ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_flip_bit_eq
% 0.30/0.53 thf(fact_2570_inc_Osimps_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( inc @ ( bit1 @ X ) )
% 0.30/0.53 = ( bit0 @ ( inc @ X ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % inc.simps(3)
% 0.30/0.53 thf(fact_2571_inc_Osimps_I2_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( inc @ ( bit0 @ X ) )
% 0.30/0.53 = ( bit1 @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % inc.simps(2)
% 0.30/0.53 thf(fact_2572_numeral__Bit1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( numeral @ A )
% 0.30/0.53 => ! [N: num] :
% 0.30/0.53 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 0.30/0.53 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_Bit1
% 0.30/0.53 thf(fact_2573_take__bit__signed__take__bit,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [M: nat,N: nat,A3: A] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
% 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_signed_take_bit
% 0.30/0.53 thf(fact_2574_and__less__eq,axiom,
% 0.30/0.53 ! [L: int,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_less_eq
% 0.30/0.53 thf(fact_2575_AND__upper1_H_H,axiom,
% 0.30/0.53 ! [Y3: int,Z3: int,Ya: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ( ord_less @ int @ Y3 @ Z3 )
% 0.30/0.53 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper1''
% 0.30/0.53 thf(fact_2576_AND__upper2_H_H,axiom,
% 0.30/0.53 ! [Y3: int,Z3: int,X: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ( ord_less @ int @ Y3 @ Z3 )
% 0.30/0.53 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % AND_upper2''
% 0.30/0.53 thf(fact_2577_eval__nat__numeral_I3_J,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ( ( numeral_numeral @ nat @ ( bit1 @ N ) )
% 0.30/0.53 = ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eval_nat_numeral(3)
% 0.30/0.53 thf(fact_2578_power__minus__Bit1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( ring_1 @ A )
% 0.30/0.53 => ! [X: A,K: num] :
% 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
% 0.30/0.53 = ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_minus_Bit1
% 0.30/0.53 thf(fact_2579_take__bit__decr__eq,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.53 != ( zero_zero @ int ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
% 0.30/0.53 = ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_decr_eq
% 0.30/0.53 thf(fact_2580_take__bit__Suc__bit1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 0.30/0.53 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_bit1
% 0.30/0.53 thf(fact_2581_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [N: num,Q4: num] :
% 0.30/0.53 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 0.30/0.53 != ( zero_zero @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(3)
% 0.30/0.53 thf(fact_2582_numeral__Bit1__div__2,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.53 => ! [N: num] :
% 0.30/0.53 ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_Bit1_div_2
% 0.30/0.53 thf(fact_2583_power3__eq__cube,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( monoid_mult @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.30/0.53 = ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power3_eq_cube
% 0.30/0.53 thf(fact_2584_numeral__3__eq__3,axiom,
% 0.30/0.53 ( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
% 0.30/0.53 = ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_3_eq_3
% 0.30/0.53 thf(fact_2585_Suc3__eq__add__3,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.30/0.53 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc3_eq_add_3
% 0.30/0.53 thf(fact_2586_take__bit__Suc__minus__bit1,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_minus_bit1
% 0.30/0.53 thf(fact_2587_take__bit__Suc__bit0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 0.30/0.53 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_bit0
% 0.30/0.53 thf(fact_2588_and__one__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.53 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_one_eq
% 0.30/0.53 thf(fact_2589_one__and__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.53 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_and_eq
% 0.30/0.53 thf(fact_2590_take__bit__nat__eq__self,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
% 0.30/0.53 = M ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_eq_self
% 0.30/0.53 thf(fact_2591_take__bit__nat__less__exp,axiom,
% 0.30/0.53 ! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_less_exp
% 0.30/0.53 thf(fact_2592_take__bit__nat__eq__self__iff,axiom,
% 0.30/0.53 ! [N: nat,M: nat] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
% 0.30/0.53 = M )
% 0.30/0.53 = ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_eq_self_iff
% 0.30/0.53 thf(fact_2593_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [Q4: num,N: num] :
% 0.30/0.53 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 0.30/0.53 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.53 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(7)
% 0.30/0.53 thf(fact_2594_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.53 => ! [M: num,Q4: num] :
% 0.30/0.53 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 0.30/0.53 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.53 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) )
% 0.30/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(11)
% 0.30/0.53 thf(fact_2595_take__bit__int__less__exp,axiom,
% 0.30/0.53 ! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_less_exp
% 0.30/0.53 thf(fact_2596_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( set_fo6178422350223883121st_nat @ A )
% 0.30/0.53 = ( ^ [F2: nat > A > A,A5: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % fold_atLeastAtMost_nat.simps
% 0.30/0.53 thf(fact_2597_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.30/0.53 ! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
% 0.30/0.53 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
% 0.30/0.53 = Y3 )
% 0.30/0.53 => ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
% 0.30/0.53 => ( Y3 = Xc ) )
% 0.30/0.53 & ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
% 0.30/0.53 => ( Y3
% 0.30/0.53 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % fold_atLeastAtMost_nat.elims
% 0.30/0.53 thf(fact_2598_Suc__mod__eq__add3__mod,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.53 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_eq_add3_mod
% 0.30/0.53 thf(fact_2599_Suc__div__eq__add3__div,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.53 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_div_eq_add3_div
% 0.30/0.53 thf(fact_2600_num_Osize__gen_I1_J,axiom,
% 0.30/0.53 ( ( size_num @ one2 )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % num.size_gen(1)
% 0.30/0.53 thf(fact_2601_take__bit__eq__0__iff,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.30/0.53 = ( zero_zero @ A ) )
% 0.30/0.53 = ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_eq_0_iff
% 0.30/0.53 thf(fact_2602_take__bit__nat__less__self__iff,axiom,
% 0.30/0.53 ! [N: nat,M: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
% 0.30/0.53 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_nat_less_self_iff
% 0.30/0.53 thf(fact_2603_take__bit__Suc__minus__bit0,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_minus_bit0
% 0.30/0.53 thf(fact_2604_take__bit__int__less__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
% 0.30/0.53 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_less_self_iff
% 0.30/0.53 thf(fact_2605_take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 0.30/0.53 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_greater_eq_self_iff
% 0.30/0.53 thf(fact_2606_take__bit__int__eq__self,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.53 = K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_eq_self
% 0.30/0.53 thf(fact_2607_take__bit__int__eq__self__iff,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.53 = K )
% 0.30/0.53 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_eq_self_iff
% 0.30/0.53 thf(fact_2608_signed__take__bit__eq__take__bit__shift,axiom,
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int )
% 0.30/0.53 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_eq_take_bit_shift
% 0.30/0.53 thf(fact_2609_take__bit__incr__eq,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.53 != ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.30/0.53 = ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_incr_eq
% 0.30/0.53 thf(fact_2610_take__bit__Suc__minus__1__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [N: nat] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc_minus_1_eq
% 0.30/0.53 thf(fact_2611_take__bit__Suc,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: nat,A3: A] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.53 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_Suc
% 0.30/0.53 thf(fact_2612_take__bit__numeral__minus__1__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.53 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_minus_1_eq
% 0.30/0.53 thf(fact_2613_take__bit__int__less__eq,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
% 0.30/0.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_less_eq
% 0.30/0.53 thf(fact_2614_take__bit__int__greater__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.53 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_int_greater_eq
% 0.30/0.53 thf(fact_2615_xor__nat__unfold,axiom,
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.53 ( if @ nat
% 0.30/0.53 @ ( M2
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 @ N3
% 0.30/0.53 @ ( if @ nat
% 0.30/0.53 @ ( N3
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 @ M2
% 0.30/0.53 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( 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 @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_unfold
% 0.30/0.53 thf(fact_2616_and__int__rec,axiom,
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int )
% 0.30/0.53 = ( ^ [K3: int,L3: int] :
% 0.30/0.53 ( plus_plus @ int
% 0.30/0.53 @ ( zero_neq_one_of_bool @ int
% 0.30/0.53 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 0.30/0.53 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.30/0.53 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_int_rec
% 0.30/0.53 thf(fact_2617_xor__nat__rec,axiom,
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.53 ( plus_plus @ nat
% 0.30/0.53 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.53 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
% 0.30/0.53 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
% 0.30/0.53 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_rec
% 0.30/0.53 thf(fact_2618_xor__one__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.53 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 0.30/0.53 @ ( zero_neq_one_of_bool @ A
% 0.30/0.53 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_one_eq
% 0.30/0.53 thf(fact_2619_one__xor__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.53 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 0.30/0.53 @ ( zero_neq_one_of_bool @ A
% 0.30/0.53 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_xor_eq
% 0.30/0.53 thf(fact_2620_stable__imp__take__bit__eq,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [A3: A,N: nat] :
% 0.30/0.53 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = A3 )
% 0.30/0.53 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.30/0.53 = ( zero_zero @ A ) ) )
% 0.30/0.53 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.30/0.53 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % stable_imp_take_bit_eq
% 0.30/0.53 thf(fact_2621_take__bit__minus__small__eq,axiom,
% 0.30/0.53 ! [K: int,N: nat] :
% 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 => ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
% 0.30/0.53 = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_minus_small_eq
% 0.30/0.53 thf(fact_2622_odd__mod__4__div__2,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.30/0.53 => ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % odd_mod_4_div_2
% 0.30/0.53 thf(fact_2623_Suc__0__xor__eq,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.53 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.30/0.53 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.53 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_0_xor_eq
% 0.30/0.53 thf(fact_2624_signed__take__bit__numeral__minus__bit1,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_numeral_minus_bit1
% 0.30/0.53 thf(fact_2625_binomial__code,axiom,
% 0.30/0.53 ( binomial
% 0.30/0.53 = ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_code
% 0.30/0.53 thf(fact_2626_signed__take__bit__numeral__bit1,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_numeral_bit1
% 0.30/0.53 thf(fact_2627_take__bit__numeral__minus__bit1,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_minus_bit1
% 0.30/0.53 thf(fact_2628_take__bit__numeral__bit1,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 0.30/0.53 = ( 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 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_bit1
% 0.30/0.53 thf(fact_2629_signed__take__bit__numeral__minus__bit0,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_numeral_minus_bit0
% 0.30/0.53 thf(fact_2630_xor__nonnegative__int__iff,axiom,
% 0.30/0.53 ! [K: int,L: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
% 0.30/0.53 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nonnegative_int_iff
% 0.30/0.53 thf(fact_2631_xor__negative__int__iff,axiom,
% 0.30/0.53 ! [K: int,L: int] :
% 0.30/0.53 ( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.53 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.53 != ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_negative_int_iff
% 0.30/0.53 thf(fact_2632_binomial__Suc__n,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ ( suc @ N ) @ N )
% 0.30/0.53 = ( suc @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_Suc_n
% 0.30/0.53 thf(fact_2633_binomial__n__n,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ N @ N )
% 0.30/0.53 = ( one_one @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_n_n
% 0.30/0.53 thf(fact_2634_binomial__1,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = N ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_1
% 0.30/0.53 thf(fact_2635_binomial__0__Suc,axiom,
% 0.30/0.53 ! [K: nat] :
% 0.30/0.53 ( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_0_Suc
% 0.30/0.53 thf(fact_2636_binomial__eq__0__iff,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( ( binomial @ N @ K )
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 = ( ord_less @ nat @ N @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_eq_0_iff
% 0.30/0.53 thf(fact_2637_binomial__Suc__Suc,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.53 = ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_Suc_Suc
% 0.30/0.53 thf(fact_2638_pred__numeral__simps_I1_J,axiom,
% 0.30/0.53 ( ( pred_numeral @ one2 )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % pred_numeral_simps(1)
% 0.30/0.53 thf(fact_2639_binomial__n__0,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ N @ ( zero_zero @ nat ) )
% 0.30/0.53 = ( one_one @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_n_0
% 0.30/0.53 thf(fact_2640_eq__numeral__Suc,axiom,
% 0.30/0.53 ! [K: num,N: nat] :
% 0.30/0.53 ( ( ( numeral_numeral @ nat @ K )
% 0.30/0.53 = ( suc @ N ) )
% 0.30/0.53 = ( ( pred_numeral @ K )
% 0.30/0.53 = N ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_numeral_Suc
% 0.30/0.53 thf(fact_2641_Suc__eq__numeral,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( ( suc @ N )
% 0.30/0.53 = ( numeral_numeral @ nat @ K ) )
% 0.30/0.53 = ( N
% 0.30/0.53 = ( pred_numeral @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_eq_numeral
% 0.30/0.53 thf(fact_2642_pred__numeral__inc,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( pred_numeral @ ( inc @ K ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ K ) ) ).
% 0.30/0.53
% 0.30/0.53 % pred_numeral_inc
% 0.30/0.53 thf(fact_2643_zero__less__binomial__iff,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_binomial_iff
% 0.30/0.53 thf(fact_2644_less__Suc__numeral,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.53 = ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_Suc_numeral
% 0.30/0.53 thf(fact_2645_less__numeral__Suc,axiom,
% 0.30/0.53 ! [K: num,N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 0.30/0.53 = ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_numeral_Suc
% 0.30/0.53 thf(fact_2646_le__Suc__numeral,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_Suc_numeral
% 0.30/0.53 thf(fact_2647_le__numeral__Suc,axiom,
% 0.30/0.53 ! [K: num,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 0.30/0.53 = ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_numeral_Suc
% 0.30/0.53 thf(fact_2648_diff__numeral__Suc,axiom,
% 0.30/0.53 ! [K: num,N: nat] :
% 0.30/0.53 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 0.30/0.53 = ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_numeral_Suc
% 0.30/0.53 thf(fact_2649_diff__Suc__numeral,axiom,
% 0.30/0.53 ! [N: nat,K: num] :
% 0.30/0.53 ( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.53 = ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_Suc_numeral
% 0.30/0.53 thf(fact_2650_pred__numeral__simps_I3_J,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.30/0.53 = ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pred_numeral_simps(3)
% 0.30/0.53 thf(fact_2651_and__nat__numerals_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_numerals(3)
% 0.30/0.53 thf(fact_2652_and__nat__numerals_I1_J,axiom,
% 0.30/0.53 ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.53 = ( zero_zero @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_numerals(1)
% 0.30/0.53 thf(fact_2653_and__nat__numerals_I2_J,axiom,
% 0.30/0.53 ! [Y3: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.53 = ( one_one @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_numerals(2)
% 0.30/0.53 thf(fact_2654_and__nat__numerals_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( one_one @ nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_numerals(4)
% 0.30/0.53 thf(fact_2655_Suc__0__and__eq,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.53 = ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_0_and_eq
% 0.30/0.53 thf(fact_2656_and__Suc__0__eq,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.53 = ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_Suc_0_eq
% 0.30/0.53 thf(fact_2657_signed__take__bit__numeral__bit0,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_numeral_bit0
% 0.30/0.53 thf(fact_2658_choose__one,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ N @ ( one_one @ nat ) )
% 0.30/0.53 = N ) ).
% 0.30/0.53
% 0.30/0.53 % choose_one
% 0.30/0.53 thf(fact_2659_binomial__eq__0,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ N @ K )
% 0.30/0.53 => ( ( binomial @ N @ K )
% 0.30/0.53 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_eq_0
% 0.30/0.53 thf(fact_2660_Suc__times__binomial__eq,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_times_binomial_eq
% 0.30/0.53 thf(fact_2661_Suc__times__binomial,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.30/0.53 = ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_times_binomial
% 0.30/0.53 thf(fact_2662_XOR__lower,axiom,
% 0.30/0.53 ! [X: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.53 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % XOR_lower
% 0.30/0.53 thf(fact_2663_binomial__symmetric,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ( binomial @ N @ K )
% 0.30/0.53 = ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_symmetric
% 0.30/0.53 thf(fact_2664_choose__mult__lemma,axiom,
% 0.30/0.53 ! [M: nat,R: nat,K: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
% 0.30/0.53 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R ) @ M ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % choose_mult_lemma
% 0.30/0.53 thf(fact_2665_binomial__le__pow,axiom,
% 0.30/0.53 ! [R: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ R @ N )
% 0.30/0.53 => ( ord_less_eq @ nat @ ( binomial @ N @ R ) @ ( power_power @ nat @ N @ R ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_le_pow
% 0.30/0.53 thf(fact_2666_binomial__gbinomial,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( field_char_0 @ A )
% 0.30/0.53 => ! [N: nat,K: nat] :
% 0.30/0.53 ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_gbinomial
% 0.30/0.53 thf(fact_2667_numeral__eq__Suc,axiom,
% 0.30/0.53 ( ( numeral_numeral @ nat )
% 0.30/0.53 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % numeral_eq_Suc
% 0.30/0.53 thf(fact_2668_zero__less__binomial,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_binomial
% 0.30/0.53 thf(fact_2669_Suc__times__binomial__add,axiom,
% 0.30/0.53 ! [A3: nat,B2: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
% 0.30/0.53 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_times_binomial_add
% 0.30/0.53 thf(fact_2670_choose__mult,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.53 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.53 => ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.30/0.53 = ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % choose_mult
% 0.30/0.53 thf(fact_2671_binomial__absorb__comp,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_absorb_comp
% 0.30/0.53 thf(fact_2672_binomial__Suc__Suc__eq__times,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.53 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_Suc_Suc_eq_times
% 0.30/0.53 thf(fact_2673_and__nat__def,axiom,
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_def
% 0.30/0.53 thf(fact_2674_xor__nat__def,axiom,
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_nat_def
% 0.30/0.53 thf(fact_2675_pred__numeral__def,axiom,
% 0.30/0.53 ( pred_numeral
% 0.30/0.53 = ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pred_numeral_def
% 0.30/0.53 thf(fact_2676_binomial__absorption,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.30/0.53 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_absorption
% 0.30/0.53 thf(fact_2677_binomial__fact__lemma,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( semiring_char_0_fact @ nat @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_fact_lemma
% 0.30/0.53 thf(fact_2678_fact__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( semiring_char_0 @ A )
% 0.30/0.53 => ! [K: num] :
% 0.30/0.53 ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.53 = ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % fact_numeral
% 0.30/0.53 thf(fact_2679_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( linordered_field @ A )
% 0.30/0.53 => ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_ge_n_over_k_pow_k
% 0.30/0.53 thf(fact_2680_choose__reduce__nat,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.53 => ( ( binomial @ N @ K )
% 0.30/0.53 = ( 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 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % choose_reduce_nat
% 0.30/0.53 thf(fact_2681_times__binomial__minus1__eq,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.53 => ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % times_binomial_minus1_eq
% 0.30/0.53 thf(fact_2682_binomial__le__pow2,axiom,
% 0.30/0.53 ! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_le_pow2
% 0.30/0.53 thf(fact_2683_binomial__altdef__nat,axiom,
% 0.30/0.53 ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ( binomial @ N @ K )
% 0.30/0.53 = ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_altdef_nat
% 0.30/0.53 thf(fact_2684_binomial__addition__formula,axiom,
% 0.30/0.53 ! [N: nat,K: nat] :
% 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.53 => ( ( binomial @ N @ ( suc @ K ) )
% 0.30/0.53 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_addition_formula
% 0.30/0.53 thf(fact_2685_fact__binomial,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( field_char_0 @ A )
% 0.30/0.53 => ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
% 0.30/0.53 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % fact_binomial
% 0.30/0.53 thf(fact_2686_binomial__fact,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( field_char_0 @ A )
% 0.30/0.53 => ! [K: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.53 => ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 0.30/0.53 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % binomial_fact
% 0.30/0.53 thf(fact_2687_take__bit__numeral__bit0,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.53 => ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 0.30/0.53 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_bit0
% 0.30/0.53 thf(fact_2688_take__bit__numeral__minus__bit0,axiom,
% 0.30/0.53 ! [L: num,K: num] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.53 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_numeral_minus_bit0
% 0.30/0.53 thf(fact_2689_XOR__upper,axiom,
% 0.30/0.53 ! [X: int,N: nat,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.53 => ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.53 => ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % XOR_upper
% 0.30/0.53 thf(fact_2690_choose__two,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.53 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % choose_two
% 0.30/0.53 thf(fact_2691_and__nat__unfold,axiom,
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.53 ( if @ nat
% 0.30/0.53 @ ( ( M2
% 0.30/0.53 = ( zero_zero @ nat ) )
% 0.30/0.53 | ( N3
% 0.30/0.53 = ( zero_zero @ nat ) ) )
% 0.30/0.53 @ ( zero_zero @ nat )
% 0.30/0.53 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_unfold
% 0.30/0.53 thf(fact_2692_and__nat__rec,axiom,
% 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ nat )
% 0.30/0.53 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.53 ( plus_plus @ nat
% 0.30/0.53 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.53 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
% 0.30/0.53 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
% 0.30/0.53 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % and_nat_rec
% 0.30/0.53 thf(fact_2693_xor__int__rec,axiom,
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ int )
% 0.30/0.53 = ( ^ [K3: int,L3: int] :
% 0.30/0.53 ( plus_plus @ int
% 0.30/0.53 @ ( zero_neq_one_of_bool @ int
% 0.30/0.53 @ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
% 0.30/0.53 != ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 0.30/0.53 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_int_rec
% 0.30/0.53 thf(fact_2694_xor__int__unfold,axiom,
% 0.30/0.53 ( ( bit_se5824344971392196577ns_xor @ int )
% 0.30/0.53 = ( ^ [K3: int,L3: int] :
% 0.30/0.53 ( if @ int
% 0.30/0.53 @ ( K3
% 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 @ ( bit_ri4277139882892585799ns_not @ int @ L3 )
% 0.30/0.53 @ ( if @ int
% 0.30/0.53 @ ( L3
% 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.53 @ ( bit_ri4277139882892585799ns_not @ int @ K3 )
% 0.30/0.53 @ ( if @ int
% 0.30/0.53 @ ( K3
% 0.30/0.53 = ( zero_zero @ int ) )
% 0.30/0.53 @ L3
% 0.30/0.53 @ ( if @ int
% 0.30/0.53 @ ( L3
% 0.30/0.53 = ( zero_zero @ int ) )
% 0.30/0.53 @ K3
% 0.30/0.53 @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % xor_int_unfold
% 0.30/0.53 thf(fact_2695_signed__take__bit__eq__take__bit__minus,axiom,
% 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int )
% 0.30/0.53 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % signed_take_bit_eq_take_bit_minus
% 0.30/0.53 thf(fact_2696_mask__numeral,axiom,
% 0.30/0.53 ! [A: $tType] :
% 0.30/0.53 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.53 => ! [N: num] :
% 0.30/0.53 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.53 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mask_numeral
% 0.30/0.53 thf(fact_2697_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.53 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 0.30/0.53 = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % take_bit_eq_mask_iff_exp_dvd
% 0.30/0.53 thf(fact_2698_take__bit__Suc__from__most,axiom,
% 0.30/0.53 ! [N: nat,K: int] :
% 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
% 0.30/0.53 = ( 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 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % take_bit_Suc_from_most
% 0.30/0.54 thf(fact_2699_push__bit__numeral__minus__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_numeral_minus_1
% 0.30/0.54 thf(fact_2700_mask__nat__positive__iff,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 0.30/0.54 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_nat_positive_iff
% 0.30/0.54 thf(fact_2701_push__bit__nonnegative__int__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
% 0.30/0.54 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_nonnegative_int_iff
% 0.30/0.54 thf(fact_2702_push__bit__negative__int__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.54 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_negative_int_iff
% 0.30/0.54 thf(fact_2703_push__bit__of__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_of_0
% 0.30/0.54 thf(fact_2704_push__bit__eq__0__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( A3
% 0.30/0.54 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_eq_0_iff
% 0.30/0.54 thf(fact_2705_concat__bit__of__zero__1,axiom,
% 0.30/0.54 ! [N: nat,L: int] :
% 0.30/0.54 ( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% 0.30/0.54
% 0.30/0.54 % concat_bit_of_zero_1
% 0.30/0.54 thf(fact_2706_mask__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_0
% 0.30/0.54 thf(fact_2707_mask__eq__0__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( ( bit_se2239418461657761734s_mask @ A @ N )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_eq_0_iff
% 0.30/0.54 thf(fact_2708_bit_Oconj__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.conj_cancel_right
% 0.30/0.54 thf(fact_2709_bit_Oconj__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.conj_cancel_left
% 0.30/0.54 thf(fact_2710_bit_Ocompl__zero,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.compl_zero
% 0.30/0.54 thf(fact_2711_bit_Ocompl__one,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.compl_one
% 0.30/0.54 thf(fact_2712_bit__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [M: num,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_numeral_Bit0_Suc_iff
% 0.30/0.54 thf(fact_2713_push__bit__Suc__numeral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_Suc_numeral
% 0.30/0.54 thf(fact_2714_bit__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [M: num,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_numeral_Bit1_Suc_iff
% 0.30/0.54 thf(fact_2715_mask__Suc__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_Suc_0
% 0.30/0.54 thf(fact_2716_bit_Oxor__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.xor_cancel_right
% 0.30/0.54 thf(fact_2717_bit_Oxor__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.xor_cancel_left
% 0.30/0.54 thf(fact_2718_bit_Oxor__one__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.xor_one_right
% 0.30/0.54 thf(fact_2719_bit_Oxor__one__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.xor_one_left
% 0.30/0.54 thf(fact_2720_take__bit__minus__one__eq__mask,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % take_bit_minus_one_eq_mask
% 0.30/0.54 thf(fact_2721_not__negative__int__iff,axiom,
% 0.30/0.54 ! [K: int] :
% 0.30/0.54 ( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
% 0.30/0.54 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_negative_int_iff
% 0.30/0.54 thf(fact_2722_not__nonnegative__int__iff,axiom,
% 0.30/0.54 ! [K: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
% 0.30/0.54 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_nonnegative_int_iff
% 0.30/0.54 thf(fact_2723_minus__not__numeral__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_not_numeral_eq
% 0.30/0.54 thf(fact_2724_signed__take__bit__nonnegative__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % signed_take_bit_nonnegative_iff
% 0.30/0.54 thf(fact_2725_signed__take__bit__negative__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % signed_take_bit_negative_iff
% 0.30/0.54 thf(fact_2726_push__bit__Suc__minus__numeral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_Suc_minus_numeral
% 0.30/0.54 thf(fact_2727_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.54 ! [W2: num,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_minus_numeral_Bit0_Suc_iff
% 0.30/0.54 thf(fact_2728_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.54 ! [W2: num,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_minus_numeral_Bit1_Suc_iff
% 0.30/0.54 thf(fact_2729_push__bit__minus__one__eq__not__mask,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_minus_one_eq_not_mask
% 0.30/0.54 thf(fact_2730_not__one__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_one_eq
% 0.30/0.54 thf(fact_2731_bit__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.54 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_0
% 0.30/0.54 thf(fact_2732_push__bit__Suc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_Suc
% 0.30/0.54 thf(fact_2733_push__bit__of__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
% 0.30/0.54 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_of_1
% 0.30/0.54 thf(fact_2734_even__push__bit__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
% 0.30/0.54 = ( ( N
% 0.30/0.54 != ( zero_zero @ nat ) )
% 0.30/0.54 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % even_push_bit_iff
% 0.30/0.54 thf(fact_2735_push__bit__of__Suc__0,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_of_Suc_0
% 0.30/0.54 thf(fact_2736_push__bit__minus__numeral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [L: num,K: num] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_minus_numeral
% 0.30/0.54 thf(fact_2737_bit__minus__numeral__int_I1_J,axiom,
% 0.30/0.54 ! [W2: num,N: num] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_minus_numeral_int(1)
% 0.30/0.54 thf(fact_2738_bit__minus__numeral__int_I2_J,axiom,
% 0.30/0.54 ! [W2: num,N: num] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_minus_numeral_int(2)
% 0.30/0.54 thf(fact_2739_bit__mod__2__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
% 0.30/0.54 = ( ( N
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_mod_2_iff
% 0.30/0.54 thf(fact_2740_of__nat__mask__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 0.30/0.54 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_mask_eq
% 0.30/0.54 thf(fact_2741_bit__of__nat__iff__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [M: nat,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_of_nat_iff_bit
% 0.30/0.54 thf(fact_2742_push__bit__of__nat,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,M: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 0.30/0.54 = ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_of_nat
% 0.30/0.54 thf(fact_2743_of__nat__push__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [M: nat,N: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
% 0.30/0.54 = ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_push_bit
% 0.30/0.54 thf(fact_2744_bit__push__bit__iff__int,axiom,
% 0.30/0.54 ! [M: nat,K: int,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
% 0.30/0.54 = ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_push_bit_iff_int
% 0.30/0.54 thf(fact_2745_push__bit__minus,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_minus
% 0.30/0.54 thf(fact_2746_less__eq__mask,axiom,
% 0.30/0.54 ! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_eq_mask
% 0.30/0.54 thf(fact_2747_bit__minus__int__iff,axiom,
% 0.30/0.54 ! [K: int,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_minus_int_iff
% 0.30/0.54 thf(fact_2748_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.30/0.54 = ( ^ [A5: A,N3: nat] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
% 0.30/0.54 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_iff_and_push_bit_not_eq_0
% 0.30/0.54 thf(fact_2749_unset__bit__eq__and__not,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_se2638667681897837118et_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unset_bit_eq_and_not
% 0.30/0.54 thf(fact_2750_unset__bit__int__def,axiom,
% 0.30/0.54 ( ( bit_se2638667681897837118et_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unset_bit_int_def
% 0.30/0.54 thf(fact_2751_take__bit__not__mask__eq__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.54 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % take_bit_not_mask_eq_0
% 0.30/0.54 thf(fact_2752_not__bit__1__Suc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_bit_1_Suc
% 0.30/0.54 thf(fact_2753_bit__1__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
% 0.30/0.54 = ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_1_iff
% 0.30/0.54 thf(fact_2754_bit__numeral__simps_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_numeral_simps(1)
% 0.30/0.54 thf(fact_2755_bit__take__bit__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [M: nat,A3: A,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
% 0.30/0.54 = ( ( ord_less @ nat @ N @ M )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_take_bit_iff
% 0.30/0.54 thf(fact_2756_bit__of__bool__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [B2: $o,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N )
% 0.30/0.54 = ( B2
% 0.30/0.54 & ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_of_bool_iff
% 0.30/0.54 thf(fact_2757_bit__not__iff__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
% 0.30/0.54 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.54 != ( zero_zero @ A ) )
% 0.30/0.54 & ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_not_iff_eq
% 0.30/0.54 thf(fact_2758_mask__nonnegative__int,axiom,
% 0.30/0.54 ! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_nonnegative_int
% 0.30/0.54 thf(fact_2759_not__mask__negative__int,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_mask_negative_int
% 0.30/0.54 thf(fact_2760_flip__bit__nat__def,axiom,
% 0.30/0.54 ( ( bit_se8732182000553998342ip_bit @ nat )
% 0.30/0.54 = ( ^ [M2: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % flip_bit_nat_def
% 0.30/0.54 thf(fact_2761_flip__bit__int__def,axiom,
% 0.30/0.54 ( ( bit_se8732182000553998342ip_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % flip_bit_int_def
% 0.30/0.54 thf(fact_2762_minus__exp__eq__not__mask,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_exp_eq_not_mask
% 0.30/0.54 thf(fact_2763_minus__eq__not__plus__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( uminus_uminus @ A )
% 0.30/0.54 = ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_eq_not_plus_1
% 0.30/0.54 thf(fact_2764_minus__eq__not__minus__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( uminus_uminus @ A )
% 0.30/0.54 = ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_eq_not_minus_1
% 0.30/0.54 thf(fact_2765_not__eq__complement,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_ri4277139882892585799ns_not @ A )
% 0.30/0.54 = ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_eq_complement
% 0.30/0.54 thf(fact_2766_not__int__def,axiom,
% 0.30/0.54 ( ( bit_ri4277139882892585799ns_not @ int )
% 0.30/0.54 = ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_int_def
% 0.30/0.54 thf(fact_2767_bit__not__int__iff_H,axiom,
% 0.30/0.54 ! [K: int,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_not_int_iff'
% 0.30/0.54 thf(fact_2768_and__not__numerals_I1_J,axiom,
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( zero_zero @ int ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(1)
% 0.30/0.54 thf(fact_2769_minus__numeral__inc__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_numeral_inc_eq
% 0.30/0.54 thf(fact_2770_less__mask,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.54 => ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_mask
% 0.30/0.54 thf(fact_2771_flip__bit__eq__xor,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se8732182000553998342ip_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % flip_bit_eq_xor
% 0.30/0.54 thf(fact_2772_and__not__numerals_I4_J,axiom,
% 0.30/0.54 ! [M: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(4)
% 0.30/0.54 thf(fact_2773_and__not__numerals_I2_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( one_one @ int ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(2)
% 0.30/0.54 thf(fact_2774_bit__imp__take__bit__positive,axiom,
% 0.30/0.54 ! [N: nat,M: nat,K: int] :
% 0.30/0.54 ( ( ord_less @ nat @ N @ M )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
% 0.30/0.54 => ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_imp_take_bit_positive
% 0.30/0.54 thf(fact_2775_not__numeral__Bit0__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_Bit0_eq
% 0.30/0.54 thf(fact_2776_bit__concat__bit__iff,axiom,
% 0.30/0.54 ! [M: nat,K: int,L: int,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.30/0.54 = ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
% 0.30/0.54 | ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_concat_bit_iff
% 0.30/0.54 thf(fact_2777_push__bit__double,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.54 = ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_double
% 0.30/0.54 thf(fact_2778_signed__take__bit__eq__concat__bit,axiom,
% 0.30/0.54 ( ( bit_ri4674362597316999326ke_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % signed_take_bit_eq_concat_bit
% 0.30/0.54 thf(fact_2779_exp__eq__0__imp__not__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 => ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % exp_eq_0_imp_not_bit
% 0.30/0.54 thf(fact_2780_bit__Suc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_Suc
% 0.30/0.54 thf(fact_2781_take__bit__eq__mask__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.30/0.54 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 0.30/0.54 = ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.30/0.54 = ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % take_bit_eq_mask_iff
% 0.30/0.54 thf(fact_2782_int__bit__bound,axiom,
% 0.30/0.54 ! [K: int] :
% 0.30/0.54 ~ ! [N4: nat] :
% 0.30/0.54 ( ! [M3: nat] :
% 0.30/0.54 ( ( ord_less_eq @ nat @ N4 @ M3 )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
% 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) )
% 0.30/0.54 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % int_bit_bound
% 0.30/0.54 thf(fact_2783_and__not__numerals_I5_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(5)
% 0.30/0.54 thf(fact_2784_and__not__numerals_I7_J,axiom,
% 0.30/0.54 ! [M: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(7)
% 0.30/0.54 thf(fact_2785_and__not__numerals_I3_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( zero_zero @ int ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(3)
% 0.30/0.54 thf(fact_2786_push__bit__int__def,axiom,
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_int_def
% 0.30/0.54 thf(fact_2787_push__bit__nat__def,axiom,
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ nat )
% 0.30/0.54 = ( ^ [N3: nat,M2: nat] : ( times_times @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_nat_def
% 0.30/0.54 thf(fact_2788_push__bit__eq__mult,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se4730199178511100633sh_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_eq_mult
% 0.30/0.54 thf(fact_2789_and__exp__eq__0__iff__not__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_exp_eq_0_iff_not_bit
% 0.30/0.54 thf(fact_2790_Suc__mask__eq__exp,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 0.30/0.54 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % Suc_mask_eq_exp
% 0.30/0.54 thf(fact_2791_mask__nat__less__exp,axiom,
% 0.30/0.54 ! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_nat_less_exp
% 0.30/0.54 thf(fact_2792_and__not__numerals_I9_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(9)
% 0.30/0.54 thf(fact_2793_and__not__numerals_I6_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(6)
% 0.30/0.54 thf(fact_2794_push__bit__minus__one,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % push_bit_minus_one
% 0.30/0.54 thf(fact_2795_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) )
% 0.30/0.54 = ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_bit_operations_class.even_mask_iff
% 0.30/0.54 thf(fact_2796_even__bit__succ__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
% 0.30/0.54 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 0.30/0.54 | ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % even_bit_succ_iff
% 0.30/0.54 thf(fact_2797_odd__bit__iff__bit__pred,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 0.30/0.54 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
% 0.30/0.54 | ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % odd_bit_iff_bit_pred
% 0.30/0.54 thf(fact_2798_mask__half__int,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 0.30/0.54 = ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_half_int
% 0.30/0.54 thf(fact_2799_mask__int__def,axiom,
% 0.30/0.54 ( ( bit_se2239418461657761734s_mask @ int )
% 0.30/0.54 = ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_int_def
% 0.30/0.54 thf(fact_2800_mask__nat__def,axiom,
% 0.30/0.54 ( ( bit_se2239418461657761734s_mask @ nat )
% 0.30/0.54 = ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_nat_def
% 0.30/0.54 thf(fact_2801_bit__sum__mult__2__cases,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.54 ( ! [J2: nat] :
% 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
% 0.30/0.54 = ( ( ( N
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.30/0.54 & ( ( N
% 0.30/0.54 != ( zero_zero @ nat ) )
% 0.30/0.54 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_sum_mult_2_cases
% 0.30/0.54 thf(fact_2802_bit__rec,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_semiring_bits @ A )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.30/0.54 = ( ^ [A5: A,N3: nat] :
% 0.30/0.54 ( ( ( N3
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
% 0.30/0.54 & ( ( N3
% 0.30/0.54 != ( zero_zero @ nat ) )
% 0.30/0.54 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_rec
% 0.30/0.54 thf(fact_2803_mask__eq__exp__minus__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se2239418461657761734s_mask @ A )
% 0.30/0.54 = ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_eq_exp_minus_1
% 0.30/0.54 thf(fact_2804_and__not__numerals_I8_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_not_numerals(8)
% 0.30/0.54 thf(fact_2805_not__int__rec,axiom,
% 0.30/0.54 ( ( bit_ri4277139882892585799ns_not @ int )
% 0.30/0.54 = ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_int_rec
% 0.30/0.54 thf(fact_2806_set__bit__eq,axiom,
% 0.30/0.54 ( ( bit_se5668285175392031749et_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] :
% 0.30/0.54 ( plus_plus @ int @ K3
% 0.30/0.54 @ ( times_times @ int
% 0.30/0.54 @ ( zero_neq_one_of_bool @ int
% 0.30/0.54 @ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
% 0.30/0.54 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % set_bit_eq
% 0.30/0.54 thf(fact_2807_unset__bit__eq,axiom,
% 0.30/0.54 ( ( bit_se2638667681897837118et_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unset_bit_eq
% 0.30/0.54 thf(fact_2808_num_Osize_I6_J,axiom,
% 0.30/0.54 ! [X32: num] :
% 0.30/0.54 ( ( size_size @ num @ ( bit1 @ X32 ) )
% 0.30/0.54 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num.size(6)
% 0.30/0.54 thf(fact_2809_num_Osize_I5_J,axiom,
% 0.30/0.54 ! [X22: num] :
% 0.30/0.54 ( ( size_size @ num @ ( bit0 @ X22 ) )
% 0.30/0.54 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num.size(5)
% 0.30/0.54 thf(fact_2810_or__int__rec,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int )
% 0.30/0.54 = ( ^ [K3: int,L3: int] :
% 0.30/0.54 ( plus_plus @ int
% 0.30/0.54 @ ( zero_neq_one_of_bool @ int
% 0.30/0.54 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 0.30/0.54 | ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.30/0.54 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_int_rec
% 0.30/0.54 thf(fact_2811_drop__bit__numeral__minus__bit1,axiom,
% 0.30/0.54 ! [L: num,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_numeral_minus_bit1
% 0.30/0.54 thf(fact_2812_or__numerals_I7_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.54 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(7)
% 0.30/0.54 thf(fact_2813_or__numerals_I6_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.54 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(6)
% 0.30/0.54 thf(fact_2814_or_Oright__neutral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.54 = A3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % or.right_neutral
% 0.30/0.54 thf(fact_2815_or_Oleft__neutral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.54 = A3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % or.left_neutral
% 0.30/0.54 thf(fact_2816_drop__bit__of__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_of_0
% 0.30/0.54 thf(fact_2817_bit_Odisj__one__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.disj_one_left
% 0.30/0.54 thf(fact_2818_bit_Odisj__one__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.disj_one_right
% 0.30/0.54 thf(fact_2819_drop__bit__of__bool,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,B2: $o] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B2 ) )
% 0.30/0.54 = ( zero_neq_one_of_bool @ A
% 0.30/0.54 @ ( ( N
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 & B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_of_bool
% 0.30/0.54 thf(fact_2820_or__nonnegative__int__iff,axiom,
% 0.30/0.54 ! [K: int,L: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
% 0.30/0.54 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.54 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nonnegative_int_iff
% 0.30/0.54 thf(fact_2821_or__negative__int__iff,axiom,
% 0.30/0.54 ! [K: int,L: int] :
% 0.30/0.54 ( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.30/0.54 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.30/0.54 | ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_negative_int_iff
% 0.30/0.54 thf(fact_2822_drop__bit__nonnegative__int__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
% 0.30/0.54 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_nonnegative_int_iff
% 0.30/0.54 thf(fact_2823_drop__bit__negative__int__iff,axiom,
% 0.30/0.54 ! [N: nat,K: int] :
% 0.30/0.54 ( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 0.30/0.54 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_negative_int_iff
% 0.30/0.54 thf(fact_2824_drop__bit__minus__one,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_minus_one
% 0.30/0.54 thf(fact_2825_or__nat__numerals_I4_J,axiom,
% 0.30/0.54 ! [X: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_numerals(4)
% 0.30/0.54 thf(fact_2826_or__nat__numerals_I2_J,axiom,
% 0.30/0.54 ! [Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_numerals(2)
% 0.30/0.54 thf(fact_2827_drop__bit__Suc__bit0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_Suc_bit0
% 0.30/0.54 thf(fact_2828_or__numerals_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(2)
% 0.30/0.54 thf(fact_2829_or__numerals_I8_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(8)
% 0.30/0.54 thf(fact_2830_bit_Odisj__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.disj_cancel_left
% 0.30/0.54 thf(fact_2831_bit_Odisj__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.disj_cancel_right
% 0.30/0.54 thf(fact_2832_drop__bit__Suc__bit1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_Suc_bit1
% 0.30/0.54 thf(fact_2833_drop__bit__of__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
% 0.30/0.54 = ( zero_neq_one_of_bool @ A
% 0.30/0.54 @ ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_of_1
% 0.30/0.54 thf(fact_2834_or__numerals_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.54 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(3)
% 0.30/0.54 thf(fact_2835_or__nat__numerals_I1_J,axiom,
% 0.30/0.54 ! [Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_numerals(1)
% 0.30/0.54 thf(fact_2836_or__nat__numerals_I3_J,axiom,
% 0.30/0.54 ! [X: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_numerals(3)
% 0.30/0.54 thf(fact_2837_or__numerals_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(1)
% 0.30/0.54 thf(fact_2838_or__numerals_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(5)
% 0.30/0.54 thf(fact_2839_or__minus__numerals_I6_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(6)
% 0.30/0.54 thf(fact_2840_or__minus__numerals_I2_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(2)
% 0.30/0.54 thf(fact_2841_drop__bit__Suc__minus__bit0,axiom,
% 0.30/0.54 ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_Suc_minus_bit0
% 0.30/0.54 thf(fact_2842_drop__bit__numeral__minus__bit0,axiom,
% 0.30/0.54 ! [L: num,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_numeral_minus_bit0
% 0.30/0.54 thf(fact_2843_drop__bit__Suc__minus__bit1,axiom,
% 0.30/0.54 ! [N: nat,K: num] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_Suc_minus_bit1
% 0.30/0.54 thf(fact_2844_and__minus__minus__numerals,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % and_minus_minus_numerals
% 0.30/0.54 thf(fact_2845_or__minus__minus__numerals,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_minus_numerals
% 0.30/0.54 thf(fact_2846_or__numerals_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 0.30/0.54 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_numerals(4)
% 0.30/0.54 thf(fact_2847_size__neq__size__imp__neq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( size @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ( size_size @ A @ X )
% 0.30/0.54 != ( size_size @ A @ Y3 ) )
% 0.30/0.54 => ( X != Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % size_neq_size_imp_neq
% 0.30/0.54 thf(fact_2848_or__nat__def,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat )
% 0.30/0.54 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_def
% 0.30/0.54 thf(fact_2849_of__nat__or__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [M: nat,N: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
% 0.30/0.54 = ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_or_eq
% 0.30/0.54 thf(fact_2850_drop__bit__of__nat,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [N: nat,M: nat] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 0.30/0.54 = ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_of_nat
% 0.30/0.54 thf(fact_2851_of__nat__drop__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [M: nat,N: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_drop_bit
% 0.30/0.54 thf(fact_2852_bit_Odisj__zero__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.disj_zero_right
% 0.30/0.54 thf(fact_2853_or__eq__0__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( ( A3
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 & ( B2
% 0.30/0.54 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_eq_0_iff
% 0.30/0.54 thf(fact_2854_set__bit__nat__def,axiom,
% 0.30/0.54 ( ( bit_se5668285175392031749et_bit @ nat )
% 0.30/0.54 = ( ^ [M2: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % set_bit_nat_def
% 0.30/0.54 thf(fact_2855_bit__Suc__0__iff,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.54 = ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_Suc_0_iff
% 0.30/0.54 thf(fact_2856_not__bit__Suc__0__Suc,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_bit_Suc_0_Suc
% 0.30/0.54 thf(fact_2857_bit__push__bit__iff__nat,axiom,
% 0.30/0.54 ! [M: nat,Q4: nat,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N )
% 0.30/0.54 = ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_push_bit_iff_nat
% 0.30/0.54 thf(fact_2858_OR__lower,axiom,
% 0.30/0.54 ! [X: int,Y3: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.54 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.54 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % OR_lower
% 0.30/0.54 thf(fact_2859_or__greater__eq,axiom,
% 0.30/0.54 ! [L: int,K: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.54 => ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_greater_eq
% 0.30/0.54 thf(fact_2860_set__bit__int__def,axiom,
% 0.30/0.54 ( ( bit_se5668285175392031749et_bit @ int )
% 0.30/0.54 = ( ^ [N3: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % set_bit_int_def
% 0.30/0.54 thf(fact_2861_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.30/0.54 = A3 )
% 0.30/0.54 = ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % take_bit_eq_self_iff_drop_bit_eq_0
% 0.30/0.54 thf(fact_2862_not__bit__Suc__0__numeral,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_bit_Suc_0_numeral
% 0.30/0.54 thf(fact_2863_or__not__numerals_I1_J,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(1)
% 0.30/0.54 thf(fact_2864_set__bit__eq__or,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se5668285175392031749et_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % set_bit_eq_or
% 0.30/0.54 thf(fact_2865_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A,N: nat] :
% 0.30/0.54 ( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_push_bit_of_1_eq_drop_bit
% 0.30/0.54 thf(fact_2866_bit__iff__and__drop__bit__eq__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.30/0.54 = ( ^ [A5: A,N3: nat] :
% 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A5 ) @ ( one_one @ A ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_iff_and_drop_bit_eq_1
% 0.30/0.54 thf(fact_2867_bit_Ocomplement__unique,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [A3: A,X: A,Y3: A] :
% 0.30/0.54 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 => ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y3 )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y3 )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 => ( X = Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.complement_unique
% 0.30/0.54 thf(fact_2868_or__not__numerals_I2_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(2)
% 0.30/0.54 thf(fact_2869_or__not__numerals_I4_J,axiom,
% 0.30/0.54 ! [M: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(4)
% 0.30/0.54 thf(fact_2870_bit__nat__iff,axiom,
% 0.30/0.54 ! [K: int,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
% 0.30/0.54 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_nat_iff
% 0.30/0.54 thf(fact_2871_num_Osize_I4_J,axiom,
% 0.30/0.54 ( ( size_size @ num @ one2 )
% 0.30/0.54 = ( zero_zero @ nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % num.size(4)
% 0.30/0.54 thf(fact_2872_bit_Ocompl__unique,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y3 )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 => ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y3 )
% 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 => ( ( bit_ri4277139882892585799ns_not @ A @ X )
% 0.30/0.54 = Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.compl_unique
% 0.30/0.54 thf(fact_2873_or__not__numerals_I3_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(3)
% 0.30/0.54 thf(fact_2874_or__not__numerals_I7_J,axiom,
% 0.30/0.54 ! [M: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(7)
% 0.30/0.54 thf(fact_2875_drop__bit__Suc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat,A3: A] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
% 0.30/0.54 = ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_Suc
% 0.30/0.54 thf(fact_2876_mask__Suc__exp,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_Suc_exp
% 0.30/0.54 thf(fact_2877_or__not__numerals_I6_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(6)
% 0.30/0.54 thf(fact_2878_one__or__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.54 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_or_eq
% 0.30/0.54 thf(fact_2879_or__one__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.54 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_one_eq
% 0.30/0.54 thf(fact_2880_mask__Suc__double,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 0.30/0.54 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_Suc_double
% 0.30/0.54 thf(fact_2881_OR__upper,axiom,
% 0.30/0.54 ! [X: int,N: nat,Y3: int] :
% 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.54 => ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.54 => ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.54 => ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % OR_upper
% 0.30/0.54 thf(fact_2882_or__not__numerals_I5_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(5)
% 0.30/0.54 thf(fact_2883_signed__take__bit__def,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % signed_take_bit_def
% 0.30/0.54 thf(fact_2884_or__Suc__0__eq,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_Suc_0_eq
% 0.30/0.54 thf(fact_2885_Suc__0__or__eq,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.54 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % Suc_0_or_eq
% 0.30/0.54 thf(fact_2886_drop__bit__rec,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.54 => ( ( bit_se4197421643247451524op_bit @ A )
% 0.30/0.54 = ( ^ [N3: nat,A5: A] :
% 0.30/0.54 ( if @ A
% 0.30/0.54 @ ( N3
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 @ A5
% 0.30/0.54 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_rec
% 0.30/0.54 thf(fact_2887_or__not__numerals_I8_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(8)
% 0.30/0.54 thf(fact_2888_or__not__numerals_I9_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_not_numerals(9)
% 0.30/0.54 thf(fact_2889_or__nat__rec,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat )
% 0.30/0.54 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.54 ( plus_plus @ nat
% 0.30/0.54 @ ( zero_neq_one_of_bool @ nat
% 0.30/0.54 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
% 0.30/0.54 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
% 0.30/0.54 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_rec
% 0.30/0.54 thf(fact_2890_or__int__unfold,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int )
% 0.30/0.54 = ( ^ [K3: int,L3: int] :
% 0.30/0.54 ( if @ int
% 0.30/0.54 @ ( ( K3
% 0.30/0.54 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.54 | ( L3
% 0.30/0.54 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.30/0.54 @ ( uminus_uminus @ int @ ( one_one @ int ) )
% 0.30/0.54 @ ( if @ int
% 0.30/0.54 @ ( K3
% 0.30/0.54 = ( zero_zero @ int ) )
% 0.30/0.54 @ L3
% 0.30/0.54 @ ( if @ int
% 0.30/0.54 @ ( L3
% 0.30/0.54 = ( zero_zero @ int ) )
% 0.30/0.54 @ K3
% 0.30/0.54 @ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_int_unfold
% 0.30/0.54 thf(fact_2891_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( 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 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.30/0.54 thf(fact_2892_or__minus__numerals_I4_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(4)
% 0.30/0.54 thf(fact_2893_or__minus__numerals_I8_J,axiom,
% 0.30/0.54 ! [N: num,M: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(8)
% 0.30/0.54 thf(fact_2894_or__nat__unfold,axiom,
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ nat )
% 0.30/0.54 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.54 ( if @ nat
% 0.30/0.54 @ ( M2
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 @ N3
% 0.30/0.54 @ ( if @ nat
% 0.30/0.54 @ ( N3
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 @ M2
% 0.30/0.54 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_nat_unfold
% 0.30/0.54 thf(fact_2895_xor__minus__numerals_I1_J,axiom,
% 0.30/0.54 ! [N: num,K: int] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % xor_minus_numerals(1)
% 0.30/0.54 thf(fact_2896_max_Oright__idem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.54 = ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.right_idem
% 0.30/0.54 thf(fact_2897_max_Oleft__idem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
% 0.30/0.54 = ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.left_idem
% 0.30/0.54 thf(fact_2898_max_Oidem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( ord_max @ A @ A3 @ A3 )
% 0.30/0.54 = A3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.idem
% 0.30/0.54 thf(fact_2899_max_Oabsorb1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 => ( ( ord_max @ A @ A3 @ B2 )
% 0.30/0.54 = A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb1
% 0.30/0.54 thf(fact_2900_max_Oabsorb2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 => ( ( ord_max @ A @ A3 @ B2 )
% 0.30/0.54 = B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb2
% 0.30/0.54 thf(fact_2901_max_Obounded__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.54 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.bounded_iff
% 0.30/0.54 thf(fact_2902_max_Oabsorb3,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.54 => ( ( ord_max @ A @ A3 @ B2 )
% 0.30/0.54 = A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb3
% 0.30/0.54 thf(fact_2903_max_Oabsorb4,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.54 => ( ( ord_max @ A @ A3 @ B2 )
% 0.30/0.54 = B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb4
% 0.30/0.54 thf(fact_2904_max__less__iff__conj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( ord_less @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ( ord_less @ A @ X @ Z3 )
% 0.30/0.54 & ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_less_iff_conj
% 0.30/0.54 thf(fact_2905_of__bool__or__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_semidom @ A )
% 0.30/0.54 => ! [P: $o,Q: $o] :
% 0.30/0.54 ( ( zero_neq_one_of_bool @ A
% 0.30/0.54 @ ( P
% 0.30/0.54 | Q ) )
% 0.30/0.54 = ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_bool_or_iff
% 0.30/0.54 thf(fact_2906_max__Suc__Suc,axiom,
% 0.30/0.54 ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.54 = ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_Suc_Suc
% 0.30/0.54 thf(fact_2907_max__0R,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % max_0R
% 0.30/0.54 thf(fact_2908_max__0L,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % max_0L
% 0.30/0.54 thf(fact_2909_max__nat_Oright__neutral,axiom,
% 0.30/0.54 ! [A3: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.54 = A3 ) ).
% 0.30/0.54
% 0.30/0.54 % max_nat.right_neutral
% 0.30/0.54 thf(fact_2910_max__nat_Oneutr__eq__iff,axiom,
% 0.30/0.54 ! [A3: nat,B2: nat] :
% 0.30/0.54 ( ( ( zero_zero @ nat )
% 0.30/0.54 = ( ord_max @ nat @ A3 @ B2 ) )
% 0.30/0.54 = ( ( A3
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 & ( B2
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_nat.neutr_eq_iff
% 0.30/0.54 thf(fact_2911_max__nat_Oleft__neutral,axiom,
% 0.30/0.54 ! [A3: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.54 = A3 ) ).
% 0.30/0.54
% 0.30/0.54 % max_nat.left_neutral
% 0.30/0.54 thf(fact_2912_max__nat_Oeq__neutr__iff,axiom,
% 0.30/0.54 ! [A3: nat,B2: nat] :
% 0.30/0.54 ( ( ( ord_max @ nat @ A3 @ B2 )
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 = ( ( A3
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 & ( B2
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_nat.eq_neutr_iff
% 0.30/0.54 thf(fact_2913_max__number__of_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ U ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_number_of(1)
% 0.30/0.54 thf(fact_2914_max__0__1_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(4)
% 0.30/0.54 thf(fact_2915_max__0__1_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(3)
% 0.30/0.54 thf(fact_2916_max__0__1_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(2)
% 0.30/0.54 thf(fact_2917_max__0__1_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(1)
% 0.30/0.54 thf(fact_2918_max__0__1_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(5)
% 0.30/0.54 thf(fact_2919_max__0__1_I6_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 0.30/0.54 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_0_1(6)
% 0.30/0.54 thf(fact_2920_sub__num__simps_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ( ( neg_numeral_sub @ A @ one2 @ one2 )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(1)
% 0.30/0.54 thf(fact_2921_diff__numeral__simps_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num,N: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_simps(1)
% 0.30/0.54 thf(fact_2922_sub__num__simps_I6_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num,L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
% 0.30/0.54 = ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(6)
% 0.30/0.54 thf(fact_2923_sub__num__simps_I9_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num,L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 0.30/0.54 = ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(9)
% 0.30/0.54 thf(fact_2924_max__number__of_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( numeral_numeral @ A @ U ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_number_of(2)
% 0.30/0.54 thf(fact_2925_max__number__of_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_number_of(3)
% 0.30/0.54 thf(fact_2926_max__number__of_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_number_of(4)
% 0.30/0.54 thf(fact_2927_add__neg__numeral__simps_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num,N: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_simps(2)
% 0.30/0.54 thf(fact_2928_add__neg__numeral__simps_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num,N: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_simps(1)
% 0.30/0.54 thf(fact_2929_semiring__norm_I166_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 0.30/0.54 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W2 ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_norm(166)
% 0.30/0.54 thf(fact_2930_semiring__norm_I167_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [V: num,W2: num,Y3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
% 0.30/0.54 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_norm(167)
% 0.30/0.54 thf(fact_2931_diff__numeral__simps_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num,N: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_simps(4)
% 0.30/0.54 thf(fact_2932_max__numeral__Suc,axiom,
% 0.30/0.54 ! [K: num,N: nat] :
% 0.30/0.54 ( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 0.30/0.54 = ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_numeral_Suc
% 0.30/0.54 thf(fact_2933_max__Suc__numeral,axiom,
% 0.30/0.54 ! [N: nat,K: num] :
% 0.30/0.54 ( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.54 = ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_Suc_numeral
% 0.30/0.54 thf(fact_2934_sub__num__simps_I7_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num,L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
% 0.30/0.54 = ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(7)
% 0.30/0.54 thf(fact_2935_sub__num__simps_I8_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num,L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
% 0.30/0.54 = ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(8)
% 0.30/0.54 thf(fact_2936_drop__bit__of__Suc__0,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( bit_se4197421643247451524op_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.54 = ( zero_neq_one_of_bool @ nat
% 0.30/0.54 @ ( N
% 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % drop_bit_of_Suc_0
% 0.30/0.54 thf(fact_2937_diff__numeral__special_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_special(2)
% 0.30/0.54 thf(fact_2938_diff__numeral__special_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_special(1)
% 0.30/0.54 thf(fact_2939_sub__num__simps_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(5)
% 0.30/0.54 thf(fact_2940_not__minus__numeral__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_minus_numeral_eq
% 0.30/0.54 thf(fact_2941_add__neg__numeral__special_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_special(1)
% 0.30/0.54 thf(fact_2942_add__neg__numeral__special_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_special(2)
% 0.30/0.54 thf(fact_2943_add__neg__numeral__special_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_special(3)
% 0.30/0.54 thf(fact_2944_add__neg__numeral__special_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_numeral_special(4)
% 0.30/0.54 thf(fact_2945_minus__sub__one__diff__one,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_sub_one_diff_one
% 0.30/0.54 thf(fact_2946_diff__numeral__special_I7_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_special(7)
% 0.30/0.54 thf(fact_2947_diff__numeral__special_I8_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [M: num] :
% 0.30/0.54 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.54 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_numeral_special(8)
% 0.30/0.54 thf(fact_2948_sub__num__simps_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(3)
% 0.30/0.54 thf(fact_2949_xor__minus__numerals_I2_J,axiom,
% 0.30/0.54 ! [K: int,N: num] :
% 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % xor_minus_numerals(2)
% 0.30/0.54 thf(fact_2950_max_Ostrict__coboundedI2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less @ A @ C2 @ B2 )
% 0.30/0.54 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.strict_coboundedI2
% 0.30/0.54 thf(fact_2951_max_Ostrict__coboundedI1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less @ A @ C2 @ A3 )
% 0.30/0.54 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.strict_coboundedI1
% 0.30/0.54 thf(fact_2952_max_Ostrict__order__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less @ A )
% 0.30/0.54 = ( ^ [B3: A,A5: A] :
% 0.30/0.54 ( ( A5
% 0.30/0.54 = ( ord_max @ A @ A5 @ B3 ) )
% 0.30/0.54 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.strict_order_iff
% 0.30/0.54 thf(fact_2953_max_Ostrict__boundedE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.54 => ~ ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.54 => ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.strict_boundedE
% 0.30/0.54 thf(fact_2954_less__max__iff__disj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.54 ( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
% 0.30/0.54 = ( ( ord_less @ A @ Z3 @ X )
% 0.30/0.54 | ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_max_iff_disj
% 0.30/0.54 thf(fact_2955_max__add__distrib__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_add_distrib_left
% 0.30/0.54 thf(fact_2956_max__add__distrib__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ X @ ( ord_max @ A @ Y3 @ Z3 ) )
% 0.30/0.54 = ( ord_max @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_add_distrib_right
% 0.30/0.54 thf(fact_2957_max__diff__distrib__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordered_ab_group_add @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( minus_minus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_diff_distrib_left
% 0.30/0.54 thf(fact_2958_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,Y3: A,Z3: A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Conj @ ( Xor @ Y3 @ Z3 ) @ X )
% 0.30/0.54 = ( Xor @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.conj_xor_distrib2
% 0.30/0.54 thf(fact_2959_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ ( Compl @ X ) )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_cancel_right
% 0.30/0.54 thf(fact_2960_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Conj @ X @ ( Xor @ Y3 @ Z3 ) )
% 0.30/0.54 = ( Xor @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.conj_xor_distrib
% 0.30/0.54 thf(fact_2961_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ ( Compl @ Y3 ) )
% 0.30/0.54 = ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_compl_right
% 0.30/0.54 thf(fact_2962_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ ( Compl @ X ) @ X )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_cancel_left
% 0.30/0.54 thf(fact_2963_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ ( Compl @ X ) @ Y3 )
% 0.30/0.54 = ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_compl_left
% 0.30/0.54 thf(fact_2964_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ One2 )
% 0.30/0.54 = ( Compl @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_one_right
% 0.30/0.54 thf(fact_2965_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ ( Xor @ X @ Y3 ) )
% 0.30/0.54 = Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_left_self
% 0.30/0.54 thf(fact_2966_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ One2 @ X )
% 0.30/0.54 = ( Compl @ X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_one_left
% 0.30/0.54 thf(fact_2967_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ X )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_self
% 0.30/0.54 thf(fact_2968_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ Y3 )
% 0.30/0.54 = ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_def2
% 0.30/0.54 thf(fact_2969_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( ( Xor @ X @ Y3 )
% 0.30/0.54 = ( Disj @ ( Conj @ X @ ( Compl @ Y3 ) ) @ ( Conj @ ( Compl @ X ) @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.xor_def
% 0.30/0.54 thf(fact_2970_of__nat__max,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: nat,Y3: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y3 ) )
% 0.30/0.54 = ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_max
% 0.30/0.54 thf(fact_2971_max_Oleft__commute,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.54 ( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
% 0.30/0.54 = ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.left_commute
% 0.30/0.54 thf(fact_2972_max_Ocommute,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_max @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] : ( ord_max @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.commute
% 0.30/0.54 thf(fact_2973_max_Oassoc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.54 = ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.assoc
% 0.30/0.54 thf(fact_2974_of__int__max,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [X: int,Y3: int] :
% 0.30/0.54 ( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y3 ) )
% 0.30/0.54 = ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_int_max
% 0.30/0.54 thf(fact_2975_max__absorb2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ord @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.54 => ( ( ord_max @ A @ X @ Y3 )
% 0.30/0.54 = Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_absorb2
% 0.30/0.54 thf(fact_2976_max__absorb1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( order @ A )
% 0.30/0.54 => ! [Y3: A,X: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.54 => ( ( ord_max @ A @ X @ Y3 )
% 0.30/0.54 = X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_absorb1
% 0.30/0.54 thf(fact_2977_max_OcoboundedI2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.54 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.coboundedI2
% 0.30/0.54 thf(fact_2978_max_OcoboundedI1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.54 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.coboundedI1
% 0.30/0.54 thf(fact_2979_max_Oabsorb__iff2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] :
% 0.30/0.54 ( ( ord_max @ A @ A5 @ B3 )
% 0.30/0.54 = B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb_iff2
% 0.30/0.54 thf(fact_2980_max_Oabsorb__iff1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [B3: A,A5: A] :
% 0.30/0.54 ( ( ord_max @ A @ A5 @ B3 )
% 0.30/0.54 = A5 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.absorb_iff1
% 0.30/0.54 thf(fact_2981_le__max__iff__disj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
% 0.30/0.54 = ( ( ord_less_eq @ A @ Z3 @ X )
% 0.30/0.54 | ( ord_less_eq @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_max_iff_disj
% 0.30/0.54 thf(fact_2982_max_Ocobounded2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.cobounded2
% 0.30/0.54 thf(fact_2983_max_Ocobounded1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.cobounded1
% 0.30/0.54 thf(fact_2984_max_Oorder__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [B3: A,A5: A] :
% 0.30/0.54 ( A5
% 0.30/0.54 = ( ord_max @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.order_iff
% 0.30/0.54 thf(fact_2985_max_OboundedI,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.54 => ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.boundedI
% 0.30/0.54 thf(fact_2986_max_OboundedE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.54 => ~ ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.boundedE
% 0.30/0.54 thf(fact_2987_max_OorderI,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( A3
% 0.30/0.54 = ( ord_max @ A @ A3 @ B2 ) )
% 0.30/0.54 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.orderI
% 0.30/0.54 thf(fact_2988_max_OorderE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 => ( A3
% 0.30/0.54 = ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.orderE
% 0.30/0.54 thf(fact_2989_max_Omono,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [C2: A,A3: A,D2: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.54 => ( ( ord_less_eq @ A @ D2 @ B2 )
% 0.30/0.54 => ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max.mono
% 0.30/0.54 thf(fact_2990_max__def,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ord @ A )
% 0.30/0.54 => ( ( ord_max @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_def
% 0.30/0.54 thf(fact_2991_nat__add__max__left,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
% 0.30/0.54 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_add_max_left
% 0.30/0.54 thf(fact_2992_nat__add__max__right,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
% 0.30/0.54 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_add_max_right
% 0.30/0.54 thf(fact_2993_nat__mult__max__left,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
% 0.30/0.54 = ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_max_left
% 0.30/0.54 thf(fact_2994_nat__mult__max__right,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
% 0.30/0.54 = ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_max_right
% 0.30/0.54 thf(fact_2995_length__induct,axiom,
% 0.30/0.54 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.30/0.54 ( ! [Xs2: list @ A] :
% 0.30/0.54 ( ! [Ys: list @ A] :
% 0.30/0.54 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.30/0.54 => ( P @ Ys ) )
% 0.30/0.54 => ( P @ Xs2 ) )
% 0.30/0.54 => ( P @ Xs ) ) ).
% 0.30/0.54
% 0.30/0.54 % length_induct
% 0.30/0.54 thf(fact_2996_nat__minus__add__max,axiom,
% 0.30/0.54 ! [N: nat,M: nat] :
% 0.30/0.54 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
% 0.30/0.54 = ( ord_max @ nat @ N @ M ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_minus_add_max
% 0.30/0.54 thf(fact_2997_neg__numeral__class_Osub__def,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ( ( neg_numeral_sub @ A )
% 0.30/0.54 = ( ^ [K3: num,L3: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_numeral_class.sub_def
% 0.30/0.54 thf(fact_2998_Heap_Osize__neq,axiom,
% 0.30/0.54 ! [A: $tType,X: heap_Time_Heap @ A] :
% 0.30/0.54 ( ( size_size @ ( heap_Time_Heap @ A ) @ X )
% 0.30/0.54 != ( zero_zero @ nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % Heap.size_neq
% 0.30/0.54 thf(fact_2999_sub__non__negative,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [N: num,M: num] :
% 0.30/0.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 0.30/0.54 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_non_negative
% 0.30/0.54 thf(fact_3000_sub__non__positive,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [N: num,M: num] :
% 0.30/0.54 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_non_positive
% 0.30/0.54 thf(fact_3001_sub__negative,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [N: num,M: num] :
% 0.30/0.54 ( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( ord_less @ num @ N @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_negative
% 0.30/0.54 thf(fact_3002_sub__positive,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [N: num,M: num] :
% 0.30/0.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 0.30/0.54 = ( ord_less @ num @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_positive
% 0.30/0.54 thf(fact_3003_sub__inc__One__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
% 0.30/0.54 = ( numeral_numeral @ A @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_inc_One_eq
% 0.30/0.54 thf(fact_3004_minus__numeral__eq__not__sub__one,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 0.30/0.54 = ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_numeral_eq_not_sub_one
% 0.30/0.54 thf(fact_3005_int__numeral__or__not__num__neg,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % int_numeral_or_not_num_neg
% 0.30/0.54 thf(fact_3006_int__numeral__not__or__num__neg,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % int_numeral_not_or_num_neg
% 0.30/0.54 thf(fact_3007_numeral__or__not__num__eq,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_or_not_num_eq
% 0.30/0.54 thf(fact_3008_or__minus__numerals_I1_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(1)
% 0.30/0.54 thf(fact_3009_or__minus__numerals_I5_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(5)
% 0.30/0.54 thf(fact_3010_horner__sum__of__bool__2__less,axiom,
% 0.30/0.54 ! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % horner_sum_of_bool_2_less
% 0.30/0.54 thf(fact_3011_or__minus__numerals_I7_J,axiom,
% 0.30/0.54 ! [N: num,M: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(7)
% 0.30/0.54 thf(fact_3012_or__minus__numerals_I3_J,axiom,
% 0.30/0.54 ! [M: num,N: num] :
% 0.30/0.54 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.54 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % or_minus_numerals(3)
% 0.30/0.54 thf(fact_3013_dbl__dec__simps_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num] :
% 0.30/0.54 ( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % dbl_dec_simps(5)
% 0.30/0.54 thf(fact_3014_pred__numeral__simps_I2_J,axiom,
% 0.30/0.54 ! [K: num] :
% 0.30/0.54 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.30/0.54 = ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pred_numeral_simps(2)
% 0.30/0.54 thf(fact_3015_sub__num__simps_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [K: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
% 0.30/0.54 = ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(4)
% 0.30/0.54 thf(fact_3016_sub__num__simps_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [L: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_num_simps(2)
% 0.30/0.54 thf(fact_3017_semiring__norm_I26_J,axiom,
% 0.30/0.54 ( ( bitM @ one2 )
% 0.30/0.54 = one2 ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_norm(26)
% 0.30/0.54 thf(fact_3018_semiring__norm_I27_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bitM @ ( bit0 @ N ) )
% 0.30/0.54 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_norm(27)
% 0.30/0.54 thf(fact_3019_semiring__norm_I28_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bitM @ ( bit1 @ N ) )
% 0.30/0.54 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % semiring_norm(28)
% 0.30/0.54 thf(fact_3020_inc__BitM__eq,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( inc @ ( bitM @ N ) )
% 0.30/0.54 = ( bit0 @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % inc_BitM_eq
% 0.30/0.54 thf(fact_3021_BitM__inc__eq,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( bitM @ ( inc @ N ) )
% 0.30/0.54 = ( bit1 @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % BitM_inc_eq
% 0.30/0.54 thf(fact_3022_eval__nat__numeral_I2_J,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( numeral_numeral @ nat @ ( bit0 @ N ) )
% 0.30/0.54 = ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eval_nat_numeral(2)
% 0.30/0.54 thf(fact_3023_one__plus__BitM,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
% 0.30/0.54 = ( bit0 @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_plus_BitM
% 0.30/0.54 thf(fact_3024_BitM__plus__one,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
% 0.30/0.54 = ( bit0 @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % BitM_plus_one
% 0.30/0.54 thf(fact_3025_numeral__BitM,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( neg_numeral @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( numeral_numeral @ A @ ( bitM @ N ) )
% 0.30/0.54 = ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_BitM
% 0.30/0.54 thf(fact_3026_not__numeral__BitM__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ! [N: num] :
% 0.30/0.54 ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_BitM_eq
% 0.30/0.54 thf(fact_3027_sub__BitM__One__eq,axiom,
% 0.30/0.54 ! [N: num] :
% 0.30/0.54 ( ( neg_numeral_sub @ int @ ( bitM @ N ) @ one2 )
% 0.30/0.54 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sub_BitM_One_eq
% 0.30/0.54 thf(fact_3028_bit__horner__sum__bit__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.54 => ! [Bs: list @ $o,N: nat] :
% 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
% 0.30/0.54 = ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
% 0.30/0.54 & ( nth @ $o @ Bs @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit_horner_sum_bit_iff
% 0.30/0.54 thf(fact_3029_bit_Oabstract__boolean__algebra__axioms,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.54 => ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % bit.abstract_boolean_algebra_axioms
% 0.30/0.54 thf(fact_3030_num__of__nat_Osimps_I2_J,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.54 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.30/0.54 & ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.54 = one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat.simps(2)
% 0.30/0.54 thf(fact_3031_eq__numeral__iff__iszero_I7_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( one_one @ A ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(7)
% 0.30/0.54 thf(fact_3032_eq__numeral__iff__iszero_I8_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( ( one_one @ A )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(8)
% 0.30/0.54 thf(fact_3033_num__of__nat__numeral__eq,axiom,
% 0.30/0.54 ! [Q4: num] :
% 0.30/0.54 ( ( num_of_nat @ ( numeral_numeral @ nat @ Q4 ) )
% 0.30/0.54 = Q4 ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat_numeral_eq
% 0.30/0.54 thf(fact_3034_iszero__neg__numeral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [W2: num] :
% 0.30/0.54 ( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % iszero_neg_numeral
% 0.30/0.54 thf(fact_3035_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ ( Conj @ Y3 @ Z3 ) @ X )
% 0.30/0.54 = ( Conj @ ( Disj @ Y3 @ X ) @ ( Disj @ Z3 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_conj_distrib2
% 0.30/0.54 thf(fact_3036_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ ( Disj @ Y3 @ Z3 ) @ X )
% 0.30/0.54 = ( Disj @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_disj_distrib2
% 0.30/0.54 thf(fact_3037_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( ( Compl @ X )
% 0.30/0.54 = ( Compl @ Y3 ) )
% 0.30/0.54 = ( X = Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.compl_eq_compl_iff
% 0.30/0.54 thf(fact_3038_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ X @ ( Conj @ Y3 @ Z3 ) )
% 0.30/0.54 = ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ X @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_conj_distrib
% 0.30/0.54 thf(fact_3039_abstract__boolean__algebra_Odisj__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ X @ ( Compl @ X ) )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_cancel_right
% 0.30/0.54 thf(fact_3040_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ X @ ( Disj @ Y3 @ Z3 ) )
% 0.30/0.54 = ( Disj @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_disj_distrib
% 0.30/0.54 thf(fact_3041_abstract__boolean__algebra_Oconj__cancel__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ X @ ( Compl @ X ) )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_cancel_right
% 0.30/0.54 thf(fact_3042_abstract__boolean__algebra_Ocomplement__unique,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,A3: A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( ( Conj @ A3 @ X )
% 0.30/0.54 = Zero )
% 0.30/0.54 => ( ( ( Disj @ A3 @ X )
% 0.30/0.54 = One2 )
% 0.30/0.54 => ( ( ( Conj @ A3 @ Y3 )
% 0.30/0.54 = Zero )
% 0.30/0.54 => ( ( ( Disj @ A3 @ Y3 )
% 0.30/0.54 = One2 )
% 0.30/0.54 => ( X = Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.complement_unique
% 0.30/0.54 thf(fact_3043_abstract__boolean__algebra_Odisj__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ ( Compl @ X ) @ X )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_cancel_left
% 0.30/0.54 thf(fact_3044_abstract__boolean__algebra_Oconj__cancel__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ ( Compl @ X ) @ X )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_cancel_left
% 0.30/0.54 thf(fact_3045_abstract__boolean__algebra_Odisj__zero__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ X @ Zero )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_zero_right
% 0.30/0.54 thf(fact_3046_abstract__boolean__algebra_Oconj__zero__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ X @ Zero )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_zero_right
% 0.30/0.54 thf(fact_3047_abstract__boolean__algebra_Odisj__one__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ X @ One2 )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_one_right
% 0.30/0.54 thf(fact_3048_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Compl @ ( Disj @ X @ Y3 ) )
% 0.30/0.54 = ( Conj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.de_Morgan_disj
% 0.30/0.54 thf(fact_3049_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Compl @ ( Conj @ X @ Y3 ) )
% 0.30/0.54 = ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.de_Morgan_conj
% 0.30/0.54 thf(fact_3050_abstract__boolean__algebra_Oconj__zero__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ Zero @ X )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_zero_left
% 0.30/0.54 thf(fact_3051_abstract__boolean__algebra_Oconj__one__right,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Conj @ X @ One2 )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.conj_one_right
% 0.30/0.54 thf(fact_3052_abstract__boolean__algebra_Odisj__one__left,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Disj @ One2 @ X )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.disj_one_left
% 0.30/0.54 thf(fact_3053_abstract__boolean__algebra_Odouble__compl,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Compl @ ( Compl @ X ) )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.double_compl
% 0.30/0.54 thf(fact_3054_abstract__boolean__algebra_Ocompl__unique,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( ( Conj @ X @ Y3 )
% 0.30/0.54 = Zero )
% 0.30/0.54 => ( ( ( Disj @ X @ Y3 )
% 0.30/0.54 = One2 )
% 0.30/0.54 => ( ( Compl @ X )
% 0.30/0.54 = Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.compl_unique
% 0.30/0.54 thf(fact_3055_abstract__boolean__algebra_Ocompl__zero,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Compl @ Zero )
% 0.30/0.54 = One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.compl_zero
% 0.30/0.54 thf(fact_3056_abstract__boolean__algebra_Ocompl__one,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.54 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.54 => ( ( Compl @ One2 )
% 0.30/0.54 = Zero ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra.compl_one
% 0.30/0.54 thf(fact_3057_iszero__def,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ( ( ring_1_iszero @ A )
% 0.30/0.54 = ( ^ [Z7: A] :
% 0.30/0.54 ( Z7
% 0.30/0.54 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % iszero_def
% 0.30/0.54 thf(fact_3058_iszero__0,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % iszero_0
% 0.30/0.54 thf(fact_3059_not__iszero__numeral,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_char_0 @ A )
% 0.30/0.54 => ! [W2: num] :
% 0.30/0.54 ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_iszero_numeral
% 0.30/0.54 thf(fact_3060_not__iszero__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_iszero_1
% 0.30/0.54 thf(fact_3061_eq__iff__iszero__diff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.54 = ( ^ [X2: A,Y6: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_iff_iszero_diff
% 0.30/0.54 thf(fact_3062_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
% 0.30/0.54 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 0.30/0.54 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.54 => ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % abstract_boolean_algebra_sym_diff.axioms(1)
% 0.30/0.54 thf(fact_3063_nth__equalityI,axiom,
% 0.30/0.54 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.54 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.54 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.54 => ( ! [I4: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.54 => ( ( nth @ A @ Xs @ I4 )
% 0.30/0.54 = ( nth @ A @ Ys2 @ I4 ) ) )
% 0.30/0.54 => ( Xs = Ys2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nth_equalityI
% 0.30/0.54 thf(fact_3064_Skolem__list__nth,axiom,
% 0.30/0.54 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.30/0.54 ( ( ! [I5: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ I5 @ K )
% 0.30/0.54 => ? [X7: A] : ( P @ I5 @ X7 ) ) )
% 0.30/0.54 = ( ? [Xs3: list @ A] :
% 0.30/0.54 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.54 = K )
% 0.30/0.54 & ! [I5: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ I5 @ K )
% 0.30/0.54 => ( P @ I5 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % Skolem_list_nth
% 0.30/0.54 thf(fact_3065_list__eq__iff__nth__eq,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ^ [Y: list @ A,Z2: list @ A] : ( Y = Z2 ) )
% 0.30/0.54 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.54 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.54 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.54 & ! [I5: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.54 => ( ( nth @ A @ Xs3 @ I5 )
% 0.30/0.54 = ( nth @ A @ Ys3 @ I5 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % list_eq_iff_nth_eq
% 0.30/0.54 thf(fact_3066_eq__numeral__iff__iszero_I10_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( ( zero_zero @ A )
% 0.30/0.54 = ( numeral_numeral @ A @ Y3 ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(10)
% 0.30/0.54 thf(fact_3067_eq__numeral__iff__iszero_I9_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ( numeral_numeral @ A @ X )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(9)
% 0.30/0.54 thf(fact_3068_num__of__nat_Osimps_I1_J,axiom,
% 0.30/0.54 ( ( num_of_nat @ ( zero_zero @ nat ) )
% 0.30/0.54 = one2 ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat.simps(1)
% 0.30/0.54 thf(fact_3069_not__iszero__Numeral1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_iszero_Numeral1
% 0.30/0.54 thf(fact_3070_not__iszero__neg__1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_iszero_neg_1
% 0.30/0.54 thf(fact_3071_eq__numeral__iff__iszero_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( ( numeral_numeral @ A @ X )
% 0.30/0.54 = ( numeral_numeral @ A @ Y3 ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(1)
% 0.30/0.54 thf(fact_3072_eq__numeral__iff__iszero_I12_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( ( zero_zero @ A )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(12)
% 0.30/0.54 thf(fact_3073_eq__numeral__iff__iszero_I11_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( zero_zero @ A ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(11)
% 0.30/0.54 thf(fact_3074_not__iszero__neg__Numeral1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_iszero_neg_Numeral1
% 0.30/0.54 thf(fact_3075_numeral__num__of__nat,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 => ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
% 0.30/0.54 = N ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_num_of_nat
% 0.30/0.54 thf(fact_3076_num__of__nat__One,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
% 0.30/0.54 => ( ( num_of_nat @ N )
% 0.30/0.54 = one2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat_One
% 0.30/0.54 thf(fact_3077_eq__numeral__iff__iszero_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( numeral_numeral @ A @ Y3 ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(3)
% 0.30/0.54 thf(fact_3078_eq__numeral__iff__iszero_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( ( numeral_numeral @ A @ X )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(2)
% 0.30/0.54 thf(fact_3079_eq__numeral__iff__iszero_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num,Y3: num] :
% 0.30/0.54 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y3 @ X ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(4)
% 0.30/0.54 thf(fact_3080_numeral__num__of__nat__unfold,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( semiring_1 @ A )
% 0.30/0.54 => ! [N: nat] :
% 0.30/0.54 ( ( ( N
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 0.30/0.54 = ( one_one @ A ) ) )
% 0.30/0.54 & ( ( N
% 0.30/0.54 != ( zero_zero @ nat ) )
% 0.30/0.54 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 0.30/0.54 = ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_num_of_nat_unfold
% 0.30/0.54 thf(fact_3081_num__of__nat__double,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 => ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
% 0.30/0.54 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat_double
% 0.30/0.54 thf(fact_3082_num__of__nat__plus__distrib,axiom,
% 0.30/0.54 ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.54 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 => ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
% 0.30/0.54 = ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num_of_nat_plus_distrib
% 0.30/0.54 thf(fact_3083_eq__numeral__iff__iszero_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ( numeral_numeral @ A @ X )
% 0.30/0.54 = ( one_one @ A ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(5)
% 0.30/0.54 thf(fact_3084_eq__numeral__iff__iszero_I6_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ring_1 @ A )
% 0.30/0.54 => ! [Y3: num] :
% 0.30/0.54 ( ( ( one_one @ A )
% 0.30/0.54 = ( numeral_numeral @ A @ Y3 ) )
% 0.30/0.54 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_numeral_iff_iszero(6)
% 0.30/0.54 thf(fact_3085_nth__rotate1,axiom,
% 0.30/0.54 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.54 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.54 => ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
% 0.30/0.54 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nth_rotate1
% 0.30/0.54 thf(fact_3086_nth__update__neq,axiom,
% 0.30/0.54 ! [B: $tType,A: $tType] :
% 0.30/0.54 ( ( ( heap @ A )
% 0.30/0.54 & ( heap @ B ) )
% 0.30/0.54 => ! [A3: array @ A,B2: array @ B,J: nat,V: B,H2: heap_ext @ product_unit,I: nat] :
% 0.30/0.54 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 0.30/0.54 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H2 ) @ A3 ) @ I )
% 0.30/0.54 = ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nth_update_neq
% 0.30/0.54 thf(fact_3087_mask__mod__exp,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.54 => ! [N: nat,M: nat] :
% 0.30/0.54 ( ( 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 ) ) @ M ) )
% 0.30/0.54 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mask_mod_exp
% 0.30/0.54 thf(fact_3088_set__decode__plus__power__2,axiom,
% 0.30/0.54 ! [N: nat,Z3: nat] :
% 0.30/0.54 ( ~ ( member @ nat @ N @ ( nat_set_decode @ Z3 ) )
% 0.30/0.54 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z3 ) )
% 0.30/0.54 = ( insert @ nat @ N @ ( nat_set_decode @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % set_decode_plus_power_2
% 0.30/0.54 thf(fact_3089_nth__rotate,axiom,
% 0.30/0.54 ! [A: $tType,N: nat,Xs: list @ A,M: nat] :
% 0.30/0.54 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.54 => ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N )
% 0.30/0.54 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nth_rotate
% 0.30/0.54 thf(fact_3090_min_Oidem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A] :
% 0.30/0.54 ( ( ord_min @ A @ A3 @ A3 )
% 0.30/0.54 = A3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.idem
% 0.30/0.54 thf(fact_3091_min_Oleft__idem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
% 0.30/0.54 = ( ord_min @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.left_idem
% 0.30/0.54 thf(fact_3092_min_Oright__idem,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.54 = ( ord_min @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.right_idem
% 0.30/0.54 thf(fact_3093_min_Obounded__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 0.30/0.54 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 & ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.bounded_iff
% 0.30/0.54 thf(fact_3094_min_Oabsorb2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.54 => ( ( ord_min @ A @ A3 @ B2 )
% 0.30/0.54 = B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb2
% 0.30/0.54 thf(fact_3095_min_Oabsorb1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 => ( ( ord_min @ A @ A3 @ B2 )
% 0.30/0.54 = A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb1
% 0.30/0.54 thf(fact_3096_min__less__iff__conj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [Z3: A,X: A,Y3: A] :
% 0.30/0.54 ( ( ord_less @ A @ Z3 @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.54 = ( ( ord_less @ A @ Z3 @ X )
% 0.30/0.54 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_less_iff_conj
% 0.30/0.54 thf(fact_3097_min_Oabsorb4,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A] :
% 0.30/0.54 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.54 => ( ( ord_min @ A @ A3 @ B2 )
% 0.30/0.54 = B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb4
% 0.30/0.54 thf(fact_3098_min_Oabsorb3,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.54 => ( ( ord_min @ A @ A3 @ B2 )
% 0.30/0.54 = A3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb3
% 0.30/0.54 thf(fact_3099_max__min__same_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [Y3: A,X: A] :
% 0.30/0.54 ( ( ord_max @ A @ Y3 @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.54 = Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_same(4)
% 0.30/0.54 thf(fact_3100_max__min__same_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ Y3 )
% 0.30/0.54 = Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_same(3)
% 0.30/0.54 thf(fact_3101_max__min__same_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ X )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_same(2)
% 0.30/0.54 thf(fact_3102_max__min__same_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.54 = X ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_same(1)
% 0.30/0.54 thf(fact_3103_min__Suc__Suc,axiom,
% 0.30/0.54 ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.54 = ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_Suc_Suc
% 0.30/0.54 thf(fact_3104_min__0L,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.54 = ( zero_zero @ nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0L
% 0.30/0.54 thf(fact_3105_min__0R,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
% 0.30/0.54 = ( zero_zero @ nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0R
% 0.30/0.54 thf(fact_3106_min__number__of_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ U ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_number_of(1)
% 0.30/0.54 thf(fact_3107_min__0__1_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(3)
% 0.30/0.54 thf(fact_3108_min__0__1_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(4)
% 0.30/0.54 thf(fact_3109_min__0__1_I1_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(1)
% 0.30/0.54 thf(fact_3110_min__0__1_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 0.30/0.54 = ( zero_zero @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(2)
% 0.30/0.54 thf(fact_3111_min__0__1_I6_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(6)
% 0.30/0.54 thf(fact_3112_min__0__1_I5_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: num] :
% 0.30/0.54 ( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.30/0.54 = ( one_one @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_0_1(5)
% 0.30/0.54 thf(fact_3113_rotate__Suc,axiom,
% 0.30/0.54 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.54 ( ( rotate @ A @ ( suc @ N ) @ Xs )
% 0.30/0.54 = ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % rotate_Suc
% 0.30/0.54 thf(fact_3114_get__update__elem__neqIndex,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( heap @ A )
% 0.30/0.54 => ! [I: nat,J: nat,A3: array @ A,V: A,H2: heap_ext @ product_unit] :
% 0.30/0.54 ( ( I != J )
% 0.30/0.54 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H2 ) @ A3 ) @ I )
% 0.30/0.54 = ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % get_update_elem_neqIndex
% 0.30/0.54 thf(fact_3115_min__number__of_I4_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_number_of(4)
% 0.30/0.54 thf(fact_3116_min__number__of_I3_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 0.30/0.54 = ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_number_of(3)
% 0.30/0.54 thf(fact_3117_min__number__of_I2_J,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ( uminus @ A )
% 0.30/0.54 & ( numeral @ A )
% 0.30/0.54 & ( ord @ A ) )
% 0.30/0.54 => ! [U: num,V: num] :
% 0.30/0.54 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( numeral_numeral @ A @ U ) ) )
% 0.30/0.54 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_number_of(2)
% 0.30/0.54 thf(fact_3118_rotate__length01,axiom,
% 0.30/0.54 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.54 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.30/0.54 => ( ( rotate @ A @ N @ Xs )
% 0.30/0.54 = Xs ) ) ).
% 0.30/0.54
% 0.30/0.54 % rotate_length01
% 0.30/0.54 thf(fact_3119_rotate__id,axiom,
% 0.30/0.54 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.54 ( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.54 = ( zero_zero @ nat ) )
% 0.30/0.54 => ( ( rotate @ A @ N @ Xs )
% 0.30/0.54 = Xs ) ) ).
% 0.30/0.54
% 0.30/0.54 % rotate_id
% 0.30/0.54 thf(fact_3120_min__Suc__numeral,axiom,
% 0.30/0.54 ! [N: nat,K: num] :
% 0.30/0.54 ( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.54 = ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_Suc_numeral
% 0.30/0.54 thf(fact_3121_min__numeral__Suc,axiom,
% 0.30/0.54 ! [K: num,N: nat] :
% 0.30/0.54 ( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 0.30/0.54 = ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_numeral_Suc
% 0.30/0.54 thf(fact_3122_rotate1__length01,axiom,
% 0.30/0.54 ! [A: $tType,Xs: list @ A] :
% 0.30/0.54 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.30/0.54 => ( ( rotate1 @ A @ Xs )
% 0.30/0.54 = Xs ) ) ).
% 0.30/0.54
% 0.30/0.54 % rotate1_length01
% 0.30/0.54 thf(fact_3123_min__max__distrib2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
% 0.30/0.54 = ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_max_distrib2
% 0.30/0.54 thf(fact_3124_min__max__distrib1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.54 = ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_max_distrib1
% 0.30/0.54 thf(fact_3125_max__min__distrib2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 0.30/0.54 = ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_distrib2
% 0.30/0.54 thf(fact_3126_max__min__distrib1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.54 = ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % max_min_distrib1
% 0.30/0.54 thf(fact_3127_min__def,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ord @ A )
% 0.30/0.54 => ( ( ord_min @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_def
% 0.30/0.54 thf(fact_3128_min_Omono,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.54 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 0.30/0.54 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.mono
% 0.30/0.54 thf(fact_3129_min_OorderE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 => ( A3
% 0.30/0.54 = ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.orderE
% 0.30/0.54 thf(fact_3130_min_OorderI,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] :
% 0.30/0.54 ( ( A3
% 0.30/0.54 = ( ord_min @ A @ A3 @ B2 ) )
% 0.30/0.54 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.orderI
% 0.30/0.54 thf(fact_3131_min_OboundedE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 0.30/0.54 => ~ ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 => ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.boundedE
% 0.30/0.54 thf(fact_3132_min_OboundedI,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.54 => ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.54 => ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.boundedI
% 0.30/0.54 thf(fact_3133_min_Oorder__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] :
% 0.30/0.54 ( A5
% 0.30/0.54 = ( ord_min @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.order_iff
% 0.30/0.54 thf(fact_3134_min_Ocobounded1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.cobounded1
% 0.30/0.54 thf(fact_3135_min_Ocobounded2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.cobounded2
% 0.30/0.54 thf(fact_3136_min_Oabsorb__iff1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] :
% 0.30/0.54 ( ( ord_min @ A @ A5 @ B3 )
% 0.30/0.54 = A5 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb_iff1
% 0.30/0.54 thf(fact_3137_min_Oabsorb__iff2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less_eq @ A )
% 0.30/0.54 = ( ^ [B3: A,A5: A] :
% 0.30/0.54 ( ( ord_min @ A @ A5 @ B3 )
% 0.30/0.54 = B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.absorb_iff2
% 0.30/0.54 thf(fact_3138_min_OcoboundedI1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.54 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.coboundedI1
% 0.30/0.54 thf(fact_3139_min_OcoboundedI2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.54 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.coboundedI2
% 0.30/0.54 thf(fact_3140_min__le__iff__disj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ( ord_less_eq @ A @ X @ Z3 )
% 0.30/0.54 | ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_le_iff_disj
% 0.30/0.54 thf(fact_3141_min__absorb1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ord @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.54 => ( ( ord_min @ A @ X @ Y3 )
% 0.30/0.54 = X ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_absorb1
% 0.30/0.54 thf(fact_3142_min__absorb2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( order @ A )
% 0.30/0.54 => ! [Y3: A,X: A] :
% 0.30/0.54 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.54 => ( ( ord_min @ A @ X @ Y3 )
% 0.30/0.54 = Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_absorb2
% 0.30/0.54 thf(fact_3143_of__int__min,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linordered_idom @ A )
% 0.30/0.54 => ! [X: int,Y3: int] :
% 0.30/0.54 ( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y3 ) )
% 0.30/0.54 = ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_int_min
% 0.30/0.54 thf(fact_3144_min_Oassoc,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.54 = ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.assoc
% 0.30/0.54 thf(fact_3145_min_Ocommute,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_min @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] : ( ord_min @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.commute
% 0.30/0.54 thf(fact_3146_min_Oleft__commute,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.54 ( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
% 0.30/0.54 = ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.left_commute
% 0.30/0.54 thf(fact_3147_of__nat__min,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord181362715937106298miring @ A )
% 0.30/0.54 => ! [X: nat,Y3: nat] :
% 0.30/0.54 ( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y3 ) )
% 0.30/0.54 = ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % of_nat_min
% 0.30/0.54 thf(fact_3148_min__diff__distrib__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordered_ab_group_add @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( minus_minus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ord_min @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_diff_distrib_left
% 0.30/0.54 thf(fact_3149_min__add__distrib__right,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ X @ ( ord_min @ A @ Y3 @ Z3 ) )
% 0.30/0.54 = ( ord_min @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_add_distrib_right
% 0.30/0.54 thf(fact_3150_min__add__distrib__left,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( ordere2412721322843649153imp_le @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( plus_plus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ord_min @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_add_distrib_left
% 0.30/0.54 thf(fact_3151_min__less__iff__disj,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.54 ( ( ord_less @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.54 = ( ( ord_less @ A @ X @ Z3 )
% 0.30/0.54 | ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_less_iff_disj
% 0.30/0.54 thf(fact_3152_min_Ostrict__boundedE,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.54 ( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 0.30/0.54 => ~ ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.54 => ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.strict_boundedE
% 0.30/0.54 thf(fact_3153_min_Ostrict__order__iff,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ( ( ord_less @ A )
% 0.30/0.54 = ( ^ [A5: A,B3: A] :
% 0.30/0.54 ( ( A5
% 0.30/0.54 = ( ord_min @ A @ A5 @ B3 ) )
% 0.30/0.54 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.strict_order_iff
% 0.30/0.54 thf(fact_3154_min_Ostrict__coboundedI1,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.54 ( ( ord_less @ A @ A3 @ C2 )
% 0.30/0.54 => ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.strict_coboundedI1
% 0.30/0.54 thf(fact_3155_min_Ostrict__coboundedI2,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linorder @ A )
% 0.30/0.54 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.54 ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.54 => ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % min.strict_coboundedI2
% 0.30/0.54 thf(fact_3156_nat__mult__min__left,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q4 )
% 0.30/0.54 = ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_min_left
% 0.30/0.54 thf(fact_3157_nat__mult__min__right,axiom,
% 0.30/0.54 ! [M: nat,N: nat,Q4: nat] :
% 0.30/0.54 ( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q4 ) )
% 0.30/0.54 = ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_min_right
% 0.30/0.54 thf(fact_3158_min__diff,axiom,
% 0.30/0.54 ! [M: nat,I: nat,N: nat] :
% 0.30/0.54 ( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
% 0.30/0.54 = ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% 0.30/0.54
% 0.30/0.54 % min_diff
% 0.30/0.54 thf(fact_3159_minus__min__eq__max,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.54 = ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % minus_min_eq_max
% 0.30/0.54 thf(fact_3160_minus__max__eq__min,axiom,
% 0.30/0.54 ! [A: $tType] :
% 0.30/0.54 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.54 => ! [X: A,Y3: A] :
% 0.30/0.54 ( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y3 ) )
% 0.30/0.55 = ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_max_eq_min
% 0.30/0.55 thf(fact_3161_atLeast0__atMost__Suc,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.30/0.55 = ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeast0_atMost_Suc
% 0.30/0.55 thf(fact_3162_atLeastAtMost__insertL,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.55 => ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
% 0.30/0.55 = ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_insertL
% 0.30/0.55 thf(fact_3163_atLeastAtMostSuc__conv,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.55 => ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
% 0.30/0.55 = ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMostSuc_conv
% 0.30/0.55 thf(fact_3164_Icc__eq__insert__lb__nat,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.55 => ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
% 0.30/0.55 = ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_insert_lb_nat
% 0.30/0.55 thf(fact_3165_length__def,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( heap @ A )
% 0.30/0.55 => ( ( array_length @ A )
% 0.30/0.55 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H4 @ A5 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_def
% 0.30/0.55 thf(fact_3166_max__mult__distrib__left,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_idom @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
% 0.30/0.55 = ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
% 0.30/0.55 = ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % max_mult_distrib_left
% 0.30/0.55 thf(fact_3167_min__mult__distrib__left,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_idom @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.55 = ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
% 0.30/0.55 = ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % min_mult_distrib_left
% 0.30/0.55 thf(fact_3168_max__mult__distrib__right,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_idom @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % max_mult_distrib_right
% 0.30/0.55 thf(fact_3169_min__mult__distrib__right,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_idom @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % min_mult_distrib_right
% 0.30/0.55 thf(fact_3170_min__divide__distrib__right,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % min_divide_distrib_right
% 0.30/0.55 thf(fact_3171_max__divide__distrib__right,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [P5: A,X: A,Y3: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 0.30/0.55 => ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 0.30/0.55 = ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % max_divide_distrib_right
% 0.30/0.55 thf(fact_3172_and__int_Oelims,axiom,
% 0.30/0.55 ! [X: int,Xa3: int,Y3: int] :
% 0.30/0.55 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( uminus_uminus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) ) ) ) )
% 0.30/0.55 & ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( plus_plus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) )
% 0.30/0.55 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % and_int.elims
% 0.30/0.55 thf(fact_3173_and__int_Osimps,axiom,
% 0.30/0.55 ( ( bit_se5824344872417868541ns_and @ int )
% 0.30/0.55 = ( ^ [K3: int,L3: int] :
% 0.30/0.55 ( if @ int
% 0.30/0.55 @ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 @ ( uminus_uminus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 0.30/0.55 @ ( plus_plus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.30/0.55 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % and_int.simps
% 0.30/0.55 thf(fact_3174_nth__Cons__pos,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.55 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.30/0.55 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_Cons_pos
% 0.30/0.55 thf(fact_3175_get__set__neq,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType] :
% 0.30/0.55 ( ( ( heap @ A )
% 0.30/0.55 & ( heap @ B ) )
% 0.30/0.55 => ! [R: array @ A,S: array @ B,X: list @ B,H2: heap_ext @ product_unit] :
% 0.30/0.55 ( ( array_noteq @ A @ B @ R @ S )
% 0.30/0.55 => ( ( array_get @ A @ ( array_set @ B @ S @ X @ H2 ) @ R )
% 0.30/0.55 = ( array_get @ A @ H2 @ R ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % get_set_neq
% 0.30/0.55 thf(fact_3176_divmod__step__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [L: num,R: A,Q4: A] :
% 0.30/0.55 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 0.30/0.55 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 0.30/0.55 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_step_eq
% 0.30/0.55 thf(fact_3177_bot__apply,axiom,
% 0.30/0.55 ! [C: $tType,D: $tType] :
% 0.30/0.55 ( ( bot @ C )
% 0.30/0.55 => ( ( bot_bot @ ( D > C ) )
% 0.30/0.55 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot_apply
% 0.30/0.55 thf(fact_3178_set__same,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( heap @ A )
% 0.30/0.55 => ! [R: array @ A,X: list @ A,Y3: list @ A,H2: heap_ext @ product_unit] :
% 0.30/0.55 ( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y3 @ H2 ) )
% 0.30/0.55 = ( array_set @ A @ R @ X @ H2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_same
% 0.30/0.55 thf(fact_3179_max__bot2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_max @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.55 = X ) ) ).
% 0.30/0.55
% 0.30/0.55 % max_bot2
% 0.30/0.55 thf(fact_3180_max__bot,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_max @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.55 = X ) ) ).
% 0.30/0.55
% 0.30/0.55 % max_bot
% 0.30/0.55 thf(fact_3181_min__bot2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_min @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.55 = ( bot_bot @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % min_bot2
% 0.30/0.55 thf(fact_3182_min__bot,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_min @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.55 = ( bot_bot @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % min_bot
% 0.30/0.55 thf(fact_3183_get__set__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( heap @ A )
% 0.30/0.55 => ! [R: array @ A,X: list @ A,H2: heap_ext @ product_unit] :
% 0.30/0.55 ( ( array_get @ A @ ( array_set @ A @ R @ X @ H2 ) @ R )
% 0.30/0.55 = X ) ) ).
% 0.30/0.55
% 0.30/0.55 % get_set_eq
% 0.30/0.55 thf(fact_3184_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( preorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.55 = ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.55 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_3185_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( preorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.55 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_3186_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order @ A )
% 0.30/0.55 => ! [B2: A,A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.55 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_3187_nth__Cons__Suc,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A,N: nat] :
% 0.30/0.55 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
% 0.30/0.55 = ( nth @ A @ Xs @ N ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_Cons_Suc
% 0.30/0.55 thf(fact_3188_nth__Cons__0,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
% 0.30/0.55 = X ) ).
% 0.30/0.55
% 0.30/0.55 % nth_Cons_0
% 0.30/0.55 thf(fact_3189_subset__Compl__singleton,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,B2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.55 = ( ~ ( member @ A @ B2 @ A6 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % subset_Compl_singleton
% 0.30/0.55 thf(fact_3190_nth__Cons__numeral,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A,V: num] :
% 0.30/0.55 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
% 0.30/0.55 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_Cons_numeral
% 0.30/0.55 thf(fact_3191_bot__fun__def,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType] :
% 0.30/0.55 ( ( bot @ B )
% 0.30/0.55 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.55 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot_fun_def
% 0.30/0.55 thf(fact_3192_bot_Onot__eq__extremum,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( bot_bot @ A ) )
% 0.30/0.55 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot.not_eq_extremum
% 0.30/0.55 thf(fact_3193_bot_Oextremum__strict,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot.extremum_strict
% 0.30/0.55 thf(fact_3194_bot_Oextremum__uniqueI,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.55 => ( A3
% 0.30/0.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot.extremum_uniqueI
% 0.30/0.55 thf(fact_3195_bot_Oextremum__unique,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.55 = ( A3
% 0.30/0.55 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot.extremum_unique
% 0.30/0.55 thf(fact_3196_bot_Oextremum,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order_bot @ A )
% 0.30/0.55 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot.extremum
% 0.30/0.55 thf(fact_3197_not__psubset__empty,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A] :
% 0.30/0.55 ~ ( ord_less @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % not_psubset_empty
% 0.30/0.55 thf(fact_3198_diff__shunt__var,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.55 => ! [X: A,Y3: A] :
% 0.30/0.55 ( ( ( minus_minus @ A @ X @ Y3 )
% 0.30/0.55 = ( bot_bot @ A ) )
% 0.30/0.55 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % diff_shunt_var
% 0.30/0.55 thf(fact_3199_simp__from__to,axiom,
% 0.30/0.55 ( ( set_or1337092689740270186AtMost @ int )
% 0.30/0.55 = ( ^ [I5: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I5 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % simp_from_to
% 0.30/0.55 thf(fact_3200_length__Suc__conv,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.55 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.55 = ( suc @ N ) )
% 0.30/0.55 = ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.55 ( ( Xs
% 0.30/0.55 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.55 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 0.30/0.55 = N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_Suc_conv
% 0.30/0.55 thf(fact_3201_Suc__length__conv,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.55 ( ( ( suc @ N )
% 0.30/0.55 = ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 = ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.55 ( ( Xs
% 0.30/0.55 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.55 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 0.30/0.55 = N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_length_conv
% 0.30/0.55 thf(fact_3202_impossible__Cons,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.55 => ( Xs
% 0.30/0.55 != ( cons @ A @ X @ Ys2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % impossible_Cons
% 0.30/0.55 thf(fact_3203_subset__Compl__self__eq,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.55 = ( A6
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % subset_Compl_self_eq
% 0.30/0.55 thf(fact_3204_set__set__swap,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType] :
% 0.30/0.55 ( ( ( heap @ A )
% 0.30/0.55 & ( heap @ B ) )
% 0.30/0.55 => ! [R: array @ A,R4: array @ B,X: list @ A,X5: list @ B,H2: heap_ext @ product_unit] :
% 0.30/0.55 ( ( array_noteq @ A @ B @ R @ R4 )
% 0.30/0.55 => ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X5 @ H2 ) )
% 0.30/0.55 = ( array_set @ B @ R4 @ X5 @ ( array_set @ A @ R @ X @ H2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_set_swap
% 0.30/0.55 thf(fact_3205_Suc__le__length__iff,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 = ( ? [X2: A,Ys3: list @ A] :
% 0.30/0.55 ( ( Xs
% 0.30/0.55 = ( cons @ A @ X2 @ Ys3 ) )
% 0.30/0.55 & ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_le_length_iff
% 0.30/0.55 thf(fact_3206_Compl__insert,axiom,
% 0.30/0.55 ! [A: $tType,X: A,A6: set @ A] :
% 0.30/0.55 ( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) )
% 0.30/0.55 = ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Compl_insert
% 0.30/0.55 thf(fact_3207_list_Osize_I4_J,axiom,
% 0.30/0.55 ! [A: $tType,X21: A,X222: list @ A] :
% 0.30/0.55 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
% 0.30/0.55 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list.size(4)
% 0.30/0.55 thf(fact_3208_nth__Cons_H,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( ( N
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.30/0.55 = X ) )
% 0.30/0.55 & ( ( N
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.30/0.55 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_Cons'
% 0.30/0.55 thf(fact_3209_atLeastAtMostPlus1__int__conv,axiom,
% 0.30/0.55 ! [M: int,N: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 0.30/0.55 => ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 0.30/0.55 = ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMostPlus1_int_conv
% 0.30/0.55 thf(fact_3210_psubset__insert__iff,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,X: A,B7: set @ A] :
% 0.30/0.55 ( ( ord_less @ ( set @ A ) @ A6 @ ( insert @ A @ X @ B7 ) )
% 0.30/0.55 = ( ( ( member @ A @ X @ B7 )
% 0.30/0.55 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.55 & ( ~ ( member @ A @ X @ B7 )
% 0.30/0.55 => ( ( ( member @ A @ X @ A6 )
% 0.30/0.55 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 0.30/0.55 & ( ~ ( member @ A @ X @ A6 )
% 0.30/0.55 => ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % psubset_insert_iff
% 0.30/0.55 thf(fact_3211_nth__non__equal__first__eq,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Y3: A,Xs: list @ A,N: nat] :
% 0.30/0.55 ( ( X != Y3 )
% 0.30/0.55 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.30/0.55 = Y3 )
% 0.30/0.55 = ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.55 = Y3 )
% 0.30/0.55 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_non_equal_first_eq
% 0.30/0.55 thf(fact_3212_horner__sum__simps_I2_J,axiom,
% 0.30/0.55 ! [A: $tType,B: $tType] :
% 0.30/0.55 ( ( comm_semiring_0 @ A )
% 0.30/0.55 => ! [F: B > A,A3: A,X: B,Xs: list @ B] :
% 0.30/0.55 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
% 0.30/0.55 = ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % horner_sum_simps(2)
% 0.30/0.55 thf(fact_3213_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less @ num @ M @ N )
% 0.30/0.55 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less @ num @ M @ N )
% 0.30/0.55 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.55 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_algorithm_code(8)
% 0.30/0.55 thf(fact_3214_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less_eq @ num @ M @ N )
% 0.30/0.55 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ num @ M @ N )
% 0.30/0.55 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.55 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_algorithm_code(7)
% 0.30/0.55 thf(fact_3215_divides__aux__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [Q4: A,R: A] :
% 0.30/0.55 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 0.30/0.55 = ( R
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divides_aux_eq
% 0.30/0.55 thf(fact_3216_product__nth,axiom,
% 0.30/0.55 ! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.55 ( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
% 0.30/0.55 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
% 0.30/0.55 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % product_nth
% 0.30/0.55 thf(fact_3217_bit__0__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
% 0.30/0.55 = ( bot_bot @ ( nat > $o ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_0_eq
% 0.30/0.55 thf(fact_3218_set__decode__zero,axiom,
% 0.30/0.55 ( ( nat_set_decode @ ( zero_zero @ nat ) )
% 0.30/0.55 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_decode_zero
% 0.30/0.55 thf(fact_3219_length__product,axiom,
% 0.30/0.55 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.55 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys2 ) )
% 0.30/0.55 = ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_product
% 0.30/0.55 thf(fact_3220_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [M: num] :
% 0.30/0.55 ( ( unique8689654367752047608divmod @ A @ M @ one2 )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_algorithm_code(2)
% 0.30/0.55 thf(fact_3221_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [N: num] :
% 0.30/0.55 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_algorithm_code(3)
% 0.30/0.55 thf(fact_3222_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [N: num] :
% 0.30/0.55 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
% 0.30/0.55 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_algorithm_code(4)
% 0.30/0.55 thf(fact_3223_divmod__BitM__2__eq,axiom,
% 0.30/0.55 ! [M: num] :
% 0.30/0.55 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_BitM_2_eq
% 0.30/0.55 thf(fact_3224_bot__nat__def,axiom,
% 0.30/0.55 ( ( bot_bot @ nat )
% 0.30/0.55 = ( zero_zero @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % bot_nat_def
% 0.30/0.55 thf(fact_3225_divmod__divmod__step,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ( ( unique8689654367752047608divmod @ A )
% 0.30/0.55 = ( ^ [M2: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_divmod_step
% 0.30/0.55 thf(fact_3226_minus__one__div__numeral,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_one_div_numeral
% 0.30/0.55 thf(fact_3227_one__div__minus__numeral,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_div_minus_numeral
% 0.30/0.55 thf(fact_3228_neg__eucl__rel__int__mult__2,axiom,
% 0.30/0.55 ! [B2: int,A3: int,Q4: int,R: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
% 0.30/0.55 => ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 0.30/0.55 => ( 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 @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % neg_eucl_rel_int_mult_2
% 0.30/0.55 thf(fact_3229_Divides_Oadjust__div__eq,axiom,
% 0.30/0.55 ! [Q4: int,R: int] :
% 0.30/0.55 ( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 0.30/0.55 = ( plus_plus @ int @ Q4
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( R
% 0.30/0.55 != ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Divides.adjust_div_eq
% 0.30/0.55 thf(fact_3230_minus__numeral__div__numeral,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_numeral_div_numeral
% 0.30/0.55 thf(fact_3231_numeral__div__minus__numeral,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % numeral_div_minus_numeral
% 0.30/0.55 thf(fact_3232_eucl__rel__int__by0,axiom,
% 0.30/0.55 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int_by0
% 0.30/0.55 thf(fact_3233_eucl__rel__int__dividesI,axiom,
% 0.30/0.55 ! [L: int,K: int,Q4: int] :
% 0.30/0.55 ( ( L
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 => ( ( K
% 0.30/0.55 = ( times_times @ int @ Q4 @ L ) )
% 0.30/0.55 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int_dividesI
% 0.30/0.55 thf(fact_3234_zminus1__lemma,axiom,
% 0.30/0.55 ! [A3: int,B2: int,Q4: int,R: int] :
% 0.30/0.55 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 0.30/0.55 => ( ( B2
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 => ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
% 0.30/0.55 @ ( product_Pair @ int @ int
% 0.30/0.55 @ ( if @ int
% 0.30/0.55 @ ( R
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 @ ( uminus_uminus @ int @ Q4 )
% 0.30/0.55 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
% 0.30/0.55 @ ( if @ int
% 0.30/0.55 @ ( R
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 @ ( zero_zero @ int )
% 0.30/0.55 @ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zminus1_lemma
% 0.30/0.55 thf(fact_3235_eucl__rel__int__iff,axiom,
% 0.30/0.55 ! [K: int,L: int,Q4: int,R: int] :
% 0.30/0.55 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 0.30/0.55 = ( ( K
% 0.30/0.55 = ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R ) )
% 0.30/0.55 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.30/0.55 & ( ord_less @ int @ R @ L ) ) )
% 0.30/0.55 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.30/0.55 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.30/0.55 => ( ( ord_less @ int @ L @ R )
% 0.30/0.55 & ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.30/0.55 => ( Q4
% 0.30/0.55 = ( zero_zero @ int ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int_iff
% 0.30/0.55 thf(fact_3236_eucl__rel__int__remainderI,axiom,
% 0.30/0.55 ! [R: int,L: int,K: int,Q4: int] :
% 0.30/0.55 ( ( ( sgn_sgn @ int @ R )
% 0.30/0.55 = ( sgn_sgn @ int @ L ) )
% 0.30/0.55 => ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
% 0.30/0.55 => ( ( K
% 0.30/0.55 = ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R ) )
% 0.30/0.55 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int_remainderI
% 0.30/0.55 thf(fact_3237_eucl__rel__int_Ocases,axiom,
% 0.30/0.55 ! [A1: int,A22: int,A32: product_prod @ int @ int] :
% 0.30/0.55 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 0.30/0.55 => ( ( ( A22
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 => ( A32
% 0.30/0.55 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
% 0.30/0.55 => ( ! [Q5: int] :
% 0.30/0.55 ( ( A32
% 0.30/0.55 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
% 0.30/0.55 => ( ( A22
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 => ( A1
% 0.30/0.55 != ( times_times @ int @ Q5 @ A22 ) ) ) )
% 0.30/0.55 => ~ ! [R3: int,Q5: int] :
% 0.30/0.55 ( ( A32
% 0.30/0.55 = ( product_Pair @ int @ int @ Q5 @ R3 ) )
% 0.30/0.55 => ( ( ( sgn_sgn @ int @ R3 )
% 0.30/0.55 = ( sgn_sgn @ int @ A22 ) )
% 0.30/0.55 => ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
% 0.30/0.55 => ( A1
% 0.30/0.55 != ( plus_plus @ int @ ( times_times @ int @ Q5 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int.cases
% 0.30/0.55 thf(fact_3238_eucl__rel__int_Osimps,axiom,
% 0.30/0.55 ( eucl_rel_int
% 0.30/0.55 = ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
% 0.30/0.55 ( ? [K3: int] :
% 0.30/0.55 ( ( A12 = K3 )
% 0.30/0.55 & ( A23
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 & ( A33
% 0.30/0.55 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
% 0.30/0.55 | ? [L3: int,K3: int,Q3: int] :
% 0.30/0.55 ( ( A12 = K3 )
% 0.30/0.55 & ( A23 = L3 )
% 0.30/0.55 & ( A33
% 0.30/0.55 = ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
% 0.30/0.55 & ( L3
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 & ( K3
% 0.30/0.55 = ( times_times @ int @ Q3 @ L3 ) ) )
% 0.30/0.55 | ? [R5: int,L3: int,K3: int,Q3: int] :
% 0.30/0.55 ( ( A12 = K3 )
% 0.30/0.55 & ( A23 = L3 )
% 0.30/0.55 & ( A33
% 0.30/0.55 = ( product_Pair @ int @ int @ Q3 @ R5 ) )
% 0.30/0.55 & ( ( sgn_sgn @ int @ R5 )
% 0.30/0.55 = ( sgn_sgn @ int @ L3 ) )
% 0.30/0.55 & ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L3 ) )
% 0.30/0.55 & ( K3
% 0.30/0.55 = ( plus_plus @ int @ ( times_times @ int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eucl_rel_int.simps
% 0.30/0.55 thf(fact_3239_pos__eucl__rel__int__mult__2,axiom,
% 0.30/0.55 ! [B2: int,A3: int,Q4: int,R: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.55 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 0.30/0.55 => ( 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 @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_eucl_rel_int_mult_2
% 0.30/0.55 thf(fact_3240_and__int_Opelims,axiom,
% 0.30/0.55 ! [X: int,Xa3: int,Y3: int] :
% 0.30/0.55 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
% 0.30/0.55 => ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( uminus_uminus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) ) ) ) )
% 0.30/0.55 & ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( plus_plus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) )
% 0.30/0.55 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
% 0.30/0.55 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % and_int.pelims
% 0.30/0.55 thf(fact_3241_and__int_Opsimps,axiom,
% 0.30/0.55 ! [K: int,L: int] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
% 0.30/0.55 => ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 0.30/0.55 = ( uminus_uminus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
% 0.30/0.55 & ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 0.30/0.55 = ( plus_plus @ int
% 0.30/0.55 @ ( zero_neq_one_of_bool @ int
% 0.30/0.55 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.55 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
% 0.30/0.55 @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % and_int.psimps
% 0.30/0.55 thf(fact_3242_length__Cons,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
% 0.30/0.55 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_Cons
% 0.30/0.55 thf(fact_3243_normalize__negative,axiom,
% 0.30/0.55 ! [Q4: int,P5: int] :
% 0.30/0.55 ( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
% 0.30/0.55 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 0.30/0.55 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % normalize_negative
% 0.30/0.55 thf(fact_3244_upto__aux__rec,axiom,
% 0.30/0.55 ( upto_aux
% 0.30/0.55 = ( ^ [I5: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ Js @ ( upto_aux @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_aux_rec
% 0.30/0.55 thf(fact_3245_normalize__denom__zero,axiom,
% 0.30/0.55 ! [P5: int] :
% 0.30/0.55 ( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % normalize_denom_zero
% 0.30/0.55 thf(fact_3246_normalize__denom__pos,axiom,
% 0.30/0.55 ! [R: product_prod @ int @ int,P5: int,Q4: int] :
% 0.30/0.55 ( ( ( normalize @ R )
% 0.30/0.55 = ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 0.30/0.55 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% 0.30/0.55
% 0.30/0.55 % normalize_denom_pos
% 0.30/0.55 thf(fact_3247_normalize__crossproduct,axiom,
% 0.30/0.55 ! [Q4: int,S: int,P5: int,R: int] :
% 0.30/0.55 ( ( Q4
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 => ( ( S
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 => ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 0.30/0.55 = ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
% 0.30/0.55 => ( ( times_times @ int @ P5 @ S )
% 0.30/0.55 = ( times_times @ int @ R @ Q4 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % normalize_crossproduct
% 0.30/0.55 thf(fact_3248_and__int_Opinduct,axiom,
% 0.30/0.55 ! [A0: int,A1: int,P: int > int > $o] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 0.30/0.55 => ( ! [K2: int,L2: int] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L2 ) )
% 0.30/0.55 => ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.30/0.55 & ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.30/0.55 => ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 0.30/0.55 => ( P @ K2 @ L2 ) ) )
% 0.30/0.55 => ( P @ A0 @ A1 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % and_int.pinduct
% 0.30/0.55 thf(fact_3249_upto_Opinduct,axiom,
% 0.30/0.55 ! [A0: int,A1: int,P: int > int > $o] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 0.30/0.55 => ( ! [I4: int,J2: int] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ I4 @ J2 )
% 0.30/0.55 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) )
% 0.30/0.55 => ( P @ I4 @ J2 ) ) )
% 0.30/0.55 => ( P @ A0 @ A1 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto.pinduct
% 0.30/0.55 thf(fact_3250_bit__cut__integer__def,axiom,
% 0.30/0.55 ( code_bit_cut_integer
% 0.30/0.55 = ( ^ [K3: code_integer] :
% 0.30/0.55 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
% 0.30/0.55 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_cut_integer_def
% 0.30/0.55 thf(fact_3251_bezw__0,axiom,
% 0.30/0.55 ! [X: nat] :
% 0.30/0.55 ( ( bezw @ X @ ( zero_zero @ nat ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw_0
% 0.30/0.55 thf(fact_3252_nth__zip,axiom,
% 0.30/0.55 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.55 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.55 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
% 0.30/0.55 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_zip
% 0.30/0.55 thf(fact_3253_Gcd__0__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_Gcd @ A )
% 0.30/0.55 => ! [A6: set @ A] :
% 0.30/0.55 ( ( ( gcd_Gcd @ A @ A6 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 = ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_0_iff
% 0.30/0.55 thf(fact_3254_Gcd__empty,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_Gcd @ A )
% 0.30/0.55 => ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_empty
% 0.30/0.55 thf(fact_3255_divmod__integer_H__def,axiom,
% 0.30/0.55 ( ( unique8689654367752047608divmod @ code_integer )
% 0.30/0.55 = ( ^ [M2: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_integer'_def
% 0.30/0.55 thf(fact_3256_Gcd__nat__eq__one,axiom,
% 0.30/0.55 ! [N6: set @ nat] :
% 0.30/0.55 ( ( member @ nat @ ( one_one @ nat ) @ N6 )
% 0.30/0.55 => ( ( gcd_Gcd @ nat @ N6 )
% 0.30/0.55 = ( one_one @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_nat_eq_one
% 0.30/0.55 thf(fact_3257_Gcd__1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_Gcd @ A )
% 0.30/0.55 => ! [A6: set @ A] :
% 0.30/0.55 ( ( member @ A @ ( one_one @ A ) @ A6 )
% 0.30/0.55 => ( ( gcd_Gcd @ A @ A6 )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_1
% 0.30/0.55 thf(fact_3258_Gcd__int__greater__eq__0,axiom,
% 0.30/0.55 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_int_greater_eq_0
% 0.30/0.55 thf(fact_3259_Gcd__eq__1__I,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_Gcd @ A )
% 0.30/0.55 => ! [A3: A,A6: set @ A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.55 => ( ( gcd_Gcd @ A @ A6 )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Gcd_eq_1_I
% 0.30/0.55 thf(fact_3260_Code__Numeral_Opositive__def,axiom,
% 0.30/0.55 ( code_positive
% 0.30/0.55 = ( numeral_numeral @ code_integer ) ) ).
% 0.30/0.55
% 0.30/0.55 % Code_Numeral.positive_def
% 0.30/0.55 thf(fact_3261_integer__of__num__triv_I2_J,axiom,
% 0.30/0.55 ( ( code_integer_of_num @ ( bit0 @ one2 ) )
% 0.30/0.55 = ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % integer_of_num_triv(2)
% 0.30/0.55 thf(fact_3262_card__atLeastAtMost__int,axiom,
% 0.30/0.55 ! [L: int,U: int] :
% 0.30/0.55 ( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
% 0.30/0.55 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_atLeastAtMost_int
% 0.30/0.55 thf(fact_3263_possible__bit__def,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ( ( bit_se6407376104438227557le_bit @ A )
% 0.30/0.55 = ( ^ [Tyrep: itself @ A,N3: nat] :
% 0.30/0.55 ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
% 0.30/0.55 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % possible_bit_def
% 0.30/0.55 thf(fact_3264_abstract__boolean__algebra__sym__diff_Ointro,axiom,
% 0.30/0.55 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 0.30/0.55 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.55 => ( ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor )
% 0.30/0.55 => ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abstract_boolean_algebra_sym_diff.intro
% 0.30/0.55 thf(fact_3265_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
% 0.30/0.55 ! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
% 0.30/0.55 ( ! [X3: A,Y5: A] :
% 0.30/0.55 ( ( Xor @ X3 @ Y5 )
% 0.30/0.55 = ( Disj @ ( Conj @ X3 @ ( Compl @ Y5 ) ) @ ( Conj @ ( Compl @ X3 ) @ Y5 ) ) )
% 0.30/0.55 => ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% 0.30/0.55
% 0.30/0.55 % abstract_boolean_algebra_sym_diff_axioms.intro
% 0.30/0.55 thf(fact_3266_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( boolea5476839437570043046axioms @ A )
% 0.30/0.55 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
% 0.30/0.55 ! [X2: A,Y6: A] :
% 0.30/0.55 ( ( Xor2 @ X2 @ Y6 )
% 0.30/0.55 = ( Disj2 @ ( Conj2 @ X2 @ ( Compl2 @ Y6 ) ) @ ( Conj2 @ ( Compl2 @ X2 ) @ Y6 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abstract_boolean_algebra_sym_diff_axioms_def
% 0.30/0.55 thf(fact_3267_possible__bit__0,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % possible_bit_0
% 0.30/0.55 thf(fact_3268_possible__bit__less__imp,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [Tyrep2: itself @ A,I: nat,J: nat] :
% 0.30/0.55 ( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
% 0.30/0.55 => ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.55 => ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % possible_bit_less_imp
% 0.30/0.55 thf(fact_3269_integer__of__num__triv_I1_J,axiom,
% 0.30/0.55 ( ( code_integer_of_num @ one2 )
% 0.30/0.55 = ( one_one @ code_integer ) ) ).
% 0.30/0.55
% 0.30/0.55 % integer_of_num_triv(1)
% 0.30/0.55 thf(fact_3270_integer__of__num__def,axiom,
% 0.30/0.55 ( code_integer_of_num
% 0.30/0.55 = ( numeral_numeral @ code_integer ) ) ).
% 0.30/0.55
% 0.30/0.55 % integer_of_num_def
% 0.30/0.55 thf(fact_3271_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
% 0.30/0.55 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 0.30/0.55 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 0.30/0.55 => ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% 0.30/0.55
% 0.30/0.55 % abstract_boolean_algebra_sym_diff.axioms(2)
% 0.30/0.55 thf(fact_3272_abstract__boolean__algebra__sym__diff__def,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( boolea3799213064322606851m_diff @ A )
% 0.30/0.55 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A,Xor2: A > A > A] :
% 0.30/0.55 ( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One )
% 0.30/0.55 & ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abstract_boolean_algebra_sym_diff_def
% 0.30/0.55 thf(fact_3273_card__Diff__insert,axiom,
% 0.30/0.55 ! [A: $tType,A3: A,A6: set @ A,B7: set @ A] :
% 0.30/0.55 ( ( member @ A @ A3 @ A6 )
% 0.30/0.55 => ( ~ ( member @ A @ A3 @ B7 )
% 0.30/0.55 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ B7 ) ) )
% 0.30/0.55 = ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_Diff_insert
% 0.30/0.55 thf(fact_3274_card_Oempty,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.55 = ( zero_zero @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % card.empty
% 0.30/0.55 thf(fact_3275_card__insert__le__m1,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,Y3: set @ A,X: A] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.55 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.55 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y3 ) ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_insert_le_m1
% 0.30/0.55 thf(fact_3276_card__Diff__singleton__if,axiom,
% 0.30/0.55 ! [A: $tType,X: A,A6: set @ A] :
% 0.30/0.55 ( ( ( member @ A @ X @ A6 )
% 0.30/0.55 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.55 = ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) )
% 0.30/0.55 & ( ~ ( member @ A @ X @ A6 )
% 0.30/0.55 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.55 = ( finite_card @ A @ A6 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_Diff_singleton_if
% 0.30/0.55 thf(fact_3277_card__Diff__singleton,axiom,
% 0.30/0.55 ! [A: $tType,X: A,A6: set @ A] :
% 0.30/0.55 ( ( member @ A @ X @ A6 )
% 0.30/0.55 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.55 = ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_Diff_singleton
% 0.30/0.55 thf(fact_3278_card__atLeastAtMost,axiom,
% 0.30/0.55 ! [L: nat,U: nat] :
% 0.30/0.55 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
% 0.30/0.55 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_atLeastAtMost
% 0.30/0.55 thf(fact_3279_sgn__integer__code,axiom,
% 0.30/0.55 ( ( sgn_sgn @ code_integer )
% 0.30/0.55 = ( ^ [K3: code_integer] :
% 0.30/0.55 ( if @ code_integer
% 0.30/0.55 @ ( K3
% 0.30/0.55 = ( zero_zero @ code_integer ) )
% 0.30/0.55 @ ( zero_zero @ code_integer )
% 0.30/0.55 @ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sgn_integer_code
% 0.30/0.55 thf(fact_3280_card__insert__le,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ ( insert @ A @ X @ A6 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_insert_le
% 0.30/0.55 thf(fact_3281_card__1__singletonE,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A] :
% 0.30/0.55 ( ( ( finite_card @ A @ A6 )
% 0.30/0.55 = ( one_one @ nat ) )
% 0.30/0.55 => ~ ! [X3: A] :
% 0.30/0.55 ( A6
% 0.30/0.55 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_1_singletonE
% 0.30/0.55 thf(fact_3282_card__1__singleton__iff,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A] :
% 0.30/0.55 ( ( ( finite_card @ A @ A6 )
% 0.30/0.55 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.55 = ( ? [X2: A] :
% 0.30/0.55 ( A6
% 0.30/0.55 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_1_singleton_iff
% 0.30/0.55 thf(fact_3283_card__eq__SucD,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,K: nat] :
% 0.30/0.55 ( ( ( finite_card @ A @ A6 )
% 0.30/0.55 = ( suc @ K ) )
% 0.30/0.55 => ? [B5: A,B9: set @ A] :
% 0.30/0.55 ( ( A6
% 0.30/0.55 = ( insert @ A @ B5 @ B9 ) )
% 0.30/0.55 & ~ ( member @ A @ B5 @ B9 )
% 0.30/0.55 & ( ( finite_card @ A @ B9 )
% 0.30/0.55 = K )
% 0.30/0.55 & ( ( K
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( B9
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_eq_SucD
% 0.30/0.55 thf(fact_3284_card__Suc__eq,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,K: nat] :
% 0.30/0.55 ( ( ( finite_card @ A @ A6 )
% 0.30/0.55 = ( suc @ K ) )
% 0.30/0.55 = ( ? [B3: A,B6: set @ A] :
% 0.30/0.55 ( ( A6
% 0.30/0.55 = ( insert @ A @ B3 @ B6 ) )
% 0.30/0.55 & ~ ( member @ A @ B3 @ B6 )
% 0.30/0.55 & ( ( finite_card @ A @ B6 )
% 0.30/0.55 = K )
% 0.30/0.55 & ( ( K
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( B6
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_Suc_eq
% 0.30/0.55 thf(fact_3285_card__Diff1__le,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_Diff1_le
% 0.30/0.55 thf(fact_3286_drop__bit__exp__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.55 => ! [M: nat,N: nat] :
% 0.30/0.55 ( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.30/0.55 = ( times_times @ A
% 0.30/0.55 @ ( zero_neq_one_of_bool @ A
% 0.30/0.55 @ ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.55 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
% 0.30/0.55 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % drop_bit_exp_eq
% 0.30/0.55 thf(fact_3287_bit__minus__2__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.55 => ! [N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_minus_2_iff
% 0.30/0.55 thf(fact_3288_bit__double__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [A3: A,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
% 0.30/0.55 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.55 & ( N
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_double_iff
% 0.30/0.55 thf(fact_3289_bit__mask__sub__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [M: nat,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_mask_sub_iff
% 0.30/0.55 thf(fact_3290_bit__minus__exp__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.55 => ! [M: nat,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_minus_exp_iff
% 0.30/0.55 thf(fact_3291_bit__minus__1__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.55 => ! [N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.30/0.55 = ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_minus_1_iff
% 0.30/0.55 thf(fact_3292_plus__integer__code_I1_J,axiom,
% 0.30/0.55 ! [K: code_integer] :
% 0.30/0.55 ( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.30/0.55 = K ) ).
% 0.30/0.55
% 0.30/0.55 % plus_integer_code(1)
% 0.30/0.55 thf(fact_3293_plus__integer__code_I2_J,axiom,
% 0.30/0.55 ! [L: code_integer] :
% 0.30/0.55 ( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 0.30/0.55 = L ) ).
% 0.30/0.55
% 0.30/0.55 % plus_integer_code(2)
% 0.30/0.55 thf(fact_3294_less__integer__code_I1_J,axiom,
% 0.30/0.55 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_integer_code(1)
% 0.30/0.55 thf(fact_3295_abs__integer__code,axiom,
% 0.30/0.55 ( ( abs_abs @ code_integer )
% 0.30/0.55 = ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abs_integer_code
% 0.30/0.55 thf(fact_3296_uminus__integer__code_I1_J,axiom,
% 0.30/0.55 ( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
% 0.30/0.55 = ( zero_zero @ code_integer ) ) ).
% 0.30/0.55
% 0.30/0.55 % uminus_integer_code(1)
% 0.30/0.55 thf(fact_3297_minus__integer__code_I1_J,axiom,
% 0.30/0.55 ! [K: code_integer] :
% 0.30/0.55 ( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.30/0.55 = K ) ).
% 0.30/0.55
% 0.30/0.55 % minus_integer_code(1)
% 0.30/0.55 thf(fact_3298_minus__integer__code_I2_J,axiom,
% 0.30/0.55 ! [L: code_integer] :
% 0.30/0.55 ( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 0.30/0.55 = ( uminus_uminus @ code_integer @ L ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_integer_code(2)
% 0.30/0.55 thf(fact_3299_bit__mask__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.55 => ! [M: nat,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_mask_iff
% 0.30/0.55 thf(fact_3300_bit__of__nat__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [M: nat,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_of_nat_iff
% 0.30/0.55 thf(fact_3301_bit__minus__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.55 => ! [A3: A,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_minus_iff
% 0.30/0.55 thf(fact_3302_bit__push__bit__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.55 => ! [M: nat,A3: A,N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) @ N )
% 0.30/0.55 = ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.55 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.30/0.55 & ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_push_bit_iff
% 0.30/0.55 thf(fact_3303_fold__possible__bit,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [N: nat] :
% 0.30/0.55 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 = ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % fold_possible_bit
% 0.30/0.55 thf(fact_3304_bit__2__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( bit_semiring_bits @ A )
% 0.30/0.55 => ! [N: nat] :
% 0.30/0.55 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.55 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
% 0.30/0.55 & ( N
% 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bit_2_iff
% 0.30/0.55 thf(fact_3305_CHAR__eq0__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 = ( ! [N3: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.55 => ( ( semiring_1_of_nat @ A @ N3 )
% 0.30/0.55 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % CHAR_eq0_iff
% 0.30/0.55 thf(fact_3306_CHAR__eq__posI,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ! [C2: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
% 0.30/0.55 => ( ( ( semiring_1_of_nat @ A @ C2 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 => ( ! [X3: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
% 0.30/0.55 => ( ( ord_less @ nat @ X3 @ C2 )
% 0.30/0.55 => ( ( semiring_1_of_nat @ A @ X3 )
% 0.30/0.55 != ( zero_zero @ A ) ) ) )
% 0.30/0.55 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 0.30/0.55 = C2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % CHAR_eq_posI
% 0.30/0.55 thf(fact_3307_CHAR__pos__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
% 0.30/0.55 = ( ? [N3: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.55 & ( ( semiring_1_of_nat @ A @ N3 )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % CHAR_pos_iff
% 0.30/0.55 thf(fact_3308_of__nat__CHAR,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % of_nat_CHAR
% 0.30/0.55 thf(fact_3309_CHAR__eqI,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ! [C2: nat] :
% 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ C2 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 => ( ! [X3: nat] :
% 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ X3 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 => ( dvd_dvd @ nat @ C2 @ X3 ) )
% 0.30/0.55 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 0.30/0.55 = C2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % CHAR_eqI
% 0.30/0.55 thf(fact_3310_CHAR__eq__0,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_char_0 @ A )
% 0.30/0.55 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % CHAR_eq_0
% 0.30/0.55 thf(fact_3311_of__nat__eq__0__iff__char__dvd,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_1 @ A )
% 0.30/0.55 => ! [N: nat] :
% 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ N )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 = ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % of_nat_eq_0_iff_char_dvd
% 0.30/0.55 thf(fact_3312_divmod__integer__def,axiom,
% 0.30/0.55 ( code_divmod_integer
% 0.30/0.55 = ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L3 ) @ ( modulo_modulo @ code_integer @ K3 @ L3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_integer_def
% 0.30/0.55 thf(fact_3313_divmod__abs__def,axiom,
% 0.30/0.55 ( code_divmod_abs
% 0.30/0.55 = ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_abs_def
% 0.30/0.55 thf(fact_3314_card__greaterThanLessThan__int,axiom,
% 0.30/0.55 ! [L: int,U: int] :
% 0.30/0.55 ( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
% 0.30/0.55 = ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_greaterThanLessThan_int
% 0.30/0.55 thf(fact_3315_nth__enumerate__eq,axiom,
% 0.30/0.55 ! [A: $tType,M: nat,Xs: list @ A,N: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
% 0.30/0.55 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_enumerate_eq
% 0.30/0.55 thf(fact_3316_count__list_Osimps_I2_J,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Y3: A,Xs: list @ A] :
% 0.30/0.55 ( ( ( X = Y3 )
% 0.30/0.55 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
% 0.30/0.55 = ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y3 ) @ ( one_one @ nat ) ) ) )
% 0.30/0.55 & ( ( X != Y3 )
% 0.30/0.55 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
% 0.30/0.55 = ( count_list @ A @ Xs @ Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % count_list.simps(2)
% 0.30/0.55 thf(fact_3317_greaterThanLessThan__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ord @ A )
% 0.30/0.55 => ! [I: A,L: A,U: A] :
% 0.30/0.55 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.30/0.55 = ( ( ord_less @ A @ L @ I )
% 0.30/0.55 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_iff
% 0.30/0.55 thf(fact_3318_greaterThanLessThan__empty,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order @ A )
% 0.30/0.55 => ! [L: A,K: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ L @ K )
% 0.30/0.55 => ( ( set_or5935395276787703475ssThan @ A @ K @ L )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_empty
% 0.30/0.55 thf(fact_3319_greaterThanLessThan__empty__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.55 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_empty_iff
% 0.30/0.55 thf(fact_3320_greaterThanLessThan__empty__iff2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.55 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
% 0.30/0.55 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_empty_iff2
% 0.30/0.55 thf(fact_3321_enumerate__simps_I2_J,axiom,
% 0.30/0.55 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 0.30/0.55 ( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
% 0.30/0.55 = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % enumerate_simps(2)
% 0.30/0.55 thf(fact_3322_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_subseteq_greaterThanLessThan
% 0.30/0.55 thf(fact_3323_divmod__abs__code_I6_J,axiom,
% 0.30/0.55 ! [J: code_integer] :
% 0.30/0.55 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
% 0.30/0.55 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_abs_code(6)
% 0.30/0.55 thf(fact_3324_count__le__length,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.55
% 0.30/0.55 % count_le_length
% 0.30/0.55 thf(fact_3325_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_subseteq_atLeastAtMost_iff
% 0.30/0.55 thf(fact_3326_divmod__abs__code_I5_J,axiom,
% 0.30/0.55 ! [J: code_integer] :
% 0.30/0.55 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
% 0.30/0.55 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_abs_code(5)
% 0.30/0.55 thf(fact_3327_sqr_Osimps_I3_J,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( sqr @ ( bit1 @ N ) )
% 0.30/0.55 = ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sqr.simps(3)
% 0.30/0.55 thf(fact_3328_quotient__of__number_I5_J,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_number(5)
% 0.30/0.55 thf(fact_3329_prod__decode__aux_Oelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
% 0.30/0.55 ( ( ( nat_prod_decode_aux @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( ord_less_eq @ nat @ Xa3 @ X )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_decode_aux.elims
% 0.30/0.55 thf(fact_3330_prod__decode__aux_Osimps,axiom,
% 0.30/0.55 ( nat_prod_decode_aux
% 0.30/0.55 = ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_decode_aux.simps
% 0.30/0.55 thf(fact_3331_nat__of__integer__non__positive,axiom,
% 0.30/0.55 ! [K: code_integer] :
% 0.30/0.55 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.30/0.55 => ( ( code_nat_of_integer @ K )
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_of_integer_non_positive
% 0.30/0.55 thf(fact_3332_nat__of__integer__of__nat,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 0.30/0.55 = N ) ).
% 0.30/0.55
% 0.30/0.55 % nat_of_integer_of_nat
% 0.30/0.55 thf(fact_3333_of__nat__of__integer,axiom,
% 0.30/0.55 ! [K: code_integer] :
% 0.30/0.55 ( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
% 0.30/0.55 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % of_nat_of_integer
% 0.30/0.55 thf(fact_3334_card__greaterThanLessThan,axiom,
% 0.30/0.55 ! [L: nat,U: nat] :
% 0.30/0.55 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
% 0.30/0.55 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_greaterThanLessThan
% 0.30/0.55 thf(fact_3335_rat__one__code,axiom,
% 0.30/0.55 ( ( quotient_of @ ( one_one @ rat ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % rat_one_code
% 0.30/0.55 thf(fact_3336_rat__zero__code,axiom,
% 0.30/0.55 ( ( quotient_of @ ( zero_zero @ rat ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % rat_zero_code
% 0.30/0.55 thf(fact_3337_quotient__of__number_I4_J,axiom,
% 0.30/0.55 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_number(4)
% 0.30/0.55 thf(fact_3338_quotient__of__number_I3_J,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_number(3)
% 0.30/0.55 thf(fact_3339_prod__decode__aux_Ocases,axiom,
% 0.30/0.55 ! [X: product_prod @ nat @ nat] :
% 0.30/0.55 ~ ! [K2: nat,M6: nat] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( product_Pair @ nat @ nat @ K2 @ M6 ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_decode_aux.cases
% 0.30/0.55 thf(fact_3340_sgn__rat__def,axiom,
% 0.30/0.55 ( ( sgn_sgn @ rat )
% 0.30/0.55 = ( ^ [A5: rat] :
% 0.30/0.55 ( if @ rat
% 0.30/0.55 @ ( A5
% 0.30/0.55 = ( zero_zero @ rat ) )
% 0.30/0.55 @ ( zero_zero @ rat )
% 0.30/0.55 @ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sgn_rat_def
% 0.30/0.55 thf(fact_3341_diff__rat__def,axiom,
% 0.30/0.55 ( ( minus_minus @ rat )
% 0.30/0.55 = ( ^ [Q3: rat,R5: rat] : ( plus_plus @ rat @ Q3 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % diff_rat_def
% 0.30/0.55 thf(fact_3342_abs__rat__def,axiom,
% 0.30/0.55 ( ( abs_abs @ rat )
% 0.30/0.55 = ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abs_rat_def
% 0.30/0.55 thf(fact_3343_sqr_Osimps_I2_J,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( sqr @ ( bit0 @ N ) )
% 0.30/0.55 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sqr.simps(2)
% 0.30/0.55 thf(fact_3344_sqr_Osimps_I1_J,axiom,
% 0.30/0.55 ( ( sqr @ one2 )
% 0.30/0.55 = one2 ) ).
% 0.30/0.55
% 0.30/0.55 % sqr.simps(1)
% 0.30/0.55 thf(fact_3345_sqr__conv__mult,axiom,
% 0.30/0.55 ( sqr
% 0.30/0.55 = ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sqr_conv_mult
% 0.30/0.55 thf(fact_3346_quotient__of__denom__pos,axiom,
% 0.30/0.55 ! [R: rat,P5: int,Q4: int] :
% 0.30/0.55 ( ( ( quotient_of @ R )
% 0.30/0.55 = ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 0.30/0.55 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_denom_pos
% 0.30/0.55 thf(fact_3347_numeral__sqr,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_numeral @ A )
% 0.30/0.55 => ! [K: num] :
% 0.30/0.55 ( ( numeral_numeral @ A @ ( sqr @ K ) )
% 0.30/0.55 = ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % numeral_sqr
% 0.30/0.55 thf(fact_3348_quotient__of__int,axiom,
% 0.30/0.55 ! [A3: int] :
% 0.30/0.55 ( ( quotient_of @ ( of_int @ A3 ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_int
% 0.30/0.55 thf(fact_3349_pow_Osimps_I3_J,axiom,
% 0.30/0.55 ! [X: num,Y3: num] :
% 0.30/0.55 ( ( pow @ X @ ( bit1 @ Y3 ) )
% 0.30/0.55 = ( times_times @ num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).
% 0.30/0.55
% 0.30/0.55 % pow.simps(3)
% 0.30/0.55 thf(fact_3350_Frct__code__post_I4_J,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
% 0.30/0.55 = ( numeral_numeral @ rat @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(4)
% 0.30/0.55 thf(fact_3351_Frct__code__post_I5_J,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
% 0.30/0.55 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(5)
% 0.30/0.55 thf(fact_3352_prod__encode__prod__decode__aux,axiom,
% 0.30/0.55 ! [K: nat,M: nat] :
% 0.30/0.55 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.30/0.55 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_encode_prod_decode_aux
% 0.30/0.55 thf(fact_3353_prod__encode__eq,axiom,
% 0.30/0.55 ! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
% 0.30/0.55 ( ( ( nat_prod_encode @ X )
% 0.30/0.55 = ( nat_prod_encode @ Y3 ) )
% 0.30/0.55 = ( X = Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_encode_eq
% 0.30/0.55 thf(fact_3354_less__eq__rat__def,axiom,
% 0.30/0.55 ( ( ord_less_eq @ rat )
% 0.30/0.55 = ( ^ [X2: rat,Y6: rat] :
% 0.30/0.55 ( ( ord_less @ rat @ X2 @ Y6 )
% 0.30/0.55 | ( X2 = Y6 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_eq_rat_def
% 0.30/0.55 thf(fact_3355_Frct__code__post_I9_J,axiom,
% 0.30/0.55 ! [Q4: product_prod @ int @ int] :
% 0.30/0.55 ( ( uminus_uminus @ rat @ ( uminus_uminus @ rat @ ( frct @ Q4 ) ) )
% 0.30/0.55 = ( frct @ Q4 ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(9)
% 0.30/0.55 thf(fact_3356_pow_Osimps_I1_J,axiom,
% 0.30/0.55 ! [X: num] :
% 0.30/0.55 ( ( pow @ X @ one2 )
% 0.30/0.55 = X ) ).
% 0.30/0.55
% 0.30/0.55 % pow.simps(1)
% 0.30/0.55 thf(fact_3357_le__prod__encode__2,axiom,
% 0.30/0.55 ! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_prod_encode_2
% 0.30/0.55 thf(fact_3358_le__prod__encode__1,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_prod_encode_1
% 0.30/0.55 thf(fact_3359_Frct__code__post_I2_J,axiom,
% 0.30/0.55 ! [A3: int] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
% 0.30/0.55 = ( zero_zero @ rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(2)
% 0.30/0.55 thf(fact_3360_Frct__code__post_I1_J,axiom,
% 0.30/0.55 ! [A3: int] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
% 0.30/0.55 = ( zero_zero @ rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(1)
% 0.30/0.55 thf(fact_3361_Frct__code__post_I3_J,axiom,
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
% 0.30/0.55 = ( one_one @ rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(3)
% 0.30/0.55 thf(fact_3362_pow_Osimps_I2_J,axiom,
% 0.30/0.55 ! [X: num,Y3: num] :
% 0.30/0.55 ( ( pow @ X @ ( bit0 @ Y3 ) )
% 0.30/0.55 = ( sqr @ ( pow @ X @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pow.simps(2)
% 0.30/0.55 thf(fact_3363_Frct__code__post_I8_J,axiom,
% 0.30/0.55 ! [A3: int,B2: int] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
% 0.30/0.55 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(8)
% 0.30/0.55 thf(fact_3364_Frct__code__post_I7_J,axiom,
% 0.30/0.55 ! [A3: int,B2: int] :
% 0.30/0.55 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
% 0.30/0.55 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Frct_code_post(7)
% 0.30/0.55 thf(fact_3365_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.55 ! [N: nat,J: nat,I: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
% 0.30/0.55 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
% 0.30/0.55 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_sorted_list_of_set_greaterThanLessThan
% 0.30/0.55 thf(fact_3366_rat__sgn__code,axiom,
% 0.30/0.55 ! [P5: rat] :
% 0.30/0.55 ( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % rat_sgn_code
% 0.30/0.55 thf(fact_3367_one__div__numeral,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [N: num] :
% 0.30/0.55 ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.55 = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_div_numeral
% 0.30/0.55 thf(fact_3368_in__measures_I2_J,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Y3: A,F: A > nat,Fs: list @ ( A > nat )] :
% 0.30/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
% 0.30/0.55 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.55 | ( ( ( F @ X )
% 0.30/0.55 = ( F @ Y3 ) )
% 0.30/0.55 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % in_measures(2)
% 0.30/0.55 thf(fact_3369_Cons__lenlex__iff,axiom,
% 0.30/0.55 ! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.55 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
% 0.30/0.55 = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
% 0.30/0.55 | ( ( ( size_size @ ( list @ A ) @ Ms )
% 0.30/0.55 = ( size_size @ ( list @ A ) @ Ns ) )
% 0.30/0.55 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
% 0.30/0.55 | ( ( M = N )
% 0.30/0.55 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Cons_lenlex_iff
% 0.30/0.55 thf(fact_3370_fst__divmod__integer,axiom,
% 0.30/0.55 ! [K: code_integer,L: code_integer] :
% 0.30/0.55 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 0.30/0.55 = ( divide_divide @ code_integer @ K @ L ) ) ).
% 0.30/0.55
% 0.30/0.55 % fst_divmod_integer
% 0.30/0.55 thf(fact_3371_fst__divmod__abs,axiom,
% 0.30/0.55 ! [K: code_integer,L: code_integer] :
% 0.30/0.55 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 0.30/0.55 = ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % fst_divmod_abs
% 0.30/0.55 thf(fact_3372_Suc__0__div__numeral,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.55 = ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_0_div_numeral
% 0.30/0.55 thf(fact_3373_lenlex__length,axiom,
% 0.30/0.55 ! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.55 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
% 0.30/0.55 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % lenlex_length
% 0.30/0.55 thf(fact_3374_measures__less,axiom,
% 0.30/0.55 ! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
% 0.30/0.55 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % measures_less
% 0.30/0.55 thf(fact_3375_measures__lesseq,axiom,
% 0.30/0.55 ! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) )
% 0.30/0.55 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % measures_lesseq
% 0.30/0.55 thf(fact_3376_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.55 ! [I: nat,J: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( suc @ I ) @ J )
% 0.30/0.55 => ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
% 0.30/0.55 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sorted_list_of_set_greaterThanLessThan
% 0.30/0.55 thf(fact_3377_bezw_Osimps,axiom,
% 0.30/0.55 ( bezw
% 0.30/0.55 = ( ^ [X2: nat,Y6: nat] :
% 0.30/0.55 ( if @ ( product_prod @ int @ int )
% 0.30/0.55 @ ( Y6
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
% 0.30/0.55 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw.simps
% 0.30/0.55 thf(fact_3378_bezw_Oelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
% 0.30/0.55 ( ( ( bezw @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( Xa3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 0.30/0.55 & ( ( Xa3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw.elims
% 0.30/0.55 thf(fact_3379_bezw__non__0,axiom,
% 0.30/0.55 ! [Y3: nat,X: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y3 )
% 0.30/0.55 => ( ( bezw @ X @ Y3 )
% 0.30/0.55 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw_non_0
% 0.30/0.55 thf(fact_3380_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.55 ! [N: nat,J: nat,I: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
% 0.30/0.55 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
% 0.30/0.55 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_sorted_list_of_set_greaterThanAtMost
% 0.30/0.55 thf(fact_3381_listrel__iff__nth,axiom,
% 0.30/0.55 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.55 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 0.30/0.55 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.55 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.55 & ! [N3: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N3 ) @ ( nth @ B @ Ys2 @ N3 ) ) @ R ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % listrel_iff_nth
% 0.30/0.55 thf(fact_3382_greaterThanAtMost__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ord @ A )
% 0.30/0.55 => ! [I: A,L: A,U: A] :
% 0.30/0.55 ( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.30/0.55 = ( ( ord_less @ A @ L @ I )
% 0.30/0.55 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_iff
% 0.30/0.55 thf(fact_3383_greaterThanAtMost__empty,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( order @ A )
% 0.30/0.55 => ! [L: A,K: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ L @ K )
% 0.30/0.55 => ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_empty
% 0.30/0.55 thf(fact_3384_greaterThanAtMost__empty__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( preorder @ A )
% 0.30/0.55 => ! [K: A,L: A] :
% 0.30/0.55 ( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 0.30/0.55 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.55 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_empty_iff
% 0.30/0.55 thf(fact_3385_greaterThanAtMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( preorder @ A )
% 0.30/0.55 => ! [K: A,L: A] :
% 0.30/0.55 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.55 = ( set_or3652927894154168847AtMost @ A @ K @ L ) )
% 0.30/0.55 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_empty_iff2
% 0.30/0.55 thf(fact_3386_one__mod__numeral,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ! [N: num] :
% 0.30/0.55 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.55 = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_mod_numeral
% 0.30/0.55 thf(fact_3387_Ioc__inj,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
% 0.30/0.55 = ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ D2 @ C2 ) )
% 0.30/0.55 | ( ( A3 = C2 )
% 0.30/0.55 & ( B2 = D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Ioc_inj
% 0.30/0.55 thf(fact_3388_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.30/0.55 ! [L: nat,U: nat] :
% 0.30/0.55 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
% 0.30/0.55 = ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastSucAtMost_greaterThanAtMost
% 0.30/0.55 thf(fact_3389_Ioc__subset__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.55 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Ioc_subset_iff
% 0.30/0.55 thf(fact_3390_divides__aux__def,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.55 => ( ( unique5940410009612947441es_aux @ A )
% 0.30/0.55 = ( ^ [Qr: product_prod @ A @ A] :
% 0.30/0.55 ( ( product_snd @ A @ A @ Qr )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divides_aux_def
% 0.30/0.55 thf(fact_3391_quotient__of__denom__pos_H,axiom,
% 0.30/0.55 ! [R: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % quotient_of_denom_pos'
% 0.30/0.55 thf(fact_3392_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.55 ! [I: nat,J: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
% 0.30/0.55 => ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
% 0.30/0.55 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sorted_list_of_set_greaterThanAtMost
% 0.30/0.55 thf(fact_3393_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_subseteq_atLeastAtMost_iff
% 0.30/0.55 thf(fact_3394_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( dense_linorder @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 0.30/0.55 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.55 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_subseteq_greaterThanAtMost_iff
% 0.30/0.55 thf(fact_3395_minus__one__mod__numeral,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_one_mod_numeral
% 0.30/0.55 thf(fact_3396_one__mod__minus__numeral,axiom,
% 0.30/0.55 ! [N: num] :
% 0.30/0.55 ( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_mod_minus_numeral
% 0.30/0.55 thf(fact_3397_numeral__mod__minus__numeral,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % numeral_mod_minus_numeral
% 0.30/0.55 thf(fact_3398_minus__numeral__mod__numeral,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_numeral_mod_numeral
% 0.30/0.55 thf(fact_3399_size__prod__simp,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType] :
% 0.30/0.55 ( ( basic_BNF_size_prod @ A @ B )
% 0.30/0.55 = ( ^ [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 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % size_prod_simp
% 0.30/0.55 thf(fact_3400_snd__divmod__integer,axiom,
% 0.30/0.55 ! [K: code_integer,L: code_integer] :
% 0.30/0.55 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 0.30/0.55 = ( modulo_modulo @ code_integer @ K @ L ) ) ).
% 0.30/0.55
% 0.30/0.55 % snd_divmod_integer
% 0.30/0.55 thf(fact_3401_snd__divmod__abs,axiom,
% 0.30/0.55 ! [K: code_integer,L: code_integer] :
% 0.30/0.55 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 0.30/0.55 = ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % snd_divmod_abs
% 0.30/0.55 thf(fact_3402_Suc__0__mod__numeral,axiom,
% 0.30/0.55 ! [K: num] :
% 0.30/0.55 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.55 = ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_0_mod_numeral
% 0.30/0.55 thf(fact_3403_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.30/0.55 ! [L: int,U: int] :
% 0.30/0.55 ( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 0.30/0.55 = ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.30/0.55 thf(fact_3404_Divides_Oadjust__mod__def,axiom,
% 0.30/0.55 ( adjust_mod
% 0.30/0.55 = ( ^ [L3: int,R5: int] :
% 0.30/0.55 ( if @ int
% 0.30/0.55 @ ( R5
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 @ ( zero_zero @ int )
% 0.30/0.55 @ ( minus_minus @ int @ L3 @ R5 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Divides.adjust_mod_def
% 0.30/0.55 thf(fact_3405_bezw_Opelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
% 0.30/0.55 ( ( ( bezw @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 0.30/0.55 => ~ ( ( ( ( Xa3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 0.30/0.55 & ( ( Xa3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) )
% 0.30/0.55 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw.pelims
% 0.30/0.55 thf(fact_3406_in__set__enumerate__eq,axiom,
% 0.30/0.55 ! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
% 0.30/0.55 ( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
% 0.30/0.55 = ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
% 0.30/0.55 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 0.30/0.55 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
% 0.30/0.55 = ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % in_set_enumerate_eq
% 0.30/0.55 thf(fact_3407_list__encode_Osimps_I2_J,axiom,
% 0.30/0.55 ! [X: nat,Xs: list @ nat] :
% 0.30/0.55 ( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
% 0.30/0.55 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_encode.simps(2)
% 0.30/0.55 thf(fact_3408_bezw__aux,axiom,
% 0.30/0.55 ! [X: nat,Y3: nat] :
% 0.30/0.55 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y3 ) )
% 0.30/0.55 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezw_aux
% 0.30/0.55 thf(fact_3409_in__set__zip,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.55 ( ( member @ ( product_prod @ A @ B ) @ P5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 0.30/0.55 = ( ? [N3: nat] :
% 0.30/0.55 ( ( ( nth @ A @ Xs @ N3 )
% 0.30/0.55 = ( product_fst @ A @ B @ P5 ) )
% 0.30/0.55 & ( ( nth @ B @ Ys2 @ N3 )
% 0.30/0.55 = ( product_snd @ A @ B @ P5 ) )
% 0.30/0.55 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % in_set_zip
% 0.30/0.55 thf(fact_3410_gcd__eq__0__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 = ( ( A3
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 & ( B2
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_eq_0_iff
% 0.30/0.55 thf(fact_3411_gcd_Obottom__left__bottom,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.55 = ( one_one @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd.bottom_left_bottom
% 0.30/0.55 thf(fact_3412_gcd_Obottom__right__bottom,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 = ( one_one @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd.bottom_right_bottom
% 0.30/0.55 thf(fact_3413_gcd__neg2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.55 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg2
% 0.30/0.55 thf(fact_3414_gcd__neg1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.55 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg1
% 0.30/0.55 thf(fact_3415_gcd__0__left__nat,axiom,
% 0.30/0.55 ! [X: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
% 0.30/0.55 = X ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_0_left_nat
% 0.30/0.55 thf(fact_3416_gcd__0__nat,axiom,
% 0.30/0.55 ! [X: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
% 0.30/0.55 = X ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_0_nat
% 0.30/0.55 thf(fact_3417_gcd__nat_Oright__neutral,axiom,
% 0.30/0.55 ! [A3: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.30/0.55 = A3 ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.right_neutral
% 0.30/0.55 thf(fact_3418_gcd__nat_Oneutr__eq__iff,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat] :
% 0.30/0.55 ( ( ( zero_zero @ nat )
% 0.30/0.55 = ( gcd_gcd @ nat @ A3 @ B2 ) )
% 0.30/0.55 = ( ( A3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 & ( B2
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.neutr_eq_iff
% 0.30/0.55 thf(fact_3419_gcd__nat_Oleft__neutral,axiom,
% 0.30/0.55 ! [A3: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
% 0.30/0.55 = A3 ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.left_neutral
% 0.30/0.55 thf(fact_3420_gcd__nat_Oeq__neutr__iff,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat] :
% 0.30/0.55 ( ( ( gcd_gcd @ nat @ A3 @ B2 )
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 = ( ( A3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 & ( B2
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.eq_neutr_iff
% 0.30/0.55 thf(fact_3421_gcd__1__nat,axiom,
% 0.30/0.55 ! [M: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
% 0.30/0.55 = ( one_one @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_1_nat
% 0.30/0.55 thf(fact_3422_gcd__neg__numeral__2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ring_gcd @ A )
% 0.30/0.55 => ! [A3: A,N: num] :
% 0.30/0.55 ( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.55 = ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg_numeral_2
% 0.30/0.55 thf(fact_3423_gcd__neg__numeral__1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( ring_gcd @ A )
% 0.30/0.55 => ! [N: num,A3: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
% 0.30/0.55 = ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg_numeral_1
% 0.30/0.55 thf(fact_3424_is__unit__gcd__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.30/0.55 = ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % is_unit_gcd_iff
% 0.30/0.55 thf(fact_3425_gcd__Suc__0,axiom,
% 0.30/0.55 ! [M: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.55 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_Suc_0
% 0.30/0.55 thf(fact_3426_gcd__pos__nat,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N ) )
% 0.30/0.55 = ( ( M
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 | ( N
% 0.30/0.55 != ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_pos_nat
% 0.30/0.55 thf(fact_3427_count__notin,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.55 ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( ( count_list @ A @ Xs @ X )
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % count_notin
% 0.30/0.55 thf(fact_3428_list__encode__eq,axiom,
% 0.30/0.55 ! [X: list @ nat,Y3: list @ nat] :
% 0.30/0.55 ( ( ( nat_list_encode @ X )
% 0.30/0.55 = ( nat_list_encode @ Y3 ) )
% 0.30/0.55 = ( X = Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_encode_eq
% 0.30/0.55 thf(fact_3429_gcd__mult__distrib__nat,axiom,
% 0.30/0.55 ! [K: nat,M: nat,N: nat] :
% 0.30/0.55 ( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N ) )
% 0.30/0.55 = ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_mult_distrib_nat
% 0.30/0.55 thf(fact_3430_gcd__add__mult,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [M: A,K: A,N: A] :
% 0.30/0.55 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N ) )
% 0.30/0.55 = ( gcd_gcd @ A @ M @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_add_mult
% 0.30/0.55 thf(fact_3431_gcd__dvd__prod,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_dvd_prod
% 0.30/0.55 thf(fact_3432_gcd__le1__nat,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_le1_nat
% 0.30/0.55 thf(fact_3433_gcd__le2__nat,axiom,
% 0.30/0.55 ! [B2: nat,A3: nat] :
% 0.30/0.55 ( ( B2
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_le2_nat
% 0.30/0.55 thf(fact_3434_gcd__diff2__nat,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.55 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M ) @ N )
% 0.30/0.55 = ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_diff2_nat
% 0.30/0.55 thf(fact_3435_gcd__diff1__nat,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.55 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N ) @ N )
% 0.30/0.55 = ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_diff1_nat
% 0.30/0.55 thf(fact_3436_gcd__non__0__nat,axiom,
% 0.30/0.55 ! [Y3: nat,X: nat] :
% 0.30/0.55 ( ( Y3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( ( gcd_gcd @ nat @ X @ Y3 )
% 0.30/0.55 = ( gcd_gcd @ nat @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_non_0_nat
% 0.30/0.55 thf(fact_3437_gcd__nat_Osimps,axiom,
% 0.30/0.55 ( ( gcd_gcd @ nat )
% 0.30/0.55 = ( ^ [X2: nat,Y6: nat] :
% 0.30/0.55 ( if @ nat
% 0.30/0.55 @ ( Y6
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 @ X2
% 0.30/0.55 @ ( gcd_gcd @ nat @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.simps
% 0.30/0.55 thf(fact_3438_gcd__nat_Oelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: nat] :
% 0.30/0.55 ( ( ( gcd_gcd @ nat @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( Xa3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3 = X ) )
% 0.30/0.55 & ( ( Xa3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.elims
% 0.30/0.55 thf(fact_3439_gcd__mult__unit1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 0.30/0.55 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_mult_unit1
% 0.30/0.55 thf(fact_3440_gcd__mult__unit2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 0.30/0.55 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_mult_unit2
% 0.30/0.55 thf(fact_3441_gcd__div__unit1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 0.30/0.55 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_div_unit1
% 0.30/0.55 thf(fact_3442_gcd__div__unit2,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( semiring_gcd @ A )
% 0.30/0.55 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.55 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 0.30/0.55 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_div_unit2
% 0.30/0.55 thf(fact_3443_bezout__nat,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ? [X3: nat,Y5: nat] :
% 0.30/0.55 ( ( times_times @ nat @ A3 @ X3 )
% 0.30/0.55 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezout_nat
% 0.30/0.55 thf(fact_3444_bezout__gcd__nat_H,axiom,
% 0.30/0.55 ! [B2: nat,A3: nat] :
% 0.30/0.55 ? [X3: nat,Y5: nat] :
% 0.30/0.55 ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( times_times @ nat @ A3 @ X3 ) )
% 0.30/0.55 & ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
% 0.30/0.55 = ( gcd_gcd @ nat @ A3 @ B2 ) ) )
% 0.30/0.55 | ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ ( times_times @ nat @ B2 @ X3 ) )
% 0.30/0.55 & ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
% 0.30/0.55 = ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezout_gcd_nat'
% 0.30/0.55 thf(fact_3445_length__pos__if__in__set,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_pos_if_in_set
% 0.30/0.55 thf(fact_3446_card__length,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.55
% 0.30/0.55 % card_length
% 0.30/0.55 thf(fact_3447_all__set__conv__all__nth,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.30/0.55 ( ( ! [X2: A] :
% 0.30/0.55 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( P @ X2 ) ) )
% 0.30/0.55 = ( ! [I5: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( P @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % all_set_conv_all_nth
% 0.30/0.55 thf(fact_3448_all__nth__imp__all__set,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
% 0.30/0.55 ( ! [I4: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( P @ ( nth @ A @ Xs @ I4 ) ) )
% 0.30/0.55 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( P @ X ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % all_nth_imp_all_set
% 0.30/0.55 thf(fact_3449_in__set__conv__nth,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.55 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.55 = ( ? [I5: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 & ( ( nth @ A @ Xs @ I5 )
% 0.30/0.55 = X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % in_set_conv_nth
% 0.30/0.55 thf(fact_3450_list__ball__nth,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 0.30/0.55 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( ! [X3: A] :
% 0.30/0.55 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( P @ X3 ) )
% 0.30/0.55 => ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_ball_nth
% 0.30/0.55 thf(fact_3451_nth__mem,axiom,
% 0.30/0.55 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.55 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_mem
% 0.30/0.55 thf(fact_3452_nth__equal__first__eq,axiom,
% 0.30/0.55 ! [A: $tType,X: A,Xs: list @ A,N: nat] :
% 0.30/0.55 ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.55 => ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.30/0.55 = X )
% 0.30/0.55 = ( N
% 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_equal_first_eq
% 0.30/0.55 thf(fact_3453_prod__decode__aux_Opelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
% 0.30/0.55 ( ( ( nat_prod_decode_aux @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 0.30/0.55 => ~ ( ( ( ( ord_less_eq @ nat @ Xa3 @ X )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
% 0.30/0.55 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % prod_decode_aux.pelims
% 0.30/0.55 thf(fact_3454_gcd__nat_Opelims,axiom,
% 0.30/0.55 ! [X: nat,Xa3: nat,Y3: nat] :
% 0.30/0.55 ( ( ( gcd_gcd @ nat @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 0.30/0.55 => ~ ( ( ( ( Xa3
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3 = X ) )
% 0.30/0.55 & ( ( Xa3
% 0.30/0.55 != ( zero_zero @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) )
% 0.30/0.55 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat.pelims
% 0.30/0.55 thf(fact_3455_list__encode_Oelims,axiom,
% 0.30/0.55 ! [X: list @ nat,Y3: nat] :
% 0.30/0.55 ( ( ( nat_list_encode @ X )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( X
% 0.30/0.55 = ( nil @ nat ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 != ( zero_zero @ nat ) ) )
% 0.30/0.55 => ~ ! [X3: nat,Xs2: list @ nat] :
% 0.30/0.55 ( ( X
% 0.30/0.55 = ( cons @ nat @ X3 @ Xs2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 != ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_encode.elims
% 0.30/0.55 thf(fact_3456_zero__rat__def,axiom,
% 0.30/0.55 ( ( zero_zero @ rat )
% 0.30/0.55 = ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_rat_def
% 0.30/0.55 thf(fact_3457_pair__lessI2,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less @ nat @ S @ T2 )
% 0.30/0.55 => ( 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 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pair_lessI2
% 0.30/0.55 thf(fact_3458_gcd__1__int,axiom,
% 0.30/0.55 ! [M: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
% 0.30/0.55 = ( one_one @ int ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_1_int
% 0.30/0.55 thf(fact_3459_gcd__neg2__int,axiom,
% 0.30/0.55 ! [X: int,Y3: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
% 0.30/0.55 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg2_int
% 0.30/0.55 thf(fact_3460_gcd__neg1__int,axiom,
% 0.30/0.55 ! [X: int,Y3: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
% 0.30/0.55 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg1_int
% 0.30/0.55 thf(fact_3461_gcd__pos__int,axiom,
% 0.30/0.55 ! [M: int,N: int] :
% 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N ) )
% 0.30/0.55 = ( ( M
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 | ( N
% 0.30/0.55 != ( zero_zero @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_pos_int
% 0.30/0.55 thf(fact_3462_gcd__neg__numeral__1__int,axiom,
% 0.30/0.55 ! [N: num,X: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
% 0.30/0.55 = ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg_numeral_1_int
% 0.30/0.55 thf(fact_3463_gcd__neg__numeral__2__int,axiom,
% 0.30/0.55 ! [X: int,N: num] :
% 0.30/0.55 ( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_neg_numeral_2_int
% 0.30/0.55 thf(fact_3464_gcd__0__left__int,axiom,
% 0.30/0.55 ! [X: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.55 = ( abs_abs @ int @ X ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_0_left_int
% 0.30/0.55 thf(fact_3465_gcd__0__int,axiom,
% 0.30/0.55 ! [X: int] :
% 0.30/0.55 ( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.55 = ( abs_abs @ int @ X ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_0_int
% 0.30/0.55 thf(fact_3466_gcd__int__int__eq,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.55 = ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_int_int_eq
% 0.30/0.55 thf(fact_3467_length__0__conv,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A] :
% 0.30/0.55 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.55 = ( zero_zero @ nat ) )
% 0.30/0.55 = ( Xs
% 0.30/0.55 = ( nil @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_0_conv
% 0.30/0.55 thf(fact_3468_horner__sum__simps_I1_J,axiom,
% 0.30/0.55 ! [B: $tType,A: $tType] :
% 0.30/0.55 ( ( comm_semiring_0 @ A )
% 0.30/0.55 => ! [F: B > A,A3: A] :
% 0.30/0.55 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( nil @ B ) )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % horner_sum_simps(1)
% 0.30/0.55 thf(fact_3469_length__greater__0__conv,axiom,
% 0.30/0.55 ! [A: $tType,Xs: list @ A] :
% 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.55 = ( Xs
% 0.30/0.55 != ( nil @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_greater_0_conv
% 0.30/0.55 thf(fact_3470_gcd__nat__abs__right__eq,axiom,
% 0.30/0.55 ! [N: nat,K: int] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 0.30/0.55 = ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat_abs_right_eq
% 0.30/0.55 thf(fact_3471_gcd__nat__abs__left__eq,axiom,
% 0.30/0.55 ! [K: int,N: nat] :
% 0.30/0.55 ( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 0.30/0.55 = ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_nat_abs_left_eq
% 0.30/0.55 thf(fact_3472_pair__less__iff1,axiom,
% 0.30/0.55 ! [X: nat,Y3: nat,Z3: nat] :
% 0.30/0.55 ( ( 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 @ Y3 ) @ ( product_Pair @ nat @ nat @ X @ Z3 ) ) @ fun_pair_less )
% 0.30/0.55 = ( ord_less @ nat @ Y3 @ Z3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % pair_less_iff1
% 0.30/0.55 thf(fact_3473_list__encode_Ocases,axiom,
% 0.30/0.55 ! [X: list @ nat] :
% 0.30/0.55 ( ( X
% 0.30/0.55 != ( nil @ nat ) )
% 0.30/0.55 => ~ ! [X3: nat,Xs2: list @ nat] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( cons @ nat @ X3 @ Xs2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_encode.cases
% 0.30/0.55 thf(fact_3474_gcd__ge__0__int,axiom,
% 0.30/0.55 ! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_ge_0_int
% 0.30/0.55 thf(fact_3475_bezout__int,axiom,
% 0.30/0.55 ! [X: int,Y3: int] :
% 0.30/0.55 ? [U2: int,V4: int] :
% 0.30/0.55 ( ( plus_plus @ int @ ( times_times @ int @ U2 @ X ) @ ( times_times @ int @ V4 @ Y3 ) )
% 0.30/0.55 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % bezout_int
% 0.30/0.55 thf(fact_3476_gcd__mult__distrib__int,axiom,
% 0.30/0.55 ! [K: int,M: int,N: int] :
% 0.30/0.55 ( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N ) )
% 0.30/0.55 = ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_mult_distrib_int
% 0.30/0.55 thf(fact_3477_list_Osize_I3_J,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 0.30/0.55 = ( zero_zero @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % list.size(3)
% 0.30/0.55 thf(fact_3478_list__encode_Osimps_I1_J,axiom,
% 0.30/0.55 ( ( nat_list_encode @ ( nil @ nat ) )
% 0.30/0.55 = ( zero_zero @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_encode.simps(1)
% 0.30/0.55 thf(fact_3479_count__list_Osimps_I1_J,axiom,
% 0.30/0.55 ! [A: $tType,Y3: A] :
% 0.30/0.55 ( ( count_list @ A @ ( nil @ A ) @ Y3 )
% 0.30/0.55 = ( zero_zero @ nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % count_list.simps(1)
% 0.30/0.55 thf(fact_3480_gcd__le1__int,axiom,
% 0.30/0.55 ! [A3: int,B2: int] :
% 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
% 0.30/0.55 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ A3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_le1_int
% 0.30/0.55 thf(fact_3481_gcd__le2__int,axiom,
% 0.30/0.55 ! [B2: int,A3: int] :
% 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.55 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ B2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_le2_int
% 0.30/0.55 thf(fact_3482_gcd__cases__int,axiom,
% 0.30/0.55 ! [X: int,Y3: int,P: int > $o] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.55 => ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.55 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 0.30/0.55 => ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.55 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.55 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 0.30/0.55 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 0.30/0.55 => ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_cases_int
% 0.30/0.55 thf(fact_3483_gcd__unique__int,axiom,
% 0.30/0.55 ! [D2: int,A3: int,B2: int] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.55 & ( dvd_dvd @ int @ D2 @ A3 )
% 0.30/0.55 & ( dvd_dvd @ int @ D2 @ B2 )
% 0.30/0.55 & ! [E3: int] :
% 0.30/0.55 ( ( ( dvd_dvd @ int @ E3 @ A3 )
% 0.30/0.55 & ( dvd_dvd @ int @ E3 @ B2 ) )
% 0.30/0.55 => ( dvd_dvd @ int @ E3 @ D2 ) ) )
% 0.30/0.55 = ( D2
% 0.30/0.55 = ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_unique_int
% 0.30/0.55 thf(fact_3484_gcd__non__0__int,axiom,
% 0.30/0.55 ! [Y3: int,X: int] :
% 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.55 => ( ( gcd_gcd @ int @ X @ Y3 )
% 0.30/0.55 = ( gcd_gcd @ int @ Y3 @ ( modulo_modulo @ int @ X @ Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_non_0_int
% 0.30/0.55 thf(fact_3485_gcd__code__int,axiom,
% 0.30/0.55 ( ( gcd_gcd @ int )
% 0.30/0.55 = ( ^ [K3: int,L3: int] :
% 0.30/0.55 ( abs_abs @ int
% 0.30/0.55 @ ( if @ int
% 0.30/0.55 @ ( L3
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 @ K3
% 0.30/0.55 @ ( gcd_gcd @ int @ L3 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_code_int
% 0.30/0.55 thf(fact_3486_pair__lessI1,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ A3 @ B2 )
% 0.30/0.55 => ( 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 ) ) ).
% 0.30/0.55
% 0.30/0.55 % pair_lessI1
% 0.30/0.55 thf(fact_3487_gcd__int__def,axiom,
% 0.30/0.55 ( ( gcd_gcd @ int )
% 0.30/0.55 = ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gcd_int_def
% 0.30/0.55 thf(fact_3488_one__rat__def,axiom,
% 0.30/0.55 ( ( one_one @ rat )
% 0.30/0.55 = ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_rat_def
% 0.30/0.55 thf(fact_3489_plus__rat_Oabs__eq,axiom,
% 0.30/0.55 ! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
% 0.30/0.55 ( ( ratrel @ Xa3 @ Xa3 )
% 0.30/0.55 => ( ( ratrel @ X @ X )
% 0.30/0.55 => ( ( plus_plus @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
% 0.30/0.55 = ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % plus_rat.abs_eq
% 0.30/0.55 thf(fact_3490_pair__leqI2,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.30/0.55 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ nat @ S @ T2 )
% 0.30/0.55 => ( 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 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pair_leqI2
% 0.30/0.55 thf(fact_3491_pair__leqI1,axiom,
% 0.30/0.55 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.30/0.55 ( ( ord_less @ nat @ A3 @ B2 )
% 0.30/0.55 => ( 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 ) ) ).
% 0.30/0.55
% 0.30/0.55 % pair_leqI1
% 0.30/0.55 thf(fact_3492_smax__insertI,axiom,
% 0.30/0.55 ! [Y3: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.55 ( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y7 )
% 0.30/0.55 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
% 0.30/0.55 => ( ( 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 ) ) @ X8 @ Y7 ) @ fun_max_strict )
% 0.30/0.55 => ( 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % smax_insertI
% 0.30/0.55 thf(fact_3493_smin__insertI,axiom,
% 0.30/0.55 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.55 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 0.30/0.55 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
% 0.30/0.55 => ( ( 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 )
% 0.30/0.55 => ( 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % smin_insertI
% 0.30/0.55 thf(fact_3494_ratrel__iff,axiom,
% 0.30/0.55 ( ratrel
% 0.30/0.55 = ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
% 0.30/0.55 ( ( ( product_snd @ int @ int @ X2 )
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 & ( ( product_snd @ int @ int @ Y6 )
% 0.30/0.55 != ( zero_zero @ int ) )
% 0.30/0.55 & ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
% 0.30/0.55 = ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % ratrel_iff
% 0.30/0.55 thf(fact_3495_smin__emptyI,axiom,
% 0.30/0.55 ! [X8: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.55 ( ( X8
% 0.30/0.55 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 0.30/0.55 => ( 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% 0.30/0.55
% 0.30/0.55 % smin_emptyI
% 0.30/0.55 thf(fact_3496_one__rat_Orsp,axiom,
% 0.30/0.55 ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_rat.rsp
% 0.30/0.55 thf(fact_3497_zero__rat_Orsp,axiom,
% 0.30/0.55 ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_rat.rsp
% 0.30/0.55 thf(fact_3498_times__rat_Oabs__eq,axiom,
% 0.30/0.55 ! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
% 0.30/0.55 ( ( ratrel @ Xa3 @ Xa3 )
% 0.30/0.55 => ( ( ratrel @ X @ X )
% 0.30/0.55 => ( ( times_times @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
% 0.30/0.55 = ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % times_rat.abs_eq
% 0.30/0.55 thf(fact_3499_uminus__rat_Oabs__eq,axiom,
% 0.30/0.55 ! [X: product_prod @ int @ int] :
% 0.30/0.55 ( ( ratrel @ X @ X )
% 0.30/0.55 => ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
% 0.30/0.55 = ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % uminus_rat.abs_eq
% 0.30/0.55 thf(fact_3500_wmin__insertI,axiom,
% 0.30/0.55 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.55 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 0.30/0.55 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
% 0.30/0.55 => ( ( 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 )
% 0.30/0.55 => ( 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % wmin_insertI
% 0.30/0.55 thf(fact_3501_wmax__insertI,axiom,
% 0.30/0.55 ! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.55 ( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ YS )
% 0.30/0.55 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
% 0.30/0.55 => ( ( 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 )
% 0.30/0.55 => ( 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % wmax_insertI
% 0.30/0.55 thf(fact_3502_positive_Oabs__eq,axiom,
% 0.30/0.55 ! [X: product_prod @ int @ int] :
% 0.30/0.55 ( ( ratrel @ X @ X )
% 0.30/0.55 => ( ( positive @ ( abs_Rat @ X ) )
% 0.30/0.55 = ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % positive.abs_eq
% 0.30/0.55 thf(fact_3503_upto_Opelims,axiom,
% 0.30/0.55 ! [X: int,Xa3: int,Y3: list @ int] :
% 0.30/0.55 ( ( ( upto @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
% 0.30/0.55 => ~ ( ( ( ( ord_less_eq @ int @ X @ Xa3 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ X @ Xa3 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( nil @ int ) ) ) )
% 0.30/0.55 => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto.pelims
% 0.30/0.55 thf(fact_3504_upto_Opsimps,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.55 => ( ( upto @ I @ J )
% 0.30/0.55 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ I @ J )
% 0.30/0.55 => ( ( upto @ I @ J )
% 0.30/0.55 = ( nil @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto.psimps
% 0.30/0.55 thf(fact_3505_upto__Nil,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( ( upto @ I @ J )
% 0.30/0.55 = ( nil @ int ) )
% 0.30/0.55 = ( ord_less @ int @ J @ I ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_Nil
% 0.30/0.55 thf(fact_3506_upto__Nil2,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( ( nil @ int )
% 0.30/0.55 = ( upto @ I @ J ) )
% 0.30/0.55 = ( ord_less @ int @ J @ I ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_Nil2
% 0.30/0.55 thf(fact_3507_upto__empty,axiom,
% 0.30/0.55 ! [J: int,I: int] :
% 0.30/0.55 ( ( ord_less @ int @ J @ I )
% 0.30/0.55 => ( ( upto @ I @ J )
% 0.30/0.55 = ( nil @ int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_empty
% 0.30/0.55 thf(fact_3508_nth__upto,axiom,
% 0.30/0.55 ! [I: int,K: nat,J: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
% 0.30/0.55 => ( ( nth @ int @ ( upto @ I @ J ) @ K )
% 0.30/0.55 = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_upto
% 0.30/0.55 thf(fact_3509_length__upto,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
% 0.30/0.55 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_upto
% 0.30/0.55 thf(fact_3510_upto__rec__numeral_I1_J,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( nil @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec_numeral(1)
% 0.30/0.55 thf(fact_3511_upto__rec__numeral_I4_J,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( nil @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec_numeral(4)
% 0.30/0.55 thf(fact_3512_upto__rec__numeral_I3_J,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.55 = ( nil @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec_numeral(3)
% 0.30/0.55 thf(fact_3513_upto__rec__numeral_I2_J,axiom,
% 0.30/0.55 ! [M: num,N: num] :
% 0.30/0.55 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.55 = ( nil @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec_numeral(2)
% 0.30/0.55 thf(fact_3514_less__by__empty,axiom,
% 0.30/0.55 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
% 0.30/0.55 ( ( A6
% 0.30/0.55 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_by_empty
% 0.30/0.55 thf(fact_3515_wmin__emptyI,axiom,
% 0.30/0.55 ! [X8: 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% 0.30/0.55
% 0.30/0.55 % wmin_emptyI
% 0.30/0.55 thf(fact_3516_positive__minus,axiom,
% 0.30/0.55 ! [X: rat] :
% 0.30/0.55 ( ~ ( positive @ X )
% 0.30/0.55 => ( ( X
% 0.30/0.55 != ( zero_zero @ rat ) )
% 0.30/0.55 => ( positive @ ( uminus_uminus @ rat @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % positive_minus
% 0.30/0.55 thf(fact_3517_greaterThanAtMost__upto,axiom,
% 0.30/0.55 ( ( set_or3652927894154168847AtMost @ int )
% 0.30/0.55 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanAtMost_upto
% 0.30/0.55 thf(fact_3518_less__rat__def,axiom,
% 0.30/0.55 ( ( ord_less @ rat )
% 0.30/0.55 = ( ^ [X2: rat,Y6: rat] : ( positive @ ( minus_minus @ rat @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_rat_def
% 0.30/0.55 thf(fact_3519_upto__rec1,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.55 => ( ( upto @ I @ J )
% 0.30/0.55 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec1
% 0.30/0.55 thf(fact_3520_upto_Oelims,axiom,
% 0.30/0.55 ! [X: int,Xa3: int,Y3: list @ int] :
% 0.30/0.55 ( ( ( upto @ X @ Xa3 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ( ord_less_eq @ int @ X @ Xa3 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq @ int @ X @ Xa3 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( nil @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto.elims
% 0.30/0.55 thf(fact_3521_upto_Osimps,axiom,
% 0.30/0.55 ( upto
% 0.30/0.55 = ( ^ [I5: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I5 @ J3 ) @ ( cons @ int @ I5 @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto.simps
% 0.30/0.55 thf(fact_3522_greaterThanLessThan__upto,axiom,
% 0.30/0.55 ( ( set_or5935395276787703475ssThan @ int )
% 0.30/0.55 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % greaterThanLessThan_upto
% 0.30/0.55 thf(fact_3523_positive_Orep__eq,axiom,
% 0.30/0.55 ( positive
% 0.30/0.55 = ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % positive.rep_eq
% 0.30/0.55 thf(fact_3524_inverse__rat_Oabs__eq,axiom,
% 0.30/0.55 ! [X: product_prod @ int @ int] :
% 0.30/0.55 ( ( ratrel @ X @ X )
% 0.30/0.55 => ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
% 0.30/0.55 = ( abs_Rat
% 0.30/0.55 @ ( if @ ( product_prod @ int @ int )
% 0.30/0.55 @ ( ( product_fst @ int @ int @ X )
% 0.30/0.55 = ( zero_zero @ int ) )
% 0.30/0.55 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.55 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_rat.abs_eq
% 0.30/0.55 thf(fact_3525_min__rpair__set,axiom,
% 0.30/0.55 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 ) ).
% 0.30/0.55
% 0.30/0.55 % min_rpair_set
% 0.30/0.55 thf(fact_3526_max__rpair__set,axiom,
% 0.30/0.55 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 ) ).
% 0.30/0.55
% 0.30/0.55 % max_rpair_set
% 0.30/0.55 thf(fact_3527_upto__rec2,axiom,
% 0.30/0.55 ! [I: int,J: int] :
% 0.30/0.55 ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.55 => ( ( upto @ I @ J )
% 0.30/0.55 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % upto_rec2
% 0.30/0.55 thf(fact_3528_inverse__eq__iff__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( A3 = B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_iff_eq
% 0.30/0.55 thf(fact_3529_inverse__inverse__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = A3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_inverse_eq
% 0.30/0.55 thf(fact_3530_inverse__zero,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_zero
% 0.30/0.55 thf(fact_3531_inverse__nonzero__iff__nonzero,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 = ( A3
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_nonzero_iff_nonzero
% 0.30/0.55 thf(fact_3532_inverse__mult__distrib,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.55 = ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_mult_distrib
% 0.30/0.55 thf(fact_3533_inverse__1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 0.30/0.55 = ( one_one @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_1
% 0.30/0.55 thf(fact_3534_inverse__eq__1__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ X )
% 0.30/0.55 = ( one_one @ A ) )
% 0.30/0.55 = ( X
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_1_iff
% 0.30/0.55 thf(fact_3535_inverse__minus__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.55 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_minus_eq
% 0.30/0.55 thf(fact_3536_inverse__divide,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.55 = ( divide_divide @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_divide
% 0.30/0.55 thf(fact_3537_abs__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field_abs_sgn @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % abs_inverse
% 0.30/0.55 thf(fact_3538_sgn__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field_abs_sgn @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sgn_inverse
% 0.30/0.55 thf(fact_3539_inverse__sgn,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.30/0.55 = ( sgn_sgn @ A @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_sgn
% 0.30/0.55 thf(fact_3540_inverse__nonnegative__iff__nonnegative,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_nonnegative_iff_nonnegative
% 0.30/0.55 thf(fact_3541_inverse__nonpositive__iff__nonpositive,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.55 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_nonpositive_iff_nonpositive
% 0.30/0.55 thf(fact_3542_inverse__less__iff__less,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.55 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_less_iff_less
% 0.30/0.55 thf(fact_3543_inverse__less__iff__less__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_less_iff_less_neg
% 0.30/0.55 thf(fact_3544_inverse__negative__iff__negative,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.55 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_negative_iff_negative
% 0.30/0.55 thf(fact_3545_inverse__positive__iff__positive,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_positive_iff_positive
% 0.30/0.55 thf(fact_3546_inverse__le__iff__le,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.55 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_le_iff_le
% 0.30/0.55 thf(fact_3547_inverse__le__iff__le__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_le_iff_le_neg
% 0.30/0.55 thf(fact_3548_left__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % left_inverse
% 0.30/0.55 thf(fact_3549_right__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % right_inverse
% 0.30/0.55 thf(fact_3550_inverse__eq__divide__numeral,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [W2: num] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
% 0.30/0.55 = ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_divide_numeral
% 0.30/0.55 thf(fact_3551_inverse__eq__divide__neg__numeral,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [W2: num] :
% 0.30/0.55 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 0.30/0.55 = ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_divide_neg_numeral
% 0.30/0.55 thf(fact_3552_mult__commute__imp__mult__inverse__commute,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [Y3: A,X: A] :
% 0.30/0.55 ( ( ( times_times @ A @ Y3 @ X )
% 0.30/0.55 = ( times_times @ A @ X @ Y3 ) )
% 0.30/0.55 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y3 ) @ X )
% 0.30/0.55 = ( times_times @ A @ X @ ( inverse_inverse @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_commute_imp_mult_inverse_commute
% 0.30/0.55 thf(fact_3553_inverse__eq__imp__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( A3 = B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_imp_eq
% 0.30/0.55 thf(fact_3554_field__class_Ofield__inverse__zero,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.30/0.55 = ( zero_zero @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % field_class.field_inverse_zero
% 0.30/0.55 thf(fact_3555_inverse__zero__imp__zero,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = ( zero_zero @ A ) )
% 0.30/0.55 => ( A3
% 0.30/0.55 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_zero_imp_zero
% 0.30/0.55 thf(fact_3556_nonzero__inverse__eq__imp__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( B2
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( A3 = B2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_inverse_eq_imp_eq
% 0.30/0.55 thf(fact_3557_nonzero__inverse__inverse__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = A3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_inverse_inverse_eq
% 0.30/0.55 thf(fact_3558_nonzero__imp__inverse__nonzero,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_imp_inverse_nonzero
% 0.30/0.55 thf(fact_3559_inverse__less__imp__less,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_less_imp_less
% 0.30/0.55 thf(fact_3560_less__imp__inverse__less,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_imp_inverse_less
% 0.30/0.55 thf(fact_3561_inverse__less__imp__less__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_less_imp_less_neg
% 0.30/0.55 thf(fact_3562_less__imp__inverse__less__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_imp_inverse_less_neg
% 0.30/0.55 thf(fact_3563_inverse__negative__imp__negative,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_negative_imp_negative
% 0.30/0.55 thf(fact_3564_inverse__positive__imp__positive,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 => ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_positive_imp_positive
% 0.30/0.55 thf(fact_3565_negative__imp__inverse__negative,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % negative_imp_inverse_negative
% 0.30/0.55 thf(fact_3566_positive__imp__inverse__positive,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % positive_imp_inverse_positive
% 0.30/0.55 thf(fact_3567_nonzero__inverse__mult__distrib,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( B2
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.55 = ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_inverse_mult_distrib
% 0.30/0.55 thf(fact_3568_nonzero__inverse__minus__eq,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.55 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_inverse_minus_eq
% 0.30/0.55 thf(fact_3569_inverse__unique,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ( times_times @ A @ A3 @ B2 )
% 0.30/0.55 = ( one_one @ A ) )
% 0.30/0.55 => ( ( inverse_inverse @ A @ A3 )
% 0.30/0.55 = B2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_unique
% 0.30/0.55 thf(fact_3570_inverse__numeral__1,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
% 0.30/0.55 = ( numeral_numeral @ A @ one2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_numeral_1
% 0.30/0.55 thf(fact_3571_divide__inverse__commute,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ( ( divide_divide @ A )
% 0.30/0.55 = ( ^ [A5: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A5 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_inverse_commute
% 0.30/0.55 thf(fact_3572_divide__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ( ( divide_divide @ A )
% 0.30/0.55 = ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_inverse
% 0.30/0.55 thf(fact_3573_field__class_Ofield__divide__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ( ( divide_divide @ A )
% 0.30/0.55 = ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % field_class.field_divide_inverse
% 0.30/0.55 thf(fact_3574_inverse__eq__divide,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ( ( inverse_inverse @ A )
% 0.30/0.55 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_eq_divide
% 0.30/0.55 thf(fact_3575_power__mult__power__inverse__commute,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [X: A,M: nat,N: nat] :
% 0.30/0.55 ( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
% 0.30/0.55 = ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mult_power_inverse_commute
% 0.30/0.55 thf(fact_3576_power__mult__inverse__distrib,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [X: A,M: nat] :
% 0.30/0.55 ( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
% 0.30/0.55 = ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mult_inverse_distrib
% 0.30/0.55 thf(fact_3577_mult__inverse__of__nat__commute,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [Xa3: nat,X: A] :
% 0.30/0.55 ( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) @ X )
% 0.30/0.55 = ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_inverse_of_nat_commute
% 0.30/0.55 thf(fact_3578_nonzero__abs__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
% 0.30/0.55 = ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_abs_inverse
% 0.30/0.55 thf(fact_3579_mult__inverse__of__int__commute,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [Xa3: int,X: A] :
% 0.30/0.55 ( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) @ X )
% 0.30/0.55 = ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_inverse_of_int_commute
% 0.30/0.55 thf(fact_3580_inverse__le__imp__le,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_le_imp_le
% 0.30/0.55 thf(fact_3581_le__imp__inverse__le,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_imp_inverse_le
% 0.30/0.55 thf(fact_3582_inverse__le__imp__le__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_le_imp_le_neg
% 0.30/0.55 thf(fact_3583_le__imp__inverse__le__neg,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.55 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.30/0.55 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_imp_inverse_le_neg
% 0.30/0.55 thf(fact_3584_inverse__le__1__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 0.30/0.55 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.55 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_le_1_iff
% 0.30/0.55 thf(fact_3585_one__less__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.55 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.55 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_less_inverse
% 0.30/0.55 thf(fact_3586_one__less__inverse__iff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( linordered_field @ A )
% 0.30/0.55 => ! [X: A] :
% 0.30/0.55 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 0.30/0.55 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.55 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_less_inverse_iff
% 0.30/0.55 thf(fact_3587_inverse__add,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( B2
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inverse_add
% 0.30/0.55 thf(fact_3588_division__ring__inverse__add,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( B2
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.55 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % division_ring_inverse_add
% 0.30/0.55 thf(fact_3589_field__class_Ofield__inverse,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( field @ A )
% 0.30/0.55 => ! [A3: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.55 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 0.30/0.55 = ( one_one @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % field_class.field_inverse
% 0.30/0.55 thf(fact_3590_division__ring__inverse__diff,axiom,
% 0.30/0.55 ! [A: $tType] :
% 0.30/0.55 ( ( division_ring @ A )
% 0.30/0.55 => ! [A3: A,B2: A] :
% 0.30/0.55 ( ( A3
% 0.30/0.55 != ( zero_zero @ A ) )
% 0.30/0.56 => ( ( B2
% 0.30/0.56 != ( zero_zero @ A ) )
% 0.30/0.56 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.56 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % division_ring_inverse_diff
% 0.30/0.56 thf(fact_3591_nonzero__inverse__eq__divide,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( division_ring @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( A3
% 0.30/0.56 != ( zero_zero @ A ) )
% 0.30/0.56 => ( ( inverse_inverse @ A @ A3 )
% 0.30/0.56 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nonzero_inverse_eq_divide
% 0.30/0.56 thf(fact_3592_inverse__le__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.56 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.56 => ( ord_less_eq @ A @ B2 @ A3 ) )
% 0.30/0.56 & ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.56 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inverse_le_iff
% 0.30/0.56 thf(fact_3593_inverse__less__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.30/0.56 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.56 => ( ord_less @ A @ B2 @ A3 ) )
% 0.30/0.56 & ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 0.30/0.56 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inverse_less_iff
% 0.30/0.56 thf(fact_3594_one__le__inverse,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.56 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % one_le_inverse
% 0.30/0.56 thf(fact_3595_inverse__less__1__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.30/0.56 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inverse_less_1_iff
% 0.30/0.56 thf(fact_3596_one__le__inverse__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 0.30/0.56 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.56 & ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % one_le_inverse_iff
% 0.30/0.56 thf(fact_3597_reals__Archimedean,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( archim462609752435547400_field @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.56 => ? [N4: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % reals_Archimedean
% 0.30/0.56 thf(fact_3598_length__append__singleton,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.30/0.56 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_append_singleton
% 0.30/0.56 thf(fact_3599_length__Suc__conv__rev,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.56 = ( suc @ N ) )
% 0.30/0.56 = ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.56 ( ( Xs
% 0.30/0.56 = ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
% 0.30/0.56 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 0.30/0.56 = N ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_Suc_conv_rev
% 0.30/0.56 thf(fact_3600_nth__append,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.56 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
% 0.30/0.56 = ( nth @ A @ Xs @ N ) ) )
% 0.30/0.56 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
% 0.30/0.56 = ( nth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_append
% 0.30/0.56 thf(fact_3601_upto__split2,axiom,
% 0.30/0.56 ! [I: int,J: int,K: int] :
% 0.30/0.56 ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.56 => ( ( ord_less_eq @ int @ J @ K )
% 0.30/0.56 => ( ( upto @ I @ K )
% 0.30/0.56 = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upto_split2
% 0.30/0.56 thf(fact_3602_upto__split1,axiom,
% 0.30/0.56 ! [I: int,J: int,K: int] :
% 0.30/0.56 ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.56 => ( ( ord_less_eq @ int @ J @ K )
% 0.30/0.56 => ( ( upto @ I @ K )
% 0.30/0.56 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upto_split1
% 0.30/0.56 thf(fact_3603_ex__inverse__of__nat__less,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( archim462609752435547400_field @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.56 => ? [N4: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.56 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) ) @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ex_inverse_of_nat_less
% 0.30/0.56 thf(fact_3604_power__diff__conv__inverse,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( division_ring @ A )
% 0.30/0.56 => ! [X: A,M: nat,N: nat] :
% 0.30/0.56 ( ( X
% 0.30/0.56 != ( zero_zero @ A ) )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.56 => ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.56 = ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % power_diff_conv_inverse
% 0.30/0.56 thf(fact_3605_horner__sum__append,axiom,
% 0.30/0.56 ! [A: $tType,B: $tType] :
% 0.30/0.56 ( ( comm_semiring_1 @ A )
% 0.30/0.56 => ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
% 0.30/0.56 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
% 0.30/0.56 = ( 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 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % horner_sum_append
% 0.30/0.56 thf(fact_3606_upto__split3,axiom,
% 0.30/0.56 ! [I: int,J: int,K: int] :
% 0.30/0.56 ( ( ord_less_eq @ int @ I @ J )
% 0.30/0.56 => ( ( ord_less_eq @ int @ J @ K )
% 0.30/0.56 => ( ( upto @ I @ K )
% 0.30/0.56 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upto_split3
% 0.30/0.56 thf(fact_3607_list__encode_Opelims,axiom,
% 0.30/0.56 ! [X: list @ nat,Y3: nat] :
% 0.30/0.56 ( ( ( nat_list_encode @ X )
% 0.30/0.56 = Y3 )
% 0.30/0.56 => ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
% 0.30/0.56 => ( ( ( X
% 0.30/0.56 = ( nil @ nat ) )
% 0.30/0.56 => ( ( Y3
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
% 0.30/0.56 => ~ ! [X3: nat,Xs2: list @ nat] :
% 0.30/0.56 ( ( X
% 0.30/0.56 = ( cons @ nat @ X3 @ Xs2 ) )
% 0.30/0.56 => ( ( Y3
% 0.30/0.56 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 0.30/0.56 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_encode.pelims
% 0.30/0.56 thf(fact_3608_rp__inv__image__rp,axiom,
% 0.30/0.56 ! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.30/0.56 ( ( fun_reduction_pair @ A @ P )
% 0.30/0.56 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % rp_inv_image_rp
% 0.30/0.56 thf(fact_3609_take__Suc__conv__app__nth,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( take @ A @ ( suc @ I ) @ Xs )
% 0.30/0.56 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_Suc_conv_app_nth
% 0.30/0.56 thf(fact_3610_min__strict__def,axiom,
% 0.30/0.56 ( fun_min_strict
% 0.30/0.56 = ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% 0.30/0.56
% 0.30/0.56 % min_strict_def
% 0.30/0.56 thf(fact_3611_upt__rec__numeral,axiom,
% 0.30/0.56 ! [M: num,N: num] :
% 0.30/0.56 ( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.56 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.56 = ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
% 0.30/0.56 & ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.56 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.56 = ( nil @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_rec_numeral
% 0.30/0.56 thf(fact_3612_take__Suc__Cons,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( cons @ A @ X @ ( take @ A @ N @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_Suc_Cons
% 0.30/0.56 thf(fact_3613_take__eq__Nil2,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ( nil @ A )
% 0.30/0.56 = ( take @ A @ N @ Xs ) )
% 0.30/0.56 = ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 | ( Xs
% 0.30/0.56 = ( nil @ A ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_eq_Nil2
% 0.30/0.56 thf(fact_3614_take__eq__Nil,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ( take @ A @ N @ Xs )
% 0.30/0.56 = ( nil @ A ) )
% 0.30/0.56 = ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 | ( Xs
% 0.30/0.56 = ( nil @ A ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_eq_Nil
% 0.30/0.56 thf(fact_3615_take0,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( take @ A @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take0
% 0.30/0.56 thf(fact_3616_take__all,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.30/0.56 => ( ( take @ A @ N @ Xs )
% 0.30/0.56 = Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_all
% 0.30/0.56 thf(fact_3617_take__all__iff,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ( take @ A @ N @ Xs )
% 0.30/0.56 = Xs )
% 0.30/0.56 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_all_iff
% 0.30/0.56 thf(fact_3618_take__upt,axiom,
% 0.30/0.56 ! [I: nat,M: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N )
% 0.30/0.56 => ( ( take @ nat @ M @ ( upt @ I @ N ) )
% 0.30/0.56 = ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_upt
% 0.30/0.56 thf(fact_3619_nth__take,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ N )
% 0.30/0.56 => ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
% 0.30/0.56 = ( nth @ A @ Xs @ I ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_take
% 0.30/0.56 thf(fact_3620_upt__conv__Nil,axiom,
% 0.30/0.56 ! [J: nat,I: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ J @ I )
% 0.30/0.56 => ( ( upt @ I @ J )
% 0.30/0.56 = ( nil @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_conv_Nil
% 0.30/0.56 thf(fact_3621_upt__eq__Nil__conv,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( ( upt @ I @ J )
% 0.30/0.56 = ( nil @ nat ) )
% 0.30/0.56 = ( ( J
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 | ( ord_less_eq @ nat @ J @ I ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_eq_Nil_conv
% 0.30/0.56 thf(fact_3622_nth__upt,axiom,
% 0.30/0.56 ! [I: nat,K: nat,J: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
% 0.30/0.56 => ( ( nth @ nat @ ( upt @ I @ J ) @ K )
% 0.30/0.56 = ( plus_plus @ nat @ I @ K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_upt
% 0.30/0.56 thf(fact_3623_take__Cons__numeral,axiom,
% 0.30/0.56 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_Cons_numeral
% 0.30/0.56 thf(fact_3624_upt__add__eq__append,axiom,
% 0.30/0.56 ! [I: nat,J: nat,K: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
% 0.30/0.56 = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_add_eq_append
% 0.30/0.56 thf(fact_3625_take__0,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( take @ A @ ( zero_zero @ nat ) @ Xs )
% 0.30/0.56 = ( nil @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_0
% 0.30/0.56 thf(fact_3626_atLeastAtMost__upt,axiom,
% 0.30/0.56 ( ( set_or1337092689740270186AtMost @ nat )
% 0.30/0.56 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastAtMost_upt
% 0.30/0.56 thf(fact_3627_upt__Suc,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.56 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
% 0.30/0.56 & ( ~ ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.56 = ( nil @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_Suc
% 0.30/0.56 thf(fact_3628_upt__Suc__append,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.56 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_Suc_append
% 0.30/0.56 thf(fact_3629_upt__conv__Cons__Cons,axiom,
% 0.30/0.56 ! [M: nat,N: nat,Ns: list @ nat,Q4: nat] :
% 0.30/0.56 ( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
% 0.30/0.56 = ( upt @ M @ Q4 ) )
% 0.30/0.56 = ( ( cons @ nat @ N @ Ns )
% 0.30/0.56 = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_conv_Cons_Cons
% 0.30/0.56 thf(fact_3630_upt__0,axiom,
% 0.30/0.56 ! [I: nat] :
% 0.30/0.56 ( ( upt @ I @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( nil @ nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_0
% 0.30/0.56 thf(fact_3631_greaterThanAtMost__upt,axiom,
% 0.30/0.56 ( ( set_or3652927894154168847AtMost @ nat )
% 0.30/0.56 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % greaterThanAtMost_upt
% 0.30/0.56 thf(fact_3632_greaterThanLessThan__upt,axiom,
% 0.30/0.56 ( ( set_or5935395276787703475ssThan @ nat )
% 0.30/0.56 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % greaterThanLessThan_upt
% 0.30/0.56 thf(fact_3633_set__take__subset__set__take,axiom,
% 0.30/0.56 ! [A: $tType,M: nat,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.56 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_take_subset_set_take
% 0.30/0.56 thf(fact_3634_upt__conv__Cons,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ J )
% 0.30/0.56 => ( ( upt @ I @ J )
% 0.30/0.56 = ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_conv_Cons
% 0.30/0.56 thf(fact_3635_nth__take__lemma,axiom,
% 0.30/0.56 ! [A: $tType,K: nat,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.56 => ( ! [I4: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I4 @ K )
% 0.30/0.56 => ( ( nth @ A @ Xs @ I4 )
% 0.30/0.56 = ( nth @ A @ Ys2 @ I4 ) ) )
% 0.30/0.56 => ( ( take @ A @ K @ Xs )
% 0.30/0.56 = ( take @ A @ K @ Ys2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_take_lemma
% 0.30/0.56 thf(fact_3636_upt__eq__Cons__conv,axiom,
% 0.30/0.56 ! [I: nat,J: nat,X: nat,Xs: list @ nat] :
% 0.30/0.56 ( ( ( upt @ I @ J )
% 0.30/0.56 = ( cons @ nat @ X @ Xs ) )
% 0.30/0.56 = ( ( ord_less @ nat @ I @ J )
% 0.30/0.56 & ( I = X )
% 0.30/0.56 & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
% 0.30/0.56 = Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_eq_Cons_conv
% 0.30/0.56 thf(fact_3637_upt__rec,axiom,
% 0.30/0.56 ( upt
% 0.30/0.56 = ( ^ [I5: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I5 @ J3 ) @ ( cons @ nat @ I5 @ ( upt @ ( suc @ I5 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upt_rec
% 0.30/0.56 thf(fact_3638_take__Cons_H,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( nil @ A ) ) )
% 0.30/0.56 & ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_Cons'
% 0.30/0.56 thf(fact_3639_horner__sum__bit__eq__take__bit,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.56 => ! [A3: A,N: nat] :
% 0.30/0.56 ( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 0.30/0.56 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % horner_sum_bit_eq_take_bit
% 0.30/0.56 thf(fact_3640_lexord__take__index__conv,axiom,
% 0.30/0.56 ! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.56 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
% 0.30/0.56 = ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) )
% 0.30/0.56 & ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y3 )
% 0.30/0.56 = X ) )
% 0.30/0.56 | ? [I5: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
% 0.30/0.56 & ( ( take @ A @ I5 @ X )
% 0.30/0.56 = ( take @ A @ I5 @ Y3 ) )
% 0.30/0.56 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I5 ) @ ( nth @ A @ Y3 @ I5 ) ) @ R ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lexord_take_index_conv
% 0.30/0.56 thf(fact_3641_lex__take__index,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.56 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 0.30/0.56 => ~ ! [I4: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.30/0.56 => ( ( ( take @ A @ I4 @ Xs )
% 0.30/0.56 = ( take @ A @ I4 @ Ys2 ) )
% 0.30/0.56 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys2 @ I4 ) ) @ R ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lex_take_index
% 0.30/0.56 thf(fact_3642_min__weak__def,axiom,
% 0.30/0.56 ( fun_min_weak
% 0.30/0.56 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % min_weak_def
% 0.30/0.56 thf(fact_3643_sup_Oidem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ A3 @ A3 )
% 0.30/0.56 = A3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.idem
% 0.30/0.56 thf(fact_3644_sup__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ X )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_idem
% 0.30/0.56 thf(fact_3645_sup_Oleft__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
% 0.30/0.56 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.left_idem
% 0.30/0.56 thf(fact_3646_sup__left__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.56 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_left_idem
% 0.30/0.56 thf(fact_3647_sup_Oright__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.56 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.right_idem
% 0.30/0.56 thf(fact_3648_sup__apply,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ B )
% 0.30/0.56 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.56 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_apply
% 0.30/0.56 thf(fact_3649_le__sup__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.56 = ( ( ord_less_eq @ A @ X @ Z3 )
% 0.30/0.56 & ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_sup_iff
% 0.30/0.56 thf(fact_3650_sup_Obounded__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.56 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.bounded_iff
% 0.30/0.56 thf(fact_3651_sup__bot__left,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot_left
% 0.30/0.56 thf(fact_3652_sup__bot__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot_right
% 0.30/0.56 thf(fact_3653_bot__eq__sup__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( ( bot_bot @ A )
% 0.30/0.56 = ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.56 = ( ( X
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 & ( Y3
% 0.30/0.56 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bot_eq_sup_iff
% 0.30/0.56 thf(fact_3654_sup__eq__bot__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 = ( ( X
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 & ( Y3
% 0.30/0.56 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_eq_bot_iff
% 0.30/0.56 thf(fact_3655_sup__bot_Oeq__neutr__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ( sup_sup @ A @ A3 @ B2 )
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 = ( ( A3
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 & ( B2
% 0.30/0.56 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot.eq_neutr_iff
% 0.30/0.56 thf(fact_3656_sup__bot_Oleft__neutral,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 0.30/0.56 = A3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot.left_neutral
% 0.30/0.56 thf(fact_3657_sup__bot_Oneutr__eq__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ( bot_bot @ A )
% 0.30/0.56 = ( sup_sup @ A @ A3 @ B2 ) )
% 0.30/0.56 = ( ( A3
% 0.30/0.56 = ( bot_bot @ A ) )
% 0.30/0.56 & ( B2
% 0.30/0.56 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot.neutr_eq_iff
% 0.30/0.56 thf(fact_3658_sup__bot_Oright__neutral,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.56 = A3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_bot.right_neutral
% 0.30/0.56 thf(fact_3659_Compl__Diff__eq,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.56 ( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.56 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ B7 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Compl_Diff_eq
% 0.30/0.56 thf(fact_3660_nth__map,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: A > B] :
% 0.30/0.56 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ B @ ( map @ A @ B @ F @ Xs ) @ N )
% 0.30/0.56 = ( F @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_map
% 0.30/0.56 thf(fact_3661_inf__sup__aci_I8_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.56 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(8)
% 0.30/0.56 thf(fact_3662_inf__sup__aci_I7_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(7)
% 0.30/0.56 thf(fact_3663_inf__sup__aci_I6_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(6)
% 0.30/0.56 thf(fact_3664_inf__sup__aci_I5_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ( ( sup_sup @ A )
% 0.30/0.56 = ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(5)
% 0.30/0.56 thf(fact_3665_sup_Oassoc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.56 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.assoc
% 0.30/0.56 thf(fact_3666_sup__assoc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_assoc
% 0.30/0.56 thf(fact_3667_sup_Ocommute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( sup_sup @ A )
% 0.30/0.56 = ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.commute
% 0.30/0.56 thf(fact_3668_sup__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( sup_sup @ A )
% 0.30/0.56 = ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_commute
% 0.30/0.56 thf(fact_3669_sup_Oleft__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.56 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
% 0.30/0.56 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.left_commute
% 0.30/0.56 thf(fact_3670_sup__left__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_left_commute
% 0.30/0.56 thf(fact_3671_sup__fun__def,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ B )
% 0.30/0.56 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.56 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_fun_def
% 0.30/0.56 thf(fact_3672_boolean__algebra__cancel_Osup1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A6: A,K: A,A3: A,B2: A] :
% 0.30/0.56 ( ( A6
% 0.30/0.56 = ( sup_sup @ A @ K @ A3 ) )
% 0.30/0.56 => ( ( sup_sup @ A @ A6 @ B2 )
% 0.30/0.56 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra_cancel.sup1
% 0.30/0.56 thf(fact_3673_boolean__algebra__cancel_Osup2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B7: A,K: A,B2: A,A3: A] :
% 0.30/0.56 ( ( B7
% 0.30/0.56 = ( sup_sup @ A @ K @ B2 ) )
% 0.30/0.56 => ( ( sup_sup @ A @ A3 @ B7 )
% 0.30/0.56 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra_cancel.sup2
% 0.30/0.56 thf(fact_3674_less__supI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ X @ A3 )
% 0.30/0.56 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_supI1
% 0.30/0.56 thf(fact_3675_less__supI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ X @ B2 )
% 0.30/0.56 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_supI2
% 0.30/0.56 thf(fact_3676_sup_Oabsorb3,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.56 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.30/0.56 = A3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb3
% 0.30/0.56 thf(fact_3677_sup_Oabsorb4,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.30/0.56 = B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb4
% 0.30/0.56 thf(fact_3678_sup_Ostrict__boundedE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.56 => ~ ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.56 => ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.strict_boundedE
% 0.30/0.56 thf(fact_3679_sup_Ostrict__order__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( ord_less @ A )
% 0.30/0.56 = ( ^ [B3: A,A5: A] :
% 0.30/0.56 ( ( A5
% 0.30/0.56 = ( sup_sup @ A @ A5 @ B3 ) )
% 0.30/0.56 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.strict_order_iff
% 0.30/0.56 thf(fact_3680_sup_Ostrict__coboundedI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ C2 @ A3 )
% 0.30/0.56 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.strict_coboundedI1
% 0.30/0.56 thf(fact_3681_sup_Ostrict__coboundedI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ C2 @ B2 )
% 0.30/0.56 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.strict_coboundedI2
% 0.30/0.56 thf(fact_3682_sup__max,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ( semilattice_sup @ A )
% 0.30/0.56 & ( linorder @ A ) )
% 0.30/0.56 => ( ( sup_sup @ A )
% 0.30/0.56 = ( ord_max @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_max
% 0.30/0.56 thf(fact_3683_sup_OcoboundedI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [C2: A,B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.30/0.56 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.coboundedI2
% 0.30/0.56 thf(fact_3684_sup_OcoboundedI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.coboundedI1
% 0.30/0.56 thf(fact_3685_sup_Oabsorb__iff2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [A5: A,B3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ A5 @ B3 )
% 0.30/0.56 = B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb_iff2
% 0.30/0.56 thf(fact_3686_sup_Oabsorb__iff1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [B3: A,A5: A] :
% 0.30/0.56 ( ( sup_sup @ A @ A5 @ B3 )
% 0.30/0.56 = A5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb_iff1
% 0.30/0.56 thf(fact_3687_sup_Ocobounded2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.cobounded2
% 0.30/0.56 thf(fact_3688_sup_Ocobounded1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.cobounded1
% 0.30/0.56 thf(fact_3689_sup_Oorder__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [B3: A,A5: A] :
% 0.30/0.56 ( A5
% 0.30/0.56 = ( sup_sup @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.order_iff
% 0.30/0.56 thf(fact_3690_sup_OboundedI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.boundedI
% 0.30/0.56 thf(fact_3691_sup_OboundedE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 0.30/0.56 => ~ ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.boundedE
% 0.30/0.56 thf(fact_3692_sup__absorb2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.56 => ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.56 = Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_absorb2
% 0.30/0.56 thf(fact_3693_sup__absorb1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [Y3: A,X: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.56 => ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.56 = X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_absorb1
% 0.30/0.56 thf(fact_3694_sup_Oabsorb2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.30/0.56 = B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb2
% 0.30/0.56 thf(fact_3695_sup_Oabsorb1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.30/0.56 = A3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.absorb1
% 0.30/0.56 thf(fact_3696_sup__unique,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [F: A > A > A,X: A,Y3: A] :
% 0.30/0.56 ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y5 ) )
% 0.30/0.56 => ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X3 @ Y5 ) )
% 0.30/0.56 => ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ Y5 @ X3 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ Z4 @ X3 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
% 0.30/0.56 => ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.56 = ( F @ X @ Y3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_unique
% 0.30/0.56 thf(fact_3697_sup_OorderI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( A3
% 0.30/0.56 = ( sup_sup @ A @ A3 @ B2 ) )
% 0.30/0.56 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.orderI
% 0.30/0.56 thf(fact_3698_sup_OorderE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 => ( A3
% 0.30/0.56 = ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.orderE
% 0.30/0.56 thf(fact_3699_le__iff__sup,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [X2: A,Y6: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X2 @ Y6 )
% 0.30/0.56 = Y6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_iff_sup
% 0.30/0.56 thf(fact_3700_sup__least,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [Y3: A,X: A,Z3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.56 => ( ( ord_less_eq @ A @ Z3 @ X )
% 0.30/0.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_least
% 0.30/0.56 thf(fact_3701_sup__mono,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_mono
% 0.30/0.56 thf(fact_3702_sup_Omono,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [C2: A,A3: A,D2: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ D2 @ B2 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup.mono
% 0.30/0.56 thf(fact_3703_le__supI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ X @ B2 )
% 0.30/0.56 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_supI2
% 0.30/0.56 thf(fact_3704_le__supI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ X @ A3 )
% 0.30/0.56 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_supI1
% 0.30/0.56 thf(fact_3705_sup__ge2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_ge2
% 0.30/0.56 thf(fact_3706_sup__ge1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_ge1
% 0.30/0.56 thf(fact_3707_le__supI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,X: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ A3 @ X )
% 0.30/0.56 => ( ( ord_less_eq @ A @ B2 @ X )
% 0.30/0.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_supI
% 0.30/0.56 thf(fact_3708_le__supE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_sup @ A )
% 0.30/0.56 => ! [A3: A,B2: A,X: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
% 0.30/0.56 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 0.30/0.56 => ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_supE
% 0.30/0.56 thf(fact_3709_inf__sup__ord_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_ord(3)
% 0.30/0.56 thf(fact_3710_inf__sup__ord_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_ord(4)
% 0.30/0.56 thf(fact_3711_boolean__algebra_Odisj__zero__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.disj_zero_right
% 0.30/0.56 thf(fact_3712_ivl__disj__un__two__touch_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two_touch(4)
% 0.30/0.56 thf(fact_3713_ivl__disj__un__two_I6_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(6)
% 0.30/0.56 thf(fact_3714_card__Un__le,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,B7: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Un_le
% 0.30/0.56 thf(fact_3715_ivl__disj__un__two_I8_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(8)
% 0.30/0.56 thf(fact_3716_ivl__disj__un__two__touch_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two_touch(3)
% 0.30/0.56 thf(fact_3717_ivl__disj__un__two_I2_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(2)
% 0.30/0.56 thf(fact_3718_nth__map__upt,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,N: nat,M: nat,F: nat > A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
% 0.30/0.56 => ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M @ N ) ) @ I )
% 0.30/0.56 = ( F @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_map_upt
% 0.30/0.56 thf(fact_3719_lexord__sufI,axiom,
% 0.30/0.56 ! [A: $tType,U: list @ A,W2: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z3: list @ A] :
% 0.30/0.56 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R ) )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
% 0.30/0.56 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W2 @ Z3 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lexord_sufI
% 0.30/0.56 thf(fact_3720_ivl__disj__un__singleton_I5_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_singleton(5)
% 0.30/0.56 thf(fact_3721_ivl__disj__un__two_I5_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(5)
% 0.30/0.56 thf(fact_3722_ivl__disj__un__singleton_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.56 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_singleton(4)
% 0.30/0.56 thf(fact_3723_map__upt__eqI,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,N: nat,M: nat,F: nat > A] :
% 0.30/0.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.56 = ( minus_minus @ nat @ N @ M ) )
% 0.30/0.56 => ( ! [I4: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ A @ Xs @ I4 )
% 0.30/0.56 = ( F @ ( plus_plus @ nat @ M @ I4 ) ) ) )
% 0.30/0.56 => ( ( map @ nat @ A @ F @ ( upt @ M @ N ) )
% 0.30/0.56 = Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % map_upt_eqI
% 0.30/0.56 thf(fact_3724_max__weak__def,axiom,
% 0.30/0.56 ( fun_max_weak
% 0.30/0.56 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % max_weak_def
% 0.30/0.56 thf(fact_3725_sorted__list__of__set__atMost__Suc,axiom,
% 0.30/0.56 ! [K: nat] :
% 0.30/0.56 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
% 0.30/0.56 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sorted_list_of_set_atMost_Suc
% 0.30/0.56 thf(fact_3726_sorted__list__of__set__lessThan__Suc,axiom,
% 0.30/0.56 ! [K: nat] :
% 0.30/0.56 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
% 0.30/0.56 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sorted_list_of_set_lessThan_Suc
% 0.30/0.56 thf(fact_3727_id__take__nth__drop,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( Xs
% 0.30/0.56 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % id_take_nth_drop
% 0.30/0.56 thf(fact_3728_atMost__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ord @ A )
% 0.30/0.56 => ! [I: A,K: A] :
% 0.30/0.56 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 0.30/0.56 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_iff
% 0.30/0.56 thf(fact_3729_lessThan__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ord @ A )
% 0.30/0.56 => ! [I: A,K: A] :
% 0.30/0.56 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 0.30/0.56 = ( ord_less @ A @ I @ K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_iff
% 0.30/0.56 thf(fact_3730_drop0,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( drop @ A @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( ^ [X2: list @ A] : X2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop0
% 0.30/0.56 thf(fact_3731_atMost__subset__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( preorder @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y3 ) )
% 0.30/0.56 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_subset_iff
% 0.30/0.56 thf(fact_3732_lessThan__subset__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y3 ) )
% 0.30/0.56 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_subset_iff
% 0.30/0.56 thf(fact_3733_drop__Suc__Cons,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( drop @ A @ N @ Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_Suc_Cons
% 0.30/0.56 thf(fact_3734_lessThan__0,axiom,
% 0.30/0.56 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_0
% 0.30/0.56 thf(fact_3735_card__atMost,axiom,
% 0.30/0.56 ! [U: nat] :
% 0.30/0.56 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
% 0.30/0.56 = ( suc @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_atMost
% 0.30/0.56 thf(fact_3736_Icc__subset__Iic__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( preorder @ A )
% 0.30/0.56 => ! [L: A,H2: A,H: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H ) )
% 0.30/0.56 = ( ~ ( ord_less_eq @ A @ L @ H2 )
% 0.30/0.56 | ( ord_less_eq @ A @ H2 @ H ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Icc_subset_Iic_iff
% 0.30/0.56 thf(fact_3737_drop__eq__Nil2,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ( nil @ A )
% 0.30/0.56 = ( drop @ A @ N @ Xs ) )
% 0.30/0.56 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_eq_Nil2
% 0.30/0.56 thf(fact_3738_drop__eq__Nil,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ( drop @ A @ N @ Xs )
% 0.30/0.56 = ( nil @ A ) )
% 0.30/0.56 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_eq_Nil
% 0.30/0.56 thf(fact_3739_drop__all,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.30/0.56 => ( ( drop @ A @ N @ Xs )
% 0.30/0.56 = ( nil @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_all
% 0.30/0.56 thf(fact_3740_atMost__0,axiom,
% 0.30/0.56 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_0
% 0.30/0.56 thf(fact_3741_drop__Cons__numeral,axiom,
% 0.30/0.56 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_Cons_numeral
% 0.30/0.56 thf(fact_3742_nth__drop,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 0.30/0.56 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_drop
% 0.30/0.56 thf(fact_3743_sup__nat__def,axiom,
% 0.30/0.56 ( ( sup_sup @ nat )
% 0.30/0.56 = ( ord_max @ nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_nat_def
% 0.30/0.56 thf(fact_3744_Iic__subset__Iio__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 0.30/0.56 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Iic_subset_Iio_iff
% 0.30/0.56 thf(fact_3745_drop__0,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
% 0.30/0.56 = Xs ) ).
% 0.30/0.56
% 0.30/0.56 % drop_0
% 0.30/0.56 thf(fact_3746_lessThan__Suc__atMost,axiom,
% 0.30/0.56 ! [K: nat] :
% 0.30/0.56 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 0.30/0.56 = ( set_ord_atMost @ nat @ K ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_Suc_atMost
% 0.30/0.56 thf(fact_3747_ivl__disj__un__one_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 0.30/0.56 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_one(4)
% 0.30/0.56 thf(fact_3748_ivl__disj__un__one_I1_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.30/0.56 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_one(1)
% 0.30/0.56 thf(fact_3749_sup__int__def,axiom,
% 0.30/0.56 ( ( sup_sup @ int )
% 0.30/0.56 = ( ord_max @ int ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_int_def
% 0.30/0.56 thf(fact_3750_atMost__atLeast0,axiom,
% 0.30/0.56 ( ( set_ord_atMost @ nat )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_atLeast0
% 0.30/0.56 thf(fact_3751_map__Suc__upt,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
% 0.30/0.56 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % map_Suc_upt
% 0.30/0.56 thf(fact_3752_lessThan__Suc,axiom,
% 0.30/0.56 ! [K: nat] :
% 0.30/0.56 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 0.30/0.56 = ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_Suc
% 0.30/0.56 thf(fact_3753_lessThan__empty__iff,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( ( set_ord_lessThan @ nat @ N )
% 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.56 = ( N
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_empty_iff
% 0.30/0.56 thf(fact_3754_atMost__Suc,axiom,
% 0.30/0.56 ! [K: nat] :
% 0.30/0.56 ( ( set_ord_atMost @ nat @ ( suc @ K ) )
% 0.30/0.56 = ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_Suc
% 0.30/0.56 thf(fact_3755_lessThan__strict__subset__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [M: A,N: A] :
% 0.30/0.56 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 0.30/0.56 = ( ord_less @ A @ M @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_strict_subset_iff
% 0.30/0.56 thf(fact_3756_set__drop__subset__set__drop,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,M: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.56 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_drop_subset_set_drop
% 0.30/0.56 thf(fact_3757_atLeast__upt,axiom,
% 0.30/0.56 ( ( set_ord_lessThan @ nat )
% 0.30/0.56 = ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeast_upt
% 0.30/0.56 thf(fact_3758_ivl__disj__un__one_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.30/0.56 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_one(3)
% 0.30/0.56 thf(fact_3759_drop__Cons_H,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.30/0.56 ( ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( cons @ A @ X @ Xs ) ) )
% 0.30/0.56 & ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.30/0.56 = ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_Cons'
% 0.30/0.56 thf(fact_3760_append__eq__append__conv__if,axiom,
% 0.30/0.56 ! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
% 0.30/0.56 ( ( ( append @ A @ Xs_1 @ Xs_2 )
% 0.30/0.56 = ( append @ A @ Ys_1 @ Ys_2 ) )
% 0.30/0.56 = ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
% 0.30/0.56 => ( ( Xs_1
% 0.30/0.56 = ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
% 0.30/0.56 & ( Xs_2
% 0.30/0.56 = ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
% 0.30/0.56 & ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
% 0.30/0.56 => ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
% 0.30/0.56 = Ys_1 )
% 0.30/0.56 & ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
% 0.30/0.56 = Ys_2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % append_eq_append_conv_if
% 0.30/0.56 thf(fact_3761_atMost__upto,axiom,
% 0.30/0.56 ( ( set_ord_atMost @ nat )
% 0.30/0.56 = ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atMost_upto
% 0.30/0.56 thf(fact_3762_max__strict__def,axiom,
% 0.30/0.56 ( fun_max_strict
% 0.30/0.56 = ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% 0.30/0.56
% 0.30/0.56 % max_strict_def
% 0.30/0.56 thf(fact_3763_Cons__nth__drop__Suc,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
% 0.30/0.56 = ( drop @ A @ I @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Cons_nth_drop_Suc
% 0.30/0.56 thf(fact_3764_atLeast1__atMost__eq__remove0,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.56 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeast1_atMost_eq_remove0
% 0.30/0.56 thf(fact_3765_take__hd__drop,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
% 0.30/0.56 = ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_hd_drop
% 0.30/0.56 thf(fact_3766_upd__conv__take__nth__drop,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,A3: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( list_update @ A @ Xs @ I @ A3 )
% 0.30/0.56 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % upd_conv_take_nth_drop
% 0.30/0.56 thf(fact_3767_effect__makeE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.30/0.56 ( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ R @ N2 )
% 0.30/0.56 => ~ ( ( R
% 0.30/0.56 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
% 0.30/0.56 => ( ( H
% 0.30/0.56 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
% 0.30/0.56 => ( ( ( array_get @ A @ H @ R )
% 0.30/0.56 = ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 0.30/0.56 => ( ( array_present @ A @ H @ R )
% 0.30/0.56 => ( ~ ( array_present @ A @ H2 @ R )
% 0.30/0.56 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 0.30/0.56 != N2 ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_makeE
% 0.30/0.56 thf(fact_3768_n__lists__Nil,axiom,
% 0.30/0.56 ! [A: $tType,N: nat] :
% 0.30/0.56 ( ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( n_lists @ A @ N @ ( nil @ A ) )
% 0.30/0.56 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
% 0.30/0.56 & ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( n_lists @ A @ N @ ( nil @ A ) )
% 0.30/0.56 = ( nil @ ( list @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % n_lists_Nil
% 0.30/0.56 thf(fact_3769_hd__upt,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ J )
% 0.30/0.56 => ( ( hd @ nat @ ( upt @ I @ J ) )
% 0.30/0.56 = I ) ) ).
% 0.30/0.56
% 0.30/0.56 % hd_upt
% 0.30/0.56 thf(fact_3770_list__update__beyond,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,I: nat,X: A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 0.30/0.56 => ( ( list_update @ A @ Xs @ I @ X )
% 0.30/0.56 = Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_beyond
% 0.30/0.56 thf(fact_3771_take__update__cancel,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,M: nat,Xs: list @ A,Y3: A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.56 => ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y3 ) )
% 0.30/0.56 = ( take @ A @ N @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % take_update_cancel
% 0.30/0.56 thf(fact_3772_drop__update__cancel,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,M: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ N @ M )
% 0.30/0.56 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
% 0.30/0.56 = ( drop @ A @ M @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_update_cancel
% 0.30/0.56 thf(fact_3773_get__update__eq,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [A3: array @ A,I: nat,V: A,H2: heap_ext @ product_unit] :
% 0.30/0.56 ( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H2 ) @ A3 )
% 0.30/0.56 = ( list_update @ A @ ( array_get @ A @ H2 @ A3 ) @ I @ V ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % get_update_eq
% 0.30/0.56 thf(fact_3774_nth__list__update__eq,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_list_update_eq
% 0.30/0.56 thf(fact_3775_hd__take,axiom,
% 0.30/0.56 ! [A: $tType,J: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
% 0.30/0.56 => ( ( hd @ A @ ( take @ A @ J @ Xs ) )
% 0.30/0.56 = ( hd @ A @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % hd_take
% 0.30/0.56 thf(fact_3776_set__swap,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
% 0.30/0.56 = ( set2 @ A @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_swap
% 0.30/0.56 thf(fact_3777_get__alloc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
% 0.30/0.56 ( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
% 0.30/0.56 = Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % get_alloc
% 0.30/0.56 thf(fact_3778_set,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [Ls: list @ A,H2: heap_ext @ product_unit,New_ls: list @ A] :
% 0.30/0.56 ( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) )
% 0.30/0.56 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set
% 0.30/0.56 thf(fact_3779_not__present__alloc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [H2: heap_ext @ product_unit,Xs: list @ A] :
% 0.30/0.56 ~ ( array_present @ A @ H2 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_present_alloc
% 0.30/0.56 thf(fact_3780_present__alloc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % present_alloc
% 0.30/0.56 thf(fact_3781_list__update__code_I3_J,axiom,
% 0.30/0.56 ! [A: $tType,X: A,Xs: list @ A,I: nat,Y3: A] :
% 0.30/0.56 ( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y3 )
% 0.30/0.56 = ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_code(3)
% 0.30/0.56 thf(fact_3782_list__update__code_I2_J,axiom,
% 0.30/0.56 ! [A: $tType,X: A,Xs: list @ A,Y3: A] :
% 0.30/0.56 ( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y3 )
% 0.30/0.56 = ( cons @ A @ Y3 @ Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_code(2)
% 0.30/0.56 thf(fact_3783_set__update__memI,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_update_memI
% 0.30/0.56 thf(fact_3784_list__update__append1,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,Ys2: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ I @ X )
% 0.30/0.56 = ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_append1
% 0.30/0.56 thf(fact_3785_length__alloc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
% 0.30/0.56 ( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
% 0.30/0.56 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_alloc
% 0.30/0.56 thf(fact_3786_list__update__same__conv,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ( list_update @ A @ Xs @ I @ X )
% 0.30/0.56 = Xs )
% 0.30/0.56 = ( ( nth @ A @ Xs @ I )
% 0.30/0.56 = X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_same_conv
% 0.30/0.56 thf(fact_3787_nth__list__update,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ( I = J )
% 0.30/0.56 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
% 0.30/0.56 = X ) )
% 0.30/0.56 & ( ( I != J )
% 0.30/0.56 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
% 0.30/0.56 = ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_list_update
% 0.30/0.56 thf(fact_3788_hd__conv__nth,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( Xs
% 0.30/0.56 != ( nil @ A ) )
% 0.30/0.56 => ( ( hd @ A @ Xs )
% 0.30/0.56 = ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % hd_conv_nth
% 0.30/0.56 thf(fact_3789_drop__update__swap,axiom,
% 0.30/0.56 ! [A: $tType,M: nat,N: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.56 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
% 0.30/0.56 = ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % drop_update_swap
% 0.30/0.56 thf(fact_3790_present__alloc__noteq,axiom,
% 0.30/0.56 ! [A: $tType,B: $tType] :
% 0.30/0.56 ( ( ( heap @ B )
% 0.30/0.56 & ( heap @ A ) )
% 0.30/0.56 => ! [H2: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
% 0.30/0.56 ( ( array_present @ A @ H2 @ A3 )
% 0.30/0.56 => ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % present_alloc_noteq
% 0.30/0.56 thf(fact_3791_update__def,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ( ( array_update @ A )
% 0.30/0.56 = ( ^ [A5: array @ A,I5: nat,X2: A,H4: heap_ext @ product_unit] : ( array_set @ A @ A5 @ ( list_update @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 @ X2 ) @ H4 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % update_def
% 0.30/0.56 thf(fact_3792_list__update__append,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A,X: A] :
% 0.30/0.56 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
% 0.30/0.56 = ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys2 ) ) )
% 0.30/0.56 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
% 0.30/0.56 = ( append @ A @ Xs @ ( list_update @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % list_update_append
% 0.30/0.56 thf(fact_3793_n__lists_Osimps_I1_J,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
% 0.30/0.56 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % n_lists.simps(1)
% 0.30/0.56 thf(fact_3794_effect__makeI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [A3: array @ A,H: heap_ext @ product_unit,F: nat > A,N: nat,H2: heap_ext @ product_unit] :
% 0.30/0.56 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 0.30/0.56 = ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) )
% 0.30/0.56 => ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_makeI
% 0.30/0.56 thf(fact_3795_hd__drop__conv__nth,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
% 0.30/0.56 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % hd_drop_conv_nth
% 0.30/0.56 thf(fact_3796_effect__of__listE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [Xs: list @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.30/0.56 ( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ R @ N2 )
% 0.30/0.56 => ~ ( ( R
% 0.30/0.56 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
% 0.30/0.56 => ( ( H
% 0.30/0.56 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
% 0.30/0.56 => ( ( ( array_get @ A @ H @ R )
% 0.30/0.56 = Xs )
% 0.30/0.56 => ( ( array_present @ A @ H @ R )
% 0.30/0.56 => ( ~ ( array_present @ A @ H2 @ R )
% 0.30/0.56 => ( N2
% 0.30/0.56 != ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_of_listE
% 0.30/0.56 thf(fact_3797_effect__of__listI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [A3: array @ A,H: heap_ext @ product_unit,Xs: list @ A,H2: heap_ext @ product_unit] :
% 0.30/0.56 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 0.30/0.56 = ( array_alloc @ A @ Xs @ H2 ) )
% 0.30/0.56 => ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_of_listI
% 0.30/0.56 thf(fact_3798_set__update__distinct,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,N: nat,X: A] :
% 0.30/0.56 ( ( distinct @ A @ Xs )
% 0.30/0.56 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
% 0.30/0.56 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_update_distinct
% 0.30/0.56 thf(fact_3799_effect__newE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [N: nat,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.30/0.56 ( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ R @ N2 )
% 0.30/0.56 => ~ ( ( R
% 0.30/0.56 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
% 0.30/0.56 => ( ( H
% 0.30/0.56 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
% 0.30/0.56 => ( ( ( array_get @ A @ H @ R )
% 0.30/0.56 = ( replicate @ A @ N @ X ) )
% 0.30/0.56 => ( ( array_present @ A @ H @ R )
% 0.30/0.56 => ( ~ ( array_present @ A @ H2 @ R )
% 0.30/0.56 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 0.30/0.56 != N2 ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_newE
% 0.30/0.56 thf(fact_3800_replicate__eq__replicate,axiom,
% 0.30/0.56 ! [A: $tType,M: nat,X: A,N: nat,Y3: A] :
% 0.30/0.56 ( ( ( replicate @ A @ M @ X )
% 0.30/0.56 = ( replicate @ A @ N @ Y3 ) )
% 0.30/0.56 = ( ( M = N )
% 0.30/0.56 & ( ( M
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( X = Y3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % replicate_eq_replicate
% 0.30/0.56 thf(fact_3801_replicate__empty,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( ( replicate @ A @ N @ X )
% 0.30/0.56 = ( nil @ A ) )
% 0.30/0.56 = ( N
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % replicate_empty
% 0.30/0.56 thf(fact_3802_empty__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( ( nil @ A )
% 0.30/0.56 = ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( N
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % empty_replicate
% 0.30/0.56 thf(fact_3803_Ball__set__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,A3: A,P: A > $o] :
% 0.30/0.56 ( ( ! [X2: A] :
% 0.30/0.56 ( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
% 0.30/0.56 => ( P @ X2 ) ) )
% 0.30/0.56 = ( ( P @ A3 )
% 0.30/0.56 | ( N
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Ball_set_replicate
% 0.30/0.56 thf(fact_3804_Bex__set__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,A3: A,P: A > $o] :
% 0.30/0.56 ( ( ? [X2: A] :
% 0.30/0.56 ( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
% 0.30/0.56 & ( P @ X2 ) ) )
% 0.30/0.56 = ( ( P @ A3 )
% 0.30/0.56 & ( N
% 0.30/0.56 != ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Bex_set_replicate
% 0.30/0.56 thf(fact_3805_in__set__replicate,axiom,
% 0.30/0.56 ! [A: $tType,X: A,N: nat,Y3: A] :
% 0.30/0.56 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y3 ) ) )
% 0.30/0.56 = ( ( X = Y3 )
% 0.30/0.56 & ( N
% 0.30/0.56 != ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_set_replicate
% 0.30/0.56 thf(fact_3806_nth__replicate,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,N: nat,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ N )
% 0.30/0.56 => ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_replicate
% 0.30/0.56 thf(fact_3807_hd__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( hd @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % hd_replicate
% 0.30/0.56 thf(fact_3808_set__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_replicate
% 0.30/0.56 thf(fact_3809_distinct__swap,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
% 0.30/0.56 = ( distinct @ A @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % distinct_swap
% 0.30/0.56 thf(fact_3810_replicate__Suc,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( replicate @ A @ ( suc @ N ) @ X )
% 0.30/0.56 = ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % replicate_Suc
% 0.30/0.56 thf(fact_3811_replicate__0,axiom,
% 0.30/0.56 ! [A: $tType,X: A] :
% 0.30/0.56 ( ( replicate @ A @ ( zero_zero @ nat ) @ X )
% 0.30/0.56 = ( nil @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % replicate_0
% 0.30/0.56 thf(fact_3812_distinct__conv__nth,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( distinct @ A )
% 0.30/0.56 = ( ^ [Xs3: list @ A] :
% 0.30/0.56 ! [I5: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.56 => ! [J3: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.56 => ( ( I5 != J3 )
% 0.30/0.56 => ( ( nth @ A @ Xs3 @ I5 )
% 0.30/0.56 != ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % distinct_conv_nth
% 0.30/0.56 thf(fact_3813_nth__eq__iff__index__eq,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,I: nat,J: nat] :
% 0.30/0.56 ( ( distinct @ A @ Xs )
% 0.30/0.56 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( ( nth @ A @ Xs @ I )
% 0.30/0.56 = ( nth @ A @ Xs @ J ) )
% 0.30/0.56 = ( I = J ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_eq_iff_index_eq
% 0.30/0.56 thf(fact_3814_effect__newI,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( heap @ A )
% 0.30/0.56 => ! [A3: array @ A,H: heap_ext @ product_unit,N: nat,X: A,H2: heap_ext @ product_unit] :
% 0.30/0.56 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 0.30/0.56 = ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) )
% 0.30/0.56 => ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % effect_newI
% 0.30/0.56 thf(fact_3815_distinct__Ex1,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( distinct @ A @ Xs )
% 0.30/0.56 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.56 => ? [X3: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 & ( ( nth @ A @ Xs @ X3 )
% 0.30/0.56 = X )
% 0.30/0.56 & ! [Y4: nat] :
% 0.30/0.56 ( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 & ( ( nth @ A @ Xs @ Y4 )
% 0.30/0.56 = X ) )
% 0.30/0.56 => ( Y4 = X3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % distinct_Ex1
% 0.30/0.56 thf(fact_3816_set__replicate__Suc,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
% 0.30/0.56 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_replicate_Suc
% 0.30/0.56 thf(fact_3817_set__replicate__conv__if,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.56 & ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_replicate_conv_if
% 0.30/0.56 thf(fact_3818_Cons__replicate__eq,axiom,
% 0.30/0.56 ! [A: $tType,X: A,Xs: list @ A,N: nat,Y3: A] :
% 0.30/0.56 ( ( ( cons @ A @ X @ Xs )
% 0.30/0.56 = ( replicate @ A @ N @ Y3 ) )
% 0.30/0.56 = ( ( X = Y3 )
% 0.30/0.56 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.56 & ( Xs
% 0.30/0.56 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Cons_replicate_eq
% 0.30/0.56 thf(fact_3819_atLeast1__lessThan__eq__remove0,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.30/0.56 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeast1_lessThan_eq_remove0
% 0.30/0.56 thf(fact_3820_listrel1__iff__update,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.56 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.30/0.56 = ( ? [Y6: A,N3: nat] :
% 0.30/0.56 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N3 ) @ Y6 ) @ R )
% 0.30/0.56 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 & ( Ys2
% 0.30/0.56 = ( list_update @ A @ Xs @ N3 @ Y6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % listrel1_iff_update
% 0.30/0.56 thf(fact_3821_Gcd__remove0__nat,axiom,
% 0.30/0.56 ! [M7: set @ nat] :
% 0.30/0.56 ( ( finite_finite2 @ nat @ M7 )
% 0.30/0.56 => ( ( gcd_Gcd @ nat @ M7 )
% 0.30/0.56 = ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Gcd_remove0_nat
% 0.30/0.56 thf(fact_3822_remdups__adj__singleton__iff,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
% 0.30/0.56 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.56 = ( ( Xs
% 0.30/0.56 != ( nil @ A ) )
% 0.30/0.56 & ( Xs
% 0.30/0.56 = ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % remdups_adj_singleton_iff
% 0.30/0.56 thf(fact_3823_atLeastLessThan__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ord @ A )
% 0.30/0.56 => ! [I: A,L: A,U: A] :
% 0.30/0.56 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ L @ I )
% 0.30/0.56 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_iff
% 0.30/0.56 thf(fact_3824_infinite__Icc__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
% 0.30/0.56 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc_iff
% 0.30/0.56 thf(fact_3825_atLeastLessThan__empty,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_empty
% 0.30/0.56 thf(fact_3826_atLeastLessThan__empty__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( preorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_empty_iff
% 0.30/0.56 thf(fact_3827_atLeastLessThan__empty__iff2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( preorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 0.30/0.56 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_empty_iff2
% 0.30/0.56 thf(fact_3828_ivl__subset,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [I: A,J: A,M: A,N: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ J @ I )
% 0.30/0.56 | ( ( ord_less_eq @ A @ M @ I )
% 0.30/0.56 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_subset
% 0.30/0.56 thf(fact_3829_infinite__Ico__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
% 0.30/0.56 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ico_iff
% 0.30/0.56 thf(fact_3830_card_Oinfinite,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ A6 )
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card.infinite
% 0.30/0.56 thf(fact_3831_ivl__diff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [I: A,N: A,M: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ I @ N )
% 0.30/0.56 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_diff
% 0.30/0.56 thf(fact_3832_infinite__Ioc__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
% 0.30/0.56 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ioc_iff
% 0.30/0.56 thf(fact_3833_infinite__Ioo__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
% 0.30/0.56 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ioo_iff
% 0.30/0.56 thf(fact_3834_card__0__eq,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( ( finite_card @ A @ A6 )
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 = ( A6
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_0_eq
% 0.30/0.56 thf(fact_3835_card__insert__disjoint,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.56 = ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_insert_disjoint
% 0.30/0.56 thf(fact_3836_atLeastLessThan__singleton,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
% 0.30/0.56 = ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_singleton
% 0.30/0.56 thf(fact_3837_finite__psubset__induct,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ! [A9: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.56 => ( ! [B10: set @ A] :
% 0.30/0.56 ( ( ord_less @ ( set @ A ) @ B10 @ A9 )
% 0.30/0.56 => ( P @ B10 ) )
% 0.30/0.56 => ( P @ A9 ) ) )
% 0.30/0.56 => ( P @ A6 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_psubset_induct
% 0.30/0.56 thf(fact_3838_atLeastLessThan__eq__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.56 => ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 = ( ( A3 = C2 )
% 0.30/0.56 & ( B2 = D2 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_eq_iff
% 0.30/0.56 thf(fact_3839_atLeastLessThan__inj_I1_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 => ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.56 => ( A3 = C2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_inj(1)
% 0.30/0.56 thf(fact_3840_atLeastLessThan__inj_I2_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 => ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less @ A @ C2 @ D2 )
% 0.30/0.56 => ( B2 = D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_inj(2)
% 0.30/0.56 thf(fact_3841_infinite__Ico,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ico
% 0.30/0.56 thf(fact_3842_finite__nat__set__iff__bounded,axiom,
% 0.30/0.56 ( ( finite_finite2 @ nat )
% 0.30/0.56 = ( ^ [N7: set @ nat] :
% 0.30/0.56 ? [M2: nat] :
% 0.30/0.56 ! [X2: nat] :
% 0.30/0.56 ( ( member @ nat @ X2 @ N7 )
% 0.30/0.56 => ( ord_less @ nat @ X2 @ M2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_nat_set_iff_bounded
% 0.30/0.56 thf(fact_3843_bounded__nat__set__is__finite,axiom,
% 0.30/0.56 ! [N6: set @ nat,N: nat] :
% 0.30/0.56 ( ! [X3: nat] :
% 0.30/0.56 ( ( member @ nat @ X3 @ N6 )
% 0.30/0.56 => ( ord_less @ nat @ X3 @ N ) )
% 0.30/0.56 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.30/0.56
% 0.30/0.56 % bounded_nat_set_is_finite
% 0.30/0.56 thf(fact_3844_finite__nat__set__iff__bounded__le,axiom,
% 0.30/0.56 ( ( finite_finite2 @ nat )
% 0.30/0.56 = ( ^ [N7: set @ nat] :
% 0.30/0.56 ? [M2: nat] :
% 0.30/0.56 ! [X2: nat] :
% 0.30/0.56 ( ( member @ nat @ X2 @ N7 )
% 0.30/0.56 => ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_nat_set_iff_bounded_le
% 0.30/0.56 thf(fact_3845_finite__has__minimal2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [A6: set @ A,A3: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.56 => ? [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ A6 )
% 0.30/0.56 & ( ord_less_eq @ A @ X3 @ A3 )
% 0.30/0.56 & ! [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ A6 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 0.30/0.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_has_minimal2
% 0.30/0.56 thf(fact_3846_finite__has__maximal2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [A6: set @ A,A3: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.56 => ? [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ A6 )
% 0.30/0.56 & ( ord_less_eq @ A @ A3 @ X3 )
% 0.30/0.56 & ! [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ A6 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 0.30/0.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_has_maximal2
% 0.30/0.56 thf(fact_3847_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.30/0.56 ! [N6: set @ nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.56 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq_atLeast0_lessThan_finite
% 0.30/0.56 thf(fact_3848_finite__set__decode,axiom,
% 0.30/0.56 ! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_set_decode
% 0.30/0.56 thf(fact_3849_atLeastLessThan__subset__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.56 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_subset_iff
% 0.30/0.56 thf(fact_3850_ivl__disj__un__two_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(3)
% 0.30/0.56 thf(fact_3851_ex__nat__less__eq,axiom,
% 0.30/0.56 ! [N: nat,P: nat > $o] :
% 0.30/0.56 ( ( ? [M2: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ M2 @ N )
% 0.30/0.56 & ( P @ M2 ) ) )
% 0.30/0.56 = ( ? [X2: nat] :
% 0.30/0.56 ( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.56 & ( P @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ex_nat_less_eq
% 0.30/0.56 thf(fact_3852_all__nat__less__eq,axiom,
% 0.30/0.56 ! [N: nat,P: nat > $o] :
% 0.30/0.56 ( ( ! [M2: nat] :
% 0.30/0.56 ( ( ord_less @ nat @ M2 @ N )
% 0.30/0.56 => ( P @ M2 ) ) )
% 0.30/0.56 = ( ! [X2: nat] :
% 0.30/0.56 ( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.56 => ( P @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % all_nat_less_eq
% 0.30/0.56 thf(fact_3853_atLeastLessThan__add__Un,axiom,
% 0.30/0.56 ! [I: nat,J: nat,K: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
% 0.30/0.56 = ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_add_Un
% 0.30/0.56 thf(fact_3854_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.30/0.56 ! [L: nat,U: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThanSuc_atLeastAtMost
% 0.30/0.56 thf(fact_3855_finite__has__minimal,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( A6
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ? [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ A6 )
% 0.30/0.56 & ! [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ A6 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 0.30/0.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_has_minimal
% 0.30/0.56 thf(fact_3856_finite__has__maximal,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( A6
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ? [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ A6 )
% 0.30/0.56 & ! [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ A6 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 0.30/0.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_has_maximal
% 0.30/0.56 thf(fact_3857_infinite__growing,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [X8: set @ A] :
% 0.30/0.56 ( ( X8
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( ! [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ X8 )
% 0.30/0.56 => ? [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ X8 )
% 0.30/0.56 & ( ord_less @ A @ X3 @ Xa ) ) )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_growing
% 0.30/0.56 thf(fact_3858_ex__min__if__finite,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( order @ A )
% 0.30/0.56 => ! [S3: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.56 => ( ( S3
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ? [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ S3 )
% 0.30/0.56 & ~ ? [Xa: A] :
% 0.30/0.56 ( ( member @ A @ Xa @ S3 )
% 0.30/0.56 & ( ord_less @ A @ Xa @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ex_min_if_finite
% 0.30/0.56 thf(fact_3859_infinite__Icc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc
% 0.30/0.56 thf(fact_3860_lessThan__atLeast0,axiom,
% 0.30/0.56 ( ( set_ord_lessThan @ nat )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % lessThan_atLeast0
% 0.30/0.56 thf(fact_3861_card__le__if__inj__on__rel,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType,B7: set @ A,A6: set @ B,R: B > A > $o] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ! [A8: B] :
% 0.30/0.56 ( ( member @ B @ A8 @ A6 )
% 0.30/0.56 => ? [B11: A] :
% 0.30/0.56 ( ( member @ A @ B11 @ B7 )
% 0.30/0.56 & ( R @ A8 @ B11 ) ) )
% 0.30/0.56 => ( ! [A13: B,A24: B,B5: A] :
% 0.30/0.56 ( ( member @ B @ A13 @ A6 )
% 0.30/0.56 => ( ( member @ B @ A24 @ A6 )
% 0.30/0.56 => ( ( member @ A @ B5 @ B7 )
% 0.30/0.56 => ( ( R @ A13 @ B5 )
% 0.30/0.56 => ( ( R @ A24 @ B5 )
% 0.30/0.56 => ( A13 = A24 ) ) ) ) ) )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_le_if_inj_on_rel
% 0.30/0.56 thf(fact_3862_remdups__adj__length,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.56
% 0.30/0.56 % remdups_adj_length
% 0.30/0.56 thf(fact_3863_infinite__Ioc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ioc
% 0.30/0.56 thf(fact_3864_atLeastLessThan0,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
% 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan0
% 0.30/0.56 thf(fact_3865_infinite__Ioo,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Ioo
% 0.30/0.56 thf(fact_3866_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.30/0.56 ! [L: nat,U: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastSucLessThan_greaterThanLessThan
% 0.30/0.56 thf(fact_3867_ivl__disj__un__two_I7_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(7)
% 0.30/0.56 thf(fact_3868_ivl__disj__un__one_I2_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 0.30/0.56 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_one(2)
% 0.30/0.56 thf(fact_3869_finite__ranking__induct,axiom,
% 0.30/0.56 ! [A: $tType,B: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [S3: set @ B,P: ( set @ B ) > $o,F: B > A] :
% 0.30/0.56 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.56 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.56 => ( ! [X3: B,S4: set @ B] :
% 0.30/0.56 ( ( finite_finite2 @ B @ S4 )
% 0.30/0.56 => ( ! [Y4: B] :
% 0.30/0.56 ( ( member @ B @ Y4 @ S4 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 0.30/0.56 => ( ( P @ S4 )
% 0.30/0.56 => ( P @ ( insert @ B @ X3 @ S4 ) ) ) ) )
% 0.30/0.56 => ( P @ S3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_ranking_induct
% 0.30/0.56 thf(fact_3870_finite__linorder__max__induct,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A6: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( ! [B5: A,A9: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.56 => ( ! [X4: A] :
% 0.30/0.56 ( ( member @ A @ X4 @ A9 )
% 0.30/0.56 => ( ord_less @ A @ X4 @ B5 ) )
% 0.30/0.56 => ( ( P @ A9 )
% 0.30/0.56 => ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
% 0.30/0.56 => ( P @ A6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_linorder_max_induct
% 0.30/0.56 thf(fact_3871_finite__linorder__min__induct,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [A6: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( ! [B5: A,A9: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.56 => ( ! [X4: A] :
% 0.30/0.56 ( ( member @ A @ X4 @ A9 )
% 0.30/0.56 => ( ord_less @ A @ B5 @ X4 ) )
% 0.30/0.56 => ( ( P @ A9 )
% 0.30/0.56 => ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
% 0.30/0.56 => ( P @ A6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_linorder_min_induct
% 0.30/0.56 thf(fact_3872_card__eq__0__iff,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( ( finite_card @ A @ A6 )
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 = ( ( A6
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 | ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_eq_0_iff
% 0.30/0.56 thf(fact_3873_atLeast0__lessThan__Suc,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.30/0.56 = ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeast0_lessThan_Suc
% 0.30/0.56 thf(fact_3874_card__Suc__eq__finite,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,K: nat] :
% 0.30/0.56 ( ( ( finite_card @ A @ A6 )
% 0.30/0.56 = ( suc @ K ) )
% 0.30/0.56 = ( ? [B3: A,B6: set @ A] :
% 0.30/0.56 ( ( A6
% 0.30/0.56 = ( insert @ A @ B3 @ B6 ) )
% 0.30/0.56 & ~ ( member @ A @ B3 @ B6 )
% 0.30/0.56 & ( ( finite_card @ A @ B6 )
% 0.30/0.56 = K )
% 0.30/0.56 & ( finite_finite2 @ A @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Suc_eq_finite
% 0.30/0.56 thf(fact_3875_card__insert__if,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.56 = ( finite_card @ A @ A6 ) ) )
% 0.30/0.56 & ( ~ ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.56 = ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_insert_if
% 0.30/0.56 thf(fact_3876_card__ge__0__finite,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
% 0.30/0.56 => ( finite_finite2 @ A @ A6 ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_ge_0_finite
% 0.30/0.56 thf(fact_3877_obtain__subset__with__card__n,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,S3: set @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
% 0.30/0.56 => ~ ! [T4: set @ A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
% 0.30/0.56 => ( ( ( finite_card @ A @ T4 )
% 0.30/0.56 = N )
% 0.30/0.56 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % obtain_subset_with_card_n
% 0.30/0.56 thf(fact_3878_finite__if__finite__subsets__card__bdd,axiom,
% 0.30/0.56 ! [A: $tType,F4: set @ A,C6: nat] :
% 0.30/0.56 ( ! [G3: set @ A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
% 0.30/0.56 => ( ( finite_finite2 @ A @ G3 )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C6 ) ) )
% 0.30/0.56 => ( ( finite_finite2 @ A @ F4 )
% 0.30/0.56 & ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C6 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_if_finite_subsets_card_bdd
% 0.30/0.56 thf(fact_3879_card__seteq,axiom,
% 0.30/0.56 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B7 ) @ ( finite_card @ A @ A6 ) )
% 0.30/0.56 => ( A6 = B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_seteq
% 0.30/0.56 thf(fact_3880_card__mono,axiom,
% 0.30/0.56 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_mono
% 0.30/0.56 thf(fact_3881_card__less__sym__Diff,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 0.30/0.56 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_less_sym_Diff
% 0.30/0.56 thf(fact_3882_card__le__sym__Diff,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_le_sym_Diff
% 0.30/0.56 thf(fact_3883_psubset__card__mono,axiom,
% 0.30/0.56 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.56 => ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_card_mono
% 0.30/0.56 thf(fact_3884_subset__eq__atLeast0__atMost__finite,axiom,
% 0.30/0.56 ! [N6: set @ nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.56 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq_atLeast0_atMost_finite
% 0.30/0.56 thf(fact_3885_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( preorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastAtMost_subseteq_atLeastLessThan_iff
% 0.30/0.56 thf(fact_3886_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 0.30/0.56 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_subseteq_atLeastAtMost_iff
% 0.30/0.56 thf(fact_3887_ivl__disj__un__two__touch_I2_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two_touch(2)
% 0.30/0.56 thf(fact_3888_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % greaterThanAtMost_subseteq_atLeastLessThan_iff
% 0.30/0.56 thf(fact_3889_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( dense_linorder @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 0.30/0.56 = ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.56 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % greaterThanLessThan_subseteq_atLeastLessThan_iff
% 0.30/0.56 thf(fact_3890_ivl__disj__un__two_I1_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less_eq @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(1)
% 0.30/0.56 thf(fact_3891_card__gt__0__iff,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
% 0.30/0.56 = ( ( A6
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 & ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_gt_0_iff
% 0.30/0.56 thf(fact_3892_card__le__Suc0__iff__eq,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.56 = ( ! [X2: A] :
% 0.30/0.56 ( ( member @ A @ X2 @ A6 )
% 0.30/0.56 => ! [Y6: A] :
% 0.30/0.56 ( ( member @ A @ Y6 @ A6 )
% 0.30/0.56 => ( X2 = Y6 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_le_Suc0_iff_eq
% 0.30/0.56 thf(fact_3893_card__le__Suc__iff,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,A6: set @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A6 ) )
% 0.30/0.56 = ( ? [A5: A,B6: set @ A] :
% 0.30/0.56 ( ( A6
% 0.30/0.56 = ( insert @ A @ A5 @ B6 ) )
% 0.30/0.56 & ~ ( member @ A @ A5 @ B6 )
% 0.30/0.56 & ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
% 0.30/0.56 & ( finite_finite2 @ A @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_le_Suc_iff
% 0.30/0.56 thf(fact_3894_ivl__disj__un__two__touch_I1_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two_touch(1)
% 0.30/0.56 thf(fact_3895_finite__induct__select,axiom,
% 0.30/0.56 ! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.56 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( ! [T4: set @ A] :
% 0.30/0.56 ( ( ord_less @ ( set @ A ) @ T4 @ S3 )
% 0.30/0.56 => ( ( P @ T4 )
% 0.30/0.56 => ? [X4: A] :
% 0.30/0.56 ( ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) )
% 0.30/0.56 & ( P @ ( insert @ A @ X4 @ T4 ) ) ) ) )
% 0.30/0.56 => ( P @ S3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_induct_select
% 0.30/0.56 thf(fact_3896_card__psubset,axiom,
% 0.30/0.56 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.56 => ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 0.30/0.56 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_psubset
% 0.30/0.56 thf(fact_3897_diff__card__le__card__Diff,axiom,
% 0.30/0.56 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % diff_card_le_card_Diff
% 0.30/0.56 thf(fact_3898_remdups__adj__adjacent,axiom,
% 0.30/0.56 ! [A: $tType,I: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
% 0.30/0.56 => ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
% 0.30/0.56 != ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % remdups_adj_adjacent
% 0.30/0.56 thf(fact_3899_remdups__adj__replicate,axiom,
% 0.30/0.56 ! [A: $tType,N: nat,X: A] :
% 0.30/0.56 ( ( ( N
% 0.30/0.56 = ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( nil @ A ) ) )
% 0.30/0.56 & ( ( N
% 0.30/0.56 != ( zero_zero @ nat ) )
% 0.30/0.56 => ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.56 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % remdups_adj_replicate
% 0.30/0.56 thf(fact_3900_atLeastLessThanSuc,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
% 0.30/0.56 = ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
% 0.30/0.56 & ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
% 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThanSuc
% 0.30/0.56 thf(fact_3901_subset__eq__atLeast0__lessThan__card,axiom,
% 0.30/0.56 ! [N6: set @ nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq_atLeast0_lessThan_card
% 0.30/0.56 thf(fact_3902_ivl__disj__un__singleton_I6_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_singleton(6)
% 0.30/0.56 thf(fact_3903_ivl__disj__un__two_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,M: A,U: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ L @ M )
% 0.30/0.56 => ( ( ord_less @ A @ M @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_two(4)
% 0.30/0.56 thf(fact_3904_ivl__disj__un__singleton_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linorder @ A )
% 0.30/0.56 => ! [L: A,U: A] :
% 0.30/0.56 ( ( ord_less @ A @ L @ U )
% 0.30/0.56 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % ivl_disj_un_singleton(3)
% 0.30/0.56 thf(fact_3905_card_Oremove,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ A6 )
% 0.30/0.56 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card.remove
% 0.30/0.56 thf(fact_3906_card_Oinsert__remove,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.56 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card.insert_remove
% 0.30/0.56 thf(fact_3907_card__Suc__Diff1,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.30/0.56 = ( finite_card @ A @ A6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Suc_Diff1
% 0.30/0.56 thf(fact_3908_card__Diff1__less,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Diff1_less
% 0.30/0.56 thf(fact_3909_card__Diff2__less,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A,Y3: A] :
% 0.30/0.56 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 => ( ( member @ A @ X @ A6 )
% 0.30/0.56 => ( ( member @ A @ Y3 @ A6 )
% 0.30/0.56 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Diff2_less
% 0.30/0.56 thf(fact_3910_card__Diff1__less__iff,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) )
% 0.30/0.56 = ( ( finite_finite2 @ A @ A6 )
% 0.30/0.56 & ( member @ A @ X @ A6 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_Diff1_less_iff
% 0.30/0.56 thf(fact_3911_remdups__adj__length__ge1,axiom,
% 0.30/0.56 ! [A: $tType,Xs: list @ A] :
% 0.30/0.56 ( ( Xs
% 0.30/0.56 != ( nil @ A ) )
% 0.30/0.56 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % remdups_adj_length_ge1
% 0.30/0.56 thf(fact_3912_atLeastLessThan__nat__numeral,axiom,
% 0.30/0.56 ! [M: nat,K: num] :
% 0.30/0.56 ( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.56 = ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.30/0.56 & ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.56 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
% 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_nat_numeral
% 0.30/0.56 thf(fact_3913_set__encode__insert,axiom,
% 0.30/0.56 ! [A6: set @ nat,N: nat] :
% 0.30/0.56 ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.56 => ( ~ ( member @ nat @ N @ A6 )
% 0.30/0.56 => ( ( nat_set_encode @ ( insert @ nat @ N @ A6 ) )
% 0.30/0.56 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_encode_insert
% 0.30/0.56 thf(fact_3914_even__set__encode__iff,axiom,
% 0.30/0.56 ! [A6: set @ nat] :
% 0.30/0.56 ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.56 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A6 ) )
% 0.30/0.56 = ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % even_set_encode_iff
% 0.30/0.56 thf(fact_3915_arg__min__if__finite_I2_J,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( order @ B )
% 0.30/0.56 => ! [S3: set @ A,F: A > B] :
% 0.30/0.56 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.56 => ( ( S3
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ~ ? [X4: A] :
% 0.30/0.56 ( ( member @ A @ X4 @ S3 )
% 0.30/0.56 & ( ord_less @ B @ ( F @ X4 ) @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % arg_min_if_finite(2)
% 0.30/0.56 thf(fact_3916_arg__min__least,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( linorder @ B )
% 0.30/0.56 => ! [S3: set @ A,Y3: A,F: A > B] :
% 0.30/0.56 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.56 => ( ( S3
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( ( member @ A @ Y3 @ S3 )
% 0.30/0.56 => ( ord_less_eq @ B @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % arg_min_least
% 0.30/0.56 thf(fact_3917_set__decode__inverse,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 0.30/0.56 = N ) ).
% 0.30/0.56
% 0.30/0.56 % set_decode_inverse
% 0.30/0.56 thf(fact_3918_set__encode__empty,axiom,
% 0.30/0.56 ( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.56 = ( zero_zero @ nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_encode_empty
% 0.30/0.56 thf(fact_3919_set__encode__inverse,axiom,
% 0.30/0.56 ! [A6: set @ nat] :
% 0.30/0.56 ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.56 => ( ( nat_set_decode @ ( nat_set_encode @ A6 ) )
% 0.30/0.56 = A6 ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_encode_inverse
% 0.30/0.56 thf(fact_3920_finite__atLeastZeroLessThan__int,axiom,
% 0.30/0.56 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_atLeastZeroLessThan_int
% 0.30/0.56 thf(fact_3921_finite__maxlen,axiom,
% 0.30/0.56 ! [A: $tType,M7: set @ ( list @ A )] :
% 0.30/0.56 ( ( finite_finite2 @ ( list @ A ) @ M7 )
% 0.30/0.56 => ? [N4: nat] :
% 0.30/0.56 ! [X4: list @ A] :
% 0.30/0.56 ( ( member @ ( list @ A ) @ X4 @ M7 )
% 0.30/0.56 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N4 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_maxlen
% 0.30/0.56 thf(fact_3922_set__encode__eq,axiom,
% 0.30/0.56 ! [A6: set @ nat,B7: set @ nat] :
% 0.30/0.56 ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.56 => ( ( finite_finite2 @ nat @ B7 )
% 0.30/0.56 => ( ( ( nat_set_encode @ A6 )
% 0.30/0.56 = ( nat_set_encode @ B7 ) )
% 0.30/0.56 = ( A6 = B7 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_encode_eq
% 0.30/0.56 thf(fact_3923_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.30/0.56 ! [L: int,U: int] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.30/0.56 thf(fact_3924_card__atLeastZeroLessThan__int,axiom,
% 0.30/0.56 ! [U: int] :
% 0.30/0.56 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
% 0.30/0.56 = ( nat2 @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % card_atLeastZeroLessThan_int
% 0.30/0.56 thf(fact_3925_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.30/0.56 ! [L: int,U: int] :
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.30/0.56 thf(fact_3926_set__encode__inf,axiom,
% 0.30/0.56 ! [A6: set @ nat] :
% 0.30/0.56 ( ~ ( finite_finite2 @ nat @ A6 )
% 0.30/0.56 => ( ( nat_set_encode @ A6 )
% 0.30/0.56 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_encode_inf
% 0.30/0.56 thf(fact_3927_atLeastLessThan__upto,axiom,
% 0.30/0.56 ( ( set_or7035219750837199246ssThan @ int )
% 0.30/0.56 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastLessThan_upto
% 0.30/0.56 thf(fact_3928_wmax__emptyI,axiom,
% 0.30/0.56 ! [X8: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.56 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 )
% 0.30/0.56 => ( 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 ) ) ) @ X8 ) @ fun_max_weak ) ) ).
% 0.30/0.56
% 0.30/0.56 % wmax_emptyI
% 0.30/0.56 thf(fact_3929_smax__emptyI,axiom,
% 0.30/0.56 ! [Y7: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.56 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 )
% 0.30/0.56 => ( ( Y7
% 0.30/0.56 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 0.30/0.56 => ( 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 ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % smax_emptyI
% 0.30/0.56 thf(fact_3930_arg__min__if__finite_I1_J,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( order @ B )
% 0.30/0.56 => ! [S3: set @ A,F: A > B] :
% 0.30/0.56 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.56 => ( ( S3
% 0.30/0.56 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.56 => ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) @ S3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % arg_min_if_finite(1)
% 0.30/0.56 thf(fact_3931_nth__image,axiom,
% 0.30/0.56 ! [A: $tType,L: nat,Xs: list @ A] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
% 0.30/0.56 = ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nth_image
% 0.30/0.56 thf(fact_3932_fact__split,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semiring_char_0 @ A )
% 0.30/0.56 => ! [K: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.56 => ( ( semiring_char_0_fact @ A @ N )
% 0.30/0.56 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % fact_split
% 0.30/0.56 thf(fact_3933_sum__list__update,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ordere1170586879665033532d_diff @ A )
% 0.30/0.56 => ! [K: nat,Xs: list @ A,X: A] :
% 0.30/0.56 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.56 => ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.30/0.56 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sum_list_update
% 0.30/0.56 thf(fact_3934_set__take__disj__set__drop__if__distinct,axiom,
% 0.30/0.56 ! [A: $tType,Vs: list @ A,I: nat,J: nat] :
% 0.30/0.56 ( ( distinct @ A @ Vs )
% 0.30/0.56 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.56 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_take_disj_set_drop_if_distinct
% 0.30/0.56 thf(fact_3935_inf_Oidem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ A3 @ A3 )
% 0.30/0.56 = A3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.idem
% 0.30/0.56 thf(fact_3936_inf__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ X )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_idem
% 0.30/0.56 thf(fact_3937_inf_Oleft__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
% 0.30/0.56 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.left_idem
% 0.30/0.56 thf(fact_3938_inf__left__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.56 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_left_idem
% 0.30/0.56 thf(fact_3939_inf_Oright__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.56 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.right_idem
% 0.30/0.56 thf(fact_3940_inf__right__idem,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 )
% 0.30/0.56 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_right_idem
% 0.30/0.56 thf(fact_3941_inf__apply,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ B )
% 0.30/0.56 => ( ( inf_inf @ ( A > B ) )
% 0.30/0.56 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_apply
% 0.30/0.56 thf(fact_3942_le__inf__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.56 & ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_inf_iff
% 0.30/0.56 thf(fact_3943_inf_Obounded__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 0.30/0.56 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.56 & ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.bounded_iff
% 0.30/0.56 thf(fact_3944_boolean__algebra_Oconj__zero__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_zero_right
% 0.30/0.56 thf(fact_3945_boolean__algebra_Oconj__zero__left,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_zero_left
% 0.30/0.56 thf(fact_3946_inf__bot__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounded_lattice_bot @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_bot_right
% 0.30/0.56 thf(fact_3947_inf__bot__left,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( bounded_lattice_bot @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_bot_left
% 0.30/0.56 thf(fact_3948_inf__sup__absorb,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_absorb
% 0.30/0.56 thf(fact_3949_sup__inf__absorb,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.56 = X ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_inf_absorb
% 0.30/0.56 thf(fact_3950_image__Suc__atLeastAtMost,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_Suc_atLeastAtMost
% 0.30/0.56 thf(fact_3951_image__Suc__atLeastLessThan,axiom,
% 0.30/0.56 ! [I: nat,J: nat] :
% 0.30/0.56 ( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_Suc_atLeastLessThan
% 0.30/0.56 thf(fact_3952_Ints__prod,axiom,
% 0.30/0.56 ! [A: $tType,B: $tType] :
% 0.30/0.56 ( ( ( comm_monoid_mult @ B )
% 0.30/0.56 & ( ring_1 @ B ) )
% 0.30/0.56 => ! [A6: set @ A,F: A > B] :
% 0.30/0.56 ( ! [X3: A] :
% 0.30/0.56 ( ( member @ A @ X3 @ A6 )
% 0.30/0.56 => ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
% 0.30/0.56 => ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Ints_prod
% 0.30/0.56 thf(fact_3953_image__add__0,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( comm_monoid_add @ A )
% 0.30/0.56 => ! [S3: set @ A] :
% 0.30/0.56 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
% 0.30/0.56 = S3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_add_0
% 0.30/0.56 thf(fact_3954_inf__compl__bot__left1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_compl_bot_left1
% 0.30/0.56 thf(fact_3955_inf__compl__bot__left2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_compl_bot_left2
% 0.30/0.56 thf(fact_3956_inf__compl__bot__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ ( uminus_uminus @ A @ X ) ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_compl_bot_right
% 0.30/0.56 thf(fact_3957_boolean__algebra_Oconj__cancel__left,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_cancel_left
% 0.30/0.56 thf(fact_3958_boolean__algebra_Oconj__cancel__right,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
% 0.30/0.56 = ( bot_bot @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_cancel_right
% 0.30/0.56 thf(fact_3959_boolean__algebra_Ode__Morgan__conj,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.56 = ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.de_Morgan_conj
% 0.30/0.56 thf(fact_3960_boolean__algebra_Ode__Morgan__disj,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.56 = ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.de_Morgan_disj
% 0.30/0.56 thf(fact_3961_image__uminus__atLeastAtMost,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ordered_ab_group_add @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_uminus_atLeastAtMost
% 0.30/0.56 thf(fact_3962_sum__list_ONil,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( monoid_add @ A )
% 0.30/0.56 => ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
% 0.30/0.56 = ( zero_zero @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sum_list.Nil
% 0.30/0.56 thf(fact_3963_sum__list__eq__0__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.56 => ! [Ns: list @ A] :
% 0.30/0.56 ( ( ( groups8242544230860333062m_list @ A @ Ns )
% 0.30/0.56 = ( zero_zero @ A ) )
% 0.30/0.56 = ( ! [X2: A] :
% 0.30/0.56 ( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
% 0.30/0.56 => ( X2
% 0.30/0.56 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sum_list_eq_0_iff
% 0.30/0.56 thf(fact_3964_image__uminus__greaterThanLessThan,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ordered_ab_group_add @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 0.30/0.56 = ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_uminus_greaterThanLessThan
% 0.30/0.56 thf(fact_3965_Compl__disjoint,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Compl_disjoint
% 0.30/0.56 thf(fact_3966_Compl__disjoint2,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A] :
% 0.30/0.56 ( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
% 0.30/0.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Compl_disjoint2
% 0.30/0.56 thf(fact_3967_Diff__Compl,axiom,
% 0.30/0.56 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.56 ( ( minus_minus @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 0.30/0.56 = ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_Compl
% 0.30/0.56 thf(fact_3968_image__uminus__greaterThanAtMost,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ordered_ab_group_add @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 0.30/0.56 = ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_uminus_greaterThanAtMost
% 0.30/0.56 thf(fact_3969_image__uminus__atLeastLessThan,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ordered_ab_group_add @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 0.30/0.56 = ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_uminus_atLeastLessThan
% 0.30/0.56 thf(fact_3970_prod_OatMost__Suc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( comm_monoid_mult @ A )
% 0.30/0.56 => ! [G: nat > A,N: nat] :
% 0.30/0.56 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.56 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % prod.atMost_Suc
% 0.30/0.56 thf(fact_3971_prod_OlessThan__Suc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( comm_monoid_mult @ A )
% 0.30/0.56 => ! [G: nat > A,N: nat] :
% 0.30/0.56 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.56 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % prod.lessThan_Suc
% 0.30/0.56 thf(fact_3972_image__mult__atLeastAtMost,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( linordered_field @ A )
% 0.30/0.56 => ! [D2: A,A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.30/0.56 => ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.56 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % image_mult_atLeastAtMost
% 0.30/0.56 thf(fact_3973_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( comm_monoid_mult @ A )
% 0.30/0.56 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.56 ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.56 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.56 = ( one_one @ A ) ) )
% 0.30/0.56 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.56 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.56 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % prod.cl_ivl_Suc
% 0.30/0.56 thf(fact_3974_prod_Oop__ivl__Suc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( comm_monoid_mult @ A )
% 0.30/0.56 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.56 ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.56 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.56 = ( one_one @ A ) ) )
% 0.30/0.56 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.56 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.56 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % prod.op_ivl_Suc
% 0.30/0.56 thf(fact_3975_distrib__imp1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y5 @ Z4 ) )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( inf_inf @ A @ X3 @ Z4 ) ) )
% 0.30/0.56 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % distrib_imp1
% 0.30/0.56 thf(fact_3976_distrib__imp2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y5 @ Z4 ) )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( sup_sup @ A @ X3 @ Z4 ) ) )
% 0.30/0.56 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % distrib_imp2
% 0.30/0.56 thf(fact_3977_inf__sup__distrib1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( distrib_lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_distrib1
% 0.30/0.56 thf(fact_3978_inf__sup__distrib2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( distrib_lattice @ A )
% 0.30/0.56 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_distrib2
% 0.30/0.56 thf(fact_3979_sup__inf__distrib1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( distrib_lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_inf_distrib1
% 0.30/0.56 thf(fact_3980_sup__inf__distrib2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( distrib_lattice @ A )
% 0.30/0.56 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % sup_inf_distrib2
% 0.30/0.56 thf(fact_3981_boolean__algebra_Oconj__disj__distrib,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_disj_distrib
% 0.30/0.56 thf(fact_3982_boolean__algebra_Odisj__conj__distrib,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.disj_conj_distrib
% 0.30/0.56 thf(fact_3983_boolean__algebra_Oconj__disj__distrib2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
% 0.30/0.56 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.conj_disj_distrib2
% 0.30/0.56 thf(fact_3984_boolean__algebra_Odisj__conj__distrib2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.56 => ! [Y3: A,Z3: A,X: A] :
% 0.30/0.56 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
% 0.30/0.56 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra.disj_conj_distrib2
% 0.30/0.56 thf(fact_3985_inf__min,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( ( semilattice_inf @ A )
% 0.30/0.56 & ( linorder @ A ) )
% 0.30/0.56 => ( ( inf_inf @ A )
% 0.30/0.56 = ( ord_min @ A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_min
% 0.30/0.56 thf(fact_3986_inf__sup__aci_I4_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.56 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(4)
% 0.30/0.56 thf(fact_3987_inf__sup__aci_I3_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(3)
% 0.30/0.56 thf(fact_3988_inf__sup__aci_I2_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.56 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(2)
% 0.30/0.56 thf(fact_3989_inf__sup__aci_I1_J,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( lattice @ A )
% 0.30/0.56 => ( ( inf_inf @ A )
% 0.30/0.56 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_sup_aci(1)
% 0.30/0.56 thf(fact_3990_inf_Oassoc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.56 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.assoc
% 0.30/0.56 thf(fact_3991_inf__assoc,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.56 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_assoc
% 0.30/0.56 thf(fact_3992_inf_Ocommute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ( ( inf_inf @ A )
% 0.30/0.56 = ( ^ [A5: A,B3: A] : ( inf_inf @ A @ B3 @ A5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.commute
% 0.30/0.56 thf(fact_3993_inf__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ( ( inf_inf @ A )
% 0.30/0.56 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_commute
% 0.30/0.56 thf(fact_3994_inf_Oleft__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B2: A,A3: A,C2: A] :
% 0.30/0.56 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
% 0.30/0.56 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.left_commute
% 0.30/0.56 thf(fact_3995_inf__left__commute,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 0.30/0.56 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_left_commute
% 0.30/0.56 thf(fact_3996_inf__fun__def,axiom,
% 0.30/0.56 ! [B: $tType,A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ B )
% 0.30/0.56 => ( ( inf_inf @ ( A > B ) )
% 0.30/0.56 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_fun_def
% 0.30/0.56 thf(fact_3997_boolean__algebra__cancel_Oinf1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A6: A,K: A,A3: A,B2: A] :
% 0.30/0.56 ( ( A6
% 0.30/0.56 = ( inf_inf @ A @ K @ A3 ) )
% 0.30/0.56 => ( ( inf_inf @ A @ A6 @ B2 )
% 0.30/0.56 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra_cancel.inf1
% 0.30/0.56 thf(fact_3998_boolean__algebra__cancel_Oinf2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B7: A,K: A,B2: A,A3: A] :
% 0.30/0.56 ( ( B7
% 0.30/0.56 = ( inf_inf @ A @ K @ B2 ) )
% 0.30/0.56 => ( ( inf_inf @ A @ A3 @ B7 )
% 0.30/0.56 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % boolean_algebra_cancel.inf2
% 0.30/0.56 thf(fact_3999_less__infI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,X: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ X )
% 0.30/0.56 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_infI1
% 0.30/0.56 thf(fact_4000_less__infI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B2: A,X: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ B2 @ X )
% 0.30/0.56 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_infI2
% 0.30/0.56 thf(fact_4001_inf_Oabsorb3,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ( ( inf_inf @ A @ A3 @ B2 )
% 0.30/0.56 = A3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.absorb3
% 0.30/0.56 thf(fact_4002_inf_Oabsorb4,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ B2 @ A3 )
% 0.30/0.56 => ( ( inf_inf @ A @ A3 @ B2 )
% 0.30/0.56 = B2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.absorb4
% 0.30/0.56 thf(fact_4003_inf_Ostrict__boundedE,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 0.30/0.56 => ~ ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.56 => ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.strict_boundedE
% 0.30/0.56 thf(fact_4004_inf_Ostrict__order__iff,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ( ( ord_less @ A )
% 0.30/0.56 = ( ^ [A5: A,B3: A] :
% 0.30/0.56 ( ( A5
% 0.30/0.56 = ( inf_inf @ A @ A5 @ B3 ) )
% 0.30/0.56 & ( A5 != B3 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.strict_order_iff
% 0.30/0.56 thf(fact_4005_inf_Ostrict__coboundedI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.56 ( ( ord_less @ A @ A3 @ C2 )
% 0.30/0.56 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.strict_coboundedI1
% 0.30/0.56 thf(fact_4006_inf_Ostrict__coboundedI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.56 ( ( ord_less @ A @ B2 @ C2 )
% 0.30/0.56 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.strict_coboundedI2
% 0.30/0.56 thf(fact_4007_inf_OcoboundedI2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.coboundedI2
% 0.30/0.56 thf(fact_4008_inf_OcoboundedI1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.56 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.56 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.coboundedI1
% 0.30/0.56 thf(fact_4009_inf_Oabsorb__iff2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [B3: A,A5: A] :
% 0.30/0.56 ( ( inf_inf @ A @ A5 @ B3 )
% 0.30/0.56 = B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.absorb_iff2
% 0.30/0.56 thf(fact_4010_inf_Oabsorb__iff1,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ( ( ord_less_eq @ A )
% 0.30/0.56 = ( ^ [A5: A,B3: A] :
% 0.30/0.56 ( ( inf_inf @ A @ A5 @ B3 )
% 0.30/0.56 = A5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.absorb_iff1
% 0.30/0.56 thf(fact_4011_inf_Ocobounded2,axiom,
% 0.30/0.56 ! [A: $tType] :
% 0.30/0.56 ( ( semilattice_inf @ A )
% 0.30/0.56 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf.cobounded2
% 0.30/0.57 thf(fact_4012_inf_Ocobounded1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.cobounded1
% 0.30/0.57 thf(fact_4013_inf_Oorder__iff,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ( ( ord_less_eq @ A )
% 0.30/0.57 = ( ^ [A5: A,B3: A] :
% 0.30/0.57 ( A5
% 0.30/0.57 = ( inf_inf @ A @ A5 @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.order_iff
% 0.30/0.57 thf(fact_4014_inf__greatest,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ X @ Z3 )
% 0.30/0.57 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_greatest
% 0.30/0.57 thf(fact_4015_inf_OboundedI,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.57 => ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.boundedI
% 0.30/0.57 thf(fact_4016_inf_OboundedE,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 0.30/0.57 => ~ ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.57 => ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.boundedE
% 0.30/0.57 thf(fact_4017_inf__absorb2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [Y3: A,X: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.57 => ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.57 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_absorb2
% 0.30/0.57 thf(fact_4018_inf__absorb1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,Y3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.57 = X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_absorb1
% 0.30/0.57 thf(fact_4019_inf_Oabsorb2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [B2: A,A3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.57 => ( ( inf_inf @ A @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.absorb2
% 0.30/0.57 thf(fact_4020_inf_Oabsorb1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.57 => ( ( inf_inf @ A @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.absorb1
% 0.30/0.57 thf(fact_4021_le__iff__inf,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ( ( ord_less_eq @ A )
% 0.30/0.57 = ( ^ [X2: A,Y6: A] :
% 0.30/0.57 ( ( inf_inf @ A @ X2 @ Y6 )
% 0.30/0.57 = X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_iff_inf
% 0.30/0.57 thf(fact_4022_inf__unique,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [F: A > A > A,X: A,Y3: A] :
% 0.30/0.57 ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ X3 )
% 0.30/0.57 => ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ Y5 )
% 0.30/0.57 => ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ X3 @ Z4 )
% 0.30/0.57 => ( ord_less_eq @ A @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
% 0.30/0.57 => ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.57 = ( F @ X @ Y3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_unique
% 0.30/0.57 thf(fact_4023_inf_OorderI,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( inf_inf @ A @ A3 @ B2 ) )
% 0.30/0.57 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.orderI
% 0.30/0.57 thf(fact_4024_inf_OorderE,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.57 => ( A3
% 0.30/0.57 = ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf.orderE
% 0.30/0.57 thf(fact_4025_le__infI2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [B2: A,X: A,A3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ B2 @ X )
% 0.30/0.57 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_infI2
% 0.30/0.57 thf(fact_4026_le__infI1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,X: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ X )
% 0.30/0.57 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_infI1
% 0.30/0.57 thf(fact_4027_inf__mono,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_mono
% 0.30/0.57 thf(fact_4028_le__infI,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X @ A3 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ X @ B2 )
% 0.30/0.57 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_infI
% 0.30/0.57 thf(fact_4029_le__infE,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
% 0.30/0.57 => ~ ( ( ord_less_eq @ A @ X @ A3 )
% 0.30/0.57 => ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_infE
% 0.30/0.57 thf(fact_4030_inf__le2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_le2
% 0.30/0.57 thf(fact_4031_inf__le1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semilattice_inf @ A )
% 0.30/0.57 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_le1
% 0.30/0.57 thf(fact_4032_inf__sup__ord_I1_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( lattice @ A )
% 0.30/0.57 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_sup_ord(1)
% 0.30/0.57 thf(fact_4033_inf__sup__ord_I2_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( lattice @ A )
% 0.30/0.57 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_sup_ord(2)
% 0.30/0.57 thf(fact_4034_distrib__sup__le,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( lattice @ A )
% 0.30/0.57 => ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % distrib_sup_le
% 0.30/0.57 thf(fact_4035_distrib__inf__le,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( lattice @ A )
% 0.30/0.57 => ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % distrib_inf_le
% 0.30/0.57 thf(fact_4036_diff__eq,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ( ( minus_minus @ A )
% 0.30/0.57 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y6 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % diff_eq
% 0.30/0.57 thf(fact_4037_inf__cancel__left1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [X: A,A3: A,B2: A] :
% 0.30/0.57 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
% 0.30/0.57 = ( bot_bot @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_cancel_left1
% 0.30/0.57 thf(fact_4038_inf__cancel__left2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [X: A,A3: A,B2: A] :
% 0.30/0.57 ( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
% 0.30/0.57 = ( bot_bot @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_cancel_left2
% 0.30/0.57 thf(fact_4039_zero__notin__Suc__image,axiom,
% 0.30/0.57 ! [A6: set @ nat] :
% 0.30/0.57 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % zero_notin_Suc_image
% 0.30/0.57 thf(fact_4040_image__int__atLeastLessThan,axiom,
% 0.30/0.57 ! [A3: nat,B2: nat] :
% 0.30/0.57 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_int_atLeastLessThan
% 0.30/0.57 thf(fact_4041_member__le__sum__list,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.57 => ! [X: A,Xs: list @ A] :
% 0.30/0.57 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % member_le_sum_list
% 0.30/0.57 thf(fact_4042_Compl__Un,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.57 ( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.57 = ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Compl_Un
% 0.30/0.57 thf(fact_4043_Compl__Int,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.57 ( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.57 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Compl_Int
% 0.30/0.57 thf(fact_4044_Diff__eq,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( minus_minus @ ( set @ A ) )
% 0.30/0.57 = ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Diff_eq
% 0.30/0.57 thf(fact_4045_prod_Oivl__cong,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ( ord @ B )
% 0.30/0.57 & ( comm_monoid_mult @ A ) )
% 0.30/0.57 => ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 0.30/0.57 ( ( A3 = C2 )
% 0.30/0.57 => ( ( B2 = D2 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( ord_less_eq @ B @ C2 @ X3 )
% 0.30/0.57 => ( ( ord_less @ B @ X3 @ D2 )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( H2 @ X3 ) ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.ivl_cong
% 0.30/0.57 thf(fact_4046_prod_OatLeastLessThan__concat,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,P5: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.30/0.57 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeastLessThan_concat
% 0.30/0.57 thf(fact_4047_inf__shunt,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [X: A,Y3: A] :
% 0.30/0.57 ( ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.57 = ( bot_bot @ A ) )
% 0.30/0.57 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_shunt
% 0.30/0.57 thf(fact_4048_shunt1,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 0.30/0.57 = ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y3 ) @ Z3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % shunt1
% 0.30/0.57 thf(fact_4049_shunt2,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) @ Z3 )
% 0.30/0.57 = ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % shunt2
% 0.30/0.57 thf(fact_4050_sup__neg__inf,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.57 => ! [P5: A,Q4: A,R: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q4 @ R ) )
% 0.30/0.57 = ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q4 ) ) @ R ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sup_neg_inf
% 0.30/0.57 thf(fact_4051_card__image__le,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( finite_card @ A @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_image_le
% 0.30/0.57 thf(fact_4052_Groups__List_Osum__list__nonneg,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [Xs: list @ A] :
% 0.30/0.57 ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Groups_List.sum_list_nonneg
% 0.30/0.57 thf(fact_4053_sum__list__nonneg__eq__0__iff,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [Xs: list @ A] :
% 0.30/0.57 ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
% 0.30/0.57 => ( ( ( groups8242544230860333062m_list @ A @ Xs )
% 0.30/0.57 = ( zero_zero @ A ) )
% 0.30/0.57 = ( ! [X2: A] :
% 0.30/0.57 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( X2
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_nonneg_eq_0_iff
% 0.30/0.57 thf(fact_4054_sum__list__nonpos,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [Xs: list @ A] :
% 0.30/0.57 ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_nonpos
% 0.30/0.57 thf(fact_4055_Ioc__disjoint,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linorder @ A )
% 0.30/0.57 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.30/0.57 ( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.57 | ( ord_less_eq @ A @ D2 @ C2 )
% 0.30/0.57 | ( ord_less_eq @ A @ B2 @ C2 )
% 0.30/0.57 | ( ord_less_eq @ A @ D2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Ioc_disjoint
% 0.30/0.57 thf(fact_4056_disjoint__eq__subset__Compl,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.57 ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 = ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % disjoint_eq_subset_Compl
% 0.30/0.57 thf(fact_4057_image__int__atLeastAtMost,axiom,
% 0.30/0.57 ! [A3: nat,B2: nat] :
% 0.30/0.57 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_int_atLeastAtMost
% 0.30/0.57 thf(fact_4058_sum__list__abs,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordere166539214618696060dd_abs @ A )
% 0.30/0.57 => ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_abs
% 0.30/0.57 thf(fact_4059_sum__list__replicate,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [N: nat,C2: A] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
% 0.30/0.57 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_replicate
% 0.30/0.57 thf(fact_4060_prod_OatLeast0__atMost__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeast0_atMost_Suc
% 0.30/0.57 thf(fact_4061_prod_OatLeast0__lessThan__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeast0_lessThan_Suc
% 0.30/0.57 thf(fact_4062_prod_OatLeast__Suc__lessThan,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less @ nat @ M @ N )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeast_Suc_lessThan
% 0.30/0.57 thf(fact_4063_prod_OatLeast__Suc__atMost,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeast_Suc_atMost
% 0.30/0.57 thf(fact_4064_prod_Onat__ivl__Suc_H,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.nat_ivl_Suc'
% 0.30/0.57 thf(fact_4065_prod_OatLeastLessThan__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A3: nat,B2: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeastLessThan_Suc
% 0.30/0.57 thf(fact_4066_prod_Olast__plus,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.last_plus
% 0.30/0.57 thf(fact_4067_surj__card__le,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B,F: A > B] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ A @ B @ F @ A6 ) )
% 0.30/0.57 => ( ord_less_eq @ nat @ ( finite_card @ B @ B7 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_card_le
% 0.30/0.57 thf(fact_4068_prod__Suc__fact,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.57 = ( semiring_char_0_fact @ nat @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_Suc_fact
% 0.30/0.57 thf(fact_4069_prod__Suc__Suc__fact,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.30/0.57 = ( semiring_char_0_fact @ nat @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_Suc_Suc_fact
% 0.30/0.57 thf(fact_4070_image__Suc__atMost,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_Suc_atMost
% 0.30/0.57 thf(fact_4071_Iio__Int__singleton,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( order @ A )
% 0.30/0.57 => ! [X: A,K: A] :
% 0.30/0.57 ( ( ( ord_less @ A @ X @ K )
% 0.30/0.57 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ A @ X @ K )
% 0.30/0.57 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Iio_Int_singleton
% 0.30/0.57 thf(fact_4072_image__Suc__lessThan,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_Suc_lessThan
% 0.30/0.57 thf(fact_4073_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.30/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % atLeast0_atMost_Suc_eq_insert_0
% 0.30/0.57 thf(fact_4074_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.30/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % atLeast0_lessThan_Suc_eq_insert_0
% 0.30/0.57 thf(fact_4075_prod_Ohead,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.head
% 0.30/0.57 thf(fact_4076_atMost__Suc__eq__insert__0,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( set_ord_atMost @ nat @ ( suc @ N ) )
% 0.30/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % atMost_Suc_eq_insert_0
% 0.30/0.57 thf(fact_4077_lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( set_ord_lessThan @ nat @ ( suc @ N ) )
% 0.30/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % lessThan_Suc_eq_insert_0
% 0.30/0.57 thf(fact_4078_sum__list__strict__mono,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( monoid_add @ B )
% 0.30/0.57 & ( strict9044650504122735259up_add @ B ) )
% 0.30/0.57 => ! [Xs: list @ A,F: A > B,G: A > B] :
% 0.30/0.57 ( ( Xs
% 0.30/0.57 != ( nil @ A ) )
% 0.30/0.57 => ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.57 => ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_strict_mono
% 0.30/0.57 thf(fact_4079_elem__le__sum__list,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.57 => ! [K: nat,Ns: list @ A] :
% 0.30/0.57 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % elem_le_sum_list
% 0.30/0.57 thf(fact_4080_prod_Ohead__if,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.57 ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.head_if
% 0.30/0.57 thf(fact_4081_prod_Oub__add__nat,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A,P5: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.ub_add_nat
% 0.30/0.57 thf(fact_4082_fact__prod__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semiring_char_0 @ A )
% 0.30/0.57 => ( ( semiring_char_0_fact @ A )
% 0.30/0.57 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % fact_prod_Suc
% 0.30/0.57 thf(fact_4083_image__atLeastZeroLessThan__int,axiom,
% 0.30/0.57 ! [U: int] :
% 0.30/0.57 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 0.30/0.57 => ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
% 0.30/0.57 = ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_atLeastZeroLessThan_int
% 0.30/0.57 thf(fact_4084_image__mult__atLeastAtMost__if,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [C2: A,X: A,Y3: A] :
% 0.30/0.57 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.57 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y3 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.57 => ( ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y3 ) @ ( times_times @ A @ C2 @ X ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_mult_atLeastAtMost_if
% 0.30/0.57 thf(fact_4085_prod__pos__nat__iff,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 ) )
% 0.30/0.57 = ( ! [X2: A] :
% 0.30/0.57 ( ( member @ A @ X2 @ A6 )
% 0.30/0.57 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_pos_nat_iff
% 0.30/0.57 thf(fact_4086_prod_Oinsert,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,X: B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ~ ( member @ B @ X @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.insert
% 0.30/0.57 thf(fact_4087_prod__eq__1__iff,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 )
% 0.30/0.57 = ( one_one @ nat ) )
% 0.30/0.57 = ( ! [X2: A] :
% 0.30/0.57 ( ( member @ A @ X2 @ A6 )
% 0.30/0.57 => ( ( F @ X2 )
% 0.30/0.57 = ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_eq_1_iff
% 0.30/0.57 thf(fact_4088_prod_Oinfinite,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A] :
% 0.30/0.57 ( ~ ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.infinite
% 0.30/0.57 thf(fact_4089_prod__zero__iff,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) )
% 0.30/0.57 = ( ? [X2: B] :
% 0.30/0.57 ( ( member @ B @ X2 @ A6 )
% 0.30/0.57 & ( ( F @ X2 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_zero_iff
% 0.30/0.57 thf(fact_4090_prod_Oempty,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: B > A] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.57 = ( one_one @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.empty
% 0.30/0.57 thf(fact_4091_inf__nat__def,axiom,
% 0.30/0.57 ( ( inf_inf @ nat )
% 0.30/0.57 = ( ord_min @ nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_nat_def
% 0.30/0.57 thf(fact_4092_inf__int__def,axiom,
% 0.30/0.57 ( ( inf_inf @ int )
% 0.30/0.57 = ( ord_min @ int ) ) ).
% 0.30/0.57
% 0.30/0.57 % inf_int_def
% 0.30/0.57 thf(fact_4093_prod_Oneutral,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.neutral
% 0.30/0.57 thf(fact_4094_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: B > A,A6: set @ B] :
% 0.30/0.57 ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 != ( one_one @ A ) )
% 0.30/0.57 => ~ ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ A6 )
% 0.30/0.57 => ( ( G @ A8 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.not_neutral_contains_not_neutral
% 0.30/0.57 thf(fact_4095_prod__nonneg,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( linordered_semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_nonneg
% 0.30/0.57 thf(fact_4096_prod__mono,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( linordered_semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A,G: B > A] :
% 0.30/0.57 ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 0.30/0.57 & ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_mono
% 0.30/0.57 thf(fact_4097_prod__pos,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( linordered_semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 0.30/0.57 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_pos
% 0.30/0.57 thf(fact_4098_prod__ge__1,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( linord181362715937106298miring @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X3 ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_ge_1
% 0.30/0.57 thf(fact_4099_prod__zero,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ? [X4: B] :
% 0.30/0.57 ( ( member @ B @ X4 @ A6 )
% 0.30/0.57 & ( ( F @ X4 )
% 0.30/0.57 = ( zero_zero @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_zero
% 0.30/0.57 thf(fact_4100_prod__le__1,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( linord181362715937106298miring @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) )
% 0.30/0.57 & ( ord_less_eq @ A @ ( F @ X3 ) @ ( one_one @ A ) ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_le_1
% 0.30/0.57 thf(fact_4101_prod_Orelated,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.57 ( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.30/0.57 => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
% 0.30/0.57 ( ( ( R2 @ X1 @ X23 )
% 0.30/0.57 & ( R2 @ Y1 @ Y22 ) )
% 0.30/0.57 => ( R2 @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
% 0.30/0.57 => ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ S3 )
% 0.30/0.57 => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.57 => ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.related
% 0.30/0.57 thf(fact_4102_prod_Oinsert__if,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,X: B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( ( member @ B @ X @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ X @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.insert_if
% 0.30/0.57 thf(fact_4103_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S5 )
% 0.30/0.57 => ( ( finite_finite2 @ C @ T5 )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 0.30/0.57 => ( ( I @ ( J @ A8 ) )
% 0.30/0.57 = A8 ) )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 0.30/0.57 => ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 0.30/0.57 => ( ! [B5: C] :
% 0.30/0.57 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.57 => ( ( J @ ( I @ B5 ) )
% 0.30/0.57 = B5 ) )
% 0.30/0.57 => ( ! [B5: C] :
% 0.30/0.57 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.57 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ S5 )
% 0.30/0.57 => ( ( G @ A8 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [B5: C] :
% 0.30/0.57 ( ( member @ C @ B5 @ T5 )
% 0.30/0.57 => ( ( H2 @ B5 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ S3 )
% 0.30/0.57 => ( ( H2 @ ( J @ A8 ) )
% 0.30/0.57 = ( G @ A8 ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.reindex_bij_witness_not_neutral
% 0.30/0.57 thf(fact_4104_less__1__prod2,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( linordered_idom @ B )
% 0.30/0.57 => ! [I6: set @ A,I: A,F: A > B] :
% 0.30/0.57 ( ( finite_finite2 @ A @ I6 )
% 0.30/0.57 => ( ( member @ A @ I @ I6 )
% 0.30/0.57 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
% 0.30/0.57 => ( ! [I4: A] :
% 0.30/0.57 ( ( member @ A @ I4 @ I6 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
% 0.30/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_1_prod2
% 0.30/0.57 thf(fact_4105_less__1__prod,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( linordered_idom @ B )
% 0.30/0.57 => ! [I6: set @ A,F: A > B] :
% 0.30/0.57 ( ( finite_finite2 @ A @ I6 )
% 0.30/0.57 => ( ( I6
% 0.30/0.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ! [I4: A] :
% 0.30/0.57 ( ( member @ A @ I4 @ I6 )
% 0.30/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
% 0.30/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_1_prod
% 0.30/0.57 thf(fact_4106_prod_Osubset__diff,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [B7: set @ B,A6: set @ B,G: B > A] :
% 0.30/0.57 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.subset_diff
% 0.30/0.57 thf(fact_4107_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ S3 )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( H2 @ X3 ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.mono_neutral_cong_right
% 0.30/0.57 thf(fact_4108_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.57 => ( ( H2 @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ S3 )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( H2 @ X3 ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.mono_neutral_cong_left
% 0.30/0.57 thf(fact_4109_prod_Omono__neutral__right,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.mono_neutral_right
% 0.30/0.57 thf(fact_4110_prod_Omono__neutral__left,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.mono_neutral_left
% 0.30/0.57 thf(fact_4111_prod_Osame__carrierI,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ C6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 0.30/0.57 => ( ( G @ A8 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [B5: B] :
% 0.30/0.57 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 0.30/0.57 => ( ( H2 @ B5 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.same_carrierI
% 0.30/0.57 thf(fact_4112_prod_Osame__carrier,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ C6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 0.30/0.57 => ( ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 0.30/0.57 => ( ( G @ A8 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [B5: B] :
% 0.30/0.57 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 0.30/0.57 => ( ( H2 @ B5 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) )
% 0.30/0.57 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.same_carrier
% 0.30/0.57 thf(fact_4113_prod_Ounion__inter,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.57 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.union_inter
% 0.30/0.57 thf(fact_4114_prod_OInt__Diff,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.Int_Diff
% 0.30/0.57 thf(fact_4115_prod_Omono__neutral__cong,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.57 => ( ( H2 @ I4 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
% 0.30/0.57 => ( ( G @ I4 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( H2 @ X3 ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.mono_neutral_cong
% 0.30/0.57 thf(fact_4116_prod__mono__strict,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( linordered_semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 0.30/0.57 & ( ord_less @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.30/0.57 => ( ( A6
% 0.30/0.57 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.57 => ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_mono_strict
% 0.30/0.57 thf(fact_4117_prod_Oinsert__remove,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A,X: B] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.insert_remove
% 0.30/0.57 thf(fact_4118_prod_Oremove,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,X: B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( member @ B @ X @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.remove
% 0.30/0.57 thf(fact_4119_prod_Ounion__inter__neutral,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( one_one @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.union_inter_neutral
% 0.30/0.57 thf(fact_4120_prod_Ounion__disjoint,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.57 => ( ( ( inf_inf @ ( set @ B ) @ A6 @ B7 )
% 0.30/0.57 = ( bot_bot @ ( set @ B ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.union_disjoint
% 0.30/0.57 thf(fact_4121_prod_Ounion__diff2,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 = ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.union_diff2
% 0.30/0.57 thf(fact_4122_prod__mono2,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( linordered_idom @ B )
% 0.30/0.57 => ! [B7: set @ A,A6: set @ A,F: A > B] :
% 0.30/0.57 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.57 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.57 => ( ! [B5: A] :
% 0.30/0.57 ( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
% 0.30/0.57 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B5 ) ) )
% 0.30/0.57 => ( ! [A8: A] :
% 0.30/0.57 ( ( member @ A @ A8 @ A6 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A8 ) ) )
% 0.30/0.57 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_mono2
% 0.30/0.57 thf(fact_4123_prod__le__power,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( linordered_semidom @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A,N: A,K: nat] :
% 0.30/0.57 ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 0.30/0.57 & ( ord_less_eq @ A @ ( F @ I4 ) @ N ) ) )
% 0.30/0.57 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ K )
% 0.30/0.57 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_le_power
% 0.30/0.57 thf(fact_4124_prod__diff1,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semidom_divide @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A,A3: B] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( ( F @ A3 )
% 0.30/0.57 != ( zero_zero @ A ) )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.30/0.57 = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( F @ A3 ) ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_diff1
% 0.30/0.57 thf(fact_4125_prod__Un,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( field @ A )
% 0.30/0.57 => ! [A6: set @ B,B7: set @ B,F: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.57 => ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 => ( ( F @ X3 )
% 0.30/0.57 != ( zero_zero @ A ) ) )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 = ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_Un
% 0.30/0.57 thf(fact_4126_translation__Compl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ab_group_add @ A )
% 0.30/0.57 => ! [A3: A,T2: set @ A] :
% 0.30/0.57 ( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 0.30/0.57 = ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % translation_Compl
% 0.30/0.57 thf(fact_4127_remdups__adj__altdef,axiom,
% 0.30/0.57 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.57 ( ( ( remdups_adj @ A @ Xs )
% 0.30/0.57 = Ys2 )
% 0.30/0.57 = ( ? [F2: nat > nat] :
% 0.30/0.57 ( ( order_mono @ nat @ nat @ F2 )
% 0.30/0.57 & ( ( image @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 0.30/0.57 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 0.30/0.57 & ! [I5: nat] :
% 0.30/0.57 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.57 => ( ( nth @ A @ Xs @ I5 )
% 0.30/0.57 = ( nth @ A @ Ys2 @ ( F2 @ I5 ) ) ) )
% 0.30/0.57 & ! [I5: nat] :
% 0.30/0.57 ( ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.57 => ( ( ( nth @ A @ Xs @ I5 )
% 0.30/0.57 = ( nth @ A @ Xs @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) )
% 0.30/0.57 = ( ( F2 @ I5 )
% 0.30/0.57 = ( F2 @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % remdups_adj_altdef
% 0.30/0.57 thf(fact_4128_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ! [Z3: A,N: nat] :
% 0.30/0.57 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [K3: nat] : ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_times_pochhammer_half
% 0.30/0.57 thf(fact_4129_butlast__take,axiom,
% 0.30/0.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.57 => ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
% 0.30/0.57 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % butlast_take
% 0.30/0.57 thf(fact_4130_prod_Oneutral__const,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [Uu: B] : ( one_one @ A )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( one_one @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.neutral_const
% 0.30/0.57 thf(fact_4131_of__int__prod,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_ring_1 @ A )
% 0.30/0.57 => ! [F: B > int,A6: set @ B] :
% 0.30/0.57 ( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F @ A6 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % of_int_prod
% 0.30/0.57 thf(fact_4132_finite__Collect__less__nat,axiom,
% 0.30/0.57 ! [K: nat] :
% 0.30/0.57 ( finite_finite2 @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_less_nat
% 0.30/0.57 thf(fact_4133_finite__Collect__le__nat,axiom,
% 0.30/0.57 ! [K: nat] :
% 0.30/0.57 ( finite_finite2 @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_le_nat
% 0.30/0.57 thf(fact_4134_card__Collect__less__nat,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) )
% 0.30/0.57 = N ) ).
% 0.30/0.57
% 0.30/0.57 % card_Collect_less_nat
% 0.30/0.57 thf(fact_4135_finite__interval__int1,axiom,
% 0.30/0.57 ! [A3: int,B2: int] :
% 0.30/0.57 ( finite_finite2 @ int
% 0.30/0.57 @ ( collect @ int
% 0.30/0.57 @ ^ [I5: int] :
% 0.30/0.57 ( ( ord_less_eq @ int @ A3 @ I5 )
% 0.30/0.57 & ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int1
% 0.30/0.57 thf(fact_4136_finite__interval__int4,axiom,
% 0.30/0.57 ! [A3: int,B2: int] :
% 0.30/0.57 ( finite_finite2 @ int
% 0.30/0.57 @ ( collect @ int
% 0.30/0.57 @ ^ [I5: int] :
% 0.30/0.57 ( ( ord_less @ int @ A3 @ I5 )
% 0.30/0.57 & ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int4
% 0.30/0.57 thf(fact_4137_prod_Odelta,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( B2 @ A3 ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.delta
% 0.30/0.57 thf(fact_4138_prod_Odelta_H,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( B2 @ A3 ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.delta'
% 0.30/0.57 thf(fact_4139_sum__list__0,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( monoid_add @ A )
% 0.30/0.57 => ! [Xs: list @ B] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ A
% 0.30/0.57 @ ( map @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( zero_zero @ A )
% 0.30/0.57 @ Xs ) )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_0
% 0.30/0.57 thf(fact_4140_card__Collect__le__nat,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( ord_less_eq @ nat @ I5 @ N ) ) )
% 0.30/0.57 = ( suc @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_Collect_le_nat
% 0.30/0.57 thf(fact_4141_of__nat__prod,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ! [F: B > nat,A6: set @ B] :
% 0.30/0.57 ( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % of_nat_prod
% 0.30/0.57 thf(fact_4142_finite__interval__int2,axiom,
% 0.30/0.57 ! [A3: int,B2: int] :
% 0.30/0.57 ( finite_finite2 @ int
% 0.30/0.57 @ ( collect @ int
% 0.30/0.57 @ ^ [I5: int] :
% 0.30/0.57 ( ( ord_less_eq @ int @ A3 @ I5 )
% 0.30/0.57 & ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int2
% 0.30/0.57 thf(fact_4143_finite__interval__int3,axiom,
% 0.30/0.57 ! [A3: int,B2: int] :
% 0.30/0.57 ( finite_finite2 @ int
% 0.30/0.57 @ ( collect @ int
% 0.30/0.57 @ ^ [I5: int] :
% 0.30/0.57 ( ( ord_less @ int @ A3 @ I5 )
% 0.30/0.57 & ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int3
% 0.30/0.57 thf(fact_4144_length__butlast,axiom,
% 0.30/0.57 ! [A: $tType,Xs: list @ A] :
% 0.30/0.57 ( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
% 0.30/0.57 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % length_butlast
% 0.30/0.57 thf(fact_4145_image__divide__atLeastAtMost,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [D2: A,A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_divide_atLeastAtMost
% 0.30/0.57 thf(fact_4146_Gcd__int__eq,axiom,
% 0.30/0.57 ! [N6: set @ nat] :
% 0.30/0.57 ( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
% 0.30/0.57 = ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Gcd_int_eq
% 0.30/0.57 thf(fact_4147_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [I6: set @ B,X: B > A,Y3: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( X @ I5 )
% 0.30/0.57 != ( one_one @ A ) ) ) ) )
% 0.30/0.57 => ( ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( Y3 @ I5 )
% 0.30/0.57 != ( one_one @ A ) ) ) ) )
% 0.30/0.57 => ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( times_times @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
% 0.30/0.57 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_4148_prod_Oshift__bounds__Suc__ivl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.shift_bounds_Suc_ivl
% 0.30/0.57 thf(fact_4149_nat__seg__image__imp__finite,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,F: nat > A,N: nat] :
% 0.30/0.57 ( ( A6
% 0.30/0.57 = ( image @ nat @ A @ F
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) ) )
% 0.30/0.57 => ( finite_finite2 @ A @ A6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_seg_image_imp_finite
% 0.30/0.57 thf(fact_4150_finite__conv__nat__seg__image,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( finite_finite2 @ A )
% 0.30/0.57 = ( ^ [A7: set @ A] :
% 0.30/0.57 ? [N3: nat,F2: nat > A] :
% 0.30/0.57 ( A7
% 0.30/0.57 = ( image @ nat @ A @ F2
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_conv_nat_seg_image
% 0.30/0.57 thf(fact_4151_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [I6: set @ B,X: B > A,Y3: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( X @ I5 )
% 0.30/0.57 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.57 => ( ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( Y3 @ I5 )
% 0.30/0.57 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.57 => ( finite_finite2 @ B
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [I5: B] :
% 0.30/0.57 ( ( member @ B @ I5 @ I6 )
% 0.30/0.57 & ( ( plus_plus @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
% 0.30/0.57 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_4152_prod_Ointer__filter,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A,P: B > $o] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [X2: B] :
% 0.30/0.57 ( ( member @ B @ X2 @ A6 )
% 0.30/0.57 & ( P @ X2 ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
% 0.30/0.57 @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inter_filter
% 0.30/0.57 thf(fact_4153_prod_Odistrib,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: B > A,H2: B > A,A6: set @ B] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.distrib
% 0.30/0.57 thf(fact_4154_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.shift_bounds_cl_Suc_ivl
% 0.30/0.57 thf(fact_4155_mod__prod__eq,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.57 => ! [F: B > A,A3: A,A6: set @ B] :
% 0.30/0.57 ( ( modulo_modulo @ A
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
% 0.30/0.57 @ A6 )
% 0.30/0.57 @ A3 )
% 0.30/0.57 = ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mod_prod_eq
% 0.30/0.57 thf(fact_4156_int__prod,axiom,
% 0.30/0.57 ! [B: $tType,F: B > nat,A6: set @ B] :
% 0.30/0.57 ( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ int
% 0.30/0.57 @ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
% 0.30/0.57 @ A6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % int_prod
% 0.30/0.57 thf(fact_4157_prod__int__eq,axiom,
% 0.30/0.57 ! [I: nat,J: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ int @ int
% 0.30/0.57 @ ^ [X2: int] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_int_eq
% 0.30/0.57 thf(fact_4158_translation__subtract__Compl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ab_group_add @ A )
% 0.30/0.57 => ! [A3: A,T2: set @ A] :
% 0.30/0.57 ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
% 0.30/0.57 @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 0.30/0.57 = ( uminus_uminus @ ( set @ A )
% 0.30/0.57 @ ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
% 0.30/0.57 @ T2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % translation_subtract_Compl
% 0.30/0.57 thf(fact_4159_Collect__imp__eq,axiom,
% 0.30/0.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.57 ( ( collect @ A
% 0.30/0.57 @ ^ [X2: A] :
% 0.30/0.57 ( ( P @ X2 )
% 0.30/0.57 => ( Q @ X2 ) ) )
% 0.30/0.57 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Collect_imp_eq
% 0.30/0.57 thf(fact_4160_nat__less__as__int,axiom,
% 0.30/0.57 ( ( ord_less @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_less_as_int
% 0.30/0.57 thf(fact_4161_len__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( heap @ A )
% 0.30/0.57 => ( ( array_len @ A )
% 0.30/0.57 = ( ^ [A5: array @ A] :
% 0.30/0.57 ( heap_Time_tap @ nat
% 0.30/0.57 @ ^ [H4: heap_ext @ product_unit] : ( array_length @ A @ H4 @ A5 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % len_def
% 0.30/0.57 thf(fact_4162_lambda__one,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( monoid_mult @ A )
% 0.30/0.57 => ( ( ^ [X2: A] : X2 )
% 0.30/0.57 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % lambda_one
% 0.30/0.57 thf(fact_4163_lambda__zero,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( mult_zero @ A )
% 0.30/0.57 => ( ( ^ [H4: A] : ( zero_zero @ A ) )
% 0.30/0.57 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % lambda_zero
% 0.30/0.57 thf(fact_4164_Compl__eq,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( uminus_uminus @ ( set @ A ) )
% 0.30/0.57 = ( ^ [A7: set @ A] :
% 0.30/0.57 ( collect @ A
% 0.30/0.57 @ ^ [X2: A] :
% 0.30/0.57 ~ ( member @ A @ X2 @ A7 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Compl_eq
% 0.30/0.57 thf(fact_4165_Collect__neg__eq,axiom,
% 0.30/0.57 ! [A: $tType,P: A > $o] :
% 0.30/0.57 ( ( collect @ A
% 0.30/0.57 @ ^ [X2: A] :
% 0.30/0.57 ~ ( P @ X2 ) )
% 0.30/0.57 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Collect_neg_eq
% 0.30/0.57 thf(fact_4166_uminus__set__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( uminus_uminus @ ( set @ A ) )
% 0.30/0.57 = ( ^ [A7: set @ A] :
% 0.30/0.57 ( collect @ A
% 0.30/0.57 @ ( uminus_uminus @ ( A > $o )
% 0.30/0.57 @ ^ [X2: A] : ( member @ A @ X2 @ A7 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % uminus_set_def
% 0.30/0.57 thf(fact_4167_min__of__mono,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( linorder @ A )
% 0.30/0.57 & ( linorder @ B ) )
% 0.30/0.57 => ! [F: A > B,M: A,N: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_min @ B @ ( F @ M ) @ ( F @ N ) )
% 0.30/0.57 = ( F @ ( ord_min @ A @ M @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % min_of_mono
% 0.30/0.57 thf(fact_4168_mono__Suc,axiom,
% 0.30/0.57 order_mono @ nat @ nat @ suc ).
% 0.30/0.57
% 0.30/0.57 % mono_Suc
% 0.30/0.57 thf(fact_4169_strict__subset__divisors__dvd,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less @ ( set @ A )
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
% 0.30/0.57 = ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.57 & ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % strict_subset_divisors_dvd
% 0.30/0.57 thf(fact_4170_mono__add,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordere6658533253407199908up_add @ A )
% 0.30/0.57 => ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_add
% 0.30/0.57 thf(fact_4171_less__set__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ord_less @ ( set @ A ) )
% 0.30/0.57 = ( ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.57 ( ord_less @ ( A > $o )
% 0.30/0.57 @ ^ [X2: A] : ( member @ A @ X2 @ A7 )
% 0.30/0.57 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_set_def
% 0.30/0.57 thf(fact_4172_mono__strict__invE,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( linorder @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.57 => ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_strict_invE
% 0.30/0.57 thf(fact_4173_max__of__mono,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( linorder @ A )
% 0.30/0.57 & ( linorder @ B ) )
% 0.30/0.57 => ! [F: A > B,M: A,N: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_max @ B @ ( F @ M ) @ ( F @ N ) )
% 0.30/0.57 = ( F @ ( ord_max @ A @ M @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_of_mono
% 0.30/0.57 thf(fact_4174_numeral__code_I2_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( numeral @ A )
% 0.30/0.57 => ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 0.30/0.57 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(2)
% 0.30/0.57 thf(fact_4175_mono__def,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( order @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ( ( order_mono @ A @ B )
% 0.30/0.57 = ( ^ [F2: A > B] :
% 0.30/0.57 ! [X2: A,Y6: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_def
% 0.30/0.57 thf(fact_4176_monoI,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( order @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ! [F: A > B] :
% 0.30/0.57 ( ! [X3: A,Y5: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.57 => ( order_mono @ A @ B @ F ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % monoI
% 0.30/0.57 thf(fact_4177_monoE,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( order @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % monoE
% 0.30/0.57 thf(fact_4178_monoD,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( order @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % monoD
% 0.30/0.57 thf(fact_4179_min__def__raw,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ord @ A )
% 0.30/0.57 => ( ( ord_min @ A )
% 0.30/0.57 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % min_def_raw
% 0.30/0.57 thf(fact_4180_subset__divisors__dvd,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( ( ord_less_eq @ ( set @ A )
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
% 0.30/0.57 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % subset_divisors_dvd
% 0.30/0.57 thf(fact_4181_max__def__raw,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ord @ A )
% 0.30/0.57 => ( ( ord_max @ A )
% 0.30/0.57 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_4182_nat__leq__as__int,axiom,
% 0.30/0.57 ( ( ord_less_eq @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_leq_as_int
% 0.30/0.57 thf(fact_4183_effect__LetI,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: B,N: nat] :
% 0.30/0.57 ( ( X = T2 )
% 0.30/0.57 => ( ( heap_Time_effect @ B @ ( F @ X ) @ H2 @ H @ R @ N )
% 0.30/0.57 => ( heap_Time_effect @ B @ ( F @ T2 ) @ H2 @ H @ R @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % effect_LetI
% 0.30/0.57 thf(fact_4184_lessThan__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ord @ A )
% 0.30/0.57 => ( ( set_ord_lessThan @ A )
% 0.30/0.57 = ( ^ [U3: A] :
% 0.30/0.57 ( collect @ A
% 0.30/0.57 @ ^ [X2: A] : ( ord_less @ A @ X2 @ U3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % lessThan_def
% 0.30/0.57 thf(fact_4185_atMost__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ord @ A )
% 0.30/0.57 => ( ( set_ord_atMost @ A )
% 0.30/0.57 = ( ^ [U3: A] :
% 0.30/0.57 ( collect @ A
% 0.30/0.57 @ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % atMost_def
% 0.30/0.57 thf(fact_4186_sum__list__const__mult,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_0 @ A )
% 0.30/0.57 => ! [C2: A,F: B > A,Xs: list @ B] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ A
% 0.30/0.57 @ ( map @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( times_times @ A @ C2 @ ( F @ X2 ) )
% 0.30/0.57 @ Xs ) )
% 0.30/0.57 = ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_const_mult
% 0.30/0.57 thf(fact_4187_sum__list__mult__const,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( semiring_0 @ A )
% 0.30/0.57 => ! [F: B > A,C2: A,Xs: list @ B] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ A
% 0.30/0.57 @ ( map @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ C2 )
% 0.30/0.57 @ Xs ) )
% 0.30/0.57 = ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ C2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_mult_const
% 0.30/0.57 thf(fact_4188_finite__less__ub,axiom,
% 0.30/0.57 ! [F: nat > nat,U: nat] :
% 0.30/0.57 ( ! [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( F @ N4 ) )
% 0.30/0.57 => ( finite_finite2 @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_less_ub
% 0.30/0.57 thf(fact_4189_finite__int__segment,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.57 => ! [A3: A,B2: A] :
% 0.30/0.57 ( finite_finite2 @ A
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [X2: A] :
% 0.30/0.57 ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
% 0.30/0.57 & ( ord_less_eq @ A @ A3 @ X2 )
% 0.30/0.57 & ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_int_segment
% 0.30/0.57 thf(fact_4190_finite__M__bounded__by__nat,axiom,
% 0.30/0.57 ! [P: nat > $o,I: nat] :
% 0.30/0.57 ( finite_finite2 @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( P @ K3 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_M_bounded_by_nat
% 0.30/0.57 thf(fact_4191_finite__divisors__int,axiom,
% 0.30/0.57 ! [I: int] :
% 0.30/0.57 ( ( I
% 0.30/0.57 != ( zero_zero @ int ) )
% 0.30/0.57 => ( finite_finite2 @ int
% 0.30/0.57 @ ( collect @ int
% 0.30/0.57 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_divisors_int
% 0.30/0.57 thf(fact_4192_integer__of__num_I2_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.30/0.57 = ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % integer_of_num(2)
% 0.30/0.57 thf(fact_4193_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( numeral @ A )
% 0.30/0.57 => ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_4194_power__numeral__even,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( monoid_mult @ A )
% 0.30/0.57 => ! [Z3: A,W2: num] :
% 0.30/0.57 ( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
% 0.30/0.57 = ( times_times @ A @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_even
% 0.30/0.57 thf(fact_4195_mono__ge2__power__minus__self,axiom,
% 0.30/0.57 ! [K: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.30/0.57 => ( order_mono @ nat @ nat
% 0.30/0.57 @ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_ge2_power_minus_self
% 0.30/0.57 thf(fact_4196_prod_Ointer__restrict,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( one_one @ A ) )
% 0.30/0.57 @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inter_restrict
% 0.30/0.57 thf(fact_4197_prod_Osetdiff__irrelevant,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 0.30/0.57 @ ( minus_minus @ ( set @ B ) @ A6
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [X2: B] :
% 0.30/0.57 ( ( G @ X2 )
% 0.30/0.57 = ( one_one @ A ) ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.setdiff_irrelevant
% 0.30/0.57 thf(fact_4198_power__numeral__odd,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( monoid_mult @ A )
% 0.30/0.57 => ! [Z3: A,W2: num] :
% 0.30/0.57 ( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
% 0.30/0.57 = ( times_times @ A @ ( times_times @ A @ Z3 @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_4199_card__length__sum__list__rec,axiom,
% 0.30/0.57 ! [M: nat,N6: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
% 0.30/0.57 => ( ( finite_card @ ( list @ nat )
% 0.30/0.57 @ ( collect @ ( list @ nat )
% 0.30/0.57 @ ^ [L3: list @ nat] :
% 0.30/0.57 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.30/0.57 = M )
% 0.30/0.57 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.30/0.57 = N6 ) ) ) )
% 0.30/0.57 = ( plus_plus @ nat
% 0.30/0.57 @ ( finite_card @ ( list @ nat )
% 0.30/0.57 @ ( collect @ ( list @ nat )
% 0.30/0.57 @ ^ [L3: list @ nat] :
% 0.30/0.57 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.30/0.57 = ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
% 0.30/0.57 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.30/0.57 = N6 ) ) ) )
% 0.30/0.57 @ ( finite_card @ ( list @ nat )
% 0.30/0.57 @ ( collect @ ( list @ nat )
% 0.30/0.57 @ ^ [L3: list @ nat] :
% 0.30/0.57 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.30/0.57 = M )
% 0.30/0.57 & ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
% 0.30/0.57 = N6 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_length_sum_list_rec
% 0.30/0.57 thf(fact_4200_sum__list__mono,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( monoid_add @ B )
% 0.30/0.57 & ( ordere6658533253407199908up_add @ B ) )
% 0.30/0.57 => ! [Xs: list @ A,F: A > B,G: A > B] :
% 0.30/0.57 ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.57 => ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_mono
% 0.30/0.57 thf(fact_4201_finite__abs__int__segment,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( archim2362893244070406136eiling @ A )
% 0.30/0.57 => ! [A3: A] :
% 0.30/0.57 ( finite_finite2 @ A
% 0.30/0.57 @ ( collect @ A
% 0.30/0.57 @ ^ [K3: A] :
% 0.30/0.57 ( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
% 0.30/0.57 & ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_abs_int_segment
% 0.30/0.57 thf(fact_4202_nat__plus__as__int,axiom,
% 0.30/0.57 ( ( plus_plus @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_plus_as_int
% 0.30/0.57 thf(fact_4203_nat__times__as__int,axiom,
% 0.30/0.57 ( ( times_times @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_times_as_int
% 0.30/0.57 thf(fact_4204_card__length__sum__list,axiom,
% 0.30/0.57 ! [M: nat,N6: nat] :
% 0.30/0.57 ( ( finite_card @ ( list @ nat )
% 0.30/0.57 @ ( collect @ ( list @ nat )
% 0.30/0.57 @ ^ [L3: list @ nat] :
% 0.30/0.57 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.30/0.57 = M )
% 0.30/0.57 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.30/0.57 = N6 ) ) ) )
% 0.30/0.57 = ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_length_sum_list
% 0.30/0.57 thf(fact_4205_nat__minus__as__int,axiom,
% 0.30/0.57 ( ( minus_minus @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_minus_as_int
% 0.30/0.57 thf(fact_4206_prod_Onat__diff__reindex,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.nat_diff_reindex
% 0.30/0.57 thf(fact_4207_finite__divisors__nat,axiom,
% 0.30/0.57 ! [M: nat] :
% 0.30/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.57 => ( finite_finite2 @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_divisors_nat
% 0.30/0.57 thf(fact_4208_card__less,axiom,
% 0.30/0.57 ! [M7: set @ nat,I: nat] :
% 0.30/0.57 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.57 => ( ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( member @ nat @ K3 @ M7 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.30/0.57 != ( zero_zero @ nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_less
% 0.30/0.57 thf(fact_4209_card__less__Suc,axiom,
% 0.30/0.57 ! [M7: set @ nat,I: nat] :
% 0.30/0.57 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.57 => ( ( suc
% 0.30/0.57 @ ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( member @ nat @ ( suc @ K3 ) @ M7 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ I ) ) ) ) )
% 0.30/0.57 = ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( member @ nat @ K3 @ M7 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_less_Suc
% 0.30/0.57 thf(fact_4210_card__less__Suc2,axiom,
% 0.30/0.57 ! [M7: set @ nat,I: nat] :
% 0.30/0.57 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.57 => ( ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( member @ nat @ ( suc @ K3 ) @ M7 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ I ) ) ) )
% 0.30/0.57 = ( finite_card @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [K3: nat] :
% 0.30/0.57 ( ( member @ nat @ K3 @ M7 )
% 0.30/0.57 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_less_Suc2
% 0.30/0.57 thf(fact_4211_sum__list__Suc,axiom,
% 0.30/0.57 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ nat
% 0.30/0.57 @ ( map @ A @ nat
% 0.30/0.57 @ ^ [X2: A] : ( suc @ ( F @ X2 ) )
% 0.30/0.57 @ Xs ) )
% 0.30/0.57 = ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_Suc
% 0.30/0.57 thf(fact_4212_prod__int__plus__eq,axiom,
% 0.30/0.57 ! [I: nat,J: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ int @ int
% 0.30/0.57 @ ^ [X2: int] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_int_plus_eq
% 0.30/0.57 thf(fact_4213_nat__div__as__int,axiom,
% 0.30/0.57 ( ( divide_divide @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_div_as_int
% 0.30/0.57 thf(fact_4214_nat__mod__as__int,axiom,
% 0.30/0.57 ( ( modulo_modulo @ nat )
% 0.30/0.57 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_mod_as_int
% 0.30/0.57 thf(fact_4215_mono__invE,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( linorder @ A )
% 0.30/0.57 & ( order @ B ) )
% 0.30/0.57 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.57 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_invE
% 0.30/0.57 thf(fact_4216_mono__iff__le__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( order @ A )
% 0.30/0.57 => ( ( order_mono @ nat @ A )
% 0.30/0.57 = ( ^ [F2: nat > A] :
% 0.30/0.57 ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_iff_le_Suc
% 0.30/0.57 thf(fact_4217_image__add__int__atLeastLessThan,axiom,
% 0.30/0.57 ! [L: int,U: int] :
% 0.30/0.57 ( ( image @ int @ int
% 0.30/0.57 @ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
% 0.30/0.57 = ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_add_int_atLeastLessThan
% 0.30/0.57 thf(fact_4218_mono__inf,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( semilattice_inf @ A )
% 0.30/0.57 & ( semilattice_inf @ B ) )
% 0.30/0.57 => ! [F: A > B,A6: A,B7: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ord_less_eq @ B @ ( F @ ( inf_inf @ A @ A6 @ B7 ) ) @ ( inf_inf @ B @ ( F @ A6 ) @ ( F @ B7 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_inf
% 0.30/0.57 thf(fact_4219_map__replicate__trivial,axiom,
% 0.30/0.57 ! [A: $tType,X: A,I: nat] :
% 0.30/0.57 ( ( map @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : X
% 0.30/0.57 @ ( upt @ ( zero_zero @ nat ) @ I ) )
% 0.30/0.57 = ( replicate @ A @ I @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % map_replicate_trivial
% 0.30/0.57 thf(fact_4220_mono__sup,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ( semilattice_sup @ A )
% 0.30/0.57 & ( semilattice_sup @ B ) )
% 0.30/0.57 => ! [F: A > B,A6: A,B7: A] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A6 ) @ ( F @ B7 ) ) @ ( F @ ( sup_sup @ A @ A6 @ B7 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_sup
% 0.30/0.57 thf(fact_4221_semiring__char__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ( ( semiri4206861660011772517g_char @ A )
% 0.30/0.57 = ( ^ [Uu2: itself @ A] :
% 0.30/0.57 ( gcd_Gcd @ nat
% 0.30/0.57 @ ( collect @ nat
% 0.30/0.57 @ ^ [N3: nat] :
% 0.30/0.57 ( ( semiring_1_of_nat @ A @ N3 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_char_def
% 0.30/0.57 thf(fact_4222_map__add__upt,axiom,
% 0.30/0.57 ! [N: nat,M: nat] :
% 0.30/0.57 ( ( map @ nat @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( plus_plus @ nat @ I5 @ N )
% 0.30/0.57 @ ( upt @ ( zero_zero @ nat ) @ M ) )
% 0.30/0.57 = ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % map_add_upt
% 0.30/0.57 thf(fact_4223_integer__of__num_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % integer_of_num(3)
% 0.30/0.57 thf(fact_4224_prod_OSuc__reindex__ivl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ M )
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.Suc_reindex_ivl
% 0.30/0.57 thf(fact_4225_prod_OIf__cases,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.If_cases
% 0.30/0.57 thf(fact_4226_sum__list__triv,axiom,
% 0.30/0.57 ! [C: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_1 @ B )
% 0.30/0.57 => ! [R: B,Xs: list @ C] :
% 0.30/0.57 ( ( groups8242544230860333062m_list @ B
% 0.30/0.57 @ ( map @ C @ B
% 0.30/0.57 @ ^ [X2: C] : R
% 0.30/0.57 @ Xs ) )
% 0.30/0.57 = ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_triv
% 0.30/0.57 thf(fact_4227_prod_OatMost__Suc__shift,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atMost_Suc_shift
% 0.30/0.57 thf(fact_4228_prod_OlessThan__Suc__shift,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.lessThan_Suc_shift
% 0.30/0.57 thf(fact_4229_prod_OatLeastLessThan__rev,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat,M: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeastLessThan_rev
% 0.30/0.57 thf(fact_4230_prod_Onested__swap,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A3: nat > nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [J3: nat] :
% 0.30/0.57 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.nested_swap
% 0.30/0.57 thf(fact_4231_prod_OatLeast1__atMost__eq,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeast1_atMost_eq
% 0.30/0.57 thf(fact_4232_prod_Onat__group,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,K: nat,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.nat_group
% 0.30/0.57 thf(fact_4233_map__upt__Suc,axiom,
% 0.30/0.57 ! [A: $tType,F: nat > A,N: nat] :
% 0.30/0.57 ( ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( cons @ A @ ( F @ ( zero_zero @ nat ) )
% 0.30/0.57 @ ( map @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
% 0.30/0.57 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % map_upt_Suc
% 0.30/0.57 thf(fact_4234_prod_Onested__swap_H,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [A3: nat > nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [J3: nat] :
% 0.30/0.57 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.nested_swap'
% 0.30/0.57 thf(fact_4235_prod__atLeastAtMost__code,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [F: nat > A,A3: nat,B2: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.30/0.57 = ( set_fo6178422350223883121st_nat @ A
% 0.30/0.57 @ ^ [A5: nat] : ( times_times @ A @ ( F @ A5 ) )
% 0.30/0.57 @ A3
% 0.30/0.57 @ B2
% 0.30/0.57 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_atLeastAtMost_code
% 0.30/0.57 thf(fact_4236_map__nth,axiom,
% 0.30/0.57 ! [A: $tType,Xs: list @ A] :
% 0.30/0.57 ( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 0.30/0.57 = Xs ) ).
% 0.30/0.57
% 0.30/0.57 % map_nth
% 0.30/0.57 thf(fact_4237_ratrel__def,axiom,
% 0.30/0.57 ( ratrel
% 0.30/0.57 = ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
% 0.30/0.57 ( ( ( product_snd @ int @ int @ X2 )
% 0.30/0.57 != ( zero_zero @ int ) )
% 0.30/0.57 & ( ( product_snd @ int @ int @ Y6 )
% 0.30/0.57 != ( zero_zero @ int ) )
% 0.30/0.57 & ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
% 0.30/0.57 = ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ratrel_def
% 0.30/0.57 thf(fact_4238_fact__prod,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semiring_char_0 @ A )
% 0.30/0.57 => ( ( semiring_char_0_fact @ A )
% 0.30/0.57 = ( ^ [N3: nat] :
% 0.30/0.57 ( semiring_1_of_nat @ A
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % fact_prod
% 0.30/0.57 thf(fact_4239_map__decr__upt,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( map @ nat @ nat
% 0.30/0.57 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.57 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( upt @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % map_decr_upt
% 0.30/0.57 thf(fact_4240_prod_Odelta__remove,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.delta_remove
% 0.30/0.57 thf(fact_4241_diff__nat__eq__if,axiom,
% 0.30/0.57 ! [Z6: int,Z3: int] :
% 0.30/0.57 ( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
% 0.30/0.57 => ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
% 0.30/0.57 = ( nat2 @ Z3 ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
% 0.30/0.57 => ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
% 0.30/0.57 = ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % diff_nat_eq_if
% 0.30/0.57 thf(fact_4242_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat,M: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atLeastLessThan_rev_at_least_Suc_atMost
% 0.30/0.57 thf(fact_4243_prod_OatMost__shift,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.atMost_shift
% 0.30/0.57 thf(fact_4244_pochhammer__prod,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ( ( comm_s3205402744901411588hammer @ A )
% 0.30/0.57 = ( ^ [A5: A,N3: nat] :
% 0.30/0.57 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_prod
% 0.30/0.57 thf(fact_4245_mono__mult,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ordered_semiring @ A )
% 0.30/0.57 => ! [A3: A] :
% 0.30/0.57 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.57 => ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_mult
% 0.30/0.57 thf(fact_4246_mono__times__nat,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.57 => ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_times_nat
% 0.30/0.57 thf(fact_4247_finite__lists__length__le,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,N: nat] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( finite_finite2 @ ( list @ A )
% 0.30/0.57 @ ( collect @ ( list @ A )
% 0.30/0.57 @ ^ [Xs3: list @ A] :
% 0.30/0.57 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 0.30/0.57 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_lists_length_le
% 0.30/0.57 thf(fact_4248_mono__image__least,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ( order @ B )
% 0.30/0.57 & ( order @ A ) )
% 0.30/0.57 => ! [F: A > B,M: A,N: A,M5: B,N2: B] :
% 0.30/0.57 ( ( order_mono @ A @ B @ F )
% 0.30/0.57 => ( ( ( image @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
% 0.30/0.57 = ( set_or7035219750837199246ssThan @ B @ M5 @ N2 ) )
% 0.30/0.57 => ( ( ord_less @ A @ M @ N )
% 0.30/0.57 => ( ( F @ M )
% 0.30/0.57 = M5 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mono_image_least
% 0.30/0.57 thf(fact_4249_fact__prod__rev,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( semiring_char_0 @ A )
% 0.30/0.57 => ( ( semiring_char_0_fact @ A )
% 0.30/0.57 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % fact_prod_rev
% 0.30/0.57 thf(fact_4250_fact__eq__fact__times,axiom,
% 0.30/0.57 ! [N: nat,M: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.57 => ( ( semiring_char_0_fact @ nat @ M )
% 0.30/0.57 = ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % fact_eq_fact_times
% 0.30/0.57 thf(fact_4251_image__mult__atLeastAtMost__if_H,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [X: A,Y3: A,C2: A] :
% 0.30/0.57 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y3 @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_mult_atLeastAtMost_if'
% 0.30/0.57 thf(fact_4252_image__affinity__atLeastAtMost,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [A3: A,B2: A,M: A,C2: A] :
% 0.30/0.57 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_affinity_atLeastAtMost
% 0.30/0.57 thf(fact_4253_image__affinity__atLeastAtMost__diff,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [A3: A,B2: A,M: A,C2: A] :
% 0.30/0.57 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_affinity_atLeastAtMost_diff
% 0.30/0.57 thf(fact_4254_image__affinity__atLeastAtMost__div,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [A3: A,B2: A,M: A,C2: A] :
% 0.30/0.57 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_affinity_atLeastAtMost_div
% 0.30/0.57 thf(fact_4255_image__affinity__atLeastAtMost__div__diff,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( linordered_field @ A )
% 0.30/0.57 => ! [A3: A,B2: A,M: A,C2: A] :
% 0.30/0.57 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.57 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.30/0.57 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.57 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.57 => ( ( image @ A @ A
% 0.30/0.57 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.30/0.57 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_affinity_atLeastAtMost_div_diff
% 0.30/0.57 thf(fact_4256_prod__gen__delta,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A,C2: A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_gen_delta
% 0.30/0.57 thf(fact_4257_pochhammer__Suc__prod,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ! [A3: A,N: nat] :
% 0.30/0.57 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_Suc_prod
% 0.30/0.57 thf(fact_4258_pochhammer__prod__rev,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ( ( comm_s3205402744901411588hammer @ A )
% 0.30/0.57 = ( ^ [A5: A,N3: nat] :
% 0.30/0.57 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I5 ) ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_prod_rev
% 0.30/0.57 thf(fact_4259_set__decode__def,axiom,
% 0.30/0.57 ( nat_set_decode
% 0.30/0.57 = ( ^ [X2: nat] :
% 0.30/0.57 ( collect @ nat
% 0.30/0.57 @ ^ [N3: nat] :
% 0.30/0.57 ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_decode_def
% 0.30/0.57 thf(fact_4260_image__minus__const__atLeastLessThan__nat,axiom,
% 0.30/0.57 ! [C2: nat,Y3: nat,X: nat] :
% 0.30/0.57 ( ( ( ord_less @ nat @ C2 @ Y3 )
% 0.30/0.57 => ( ( image @ nat @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 0.30/0.57 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y3 @ C2 ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ nat @ C2 @ Y3 )
% 0.30/0.57 => ( ( ( ord_less @ nat @ X @ Y3 )
% 0.30/0.57 => ( ( image @ nat @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 0.30/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ nat @ X @ Y3 )
% 0.30/0.57 => ( ( image @ nat @ nat
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 0.30/0.57 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % image_minus_const_atLeastLessThan_nat
% 0.30/0.57 thf(fact_4261_fact__div__fact,axiom,
% 0.30/0.57 ! [N: nat,M: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ N @ M )
% 0.30/0.57 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % fact_div_fact
% 0.30/0.57 thf(fact_4262_prod_Oin__pairs,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.in_pairs
% 0.30/0.57 thf(fact_4263_prod_Oin__pairs__0,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.in_pairs_0
% 0.30/0.57 thf(fact_4264_gbinomial__altdef__of__nat,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ( ( gbinomial @ A )
% 0.30/0.57 = ( ^ [A5: A,K3: nat] :
% 0.30/0.57 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I5 ) ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_altdef_of_nat
% 0.30/0.57 thf(fact_4265_signed__take__bit__code,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.57 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 0.30/0.57 = ( ^ [N3: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % signed_take_bit_code
% 0.30/0.57 thf(fact_4266_pochhammer__Suc__prod__rev,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ! [A3: A,N: nat] :
% 0.30/0.57 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_Suc_prod_rev
% 0.30/0.57 thf(fact_4267_binomial__altdef__of__nat,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ! [K: nat,N: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ K @ N )
% 0.30/0.57 => ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I5 ) ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % binomial_altdef_of_nat
% 0.30/0.57 thf(fact_4268_gbinomial__mult__fact,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ! [K: nat,A3: A] :
% 0.30/0.57 ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_mult_fact
% 0.30/0.57 thf(fact_4269_gbinomial__mult__fact_H,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ! [A3: A,K: nat] :
% 0.30/0.57 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_mult_fact'
% 0.30/0.57 thf(fact_4270_gbinomial__prod__rev,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ( semiring_char_0 @ A )
% 0.30/0.57 & ( semidom_divide @ A ) )
% 0.30/0.57 => ( ( gbinomial @ A )
% 0.30/0.57 = ( ^ [A5: A,K3: nat] :
% 0.30/0.57 ( divide_divide @ A
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
% 0.30/0.57 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_prod_rev
% 0.30/0.57 thf(fact_4271_nth__butlast,axiom,
% 0.30/0.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
% 0.30/0.57 => ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
% 0.30/0.57 = ( nth @ A @ Xs @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_butlast
% 0.30/0.57 thf(fact_4272_take__butlast,axiom,
% 0.30/0.57 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.57 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.57 => ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
% 0.30/0.57 = ( take @ A @ N @ Xs ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % take_butlast
% 0.30/0.57 thf(fact_4273_prod_Ozero__middle,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_mult @ A )
% 0.30/0.57 => ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.30/0.57 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.30/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.zero_middle
% 0.30/0.57 thf(fact_4274_gbinomial__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( ( semiring_char_0 @ A )
% 0.30/0.57 & ( semidom_divide @ A ) )
% 0.30/0.57 => ! [A3: A,K: nat] :
% 0.30/0.57 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 0.30/0.57 = ( divide_divide @ A
% 0.30/0.57 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
% 0.30/0.57 @ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_Suc
% 0.30/0.57 thf(fact_4275_pochhammer__code,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_semiring_1 @ A )
% 0.30/0.57 => ( ( comm_s3205402744901411588hammer @ A )
% 0.30/0.57 = ( ^ [A5: A,N3: nat] :
% 0.30/0.57 ( if @ A
% 0.30/0.57 @ ( N3
% 0.30/0.57 = ( zero_zero @ nat ) )
% 0.30/0.57 @ ( one_one @ A )
% 0.30/0.57 @ ( set_fo6178422350223883121st_nat @ A
% 0.30/0.57 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
% 0.30/0.57 @ ( zero_zero @ nat )
% 0.30/0.57 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
% 0.30/0.57 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pochhammer_code
% 0.30/0.57 thf(fact_4276_normalize__def,axiom,
% 0.30/0.57 ( normalize
% 0.30/0.57 = ( ^ [P6: product_prod @ int @ int] :
% 0.30/0.57 ( 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 ) ) ) )
% 0.30/0.57 @ ( if @ ( product_prod @ int @ int )
% 0.30/0.57 @ ( ( product_snd @ int @ int @ P6 )
% 0.30/0.57 = ( zero_zero @ int ) )
% 0.30/0.57 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.57 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % normalize_def
% 0.30/0.57 thf(fact_4277_butlast__conv__take,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( butlast @ A )
% 0.30/0.57 = ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % butlast_conv_take
% 0.30/0.57 thf(fact_4278_butlast__list__update,axiom,
% 0.30/0.57 ! [A: $tType,K: nat,Xs: list @ A,X: A] :
% 0.30/0.57 ( ( ( K
% 0.30/0.57 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.30/0.57 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.30/0.57 = ( butlast @ A @ Xs ) ) )
% 0.30/0.57 & ( ( K
% 0.30/0.57 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.30/0.57 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.30/0.57 = ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % butlast_list_update
% 0.30/0.57 thf(fact_4279_gbinomial__code,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field_char_0 @ A )
% 0.30/0.57 => ( ( gbinomial @ A )
% 0.30/0.57 = ( ^ [A5: A,K3: nat] :
% 0.30/0.57 ( if @ A
% 0.30/0.57 @ ( K3
% 0.30/0.57 = ( zero_zero @ nat ) )
% 0.30/0.57 @ ( one_one @ A )
% 0.30/0.57 @ ( divide_divide @ A
% 0.30/0.57 @ ( set_fo6178422350223883121st_nat @ A
% 0.30/0.57 @ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L3 ) ) )
% 0.30/0.57 @ ( zero_zero @ nat )
% 0.30/0.57 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
% 0.30/0.57 @ ( one_one @ A ) )
% 0.30/0.57 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % gbinomial_code
% 0.30/0.57 thf(fact_4280_card__lists__distinct__length__eq_H,axiom,
% 0.30/0.57 ! [A: $tType,K: nat,A6: set @ A] :
% 0.30/0.57 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A6 ) )
% 0.30/0.57 => ( ( finite_card @ ( list @ A )
% 0.30/0.57 @ ( collect @ ( list @ A )
% 0.30/0.57 @ ^ [Xs3: list @ A] :
% 0.30/0.57 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.57 = K )
% 0.30/0.57 & ( distinct @ A @ Xs3 )
% 0.30/0.57 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_lists_distinct_length_eq'
% 0.30/0.57 thf(fact_4281_card__lists__distinct__length__eq,axiom,
% 0.30/0.57 ! [A: $tType,A6: set @ A,K: nat] :
% 0.30/0.57 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.57 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A6 ) )
% 0.30/0.57 => ( ( finite_card @ ( list @ A )
% 0.30/0.57 @ ( collect @ ( list @ A )
% 0.30/0.57 @ ^ [Xs3: list @ A] :
% 0.30/0.57 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.57 = K )
% 0.30/0.57 & ( distinct @ A @ Xs3 )
% 0.30/0.57 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
% 0.30/0.57 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % card_lists_distinct_length_eq
% 0.30/0.57 thf(fact_4282_bit__cut__integer__code,axiom,
% 0.30/0.57 ( code_bit_cut_integer
% 0.30/0.57 = ( ^ [K3: code_integer] :
% 0.30/0.57 ( if @ ( product_prod @ code_integer @ $o )
% 0.30/0.57 @ ( K3
% 0.30/0.57 = ( zero_zero @ code_integer ) )
% 0.30/0.57 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
% 0.30/0.57 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
% 0.30/0.57 @ ^ [R5: code_integer,S6: code_integer] :
% 0.30/0.57 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S6 ) )
% 0.30/0.57 @ ( S6
% 0.30/0.57 = ( one_one @ code_integer ) ) )
% 0.30/0.57 @ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % bit_cut_integer_code
% 0.30/0.57 thf(fact_4283_integer__of__int__code,axiom,
% 0.30/0.57 ( code_integer_of_int
% 0.30/0.57 = ( ^ [K3: int] :
% 0.30/0.57 ( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
% 0.30/0.57 @ ( if @ code_integer
% 0.30/0.57 @ ( K3
% 0.30/0.57 = ( zero_zero @ int ) )
% 0.30/0.57 @ ( zero_zero @ code_integer )
% 0.30/0.57 @ ( if @ code_integer
% 0.30/0.57 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 0.30/0.57 = ( zero_zero @ int ) )
% 0.30/0.57 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 0.30/0.57 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % integer_of_int_code
% 0.30/0.57 thf(fact_4284_swap__def,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( heap @ A )
% 0.30/0.57 => ( ( array_swap @ A )
% 0.30/0.57 = ( ^ [I5: nat,X2: A,A5: array @ A] :
% 0.30/0.57 ( heap_Time_guard @ A
% 0.30/0.57 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 0.30/0.57 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % swap_def
% 0.30/0.57 thf(fact_4285_choose__even__sum,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_ring_1 @ A )
% 0.30/0.57 => ! [N: nat] :
% 0.30/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.57 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 0.30/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ ( set_ord_atMost @ nat @ N ) ) )
% 0.30/0.57 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % choose_even_sum
% 0.30/0.57 thf(fact_4286_integer__of__int__eq__of__int,axiom,
% 0.30/0.57 ( code_integer_of_int
% 0.30/0.57 = ( ring_1_of_int @ code_integer ) ) ).
% 0.30/0.57
% 0.30/0.57 % integer_of_int_eq_of_int
% 0.30/0.57 thf(fact_4287_Ints__sum,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ring_1 @ B )
% 0.30/0.57 => ! [A6: set @ A,F: A > B] :
% 0.30/0.57 ( ! [X3: A] :
% 0.30/0.57 ( ( member @ A @ X3 @ A6 )
% 0.30/0.57 => ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
% 0.30/0.57 => ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Ints_sum
% 0.30/0.57 thf(fact_4288_sum_Oneutral__const,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [A6: set @ B] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [Uu: B] : ( zero_zero @ A )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.neutral_const
% 0.30/0.57 thf(fact_4289_of__nat__sum,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [F: B > nat,A6: set @ B] :
% 0.30/0.57 ( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % of_nat_sum
% 0.30/0.57 thf(fact_4290_of__int__sum,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ring_1 @ A )
% 0.30/0.57 => ! [F: B > int,A6: set @ B] :
% 0.30/0.57 ( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F @ A6 ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % of_int_sum
% 0.30/0.57 thf(fact_4291_sum_Oempty,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: B > A] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.empty
% 0.30/0.57 thf(fact_4292_sum__eq__0__iff,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( canoni5634975068530333245id_add @ A )
% 0.30/0.57 => ! [F4: set @ B,F: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ F4 )
% 0.30/0.57 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ F4 )
% 0.30/0.57 = ( zero_zero @ A ) )
% 0.30/0.57 = ( ! [X2: B] :
% 0.30/0.57 ( ( member @ B @ X2 @ F4 )
% 0.30/0.57 => ( ( F @ X2 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_eq_0_iff
% 0.30/0.57 thf(fact_4293_sum_Oinfinite,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A] :
% 0.30/0.57 ( ~ ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.infinite
% 0.30/0.57 thf(fact_4294_sum_Odelta,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( B2 @ A3 ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.delta
% 0.30/0.57 thf(fact_4295_sum_Odelta_H,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [S3: set @ B,A3: B,B2: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.57 => ( ( ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( B2 @ A3 ) ) )
% 0.30/0.57 & ( ~ ( member @ B @ A3 @ S3 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ S3 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.delta'
% 0.30/0.57 thf(fact_4296_sum__abs,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ordere166539214618696060dd_abs @ B )
% 0.30/0.57 => ! [F: A > B,A6: set @ A] :
% 0.30/0.57 ( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) )
% 0.30/0.57 @ ( groups7311177749621191930dd_sum @ A @ B
% 0.30/0.57 @ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_abs
% 0.30/0.57 thf(fact_4297_sum_OatMost__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.atMost_Suc
% 0.30/0.57 thf(fact_4298_sum_OlessThan__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: nat > A,N: nat] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.lessThan_Suc
% 0.30/0.57 thf(fact_4299_sum__abs__ge__zero,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ordere166539214618696060dd_abs @ B )
% 0.30/0.57 => ! [F: A > B,A6: set @ A] :
% 0.30/0.57 ( ord_less_eq @ B @ ( zero_zero @ B )
% 0.30/0.57 @ ( groups7311177749621191930dd_sum @ A @ B
% 0.30/0.57 @ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_abs_ge_zero
% 0.30/0.57 thf(fact_4300_sum__constant,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [Y3: A,A6: set @ B] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : Y3
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_constant
% 0.30/0.57 thf(fact_4301_sum__list__upt,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.57 => ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.57 @ ^ [X2: nat] : X2
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_list_upt
% 0.30/0.57 thf(fact_4302_sum__of__bool__mult__eq,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [A6: set @ B,P: B > $o,F: B > A] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F @ X2 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_of_bool_mult_eq
% 0.30/0.57 thf(fact_4303_sum__mult__of__bool__eq,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A,P: B > $o] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_mult_of_bool_eq
% 0.30/0.57 thf(fact_4304_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.57 ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.57 = ( zero_zero @ A ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.cl_ivl_Suc
% 0.30/0.57 thf(fact_4305_sum_Oop__ivl__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.57 ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.57 = ( zero_zero @ A ) ) )
% 0.30/0.57 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.op_ivl_Suc
% 0.30/0.57 thf(fact_4306_sum__of__bool__eq,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_1 @ A )
% 0.30/0.57 => ! [A6: set @ B,P: B > $o] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_of_bool_eq
% 0.30/0.57 thf(fact_4307_sum__zero__power,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( division_ring @ A )
% 0.30/0.57 => ! [A6: set @ nat,C2: nat > A] :
% 0.30/0.57 ( ( ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.57 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( C2 @ ( zero_zero @ nat ) ) ) )
% 0.30/0.57 & ( ~ ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.57 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_zero_power
% 0.30/0.57 thf(fact_4308_sum__zero__power_H,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( field @ A )
% 0.30/0.57 => ! [A6: set @ nat,C2: nat > A,D2: nat > A] :
% 0.30/0.57 ( ( ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.57 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
% 0.30/0.57 & ( ~ ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.57 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_zero_power'
% 0.30/0.57 thf(fact_4309_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.57 => ! [M: num,N: num] :
% 0.30/0.57 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.30/0.57 @ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
% 0.30/0.57 @ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(5)
% 0.30/0.57 thf(fact_4310_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.57 => ! [M: num,N: num] :
% 0.30/0.57 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.30/0.57 @ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
% 0.30/0.57 @ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(6)
% 0.30/0.57 thf(fact_4311_sum__mono,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [K5: set @ B,F: B > A,G: B > A] :
% 0.30/0.57 ( ! [I4: B] :
% 0.30/0.57 ( ( member @ B @ I4 @ K5 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_mono
% 0.30/0.57 thf(fact_4312_sum__negf,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ab_group_add @ A )
% 0.30/0.57 => ! [F: B > A,A6: set @ B] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( uminus_uminus @ A @ ( F @ X2 ) )
% 0.30/0.57 @ A6 )
% 0.30/0.57 = ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_negf
% 0.30/0.57 thf(fact_4313_sum__distrib__left,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_0 @ A )
% 0.30/0.57 => ! [R: A,F: B > A,A6: set @ B] :
% 0.30/0.57 ( ( times_times @ A @ R @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [N3: B] : ( times_times @ A @ R @ ( F @ N3 ) )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_distrib_left
% 0.30/0.57 thf(fact_4314_sum__distrib__right,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( semiring_0 @ A )
% 0.30/0.57 => ! [F: B > A,A6: set @ B,R: A] :
% 0.30/0.57 ( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ R )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [N3: B] : ( times_times @ A @ ( F @ N3 ) @ R )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_distrib_right
% 0.30/0.57 thf(fact_4315_sum__product,axiom,
% 0.30/0.57 ! [B: $tType,C: $tType,A: $tType] :
% 0.30/0.57 ( ( semiring_0 @ B )
% 0.30/0.57 => ! [F: A > B,A6: set @ A,G: C > B,B7: set @ C] :
% 0.30/0.57 ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ A @ B
% 0.30/0.57 @ ^ [I5: A] :
% 0.30/0.57 ( groups7311177749621191930dd_sum @ C @ B
% 0.30/0.57 @ ^ [J3: C] : ( times_times @ B @ ( F @ I5 ) @ ( G @ J3 ) )
% 0.30/0.57 @ B7 )
% 0.30/0.57 @ A6 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_product
% 0.30/0.57 thf(fact_4316_mod__sum__eq,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.57 => ! [F: B > A,A3: A,A6: set @ B] :
% 0.30/0.57 ( ( modulo_modulo @ A
% 0.30/0.57 @ ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
% 0.30/0.57 @ A6 )
% 0.30/0.57 @ A3 )
% 0.30/0.57 = ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mod_sum_eq
% 0.30/0.57 thf(fact_4317_unset__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se2638667681897837118et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % unset_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4318_bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % bit_integer.abs_eq
% 0.30/0.57 thf(fact_4319_or__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % or_integer.abs_eq
% 0.30/0.57 thf(fact_4320_and__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % and_integer.abs_eq
% 0.30/0.57 thf(fact_4321_euclidean__size__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( euclid6346220572633701492n_size @ int @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % euclidean_size_integer.abs_eq
% 0.30/0.57 thf(fact_4322_not__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % not_integer.abs_eq
% 0.30/0.57 thf(fact_4323_take__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % take_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4324_xor__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % xor_integer.abs_eq
% 0.30/0.57 thf(fact_4325_times__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( times_times @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( times_times @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % times_integer.abs_eq
% 0.30/0.57 thf(fact_4326_push__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % push_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4327_drop__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se4197421643247451524op_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % drop_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4328_mask__integer_Oabs__eq,axiom,
% 0.30/0.57 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 0.30/0.57 = ( ^ [X2: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mask_integer.abs_eq
% 0.30/0.57 thf(fact_4329_split__cong,axiom,
% 0.30/0.57 ! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
% 0.30/0.57 ( ! [X3: A,Y5: B] :
% 0.30/0.57 ( ( ( product_Pair @ A @ B @ X3 @ Y5 )
% 0.30/0.57 = Q4 )
% 0.30/0.57 => ( ( F @ X3 @ Y5 )
% 0.30/0.57 = ( G @ X3 @ Y5 ) ) )
% 0.30/0.57 => ( ( P5 = Q4 )
% 0.30/0.57 => ( ( product_case_prod @ A @ B @ C @ F @ P5 )
% 0.30/0.57 = ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % split_cong
% 0.30/0.57 thf(fact_4330_set__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se5668285175392031749et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4331_flip__bit__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: nat,X: int] :
% 0.30/0.57 ( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % flip_bit_integer.abs_eq
% 0.30/0.57 thf(fact_4332_division__segment__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % division_segment_integer.abs_eq
% 0.30/0.57 thf(fact_4333_sum_Oneutral,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ( G @ X3 )
% 0.30/0.57 = ( zero_zero @ A ) ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.neutral
% 0.30/0.57 thf(fact_4334_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: B > A,A6: set @ B] :
% 0.30/0.57 ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 0.30/0.57 != ( zero_zero @ A ) )
% 0.30/0.57 => ~ ! [A8: B] :
% 0.30/0.57 ( ( member @ B @ A8 @ A6 )
% 0.30/0.57 => ( ( G @ A8 )
% 0.30/0.57 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.not_neutral_contains_not_neutral
% 0.30/0.57 thf(fact_4335_sum__nonpos,axiom,
% 0.30/0.57 ! [B: $tType,A: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_nonpos
% 0.30/0.57 thf(fact_4336_sum__nonneg,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.57 => ! [A6: set @ B,F: B > A] :
% 0.30/0.57 ( ! [X3: B] :
% 0.30/0.57 ( ( member @ B @ X3 @ A6 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 0.30/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_nonneg
% 0.30/0.57 thf(fact_4337_sum__mono__inv,axiom,
% 0.30/0.57 ! [A: $tType,I7: $tType] :
% 0.30/0.57 ( ( ordere8940638589300402666id_add @ A )
% 0.30/0.57 => ! [F: I7 > A,I6: set @ I7,G: I7 > A,I: I7] :
% 0.30/0.57 ( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F @ I6 )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) )
% 0.30/0.57 => ( ! [I4: I7] :
% 0.30/0.57 ( ( member @ I7 @ I4 @ I6 )
% 0.30/0.57 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.30/0.57 => ( ( member @ I7 @ I @ I6 )
% 0.30/0.57 => ( ( finite_finite2 @ I7 @ I6 )
% 0.30/0.57 => ( ( F @ I )
% 0.30/0.57 = ( G @ I ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_mono_inv
% 0.30/0.57 thf(fact_4338_sum__cong__Suc,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [A6: set @ nat,F: nat > A,G: nat > A] :
% 0.30/0.57 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 0.30/0.57 => ( ! [X3: nat] :
% 0.30/0.57 ( ( member @ nat @ ( suc @ X3 ) @ A6 )
% 0.30/0.57 => ( ( F @ ( suc @ X3 ) )
% 0.30/0.57 = ( G @ ( suc @ X3 ) ) ) )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A6 )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ A6 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum_cong_Suc
% 0.30/0.57 thf(fact_4339_zero__integer__def,axiom,
% 0.30/0.57 ( ( zero_zero @ code_integer )
% 0.30/0.57 = ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % zero_integer_def
% 0.30/0.57 thf(fact_4340_less__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( ord_less @ int @ Xa3 @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_integer.abs_eq
% 0.30/0.57 thf(fact_4341_plus__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( plus_plus @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % plus_integer.abs_eq
% 0.30/0.57 thf(fact_4342_uminus__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % uminus_integer.abs_eq
% 0.30/0.57 thf(fact_4343_sum_Ointer__filter,axiom,
% 0.30/0.57 ! [A: $tType,B: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [A6: set @ B,G: B > A,P: B > $o] :
% 0.30/0.57 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.57 => ( ( groups7311177749621191930dd_sum @ B @ A @ G
% 0.30/0.57 @ ( collect @ B
% 0.30/0.57 @ ^ [X2: B] :
% 0.30/0.57 ( ( member @ B @ X2 @ A6 )
% 0.30/0.57 & ( P @ X2 ) ) ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.57 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
% 0.30/0.57 @ A6 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.inter_filter
% 0.30/0.57 thf(fact_4344_one__integer__def,axiom,
% 0.30/0.57 ( ( one_one @ code_integer )
% 0.30/0.57 = ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % one_integer_def
% 0.30/0.57 thf(fact_4345_less__eq__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( ord_less_eq @ int @ Xa3 @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_eq_integer.abs_eq
% 0.30/0.57 thf(fact_4346_minus__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( minus_minus @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % minus_integer.abs_eq
% 0.30/0.57 thf(fact_4347_abs__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % abs_integer.abs_eq
% 0.30/0.57 thf(fact_4348_divide__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( divide_divide @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divide_integer.abs_eq
% 0.30/0.57 thf(fact_4349_nat__of__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( nat2 @ X ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_of_integer.abs_eq
% 0.30/0.57 thf(fact_4350_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.shift_bounds_cl_Suc_ivl
% 0.30/0.57 thf(fact_4351_sum_Oshift__bounds__Suc__ivl,axiom,
% 0.30/0.57 ! [A: $tType] :
% 0.30/0.57 ( ( comm_monoid_add @ A )
% 0.30/0.57 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.57 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.57 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.57 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.shift_bounds_Suc_ivl
% 0.30/0.57 thf(fact_4352_modulo__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [Xa3: int,X: int] :
% 0.30/0.57 ( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( modulo_modulo @ int @ Xa3 @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % modulo_integer.abs_eq
% 0.30/0.57 thf(fact_4353_sgn__integer_Oabs__eq,axiom,
% 0.30/0.57 ! [X: int] :
% 0.30/0.57 ( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.57 = ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sgn_integer.abs_eq
% 0.30/0.57 thf(fact_4354_effect__guardE,axiom,
% 0.30/0.57 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.30/0.58 ( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N )
% 0.30/0.58 => ~ ( ( H
% 0.30/0.58 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( N
% 0.30/0.58 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( R
% 0.30/0.58 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 0.30/0.58 => ~ ( P @ H2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % effect_guardE
% 0.30/0.58 thf(fact_4355_effect__guardI,axiom,
% 0.30/0.58 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
% 0.30/0.58 ( ( P @ H2 )
% 0.30/0.58 => ( ( H
% 0.30/0.58 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( N
% 0.30/0.58 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( R
% 0.30/0.58 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 0.30/0.58 => ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % effect_guardI
% 0.30/0.58 thf(fact_4356_sum__le__included,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ S )
% 0.30/0.58 => ( ( finite_finite2 @ C @ T2 )
% 0.30/0.58 => ( ! [X3: C] :
% 0.30/0.58 ( ( member @ C @ X3 @ T2 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S )
% 0.30/0.58 => ? [Xa: C] :
% 0.30/0.58 ( ( member @ C @ Xa @ T2 )
% 0.30/0.58 & ( ( I @ Xa )
% 0.30/0.58 = X3 )
% 0.30/0.58 & ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_le_included
% 0.30/0.58 thf(fact_4357_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [A6: set @ B,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 )
% 0.30/0.58 = ( zero_zero @ A ) )
% 0.30/0.58 = ( ! [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ A6 )
% 0.30/0.58 => ( ( F @ X2 )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_nonneg_eq_0_iff
% 0.30/0.58 thf(fact_4358_sum__strict__mono__ex1,axiom,
% 0.30/0.58 ! [A: $tType,I7: $tType] :
% 0.30/0.58 ( ( ordere8940638589300402666id_add @ A )
% 0.30/0.58 => ! [A6: set @ I7,F: I7 > A,G: I7 > A] :
% 0.30/0.58 ( ( finite_finite2 @ I7 @ A6 )
% 0.30/0.58 => ( ! [X3: I7] :
% 0.30/0.58 ( ( member @ I7 @ X3 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.58 => ( ? [X4: I7] :
% 0.30/0.58 ( ( member @ I7 @ X4 @ A6 )
% 0.30/0.58 & ( ord_less @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.58 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_strict_mono_ex1
% 0.30/0.58 thf(fact_4359_sum_Orelated,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.58 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
% 0.30/0.58 => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
% 0.30/0.58 ( ( ( R2 @ X1 @ X23 )
% 0.30/0.58 & ( R2 @ Y1 @ Y22 ) )
% 0.30/0.58 => ( R2 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
% 0.30/0.58 => ( ( finite_finite2 @ B @ S3 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S3 )
% 0.30/0.58 => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.58 => ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.related
% 0.30/0.58 thf(fact_4360_sum__strict__mono,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( strict7427464778891057005id_add @ A )
% 0.30/0.58 => ! [A6: set @ B,F: B > A,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( A6
% 0.30/0.58 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ A6 )
% 0.30/0.58 => ( ord_less @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.58 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_strict_mono
% 0.30/0.58 thf(fact_4361_sum_Oivl__cong,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ( ord @ B )
% 0.30/0.58 & ( comm_monoid_add @ A ) )
% 0.30/0.58 => ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( A3 = C2 )
% 0.30/0.58 => ( ( B2 = D2 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( ord_less_eq @ B @ C2 @ X3 )
% 0.30/0.58 => ( ( ord_less @ B @ X3 @ D2 )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.ivl_cong
% 0.30/0.58 thf(fact_4362_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ S5 )
% 0.30/0.58 => ( ( finite_finite2 @ C @ T5 )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 0.30/0.58 => ( ( I @ ( J @ A8 ) )
% 0.30/0.58 = A8 ) )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 0.30/0.58 => ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 0.30/0.58 => ( ! [B5: C] :
% 0.30/0.58 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.58 => ( ( J @ ( I @ B5 ) )
% 0.30/0.58 = B5 ) )
% 0.30/0.58 => ( ! [B5: C] :
% 0.30/0.58 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.58 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ S5 )
% 0.30/0.58 => ( ( G @ A8 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [B5: C] :
% 0.30/0.58 ( ( member @ C @ B5 @ T5 )
% 0.30/0.58 => ( ( H2 @ B5 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ S3 )
% 0.30/0.58 => ( ( H2 @ ( J @ A8 ) )
% 0.30/0.58 = ( G @ A8 ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.reindex_bij_witness_not_neutral
% 0.30/0.58 thf(fact_4363_sum_OatLeastLessThan__concat,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,P5: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.30/0.58 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeastLessThan_concat
% 0.30/0.58 thf(fact_4364_sum__diff__nat__ivl,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,P5: nat,F: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.30/0.58 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_diff_nat_ivl
% 0.30/0.58 thf(fact_4365_sum__nonneg__0,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [S: set @ B,F: B > A,I: B] :
% 0.30/0.58 ( ( finite_finite2 @ B @ S )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ S )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
% 0.30/0.58 = ( zero_zero @ A ) )
% 0.30/0.58 => ( ( member @ B @ I @ S )
% 0.30/0.58 => ( ( F @ I )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_nonneg_0
% 0.30/0.58 thf(fact_4366_sum__nonneg__leq__bound,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [S: set @ B,F: B > A,B7: A,I: B] :
% 0.30/0.58 ( ( finite_finite2 @ B @ S )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ S )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
% 0.30/0.58 = B7 )
% 0.30/0.58 => ( ( member @ B @ I @ S )
% 0.30/0.58 => ( ord_less_eq @ A @ ( F @ I ) @ B7 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_nonneg_leq_bound
% 0.30/0.58 thf(fact_4367_sum_Ointer__restrict,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.58 @ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
% 0.30/0.58 @ A6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.inter_restrict
% 0.30/0.58 thf(fact_4368_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A6: set @ B,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G
% 0.30/0.58 @ ( minus_minus @ ( set @ B ) @ A6
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( G @ X2 )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.setdiff_irrelevant
% 0.30/0.58 thf(fact_4369_sum__power__add,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,M: nat,I6: set @ nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I5 ) )
% 0.30/0.58 @ I6 )
% 0.30/0.58 = ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_power_add
% 0.30/0.58 thf(fact_4370_sum_Onat__diff__reindex,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.nat_diff_reindex
% 0.30/0.58 thf(fact_4371_sum__choose__upper,axiom,
% 0.30/0.58 ! [M: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_choose_upper
% 0.30/0.58 thf(fact_4372_sum__pos2,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [I6: set @ B,I: B,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ I6 )
% 0.30/0.58 => ( ( member @ B @ I @ I6 )
% 0.30/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ I6 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 0.30/0.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_pos2
% 0.30/0.58 thf(fact_4373_sum__pos,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [I6: set @ B,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ I6 )
% 0.30/0.58 => ( ( I6
% 0.30/0.58 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ I6 )
% 0.30/0.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 0.30/0.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_pos
% 0.30/0.58 thf(fact_4374_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S3 )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_cong_right
% 0.30/0.58 thf(fact_4375_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( H2 @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S3 )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_cong_left
% 0.30/0.58 thf(fact_4376_sum_Omono__neutral__right,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_right
% 0.30/0.58 thf(fact_4377_sum_Omono__neutral__left,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_left
% 0.30/0.58 thf(fact_4378_sum_Osame__carrierI,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ C6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 0.30/0.58 => ( ( G @ A8 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [B5: B] :
% 0.30/0.58 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 0.30/0.58 => ( ( H2 @ B5 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.same_carrierI
% 0.30/0.58 thf(fact_4379_sum_Osame__carrier,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ C6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 0.30/0.58 => ( ! [A8: B] :
% 0.30/0.58 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 0.30/0.58 => ( ( G @ A8 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [B5: B] :
% 0.30/0.58 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 0.30/0.58 => ( ( H2 @ B5 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) )
% 0.30/0.58 = ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.same_carrier
% 0.30/0.58 thf(fact_4380_sum__bounded__below,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 & ( semiring_1 @ A ) )
% 0.30/0.58 => ! [A6: set @ B,K5: A,F: B > A] :
% 0.30/0.58 ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ A @ K5 @ ( F @ I4 ) ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_bounded_below
% 0.30/0.58 thf(fact_4381_sum__bounded__above,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 & ( semiring_1 @ A ) )
% 0.30/0.58 => ! [A6: set @ B,F: B > A,K5: A] :
% 0.30/0.58 ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( F @ I4 ) @ K5 ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_bounded_above
% 0.30/0.58 thf(fact_4382_sum_Omono__neutral__cong,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ( finite_finite2 @ B @ S3 )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( H2 @ I4 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
% 0.30/0.58 => ( ( G @ I4 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_cong
% 0.30/0.58 thf(fact_4383_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [F: nat > A,K: nat] :
% 0.30/0.58 ( ( ( F @ ( zero_zero @ nat ) )
% 0.30/0.58 = ( zero_zero @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_shift_lb_Suc0_0
% 0.30/0.58 thf(fact_4384_sum__shift__lb__Suc0__0__upt,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [F: nat > A,K: nat] :
% 0.30/0.58 ( ( ( F @ ( zero_zero @ nat ) )
% 0.30/0.58 = ( zero_zero @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_shift_lb_Suc0_0_upt
% 0.30/0.58 thf(fact_4385_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeast0_atMost_Suc
% 0.30/0.58 thf(fact_4386_sum_OatLeast0__lessThan__Suc,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeast0_lessThan_Suc
% 0.30/0.58 thf(fact_4387_sum_OatLeast__Suc__lessThan,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeast_Suc_lessThan
% 0.30/0.58 thf(fact_4388_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeast_Suc_atMost
% 0.30/0.58 thf(fact_4389_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.nat_ivl_Suc'
% 0.30/0.58 thf(fact_4390_sum_OatLeastLessThan__Suc,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A3: nat,B2: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeastLessThan_Suc
% 0.30/0.58 thf(fact_4391_sum_Olast__plus,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.last_plus
% 0.30/0.58 thf(fact_4392_sum_OIf__cases,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.58 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 0.30/0.58 @ A6 )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.If_cases
% 0.30/0.58 thf(fact_4393_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ M )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.Suc_reindex_ivl
% 0.30/0.58 thf(fact_4394_sum__Suc__diff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,F: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_Suc_diff
% 0.30/0.58 thf(fact_4395_sum__Suc__diff_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,F: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_Suc_diff'
% 0.30/0.58 thf(fact_4396_sum_OatMost__Suc__shift,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atMost_Suc_shift
% 0.30/0.58 thf(fact_4397_sum__telescope,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [F: nat > A,I: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ I5 ) @ ( F @ ( suc @ I5 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ I ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_telescope
% 0.30/0.58 thf(fact_4398_sum_Ohead,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.head
% 0.30/0.58 thf(fact_4399_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.lessThan_Suc_shift
% 0.30/0.58 thf(fact_4400_sum__lessThan__telescope_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [F: nat > A,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [N3: nat] : ( minus_minus @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ M ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_lessThan_telescope'
% 0.30/0.58 thf(fact_4401_sum__lessThan__telescope,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [F: nat > A,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [N3: nat] : ( minus_minus @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ M ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_lessThan_telescope
% 0.30/0.58 thf(fact_4402_sum_OatLeastLessThan__rev,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeastLessThan_rev
% 0.30/0.58 thf(fact_4403_sum_Onested__swap,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A3: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.nested_swap
% 0.30/0.58 thf(fact_4404_sum_OatLeast1__atMost__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeast1_atMost_eq
% 0.30/0.58 thf(fact_4405_sum__bounds__lt__plus1,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [F: nat > A,Mm: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ Mm ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_bounds_lt_plus1
% 0.30/0.58 thf(fact_4406_sum_Onat__group,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,K: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.nat_group
% 0.30/0.58 thf(fact_4407_sum__choose__lower,axiom,
% 0.30/0.58 ! [R: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R @ K3 ) @ K3 )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_choose_lower
% 0.30/0.58 thf(fact_4408_sum_Onested__swap_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A3: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.nested_swap'
% 0.30/0.58 thf(fact_4409_choose__rising__sum_I1_J,axiom,
% 0.30/0.58 ! [N: nat,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_rising_sum(1)
% 0.30/0.58 thf(fact_4410_choose__rising__sum_I2_J,axiom,
% 0.30/0.58 ! [N: nat,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_rising_sum(2)
% 0.30/0.58 thf(fact_4411_sum__atLeastAtMost__code,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [F: nat > A,A3: nat,B2: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.30/0.58 = ( set_fo6178422350223883121st_nat @ A
% 0.30/0.58 @ ^ [A5: nat] : ( plus_plus @ A @ ( F @ A5 ) )
% 0.30/0.58 @ A3
% 0.30/0.58 @ B2
% 0.30/0.58 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_atLeastAtMost_code
% 0.30/0.58 thf(fact_4412_card__sum__le__nat__sum,axiom,
% 0.30/0.58 ! [S3: set @ nat] :
% 0.30/0.58 ( ord_less_eq @ nat
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [X2: nat] : X2
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [X2: nat] : X2
% 0.30/0.58 @ S3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % card_sum_le_nat_sum
% 0.30/0.58 thf(fact_4413_sum__mono2,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ A )
% 0.30/0.58 => ! [B7: set @ B,A6: set @ B,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ B7 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 0.30/0.58 => ( ! [B5: B] :
% 0.30/0.58 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B5 ) ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B7 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_mono2
% 0.30/0.58 thf(fact_4414_sum_Ounion__inter__neutral,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.union_inter_neutral
% 0.30/0.58 thf(fact_4415_sum_Ohead__if,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [N: nat,M: nat,G: nat > A] :
% 0.30/0.58 ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.head_if
% 0.30/0.58 thf(fact_4416_sum_Oub__add__nat,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,G: nat > A,P5: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.ub_add_nat
% 0.30/0.58 thf(fact_4417_sum__div__partition,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( euclid4440199948858584721cancel @ A )
% 0.30/0.58 => ! [A6: set @ B,F: B > A,B2: A] :
% 0.30/0.58 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ B2 )
% 0.30/0.58 = ( plus_plus @ A
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.58 @ ^ [A5: B] : ( divide_divide @ A @ ( F @ A5 ) @ B2 )
% 0.30/0.58 @ ( inf_inf @ ( set @ B ) @ A6
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
% 0.30/0.58 @ ( divide_divide @ A
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ B @ A @ F
% 0.30/0.58 @ ( inf_inf @ ( set @ B ) @ A6
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [A5: B] :
% 0.30/0.58 ~ ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
% 0.30/0.58 @ B2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_div_partition
% 0.30/0.58 thf(fact_4418_one__diff__power__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 0.30/0.58 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % one_diff_power_eq
% 0.30/0.58 thf(fact_4419_power__diff__1__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
% 0.30/0.58 = ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_diff_1_eq
% 0.30/0.58 thf(fact_4420_geometric__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field @ A )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( X
% 0.30/0.58 != ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.58 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % geometric_sum
% 0.30/0.58 thf(fact_4421_set__encode__def,axiom,
% 0.30/0.58 ( nat_set_encode
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % set_encode_def
% 0.30/0.58 thf(fact_4422_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atLeastLessThan_rev_at_least_Suc_atMost
% 0.30/0.58 thf(fact_4423_sum_OatMost__shift,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.atMost_shift
% 0.30/0.58 thf(fact_4424_gbinomial__parallel__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [A3: A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_parallel_sum
% 0.30/0.58 thf(fact_4425_sum__list__sum__nth,axiom,
% 0.30/0.58 ! [B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ B )
% 0.30/0.58 => ( ( groups8242544230860333062m_list @ B )
% 0.30/0.58 = ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_list_sum_nth
% 0.30/0.58 thf(fact_4426_sum__choose__diagonal,axiom,
% 0.30/0.58 ! [M: nat,N: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_choose_diagonal
% 0.30/0.58 thf(fact_4427_vandermonde,axiom,
% 0.30/0.58 ! [M: nat,N: nat,R: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ R ) )
% 0.30/0.58 = ( binomial @ ( plus_plus @ nat @ M @ N ) @ R ) ) ).
% 0.30/0.58
% 0.30/0.58 % vandermonde
% 0.30/0.58 thf(fact_4428_sum__strict__mono2,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ordere8940638589300402666id_add @ B )
% 0.30/0.58 => ! [B7: set @ A,A6: set @ A,B2: A,F: A > B] :
% 0.30/0.58 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.58 => ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
% 0.30/0.58 => ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
% 0.30/0.58 => ( ! [X3: A] :
% 0.30/0.58 ( ( member @ A @ X3 @ B7 )
% 0.30/0.58 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
% 0.30/0.58 => ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_strict_mono2
% 0.30/0.58 thf(fact_4429_member__le__sum,axiom,
% 0.30/0.58 ! [B: $tType,C: $tType] :
% 0.30/0.58 ( ( ( ordere6911136660526730532id_add @ B )
% 0.30/0.58 & ( semiring_1 @ B ) )
% 0.30/0.58 => ! [I: C,A6: set @ C,F: C > B] :
% 0.30/0.58 ( ( member @ C @ I @ A6 )
% 0.30/0.58 => ( ! [X3: C] :
% 0.30/0.58 ( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A6 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
% 0.30/0.58 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
% 0.30/0.58 => ( ( finite_finite2 @ C @ A6 )
% 0.30/0.58 => ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % member_le_sum
% 0.30/0.58 thf(fact_4430_sum__bounded__above__divide,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( linordered_field @ A )
% 0.30/0.58 => ! [A6: set @ B,F: B > A,K5: A] :
% 0.30/0.58 ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) ) ) )
% 0.30/0.58 => ( ( finite_finite2 @ B @ A6 )
% 0.30/0.58 => ( ( A6
% 0.30/0.58 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ K5 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_bounded_above_divide
% 0.30/0.58 thf(fact_4431_sum__bounded__above__strict,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ( ordere8940638589300402666id_add @ A )
% 0.30/0.58 & ( semiring_1 @ A ) )
% 0.30/0.58 => ! [A6: set @ B,F: B > A,K5: A] :
% 0.30/0.58 ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ A6 )
% 0.30/0.58 => ( ord_less @ A @ ( F @ I4 ) @ K5 ) )
% 0.30/0.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A6 ) )
% 0.30/0.58 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_bounded_above_strict
% 0.30/0.58 thf(fact_4432_convex__sum__bound__le,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( linordered_idom @ B )
% 0.30/0.58 => ! [I6: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
% 0.30/0.58 ( ! [I4: A] :
% 0.30/0.58 ( ( member @ A @ I4 @ I6 )
% 0.30/0.58 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I4 ) ) )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
% 0.30/0.58 = ( one_one @ B ) )
% 0.30/0.58 => ( ! [I4: A] :
% 0.30/0.58 ( ( member @ A @ I4 @ I6 )
% 0.30/0.58 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I4 ) @ B2 ) ) @ Delta ) )
% 0.30/0.58 => ( ord_less_eq @ B
% 0.30/0.58 @ ( abs_abs @ B
% 0.30/0.58 @ ( minus_minus @ B
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ A @ B
% 0.30/0.58 @ ^ [I5: A] : ( times_times @ B @ ( A3 @ I5 ) @ ( X @ I5 ) )
% 0.30/0.58 @ I6 )
% 0.30/0.58 @ B2 ) )
% 0.30/0.58 @ Delta ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % convex_sum_bound_le
% 0.30/0.58 thf(fact_4433_sum__fun__comp,axiom,
% 0.30/0.58 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.58 ( ( semiring_1 @ C )
% 0.30/0.58 => ! [S3: set @ A,R2: set @ B,G: A > B,F: B > C] :
% 0.30/0.58 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.58 => ( ( finite_finite2 @ B @ R2 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R2 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ A @ C
% 0.30/0.58 @ ^ [X2: A] : ( F @ ( G @ X2 ) )
% 0.30/0.58 @ S3 )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ C
% 0.30/0.58 @ ^ [Y6: B] :
% 0.30/0.58 ( times_times @ C
% 0.30/0.58 @ ( semiring_1_of_nat @ C
% 0.30/0.58 @ ( finite_card @ A
% 0.30/0.58 @ ( collect @ A
% 0.30/0.58 @ ^ [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ S3 )
% 0.30/0.58 & ( ( G @ X2 )
% 0.30/0.58 = Y6 ) ) ) ) )
% 0.30/0.58 @ ( F @ Y6 ) )
% 0.30/0.58 @ R2 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_fun_comp
% 0.30/0.58 thf(fact_4434_sum__gp__basic,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
% 0.30/0.58 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp_basic
% 0.30/0.58 thf(fact_4435_sum__natinterval__diff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,F: nat > A] :
% 0.30/0.58 ( ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
% 0.30/0.58 & ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_natinterval_diff
% 0.30/0.58 thf(fact_4436_sum__gp__strict,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( division_ring @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( ( X
% 0.30/0.58 = ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.58 = ( semiring_1_of_nat @ A @ N ) ) )
% 0.30/0.58 & ( ( X
% 0.30/0.58 != ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.58 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp_strict
% 0.30/0.58 thf(fact_4437_diff__power__eq__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat,Y3: A] :
% 0.30/0.58 ( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y3 @ ( suc @ N ) ) )
% 0.30/0.58 = ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P6 ) ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % diff_power_eq_sum
% 0.30/0.58 thf(fact_4438_power__diff__sumr2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat,Y3: A] :
% 0.30/0.58 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
% 0.30/0.58 = ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) ) @ ( power_power @ A @ X @ I5 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_diff_sumr2
% 0.30/0.58 thf(fact_4439_sum__telescope_H_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ A )
% 0.30/0.58 => ! [M: nat,N: nat,F: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
% 0.30/0.58 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_telescope''
% 0.30/0.58 thf(fact_4440_sum__power__shift,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [M: nat,N: nat,X: A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_power_shift
% 0.30/0.58 thf(fact_4441_take__bit__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.58 => ( ( bit_se2584673776208193580ke_bit @ A )
% 0.30/0.58 = ( ^ [N3: nat,A5: A] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % take_bit_sum
% 0.30/0.58 thf(fact_4442_binomial,axiom,
% 0.30/0.58 ! [A3: nat,B2: nat,N: nat] :
% 0.30/0.58 ( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % binomial
% 0.30/0.58 thf(fact_4443_mask__eq__sum__exp,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( semiring_parity @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.30/0.58 @ ( collect @ nat
% 0.30/0.58 @ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mask_eq_sum_exp
% 0.30/0.58 thf(fact_4444_sum__gp__multiplied,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [M: nat,N: nat,X: A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.58 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
% 0.30/0.58 = ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp_multiplied
% 0.30/0.58 thf(fact_4445_sum_Oin__pairs,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.in_pairs
% 0.30/0.58 thf(fact_4446_one__diff__power__eq_H,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( monoid_mult @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 0.30/0.58 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % one_diff_power_eq'
% 0.30/0.58 thf(fact_4447_sum_Oin__pairs__0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.in_pairs_0
% 0.30/0.58 thf(fact_4448_gbinomial__sum__lower__neg,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [A3: A,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_sum_lower_neg
% 0.30/0.58 thf(fact_4449_binomial__ring,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_semiring_1 @ A )
% 0.30/0.58 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.58 ( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % binomial_ring
% 0.30/0.58 thf(fact_4450_pochhammer__binomial__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_ring_1 @ A )
% 0.30/0.58 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.58 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % pochhammer_binomial_sum
% 0.30/0.58 thf(fact_4451_horner__sum__eq__sum,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_semiring_1 @ A )
% 0.30/0.58 => ( ( groups4207007520872428315er_sum @ B @ A )
% 0.30/0.58 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) @ ( power_power @ A @ A5 @ N3 ) )
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % horner_sum_eq_sum
% 0.30/0.58 thf(fact_4452_choose__square__sum,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_square_sum
% 0.30/0.58 thf(fact_4453_sum_Ozero__middle,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.30/0.58 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.zero_middle
% 0.30/0.58 thf(fact_4454_mask__eq__sum__exp__nat,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.58 @ ( collect @ nat
% 0.30/0.58 @ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mask_eq_sum_exp_nat
% 0.30/0.58 thf(fact_4455_gauss__sum__nat,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [X2: nat] : X2
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gauss_sum_nat
% 0.30/0.58 thf(fact_4456_gbinomial__sum__up__index,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [K: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_sum_up_index
% 0.30/0.58 thf(fact_4457_gbinomial__partial__sum__poly,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [M: nat,A3: A,X: A,Y3: A] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_partial_sum_poly
% 0.30/0.58 thf(fact_4458_sum__power2,axiom,
% 0.30/0.58 ! [K: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
% 0.30/0.58 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_power2
% 0.30/0.58 thf(fact_4459_Sum__Ico__nat,axiom,
% 0.30/0.58 ! [M: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [X2: nat] : X2
% 0.30/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.58 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Sum_Ico_nat
% 0.30/0.58 thf(fact_4460_card__lists__length__le,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A,N: nat] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ( ( finite_card @ ( list @ A )
% 0.30/0.58 @ ( collect @ ( list @ A )
% 0.30/0.58 @ ^ [Xs3: list @ A] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 0.30/0.58 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A6 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % card_lists_length_le
% 0.30/0.58 thf(fact_4461_double__gauss__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_semiring_1 @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 0.30/0.58 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % double_gauss_sum
% 0.30/0.58 thf(fact_4462_double__arith__series,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_semiring_1 @ A )
% 0.30/0.58 => ! [A3: A,D2: A,N: nat] :
% 0.30/0.58 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 0.30/0.58 = ( 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 ) @ D2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % double_arith_series
% 0.30/0.58 thf(fact_4463_sum__gp0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( division_ring @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,N: nat] :
% 0.30/0.58 ( ( ( X
% 0.30/0.58 = ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
% 0.30/0.58 & ( ( X
% 0.30/0.58 != ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp0
% 0.30/0.58 thf(fact_4464_choose__alternating__linear__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_ring_1 @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( N
% 0.30/0.58 != ( one_one @ nat ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_alternating_linear_sum
% 0.30/0.58 thf(fact_4465_gbinomial__partial__sum__poly__xpos,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [M: nat,A3: A,X: A,Y3: A] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_partial_sum_poly_xpos
% 0.30/0.58 thf(fact_4466_gbinomial__sum__nat__pow2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_sum_nat_pow2
% 0.30/0.58 thf(fact_4467_arith__series__nat,axiom,
% 0.30/0.58 ! [A3: nat,D2: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [I5: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I5 @ D2 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % arith_series_nat
% 0.30/0.58 thf(fact_4468_Sum__Icc__nat,axiom,
% 0.30/0.58 ! [M: nat,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [X2: nat] : X2
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Sum_Icc_nat
% 0.30/0.58 thf(fact_4469_binomial__r__part__sum,axiom,
% 0.30/0.58 ! [M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % binomial_r_part_sum
% 0.30/0.58 thf(fact_4470_choose__linear__sum,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.30/0.58 @ ^ [I5: nat] : ( times_times @ nat @ I5 @ ( binomial @ N @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_linear_sum
% 0.30/0.58 thf(fact_4471_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_semiring_1 @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( 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 ) ) )
% 0.30/0.58 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % double_gauss_sum_from_Suc_0
% 0.30/0.58 thf(fact_4472_arith__series,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.58 => ! [A3: A,D2: A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( 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 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % arith_series
% 0.30/0.58 thf(fact_4473_gauss__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.58 = ( 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 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gauss_sum
% 0.30/0.58 thf(fact_4474_sum__gp__offset,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( division_ring @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [X: A,M: nat,N: nat] :
% 0.30/0.58 ( ( ( X
% 0.30/0.58 = ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
% 0.30/0.58 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
% 0.30/0.58 & ( ( X
% 0.30/0.58 != ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
% 0.30/0.58 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp_offset
% 0.30/0.58 thf(fact_4475_choose__alternating__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_ring_1 @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_alternating_sum
% 0.30/0.58 thf(fact_4476_divmod__step__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( unique1627219031080169319umeral @ A )
% 0.30/0.58 => ( ( unique1321980374590559556d_step @ A )
% 0.30/0.58 = ( ^ [L3: num] :
% 0.30/0.58 ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.30/0.58 @ ^ [Q3: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_step_def
% 0.30/0.58 thf(fact_4477_gauss__sum__from__Suc__0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( euclid5411537665997757685th_nat @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.30/0.58 = ( 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 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gauss_sum_from_Suc_0
% 0.30/0.58 thf(fact_4478_gchoose__row__sum__weighted,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [R: A,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
% 0.30/0.58 = ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R @ ( suc @ M ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gchoose_row_sum_weighted
% 0.30/0.58 thf(fact_4479_sum__gp,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ( division_ring @ A )
% 0.30/0.58 & ( comm_ring @ A ) )
% 0.30/0.58 => ! [N: nat,M: nat,X: A] :
% 0.30/0.58 ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.58 => ( ( ( X
% 0.30/0.58 = ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
% 0.30/0.58 & ( ( X
% 0.30/0.58 != ( one_one @ A ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.58 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_gp
% 0.30/0.58 thf(fact_4480_gbinomial__r__part__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_r_part_sum
% 0.30/0.58 thf(fact_4481_gbinomial__partial__row__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( field_char_0 @ A )
% 0.30/0.58 => ! [A3: A,M: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ M ) )
% 0.30/0.58 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gbinomial_partial_row_sum
% 0.30/0.58 thf(fact_4482_choose__odd__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_ring_1 @ A )
% 0.30/0.58 => ! [N: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.58 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] :
% 0.30/0.58 ( if @ A
% 0.30/0.58 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 )
% 0.30/0.58 @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) )
% 0.30/0.58 @ ( zero_zero @ A ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) )
% 0.30/0.58 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % choose_odd_sum
% 0.30/0.58 thf(fact_4483_map__entry__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_map_entry @ A )
% 0.30/0.58 = ( ^ [I5: nat,F2: A > A,A5: array @ A] :
% 0.30/0.58 ( heap_Time_guard @ ( array @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) ) @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % map_entry_def
% 0.30/0.58 thf(fact_4484_Array__Time_Onth__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_nth @ A )
% 0.30/0.58 = ( ^ [A5: array @ A,I5: nat] :
% 0.30/0.58 ( heap_Time_guard @ A
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Array_Time.nth_def
% 0.30/0.58 thf(fact_4485_upd__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_upd @ A )
% 0.30/0.58 = ( ^ [I5: nat,X2: A,A5: array @ A] :
% 0.30/0.58 ( heap_Time_guard @ ( array @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % upd_def
% 0.30/0.58 thf(fact_4486_nat__of__integer__code,axiom,
% 0.30/0.58 ( code_nat_of_integer
% 0.30/0.58 = ( ^ [K3: code_integer] :
% 0.30/0.58 ( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
% 0.30/0.58 @ ( product_case_prod @ code_integer @ code_integer @ nat
% 0.30/0.58 @ ^ [L3: code_integer,J3: code_integer] :
% 0.30/0.58 ( if @ nat
% 0.30/0.58 @ ( J3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
% 0.30/0.58 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
% 0.30/0.58 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nat_of_integer_code
% 0.30/0.58 thf(fact_4487_int__sum,axiom,
% 0.30/0.58 ! [B: $tType,F: B > nat,A6: set @ B] :
% 0.30/0.58 ( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ B @ int
% 0.30/0.58 @ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
% 0.30/0.58 @ A6 ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_sum
% 0.30/0.58 thf(fact_4488_sum_Otriangle__reindex__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.58 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.58 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.triangle_reindex_eq
% 0.30/0.58 thf(fact_4489_sum_Otriangle__reindex,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.58 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.58 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] :
% 0.30/0.58 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.triangle_reindex
% 0.30/0.58 thf(fact_4490_sum__subtractf__nat,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A,G: A > nat,F: A > nat] :
% 0.30/0.58 ( ! [X3: A] :
% 0.30/0.58 ( ( member @ A @ X3 @ A6 )
% 0.30/0.58 => ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.58 @ A6 )
% 0.30/0.58 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_subtractf_nat
% 0.30/0.58 thf(fact_4491_card__eq__sum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( finite_card @ A )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % card_eq_sum
% 0.30/0.58 thf(fact_4492_prod_Otriangle__reindex__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_mult @ A )
% 0.30/0.58 => ! [G: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.58 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.58 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.58 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] :
% 0.30/0.58 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % prod.triangle_reindex_eq
% 0.30/0.58 thf(fact_4493_sum__eq__Suc0__iff,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 0.30/0.58 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.58 = ( ? [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ A6 )
% 0.30/0.58 & ( ( F @ X2 )
% 0.30/0.58 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.58 & ! [Y6: A] :
% 0.30/0.58 ( ( member @ A @ Y6 @ A6 )
% 0.30/0.58 => ( ( X2 != Y6 )
% 0.30/0.58 => ( ( F @ Y6 )
% 0.30/0.58 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_eq_Suc0_iff
% 0.30/0.58 thf(fact_4494_sum__SucD,axiom,
% 0.30/0.58 ! [A: $tType,F: A > nat,A6: set @ A,N: nat] :
% 0.30/0.58 ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 0.30/0.58 = ( suc @ N ) )
% 0.30/0.58 => ? [X3: A] :
% 0.30/0.58 ( ( member @ A @ X3 @ A6 )
% 0.30/0.58 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_SucD
% 0.30/0.58 thf(fact_4495_sum__eq__1__iff,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 0.30/0.58 = ( one_one @ nat ) )
% 0.30/0.58 = ( ? [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ A6 )
% 0.30/0.58 & ( ( F @ X2 )
% 0.30/0.58 = ( one_one @ nat ) )
% 0.30/0.58 & ! [Y6: A] :
% 0.30/0.58 ( ( member @ A @ Y6 @ A6 )
% 0.30/0.58 => ( ( X2 != Y6 )
% 0.30/0.58 => ( ( F @ Y6 )
% 0.30/0.58 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_eq_1_iff
% 0.30/0.58 thf(fact_4496_prod_Otriangle__reindex,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( comm_monoid_mult @ A )
% 0.30/0.58 => ! [G: nat > nat > A,N: nat] :
% 0.30/0.58 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.58 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.58 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.58 = ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.58 @ ^ [K3: nat] :
% 0.30/0.58 ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.58 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.58 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % prod.triangle_reindex
% 0.30/0.58 thf(fact_4497_sum__Suc,axiom,
% 0.30/0.58 ! [A: $tType,F: A > nat,A6: set @ A] :
% 0.30/0.58 ( ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( suc @ ( F @ X2 ) )
% 0.30/0.58 @ A6 )
% 0.30/0.58 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( finite_card @ A @ A6 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_Suc
% 0.30/0.58 thf(fact_4498_sum__multicount,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType,S3: set @ A,T6: set @ B,R2: A > B > $o,K: nat] :
% 0.30/0.58 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.58 => ( ( finite_finite2 @ B @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ T6 )
% 0.30/0.58 => ( ( finite_card @ A
% 0.30/0.58 @ ( collect @ A
% 0.30/0.58 @ ^ [I5: A] :
% 0.30/0.58 ( ( member @ A @ I5 @ S3 )
% 0.30/0.58 & ( R2 @ I5 @ X3 ) ) ) )
% 0.30/0.58 = K ) )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [I5: A] :
% 0.30/0.58 ( finite_card @ B
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [J3: B] :
% 0.30/0.58 ( ( member @ B @ J3 @ T6 )
% 0.30/0.58 & ( R2 @ I5 @ J3 ) ) ) )
% 0.30/0.58 @ S3 )
% 0.30/0.58 = ( times_times @ nat @ K @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_multicount
% 0.30/0.58 thf(fact_4499_sum__diff__distrib,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( ord @ A )
% 0.30/0.58 => ! [Q: A > nat,P: A > nat,N: A] :
% 0.30/0.58 ( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 0.30/0.58 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.30/0.58 @ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_diff_distrib
% 0.30/0.58 thf(fact_4500_prod__encode__def,axiom,
% 0.30/0.58 ( nat_prod_encode
% 0.30/0.58 = ( product_case_prod @ nat @ nat @ nat
% 0.30/0.58 @ ^ [M2: nat,N3: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N3 ) ) @ M2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % prod_encode_def
% 0.30/0.58 thf(fact_4501_rat__less__code,axiom,
% 0.30/0.58 ( ( ord_less @ rat )
% 0.30/0.58 = ( ^ [P6: rat,Q3: rat] :
% 0.30/0.58 ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
% 0.30/0.58 @ ( quotient_of @ Q3 ) )
% 0.30/0.58 @ ( quotient_of @ P6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_less_code
% 0.30/0.58 thf(fact_4502_rat__less__eq__code,axiom,
% 0.30/0.58 ( ( ord_less_eq @ rat )
% 0.30/0.58 = ( ^ [P6: rat,Q3: rat] :
% 0.30/0.58 ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
% 0.30/0.58 @ ( quotient_of @ Q3 ) )
% 0.30/0.58 @ ( quotient_of @ P6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_less_eq_code
% 0.30/0.58 thf(fact_4503_rat__uminus__code,axiom,
% 0.30/0.58 ! [P5: rat] :
% 0.30/0.58 ( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_uminus_code
% 0.30/0.58 thf(fact_4504_lenlex__conv,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( lenlex @ A )
% 0.30/0.58 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 0.30/0.58 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.30/0.58 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.30/0.58 @ ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.58 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.58 | ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.58 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.58 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % lenlex_conv
% 0.30/0.58 thf(fact_4505_Divides_Oadjust__div__def,axiom,
% 0.30/0.58 ( adjust_div
% 0.30/0.58 = ( product_case_prod @ int @ int @ int
% 0.30/0.58 @ ^ [Q3: int,R5: int] :
% 0.30/0.58 ( plus_plus @ int @ Q3
% 0.30/0.58 @ ( zero_neq_one_of_bool @ int
% 0.30/0.58 @ ( R5
% 0.30/0.58 != ( zero_zero @ int ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Divides.adjust_div_def
% 0.30/0.58 thf(fact_4506_sum__list__map__eq__sum__count,axiom,
% 0.30/0.58 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 0.30/0.58 ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
% 0.30/0.58 @ ( set2 @ A @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_list_map_eq_sum_count
% 0.30/0.58 thf(fact_4507_rat__divide__code,axiom,
% 0.30/0.58 ! [P5: rat,Q4: rat] :
% 0.30/0.58 ( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q4 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) ) )
% 0.30/0.58 @ ( quotient_of @ Q4 ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_divide_code
% 0.30/0.58 thf(fact_4508_rat__times__code,axiom,
% 0.30/0.58 ! [P5: rat,Q4: rat] :
% 0.30/0.58 ( ( quotient_of @ ( times_times @ rat @ P5 @ Q4 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B3 ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 0.30/0.58 @ ( quotient_of @ Q4 ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_times_code
% 0.30/0.58 thf(fact_4509_rat__plus__code,axiom,
% 0.30/0.58 ! [P5: rat,Q4: rat] :
% 0.30/0.58 ( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q4 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 0.30/0.58 @ ( quotient_of @ Q4 ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_plus_code
% 0.30/0.58 thf(fact_4510_rat__minus__code,axiom,
% 0.30/0.58 ! [P5: rat,Q4: rat] :
% 0.30/0.58 ( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q4 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int,C4: int] :
% 0.30/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 0.30/0.58 @ ( quotient_of @ Q4 ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_minus_code
% 0.30/0.58 thf(fact_4511_sum__list__map__eq__sum__count2,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ A,X8: set @ A,F: A > nat] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
% 0.30/0.58 => ( ( finite_finite2 @ A @ X8 )
% 0.30/0.58 => ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
% 0.30/0.58 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.58 @ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
% 0.30/0.58 @ X8 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_list_map_eq_sum_count2
% 0.30/0.58 thf(fact_4512_divmod__step__nat__def,axiom,
% 0.30/0.58 ( ( unique1321980374590559556d_step @ nat )
% 0.30/0.58 = ( ^ [L3: num] :
% 0.30/0.58 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ^ [Q3: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_step_nat_def
% 0.30/0.58 thf(fact_4513_rat__inverse__code,axiom,
% 0.30/0.58 ! [P5: rat] :
% 0.30/0.58 ( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
% 0.30/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [A5: int,B3: int] :
% 0.30/0.58 ( if @ ( product_prod @ int @ int )
% 0.30/0.58 @ ( A5
% 0.30/0.58 = ( zero_zero @ int ) )
% 0.30/0.58 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.58 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B3 ) @ ( abs_abs @ int @ A5 ) ) )
% 0.30/0.58 @ ( quotient_of @ P5 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rat_inverse_code
% 0.30/0.58 thf(fact_4514_divmod__step__int__def,axiom,
% 0.30/0.58 ( ( unique1321980374590559556d_step @ int )
% 0.30/0.58 = ( ^ [L3: num] :
% 0.30/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.30/0.58 @ ^ [Q3: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_step_int_def
% 0.30/0.58 thf(fact_4515_Sum__Icc__int,axiom,
% 0.30/0.58 ! [M: int,N: int] :
% 0.30/0.58 ( ( ord_less_eq @ int @ M @ N )
% 0.30/0.58 => ( ( groups7311177749621191930dd_sum @ int @ int
% 0.30/0.58 @ ^ [X2: int] : X2
% 0.30/0.58 @ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
% 0.30/0.58 = ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Sum_Icc_int
% 0.30/0.58 thf(fact_4516_divmod__step__integer__def,axiom,
% 0.30/0.58 ( ( unique1321980374590559556d_step @ code_integer )
% 0.30/0.58 = ( ^ [L3: num] :
% 0.30/0.58 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ^ [Q3: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_step_integer_def
% 0.30/0.58 thf(fact_4517_num__of__integer__code,axiom,
% 0.30/0.58 ( code_num_of_integer
% 0.30/0.58 = ( ^ [K3: code_integer] :
% 0.30/0.58 ( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
% 0.30/0.58 @ ( product_case_prod @ code_integer @ code_integer @ num
% 0.30/0.58 @ ^ [L3: code_integer,J3: code_integer] :
% 0.30/0.58 ( if @ num
% 0.30/0.58 @ ( J3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
% 0.30/0.58 @ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
% 0.30/0.58 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % num_of_integer_code
% 0.30/0.58 thf(fact_4518_Array__Time_Omake__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_make @ A )
% 0.30/0.58 = ( ^ [N3: nat,F2: nat > A] :
% 0.30/0.58 ( heap_Time_heap @ ( array @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] :
% 0.30/0.58 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.58 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
% 0.30/0.58 @ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) @ H4 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Array_Time.make_def
% 0.30/0.58 thf(fact_4519_new__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_new @ A )
% 0.30/0.58 = ( ^ [N3: nat,X2: A] :
% 0.30/0.58 ( heap_Time_heap @ ( array @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] :
% 0.30/0.58 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.58 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
% 0.30/0.58 @ ( array_alloc @ A @ ( replicate @ A @ N3 @ X2 ) @ H4 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % new_def
% 0.30/0.58 thf(fact_4520_of__list__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_of_list @ A )
% 0.30/0.58 = ( ^ [Xs3: list @ A] :
% 0.30/0.58 ( heap_Time_heap @ ( array @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] :
% 0.30/0.58 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.58 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
% 0.30/0.58 @ ( array_alloc @ A @ Xs3 @ H4 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % of_list_def
% 0.30/0.58 thf(fact_4521_num__of__integer_Oabs__eq,axiom,
% 0.30/0.58 ! [X: int] :
% 0.30/0.58 ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
% 0.30/0.58 = ( num_of_nat @ ( nat2 @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % num_of_integer.abs_eq
% 0.30/0.58 thf(fact_4522_return__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap_Time_return @ A )
% 0.30/0.58 = ( ^ [X2: A] :
% 0.30/0.58 ( heap_Time_heap @ A
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % return_def
% 0.30/0.58 thf(fact_4523_ureturn__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap_Time_ureturn @ A )
% 0.30/0.58 = ( ^ [X2: A] :
% 0.30/0.58 ( heap_Time_heap @ A
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ureturn_def
% 0.30/0.58 thf(fact_4524_effect__heapE,axiom,
% 0.30/0.58 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.30/0.58 ( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N )
% 0.30/0.58 => ~ ( ( H
% 0.30/0.58 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( N
% 0.30/0.58 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( R
% 0.30/0.58 != ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % effect_heapE
% 0.30/0.58 thf(fact_4525_effect__heapI,axiom,
% 0.30/0.58 ! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A] :
% 0.30/0.58 ( ( N
% 0.30/0.58 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( H
% 0.30/0.58 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 0.30/0.58 => ( ( R
% 0.30/0.58 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 0.30/0.58 => ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % effect_heapI
% 0.30/0.58 thf(fact_4526_divmod__nat__if,axiom,
% 0.30/0.58 ( divmod_nat
% 0.30/0.58 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.58 ( if @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ( ( N3
% 0.30/0.58 = ( zero_zero @ nat ) )
% 0.30/0.58 | ( ord_less @ nat @ M2 @ N3 ) )
% 0.30/0.58 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.30/0.58 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.58 @ ^ [Q3: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q3 ) )
% 0.30/0.58 @ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_nat_if
% 0.30/0.58 thf(fact_4527_int__ge__less__than__def,axiom,
% 0.30/0.58 ( int_ge_less_than
% 0.30/0.58 = ( ^ [D5: int] :
% 0.30/0.58 ( collect @ ( product_prod @ int @ int )
% 0.30/0.58 @ ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [Z8: int,Z7: int] :
% 0.30/0.58 ( ( ord_less_eq @ int @ D5 @ Z8 )
% 0.30/0.58 & ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_ge_less_than_def
% 0.30/0.58 thf(fact_4528_int__ge__less__than2__def,axiom,
% 0.30/0.58 ( int_ge_less_than2
% 0.30/0.58 = ( ^ [D5: int] :
% 0.30/0.58 ( collect @ ( product_prod @ int @ int )
% 0.30/0.58 @ ( product_case_prod @ int @ int @ $o
% 0.30/0.58 @ ^ [Z8: int,Z7: int] :
% 0.30/0.58 ( ( ord_less_eq @ int @ D5 @ Z7 )
% 0.30/0.58 & ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_ge_less_than2_def
% 0.30/0.58 thf(fact_4529_divmod__integer__code,axiom,
% 0.30/0.58 ( code_divmod_integer
% 0.30/0.58 = ( ^ [K3: code_integer,L3: code_integer] :
% 0.30/0.58 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ( K3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L3 )
% 0.30/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
% 0.30/0.58 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ^ [R5: code_integer,S6: code_integer] :
% 0.30/0.58 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ( S6
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S6 ) ) )
% 0.30/0.58 @ ( code_divmod_abs @ K3 @ L3 ) ) )
% 0.30/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ( L3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
% 0.30/0.58 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
% 0.30/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L3 )
% 0.30/0.58 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ^ [R5: code_integer,S6: code_integer] :
% 0.30/0.58 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.58 @ ( S6
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S6 ) ) )
% 0.30/0.58 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divmod_integer_code
% 0.30/0.58 thf(fact_4530_freeze__def,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( heap @ A )
% 0.30/0.58 => ( ( array_freeze @ A )
% 0.30/0.58 = ( ^ [A5: array @ A] :
% 0.30/0.58 ( heap_Time_heap @ ( list @ A )
% 0.30/0.58 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H4 @ A5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H4 @ A5 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % freeze_def
% 0.30/0.58 thf(fact_4531_int__of__integer__code,axiom,
% 0.30/0.58 ( code_int_of_integer
% 0.30/0.58 = ( ^ [K3: code_integer] :
% 0.30/0.58 ( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
% 0.30/0.58 @ ( if @ int
% 0.30/0.58 @ ( K3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( zero_zero @ int )
% 0.30/0.58 @ ( product_case_prod @ code_integer @ code_integer @ int
% 0.30/0.58 @ ^ [L3: code_integer,J3: code_integer] :
% 0.30/0.58 ( if @ int
% 0.30/0.58 @ ( J3
% 0.30/0.58 = ( zero_zero @ code_integer ) )
% 0.30/0.58 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
% 0.30/0.58 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
% 0.30/0.58 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_code
% 0.30/0.58 thf(fact_4532_of__nat__code,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( semiring_1 @ A )
% 0.30/0.58 => ( ( semiring_1_of_nat @ A )
% 0.30/0.58 = ( ^ [N3: nat] :
% 0.30/0.58 ( semiri8178284476397505188at_aux @ A
% 0.30/0.58 @ ^ [I5: A] : ( plus_plus @ A @ I5 @ ( one_one @ A ) )
% 0.30/0.58 @ N3
% 0.30/0.58 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % of_nat_code
% 0.30/0.58 thf(fact_4533_inverse__rat__def,axiom,
% 0.30/0.58 ( ( inverse_inverse @ rat )
% 0.30/0.58 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 0.30/0.58 @ ^ [X2: product_prod @ int @ int] :
% 0.30/0.58 ( if @ ( product_prod @ int @ int )
% 0.30/0.58 @ ( ( product_fst @ int @ int @ X2 )
% 0.30/0.58 = ( zero_zero @ int ) )
% 0.30/0.58 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.58 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % inverse_rat_def
% 0.30/0.58 thf(fact_4534_int__of__integer__inverse,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
% 0.30/0.58 = X ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_inverse
% 0.30/0.58 thf(fact_4535_int__of__integer__integer__of__int,axiom,
% 0.30/0.58 ! [K: int] :
% 0.30/0.58 ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
% 0.30/0.58 = K ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_integer_of_int
% 0.30/0.58 thf(fact_4536_integer__of__int__int__of__integer,axiom,
% 0.30/0.58 ! [K: code_integer] :
% 0.30/0.58 ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
% 0.30/0.58 = K ) ).
% 0.30/0.58
% 0.30/0.58 % integer_of_int_int_of_integer
% 0.30/0.58 thf(fact_4537_int__of__integer__sub,axiom,
% 0.30/0.58 ! [K: num,L: num] :
% 0.30/0.58 ( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
% 0.30/0.58 = ( neg_numeral_sub @ int @ K @ L ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_sub
% 0.30/0.58 thf(fact_4538_int__of__integer__min,axiom,
% 0.30/0.58 ! [K: code_integer,L: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
% 0.30/0.58 = ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_min
% 0.30/0.58 thf(fact_4539_of__int__integer__of,axiom,
% 0.30/0.58 ! [K: code_integer] :
% 0.30/0.58 ( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
% 0.30/0.58 = K ) ).
% 0.30/0.58
% 0.30/0.58 % of_int_integer_of
% 0.30/0.58 thf(fact_4540_int__of__integer__of__int,axiom,
% 0.30/0.58 ! [K: int] :
% 0.30/0.58 ( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
% 0.30/0.58 = K ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_of_int
% 0.30/0.58 thf(fact_4541_zero__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
% 0.30/0.58 = ( zero_zero @ int ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_integer.rep_eq
% 0.30/0.58 thf(fact_4542_int__of__integer__numeral,axiom,
% 0.30/0.58 ! [K: num] :
% 0.30/0.58 ( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
% 0.30/0.58 = ( numeral_numeral @ int @ K ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_numeral
% 0.30/0.58 thf(fact_4543_plus__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % plus_integer.rep_eq
% 0.30/0.58 thf(fact_4544_uminus__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
% 0.30/0.58 = ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % uminus_integer.rep_eq
% 0.30/0.58 thf(fact_4545_times__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % times_integer.rep_eq
% 0.30/0.58 thf(fact_4546_one__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( code_int_of_integer @ ( one_one @ code_integer ) )
% 0.30/0.58 = ( one_one @ int ) ) ).
% 0.30/0.58
% 0.30/0.58 % one_integer.rep_eq
% 0.30/0.58 thf(fact_4547_minus__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % minus_integer.rep_eq
% 0.30/0.58 thf(fact_4548_abs__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
% 0.30/0.58 = ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % abs_integer.rep_eq
% 0.30/0.58 thf(fact_4549_divide__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % divide_integer.rep_eq
% 0.30/0.58 thf(fact_4550_modulo__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % modulo_integer.rep_eq
% 0.30/0.58 thf(fact_4551_sgn__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
% 0.30/0.58 = ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sgn_integer.rep_eq
% 0.30/0.58 thf(fact_4552_int__of__integer__of__nat,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 0.30/0.58 = ( semiring_1_of_nat @ int @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_of_nat
% 0.30/0.58 thf(fact_4553_division__segment__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
% 0.30/0.58 = ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % division_segment_integer.rep_eq
% 0.30/0.58 thf(fact_4554_euclidean__size__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( euclid6346220572633701492n_size @ code_integer )
% 0.30/0.58 = ( ^ [X2: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % euclidean_size_integer.rep_eq
% 0.30/0.58 thf(fact_4555_int__of__integer__max,axiom,
% 0.30/0.58 ! [K: code_integer,L: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
% 0.30/0.58 = ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_max
% 0.30/0.58 thf(fact_4556_integer__eq__iff,axiom,
% 0.30/0.58 ( ( ^ [Y: code_integer,Z2: code_integer] : ( Y = Z2 ) )
% 0.30/0.58 = ( ^ [K3: code_integer,L3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ K3 )
% 0.30/0.58 = ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_eq_iff
% 0.30/0.58 thf(fact_4557_integer__eqI,axiom,
% 0.30/0.58 ! [K: code_integer,L: code_integer] :
% 0.30/0.58 ( ( ( code_int_of_integer @ K )
% 0.30/0.58 = ( code_int_of_integer @ L ) )
% 0.30/0.58 => ( K = L ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_eqI
% 0.30/0.58 thf(fact_4558_int__of__integer__inject,axiom,
% 0.30/0.58 ! [X: code_integer,Y3: code_integer] :
% 0.30/0.58 ( ( ( code_int_of_integer @ X )
% 0.30/0.58 = ( code_int_of_integer @ Y3 ) )
% 0.30/0.58 = ( X = Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_inject
% 0.30/0.58 thf(fact_4559_less__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( ord_less @ code_integer )
% 0.30/0.58 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_integer.rep_eq
% 0.30/0.58 thf(fact_4560_integer__less__iff,axiom,
% 0.30/0.58 ( ( ord_less @ code_integer )
% 0.30/0.58 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_less_iff
% 0.30/0.58 thf(fact_4561_integer__less__eq__iff,axiom,
% 0.30/0.58 ( ( ord_less_eq @ code_integer )
% 0.30/0.58 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_less_eq_iff
% 0.30/0.58 thf(fact_4562_less__eq__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( ord_less_eq @ code_integer )
% 0.30/0.58 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_eq_integer.rep_eq
% 0.30/0.58 thf(fact_4563_nat__of__integer_Orep__eq,axiom,
% 0.30/0.58 ( code_nat_of_integer
% 0.30/0.58 = ( ^ [X2: code_integer] : ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nat_of_integer.rep_eq
% 0.30/0.58 thf(fact_4564_take__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % take_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4565_not__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
% 0.30/0.58 = ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_integer.rep_eq
% 0.30/0.58 thf(fact_4566_and__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % and_integer.rep_eq
% 0.30/0.58 thf(fact_4567_bit__integer_Orep__eq,axiom,
% 0.30/0.58 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 0.30/0.58 = ( ^ [X2: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % bit_integer.rep_eq
% 0.30/0.58 thf(fact_4568_or__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % or_integer.rep_eq
% 0.30/0.58 thf(fact_4569_xor__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: code_integer,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % xor_integer.rep_eq
% 0.30/0.58 thf(fact_4570_push__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % push_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4571_drop__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % drop_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4572_mask__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
% 0.30/0.58 = ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% 0.30/0.58
% 0.30/0.58 % mask_integer.rep_eq
% 0.30/0.58 thf(fact_4573_unset__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unset_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4574_set__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % set_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4575_flip__bit__integer_Orep__eq,axiom,
% 0.30/0.58 ! [X: nat,Xa3: code_integer] :
% 0.30/0.58 ( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa3 ) )
% 0.30/0.58 = ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % flip_bit_integer.rep_eq
% 0.30/0.58 thf(fact_4576_of__nat__aux_Osimps_I2_J,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( semiring_1 @ A )
% 0.30/0.58 => ! [Inc: A > A,N: nat,I: A] :
% 0.30/0.58 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 0.30/0.58 = ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % of_nat_aux.simps(2)
% 0.30/0.58 thf(fact_4577_of__nat__aux_Osimps_I1_J,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( semiring_1 @ A )
% 0.30/0.58 => ! [Inc: A > A,I: A] :
% 0.30/0.58 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 0.30/0.58 = I ) ) ).
% 0.30/0.58
% 0.30/0.58 % of_nat_aux.simps(1)
% 0.30/0.58 thf(fact_4578_num__of__integer_Orep__eq,axiom,
% 0.30/0.58 ( code_num_of_integer
% 0.30/0.58 = ( ^ [X2: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % num_of_integer.rep_eq
% 0.30/0.58 thf(fact_4579_uminus__rat__def,axiom,
% 0.30/0.58 ( ( uminus_uminus @ rat )
% 0.30/0.58 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 0.30/0.58 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % uminus_rat_def
% 0.30/0.58 thf(fact_4580_plus__rat__def,axiom,
% 0.30/0.58 ( ( plus_plus @ rat )
% 0.30/0.58 = ( 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 )
% 0.30/0.58 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % plus_rat_def
% 0.30/0.58 thf(fact_4581_times__rat__def,axiom,
% 0.30/0.58 ( ( times_times @ rat )
% 0.30/0.58 = ( 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 )
% 0.30/0.58 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % times_rat_def
% 0.30/0.58 thf(fact_4582_transpose__rectangle,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
% 0.30/0.58 ( ( ( Xs
% 0.30/0.58 = ( nil @ ( list @ A ) ) )
% 0.30/0.58 => ( N
% 0.30/0.58 = ( zero_zero @ nat ) ) )
% 0.30/0.58 => ( ! [I4: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 0.30/0.58 => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) )
% 0.30/0.58 = N ) )
% 0.30/0.58 => ( ( transpose @ A @ Xs )
% 0.30/0.58 = ( map @ nat @ ( list @ A )
% 0.30/0.58 @ ^ [I5: nat] :
% 0.30/0.58 ( map @ nat @ A
% 0.30/0.58 @ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I5 )
% 0.30/0.58 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.58 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % transpose_rectangle
% 0.30/0.58 thf(fact_4583_sorted__wrt__less__sum__mono__lowerbound,axiom,
% 0.30/0.58 ! [B: $tType] :
% 0.30/0.58 ( ( ordere6911136660526730532id_add @ B )
% 0.30/0.58 => ! [F: nat > B,Ns: list @ nat] :
% 0.30/0.58 ( ! [X3: nat,Y5: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ X3 @ Y5 )
% 0.30/0.58 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.58 => ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
% 0.30/0.58 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F @ Ns ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_less_sum_mono_lowerbound
% 0.30/0.58 thf(fact_4584_strict__sorted__imp__sorted,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % strict_sorted_imp_sorted
% 0.30/0.58 thf(fact_4585_sorted2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [X: A,Y3: A,Zs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
% 0.30/0.58 = ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted2
% 0.30/0.58 thf(fact_4586_sorted0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted0
% 0.30/0.58 thf(fact_4587_strict__sorted__simps_I1_J,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % strict_sorted_simps(1)
% 0.30/0.58 thf(fact_4588_strict__sorted__equal,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
% 0.30/0.58 => ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 )
% 0.30/0.58 => ( ( ( set2 @ A @ Ys2 )
% 0.30/0.58 = ( set2 @ A @ Xs ) )
% 0.30/0.58 => ( Ys2 = Xs ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % strict_sorted_equal
% 0.30/0.58 thf(fact_4589_sorted__take,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,N: nat] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_take
% 0.30/0.58 thf(fact_4590_sorted__replicate,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_replicate
% 0.30/0.58 thf(fact_4591_sorted__drop,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,N: nat] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_drop
% 0.30/0.58 thf(fact_4592_sorted__remdups__adj,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_remdups_adj
% 0.30/0.58 thf(fact_4593_sorted__wrt__upt,axiom,
% 0.30/0.58 ! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_upt
% 0.30/0.58 thf(fact_4594_sorted__upt,axiom,
% 0.30/0.58 ! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_upt
% 0.30/0.58 thf(fact_4595_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_list_of_set.sorted_sorted_key_list_of_set
% 0.30/0.58 thf(fact_4596_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_list_of_set.strict_sorted_key_list_of_set
% 0.30/0.58 thf(fact_4597_sorted__map,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [F: B > A,Xs: list @ B] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.58 = ( sorted_wrt @ B
% 0.30/0.58 @ ^ [X2: B,Y6: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y6 ) )
% 0.30/0.58 @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_map
% 0.30/0.58 thf(fact_4598_sorted1,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted1
% 0.30/0.58 thf(fact_4599_sorted__simps_I2_J,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [X: A,Ys2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys2 ) )
% 0.30/0.58 = ( ! [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
% 0.30/0.58 => ( ord_less_eq @ A @ X @ X2 ) )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_simps(2)
% 0.30/0.58 thf(fact_4600_strict__sorted__simps_I2_J,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [X: A,Ys2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys2 ) )
% 0.30/0.58 = ( ! [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
% 0.30/0.58 => ( ord_less @ A @ X @ X2 ) )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % strict_sorted_simps(2)
% 0.30/0.58 thf(fact_4601_sorted__append,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.58 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
% 0.30/0.58 & ! [X2: A] :
% 0.30/0.58 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.30/0.58 => ! [Y6: A] :
% 0.30/0.58 ( ( member @ A @ Y6 @ ( set2 @ A @ Ys2 ) )
% 0.30/0.58 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_append
% 0.30/0.58 thf(fact_4602_strict__sorted__iff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [L: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
% 0.30/0.58 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.30/0.58 & ( distinct @ A @ L ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % strict_sorted_iff
% 0.30/0.58 thf(fact_4603_sorted__distinct__set__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( ( distinct @ A @ Xs )
% 0.30/0.58 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
% 0.30/0.58 => ( ( distinct @ A @ Ys2 )
% 0.30/0.58 => ( ( ( set2 @ A @ Xs )
% 0.30/0.58 = ( set2 @ A @ Ys2 ) )
% 0.30/0.58 => ( Xs = Ys2 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_distinct_set_unique
% 0.30/0.58 thf(fact_4604_sorted__wrt01,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ A,P: A > A > $o] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.30/0.58 => ( sorted_wrt @ A @ P @ Xs ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt01
% 0.30/0.58 thf(fact_4605_sorted__wrt__iff__nth__less,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( sorted_wrt @ A )
% 0.30/0.58 = ( ^ [P4: A > A > $o,Xs3: list @ A] :
% 0.30/0.58 ! [I5: nat,J3: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ I5 @ J3 )
% 0.30/0.58 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.58 => ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_iff_nth_less
% 0.30/0.58 thf(fact_4606_sorted__wrt__nth__less,axiom,
% 0.30/0.58 ! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
% 0.30/0.58 ( ( sorted_wrt @ A @ P @ Xs )
% 0.30/0.58 => ( ( ord_less @ nat @ I @ J )
% 0.30/0.58 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_nth_less
% 0.30/0.58 thf(fact_4607_sorted__butlast,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( Xs
% 0.30/0.58 != ( nil @ A ) )
% 0.30/0.58 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_butlast
% 0.30/0.58 thf(fact_4608_sorted__enumerate,axiom,
% 0.30/0.58 ! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_enumerate
% 0.30/0.58 thf(fact_4609_sorted01,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.30/0.58 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted01
% 0.30/0.58 thf(fact_4610_finite__sorted__distinct__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [A6: set @ A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ? [X3: list @ A] :
% 0.30/0.58 ( ( ( set2 @ A @ X3 )
% 0.30/0.58 = A6 )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
% 0.30/0.58 & ( distinct @ A @ X3 )
% 0.30/0.58 & ! [Y4: list @ A] :
% 0.30/0.58 ( ( ( ( set2 @ A @ Y4 )
% 0.30/0.58 = A6 )
% 0.30/0.58 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
% 0.30/0.58 & ( distinct @ A @ Y4 ) )
% 0.30/0.58 => ( Y4 = X3 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % finite_sorted_distinct_unique
% 0.30/0.58 thf(fact_4611_sorted__iff__nth__mono__less,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 = ( ! [I5: nat,J3: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ I5 @ J3 )
% 0.30/0.58 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_iff_nth_mono_less
% 0.30/0.58 thf(fact_4612_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( ( distinct @ A @ Xs )
% 0.30/0.58 => ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
% 0.30/0.58 = Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_list_of_set.idem_if_sorted_distinct
% 0.30/0.58 thf(fact_4613_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [A6: set @ A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ~ ! [L2: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
% 0.30/0.58 => ( ( ( set2 @ A @ L2 )
% 0.30/0.58 = A6 )
% 0.30/0.58 => ( ( size_size @ ( list @ A ) @ L2 )
% 0.30/0.58 != ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_list_of_set.finite_set_strict_sorted
% 0.30/0.58 thf(fact_4614_sorted__iff__nth__Suc,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 = ( ! [I5: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_iff_nth_Suc
% 0.30/0.58 thf(fact_4615_sorted__nth__mono,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,I: nat,J: nat] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.58 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_nth_mono
% 0.30/0.58 thf(fact_4616_sorted__iff__nth__mono,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.58 = ( ! [I5: nat,J3: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ I5 @ J3 )
% 0.30/0.58 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_iff_nth_mono
% 0.30/0.58 thf(fact_4617_sorted__wrt__less__idx,axiom,
% 0.30/0.58 ! [Ns: list @ nat,I: nat] :
% 0.30/0.58 ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
% 0.30/0.58 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
% 0.30/0.58 => ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_less_idx
% 0.30/0.58 thf(fact_4618_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [A6: set @ A,L: list @ A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
% 0.30/0.58 & ( ( set2 @ A @ L )
% 0.30/0.58 = A6 )
% 0.30/0.58 & ( ( size_size @ ( list @ A ) @ L )
% 0.30/0.58 = ( finite_card @ A @ A6 ) ) )
% 0.30/0.58 = ( ( linord4507533701916653071of_set @ A @ A6 )
% 0.30/0.58 = L ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_list_of_set.sorted_key_list_of_set_unique
% 0.30/0.58 thf(fact_4619_length__transpose__sorted,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.30/0.58 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.58 => ( ( ( Xs
% 0.30/0.58 = ( nil @ ( list @ A ) ) )
% 0.30/0.58 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 0.30/0.58 = ( zero_zero @ nat ) ) )
% 0.30/0.58 & ( ( Xs
% 0.30/0.58 != ( nil @ ( list @ A ) ) )
% 0.30/0.58 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 0.30/0.58 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % length_transpose_sorted
% 0.30/0.58 thf(fact_4620_sum__diff1_H__aux,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ B )
% 0.30/0.58 => ! [F4: set @ A,I6: set @ A,F: A > B,I: A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ F4 )
% 0.30/0.58 => ( ( ord_less_eq @ ( set @ A )
% 0.30/0.58 @ ( collect @ A
% 0.30/0.58 @ ^ [I5: A] :
% 0.30/0.58 ( ( member @ A @ I5 @ I6 )
% 0.30/0.58 & ( ( F @ I5 )
% 0.30/0.58 != ( zero_zero @ B ) ) ) )
% 0.30/0.58 @ F4 )
% 0.30/0.58 => ( ( ( member @ A @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.58 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.30/0.58 & ( ~ ( member @ A @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_diff1'_aux
% 0.30/0.58 thf(fact_4621_finite__mono__remains__stable__implies__strict__prefix,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order @ A )
% 0.30/0.58 => ! [F: nat > A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) )
% 0.30/0.58 => ( ( order_mono @ nat @ A @ F )
% 0.30/0.58 => ( ! [N4: nat] :
% 0.30/0.58 ( ( ( F @ N4 )
% 0.30/0.58 = ( F @ ( suc @ N4 ) ) )
% 0.30/0.58 => ( ( F @ ( suc @ N4 ) )
% 0.30/0.58 = ( F @ ( suc @ ( suc @ N4 ) ) ) ) )
% 0.30/0.58 => ? [N8: nat] :
% 0.30/0.58 ( ! [N5: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ N5 @ N8 )
% 0.30/0.58 => ! [M3: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ M3 @ N8 )
% 0.30/0.58 => ( ( ord_less @ nat @ M3 @ N5 )
% 0.30/0.58 => ( ord_less @ A @ ( F @ M3 ) @ ( F @ N5 ) ) ) ) )
% 0.30/0.58 & ! [N5: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ N8 @ N5 )
% 0.30/0.58 => ( ( F @ N8 )
% 0.30/0.58 = ( F @ N5 ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % finite_mono_remains_stable_implies_strict_prefix
% 0.30/0.58 thf(fact_4622_top__apply,axiom,
% 0.30/0.58 ! [C: $tType,D: $tType] :
% 0.30/0.58 ( ( top @ C )
% 0.30/0.58 => ( ( top_top @ ( D > C ) )
% 0.30/0.58 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top_apply
% 0.30/0.58 thf(fact_4623_inf__top__left,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 0.30/0.58 = X ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top_left
% 0.30/0.58 thf(fact_4624_inf__top__right,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = X ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top_right
% 0.30/0.58 thf(fact_4625_inf__eq__top__iff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 = ( ( X
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 & ( Y3
% 0.30/0.58 = ( top_top @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_eq_top_iff
% 0.30/0.58 thf(fact_4626_top__eq__inf__iff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( ( top_top @ A )
% 0.30/0.58 = ( inf_inf @ A @ X @ Y3 ) )
% 0.30/0.58 = ( ( X
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 & ( Y3
% 0.30/0.58 = ( top_top @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top_eq_inf_iff
% 0.30/0.58 thf(fact_4627_inf__top_Oeq__neutr__iff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [A3: A,B2: A] :
% 0.30/0.58 ( ( ( inf_inf @ A @ A3 @ B2 )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 = ( ( A3
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 & ( B2
% 0.30/0.58 = ( top_top @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top.eq_neutr_iff
% 0.30/0.58 thf(fact_4628_inf__top_Oleft__neutral,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
% 0.30/0.58 = A3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top.left_neutral
% 0.30/0.58 thf(fact_4629_inf__top_Oneutr__eq__iff,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [A3: A,B2: A] :
% 0.30/0.58 ( ( ( top_top @ A )
% 0.30/0.58 = ( inf_inf @ A @ A3 @ B2 ) )
% 0.30/0.58 = ( ( A3
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 & ( B2
% 0.30/0.58 = ( top_top @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top.neutr_eq_iff
% 0.30/0.58 thf(fact_4630_inf__top_Oright__neutral,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
% 0.30/0.58 = A3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % inf_top.right_neutral
% 0.30/0.58 thf(fact_4631_boolean__algebra_Odisj__one__right,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.disj_one_right
% 0.30/0.58 thf(fact_4632_boolean__algebra_Odisj__one__left,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.disj_one_left
% 0.30/0.58 thf(fact_4633_sup__top__right,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounded_lattice_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_top_right
% 0.30/0.58 thf(fact_4634_sup__top__left,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( bounded_lattice_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_top_left
% 0.30/0.58 thf(fact_4635_max__top,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( ord_max @ A @ ( top_top @ A ) @ X )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % max_top
% 0.30/0.58 thf(fact_4636_max__top2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( ord_max @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % max_top2
% 0.30/0.58 thf(fact_4637_min__top2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( ord_min @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = X ) ) ).
% 0.30/0.58
% 0.30/0.58 % min_top2
% 0.30/0.58 thf(fact_4638_min__top,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( ord_min @ A @ ( top_top @ A ) @ X )
% 0.30/0.58 = X ) ) ).
% 0.30/0.58
% 0.30/0.58 % min_top
% 0.30/0.58 thf(fact_4639_boolean__algebra_Ocompl__zero,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.compl_zero
% 0.30/0.58 thf(fact_4640_boolean__algebra_Ocompl__one,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 0.30/0.58 = ( bot_bot @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.compl_one
% 0.30/0.58 thf(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.disj_cancel_right
% 0.30/0.58 thf(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.disj_cancel_left
% 0.30/0.58 thf(fact_4643_sup__compl__top__left2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_compl_top_left2
% 0.30/0.58 thf(fact_4644_sup__compl__top__left1,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y3 ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_compl_top_left1
% 0.30/0.58 thf(fact_4645_finite__compl,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.58 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.58 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % finite_compl
% 0.30/0.58 thf(fact_4646_Gcd__UNIV,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( semiring_Gcd @ A )
% 0.30/0.58 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
% 0.30/0.58 = ( one_one @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Gcd_UNIV
% 0.30/0.58 thf(fact_4647_sum_Oempty_H,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [P5: B > A] :
% 0.30/0.58 ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.58 = ( zero_zero @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.empty'
% 0.30/0.58 thf(fact_4648_sum_Oinsert_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [I6: set @ B,P5: B > A,I: B] :
% 0.30/0.58 ( ( finite_finite2 @ B
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I6 )
% 0.30/0.58 & ( ( P5 @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 => ( ( ( member @ B @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
% 0.30/0.58 & ( ~ ( member @ B @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 0.30/0.58 = ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.insert'
% 0.30/0.58 thf(fact_4649_boolean__algebra_Oconj__one__right,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A] :
% 0.30/0.58 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 0.30/0.58 = X ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.conj_one_right
% 0.30/0.58 thf(fact_4650_top__greatest,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top_greatest
% 0.30/0.58 thf(fact_4651_top_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.30/0.58 = ( A3
% 0.30/0.58 = ( top_top @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top.extremum_unique
% 0.30/0.58 thf(fact_4652_top_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.30/0.58 => ( A3
% 0.30/0.58 = ( top_top @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top.extremum_uniqueI
% 0.30/0.58 thf(fact_4653_top_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % top.extremum_strict
% 0.30/0.58 thf(fact_4654_top_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( order_top @ A )
% 0.30/0.58 => ! [A3: A] :
% 0.30/0.58 ( ( A3
% 0.30/0.58 != ( top_top @ A ) )
% 0.30/0.58 = ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % top.not_eq_extremum
% 0.30/0.58 thf(fact_4655_sum_Onon__neutral_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [G: B > A,I6: set @ B] :
% 0.30/0.58 ( ( groups1027152243600224163dd_sum @ B @ A @ G
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I6 )
% 0.30/0.58 & ( ( G @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.non_neutral'
% 0.30/0.58 thf(fact_4656_finite__fun__UNIVD1,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 0.30/0.58 => ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
% 0.30/0.58 != ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.58 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % finite_fun_UNIVD1
% 0.30/0.58 thf(fact_4657_sorted__upto,axiom,
% 0.30/0.58 ! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_upto
% 0.30/0.58 thf(fact_4658_sorted__wrt__upto,axiom,
% 0.30/0.58 ! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_wrt_upto
% 0.30/0.58 thf(fact_4659_sup__cancel__left2,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,A3: A,B2: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_cancel_left2
% 0.30/0.58 thf(fact_4660_sup__cancel__left1,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,A3: A,B2: A] :
% 0.30/0.58 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
% 0.30/0.58 = ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_cancel_left1
% 0.30/0.58 thf(fact_4661_Compl__empty__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.58 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Compl_empty_eq
% 0.30/0.58 thf(fact_4662_Compl__UNIV__eq,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.58 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Compl_UNIV_eq
% 0.30/0.58 thf(fact_4663_Compl__partition2,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A] :
% 0.30/0.58 ( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
% 0.30/0.58 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Compl_partition2
% 0.30/0.58 thf(fact_4664_Compl__partition,axiom,
% 0.30/0.58 ! [A: $tType,A6: set @ A] :
% 0.30/0.58 ( ( sup_sup @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.58 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Compl_partition
% 0.30/0.58 thf(fact_4665_Compl__eq__Diff__UNIV,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( uminus_uminus @ ( set @ A ) )
% 0.30/0.58 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % Compl_eq_Diff_UNIV
% 0.30/0.58 thf(fact_4666_sup__shunt,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sup_shunt
% 0.30/0.58 thf(fact_4667_boolean__algebra_Ocomplement__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [A3: A,X: A,Y3: A] :
% 0.30/0.58 ( ( ( inf_inf @ A @ A3 @ X )
% 0.30/0.58 = ( bot_bot @ A ) )
% 0.30/0.58 => ( ( ( sup_sup @ A @ A3 @ X )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 => ( ( ( inf_inf @ A @ A3 @ Y3 )
% 0.30/0.58 = ( bot_bot @ A ) )
% 0.30/0.58 => ( ( ( sup_sup @ A @ A3 @ Y3 )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 => ( X = Y3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.complement_unique
% 0.30/0.58 thf(fact_4668_surj__Compl__image__subset,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
% 0.30/0.58 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.58 = ( top_top @ ( set @ A ) ) )
% 0.30/0.58 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A6 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % surj_Compl_image_subset
% 0.30/0.58 thf(fact_4669_sum_Omono__neutral__left_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_left'
% 0.30/0.58 thf(fact_4670_sum_Omono__neutral__right_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_right'
% 0.30/0.58 thf(fact_4671_sum_Omono__neutral__cong__left_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [I4: B] :
% 0.30/0.58 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( H2 @ I4 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S3 )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_cong_left'
% 0.30/0.58 thf(fact_4672_sum_Omono__neutral__cong__right_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( zero_zero @ A ) ) )
% 0.30/0.58 => ( ! [X3: B] :
% 0.30/0.58 ( ( member @ B @ X3 @ S3 )
% 0.30/0.58 => ( ( G @ X3 )
% 0.30/0.58 = ( H2 @ X3 ) ) )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.mono_neutral_cong_right'
% 0.30/0.58 thf(fact_4673_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ! [X: A,Y3: A] :
% 0.30/0.58 ( ( ( inf_inf @ A @ X @ Y3 )
% 0.30/0.58 = ( bot_bot @ A ) )
% 0.30/0.58 => ( ( ( sup_sup @ A @ X @ Y3 )
% 0.30/0.58 = ( top_top @ A ) )
% 0.30/0.58 => ( ( uminus_uminus @ A @ X )
% 0.30/0.58 = Y3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.30/0.58 thf(fact_4674_sum_Odistrib_H,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ B
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I6 )
% 0.30/0.58 & ( ( G @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 => ( ( finite_finite2 @ B
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I6 )
% 0.30/0.58 & ( ( H2 @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A
% 0.30/0.58 @ ^ [I5: B] : ( plus_plus @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 0.30/0.58 @ I6 )
% 0.30/0.58 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.distrib'
% 0.30/0.58 thf(fact_4675_sum_OG__def,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType] :
% 0.30/0.58 ( ( comm_monoid_add @ A )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A )
% 0.30/0.58 = ( ^ [P6: B > A,I8: set @ B] :
% 0.30/0.58 ( if @ A
% 0.30/0.58 @ ( finite_finite2 @ B
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I8 )
% 0.30/0.58 & ( ( P6 @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 @ ( groups7311177749621191930dd_sum @ B @ A @ P6
% 0.30/0.58 @ ( collect @ B
% 0.30/0.58 @ ^ [X2: B] :
% 0.30/0.58 ( ( member @ B @ X2 @ I8 )
% 0.30/0.58 & ( ( P6 @ X2 )
% 0.30/0.58 != ( zero_zero @ A ) ) ) ) )
% 0.30/0.58 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum.G_def
% 0.30/0.58 thf(fact_4676_finite__UNIV__card__ge__0,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.30/0.58 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % finite_UNIV_card_ge_0
% 0.30/0.58 thf(fact_4677_sorted__transpose,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_transpose
% 0.30/0.58 thf(fact_4678_rev__nth,axiom,
% 0.30/0.58 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.58 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 0.30/0.58 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rev_nth
% 0.30/0.58 thf(fact_4679_rev__update,axiom,
% 0.30/0.58 ! [A: $tType,K: nat,Xs: list @ A,Y3: A] :
% 0.30/0.58 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y3 ) )
% 0.30/0.58 = ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % rev_update
% 0.30/0.58 thf(fact_4680_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.58 => ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % boolean_algebra.abstract_boolean_algebra_axioms
% 0.30/0.58 thf(fact_4681_card__range__greater__zero,axiom,
% 0.30/0.58 ! [A: $tType,B: $tType,F: B > A] :
% 0.30/0.58 ( ( finite_finite2 @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.58 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % card_range_greater_zero
% 0.30/0.58 thf(fact_4682_sorted__rev__iff__nth__Suc,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 0.30/0.58 = ( ! [I5: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I5 ) ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_rev_iff_nth_Suc
% 0.30/0.58 thf(fact_4683_sorted__rev__iff__nth__mono,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 0.30/0.58 = ( ! [I5: nat,J3: nat] :
% 0.30/0.58 ( ( ord_less_eq @ nat @ I5 @ J3 )
% 0.30/0.58 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_rev_iff_nth_mono
% 0.30/0.58 thf(fact_4684_sorted__rev__nth__mono,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [Xs: list @ A,I: nat,J: nat] :
% 0.30/0.58 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 0.30/0.58 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.58 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.58 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_rev_nth_mono
% 0.30/0.58 thf(fact_4685_range__mod,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.58 => ( ( image @ nat @ nat
% 0.30/0.58 @ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
% 0.30/0.58 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.58 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % range_mod
% 0.30/0.58 thf(fact_4686_sum__diff1_H,axiom,
% 0.30/0.58 ! [B: $tType,A: $tType] :
% 0.30/0.58 ( ( ab_group_add @ B )
% 0.30/0.58 => ! [I6: set @ A,F: A > B,I: A] :
% 0.30/0.58 ( ( finite_finite2 @ A
% 0.30/0.58 @ ( collect @ A
% 0.30/0.58 @ ^ [I5: A] :
% 0.30/0.58 ( ( member @ A @ I5 @ I6 )
% 0.30/0.58 & ( ( F @ I5 )
% 0.30/0.58 != ( zero_zero @ B ) ) ) ) )
% 0.30/0.58 => ( ( ( member @ A @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.58 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
% 0.30/0.58 & ( ~ ( member @ A @ I @ I6 )
% 0.30/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.58 = ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sum_diff1'
% 0.30/0.58 thf(fact_4687_nth__nth__transpose__sorted,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
% 0.30/0.58 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.58 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
% 0.30/0.58 => ( ( ord_less @ nat @ J
% 0.30/0.58 @ ( size_size @ ( list @ ( list @ A ) )
% 0.30/0.58 @ ( filter @ ( list @ A )
% 0.30/0.58 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.58 @ Xs ) ) )
% 0.30/0.58 => ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
% 0.30/0.58 = ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nth_nth_transpose_sorted
% 0.30/0.58 thf(fact_4688_transpose__column,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
% 0.30/0.58 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.58 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 0.30/0.58 => ( ( map @ ( list @ A ) @ A
% 0.30/0.58 @ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
% 0.30/0.58 @ ( filter @ ( list @ A )
% 0.30/0.58 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.58 @ ( transpose @ A @ Xs ) ) )
% 0.30/0.58 = ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % transpose_column
% 0.30/0.58 thf(fact_4689_transpose__column__length,axiom,
% 0.30/0.58 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
% 0.30/0.58 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.58 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 0.30/0.58 => ( ( size_size @ ( list @ ( list @ A ) )
% 0.30/0.58 @ ( filter @ ( list @ A )
% 0.30/0.58 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.58 @ ( transpose @ A @ Xs ) ) )
% 0.30/0.58 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % transpose_column_length
% 0.30/0.58 thf(fact_4690_card__UNIV__unit,axiom,
% 0.30/0.58 ( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
% 0.30/0.58 = ( one_one @ nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % card_UNIV_unit
% 0.30/0.58 thf(fact_4691_infinite__UNIV__int,axiom,
% 0.30/0.58 ~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % infinite_UNIV_int
% 0.30/0.58 thf(fact_4692_integer__of__int__cases,axiom,
% 0.30/0.58 ! [X: code_integer] :
% 0.30/0.58 ~ ! [Y5: int] :
% 0.30/0.58 ( ( X
% 0.30/0.58 = ( code_integer_of_int @ Y5 ) )
% 0.30/0.58 => ~ ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_of_int_cases
% 0.30/0.58 thf(fact_4693_integer__of__int__induct,axiom,
% 0.30/0.58 ! [P: code_integer > $o,X: code_integer] :
% 0.30/0.58 ( ! [Y5: int] :
% 0.30/0.58 ( ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ( P @ ( code_integer_of_int @ Y5 ) ) )
% 0.30/0.58 => ( P @ X ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_of_int_induct
% 0.30/0.58 thf(fact_4694_integer__of__int__inject,axiom,
% 0.30/0.58 ! [X: int,Y3: int] :
% 0.30/0.58 ( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ( ( ( code_integer_of_int @ X )
% 0.30/0.58 = ( code_integer_of_int @ Y3 ) )
% 0.30/0.58 = ( X = Y3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_of_int_inject
% 0.30/0.58 thf(fact_4695_int__of__integer,axiom,
% 0.30/0.58 ! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer
% 0.30/0.58 thf(fact_4696_int__of__integer__cases,axiom,
% 0.30/0.58 ! [Y3: int] :
% 0.30/0.58 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ~ ! [X3: code_integer] :
% 0.30/0.58 ( Y3
% 0.30/0.58 != ( code_int_of_integer @ X3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_cases
% 0.30/0.58 thf(fact_4697_int__of__integer__induct,axiom,
% 0.30/0.58 ! [Y3: int,P: int > $o] :
% 0.30/0.58 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ( ! [X3: code_integer] : ( P @ ( code_int_of_integer @ X3 ) )
% 0.30/0.58 => ( P @ Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % int_of_integer_induct
% 0.30/0.58 thf(fact_4698_length__filter__le,axiom,
% 0.30/0.58 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.58
% 0.30/0.58 % length_filter_le
% 0.30/0.58 thf(fact_4699_sorted__same,axiom,
% 0.30/0.58 ! [A: $tType] :
% 0.30/0.58 ( ( linorder @ A )
% 0.30/0.58 => ! [G: ( list @ A ) > A,Xs: list @ A] :
% 0.30/0.58 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 0.30/0.58 @ ( filter @ A
% 0.30/0.58 @ ^ [X2: A] :
% 0.30/0.58 ( X2
% 0.30/0.58 = ( G @ Xs ) )
% 0.30/0.58 @ Xs ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % sorted_same
% 0.30/0.58 thf(fact_4700_integer__of__int__inverse,axiom,
% 0.30/0.58 ! [Y3: int] :
% 0.30/0.58 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 0.30/0.58 => ( ( code_int_of_integer @ ( code_integer_of_int @ Y3 ) )
% 0.30/0.58 = Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % integer_of_int_inverse
% 0.30/0.58 thf(fact_4701_length__filter__less,axiom,
% 0.30/0.58 ! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
% 0.30/0.58 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.58 => ( ~ ( P @ X )
% 0.30/0.58 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % length_filter_less
% 0.30/0.58 thf(fact_4702_surj__list__encode,axiom,
% 0.30/0.58 ( ( image @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
% 0.30/0.58 = ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % surj_list_encode
% 0.30/0.58 thf(fact_4703_surj__prod__encode,axiom,
% 0.30/0.58 ( ( image @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 0.30/0.59 = ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % surj_prod_encode
% 0.30/0.59 thf(fact_4704_sorted__filter,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,Xs: list @ B,P: B > $o] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.59 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_filter
% 0.30/0.59 thf(fact_4705_sorted__map__same,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,G: ( list @ B ) > A,Xs: list @ B] :
% 0.30/0.59 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 0.30/0.59 @ ( map @ B @ A @ F
% 0.30/0.59 @ ( filter @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( F @ X2 )
% 0.30/0.59 = ( G @ Xs ) )
% 0.30/0.59 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_map_same
% 0.30/0.59 thf(fact_4706_sum__list__map__filter_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( monoid_add @ A )
% 0.30/0.59 => ! [F: B > A,P: B > $o,Xs: list @ B] :
% 0.30/0.59 ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
% 0.30/0.59 = ( groups8242544230860333062m_list @ A
% 0.30/0.59 @ ( map @ B @ A
% 0.30/0.59 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F @ X2 ) @ ( zero_zero @ A ) )
% 0.30/0.59 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_list_map_filter'
% 0.30/0.59 thf(fact_4707_sum__list__filter__le__nat,axiom,
% 0.30/0.59 ! [A: $tType,F: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ ( filter @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_list_filter_le_nat
% 0.30/0.59 thf(fact_4708_sum__list__map__filter,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( monoid_add @ A )
% 0.30/0.59 => ! [Xs: list @ B,P: B > $o,F: B > A] :
% 0.30/0.59 ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 0.30/0.59 => ( ~ ( P @ X3 )
% 0.30/0.59 => ( ( F @ X3 )
% 0.30/0.59 = ( zero_zero @ A ) ) ) )
% 0.30/0.59 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
% 0.30/0.59 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_list_map_filter
% 0.30/0.59 thf(fact_4709_Ints__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ( ( ring_1_Ints @ A )
% 0.30/0.59 = ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Ints_def
% 0.30/0.59 thf(fact_4710_int__in__range__abs,axiom,
% 0.30/0.59 ! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_in_range_abs
% 0.30/0.59 thf(fact_4711_length__filter__conv__card,axiom,
% 0.30/0.59 ! [A: $tType,P5: A > $o,Xs: list @ A] :
% 0.30/0.59 ( ( size_size @ ( list @ A ) @ ( filter @ A @ P5 @ Xs ) )
% 0.30/0.59 = ( finite_card @ nat
% 0.30/0.59 @ ( collect @ nat
% 0.30/0.59 @ ^ [I5: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 & ( P5 @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_filter_conv_card
% 0.30/0.59 thf(fact_4712_nth__transpose,axiom,
% 0.30/0.59 ! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
% 0.30/0.59 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
% 0.30/0.59 => ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
% 0.30/0.59 = ( map @ ( list @ A ) @ A
% 0.30/0.59 @ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
% 0.30/0.59 @ ( filter @ ( list @ A )
% 0.30/0.59 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.30/0.59 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nth_transpose
% 0.30/0.59 thf(fact_4713_transpose__max__length,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.30/0.59 ( ( foldr @ ( list @ A ) @ nat
% 0.30/0.59 @ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.59 @ ( transpose @ A @ Xs )
% 0.30/0.59 @ ( zero_zero @ nat ) )
% 0.30/0.59 = ( size_size @ ( list @ ( list @ A ) )
% 0.30/0.59 @ ( filter @ ( list @ A )
% 0.30/0.59 @ ^ [X2: list @ A] :
% 0.30/0.59 ( X2
% 0.30/0.59 != ( nil @ A ) )
% 0.30/0.59 @ Xs ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % transpose_max_length
% 0.30/0.59 thf(fact_4714_transpose__aux__max,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
% 0.30/0.59 ( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 @ ( foldr @ ( list @ B ) @ nat
% 0.30/0.59 @ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
% 0.30/0.59 @ Xss
% 0.30/0.59 @ ( zero_zero @ nat ) ) )
% 0.30/0.59 = ( suc
% 0.30/0.59 @ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.59 @ ( foldr @ ( list @ B ) @ nat
% 0.30/0.59 @ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.59 @ ( filter @ ( list @ B )
% 0.30/0.59 @ ^ [Ys3: list @ B] :
% 0.30/0.59 ( Ys3
% 0.30/0.59 != ( nil @ B ) )
% 0.30/0.59 @ Xss )
% 0.30/0.59 @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % transpose_aux_max
% 0.30/0.59 thf(fact_4715_transpose__transpose,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.30/0.59 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 0.30/0.59 => ( ( transpose @ A @ ( transpose @ A @ Xs ) )
% 0.30/0.59 = ( takeWhile @ ( list @ A )
% 0.30/0.59 @ ^ [X2: list @ A] :
% 0.30/0.59 ( X2
% 0.30/0.59 != ( nil @ A ) )
% 0.30/0.59 @ Xs ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % transpose_transpose
% 0.30/0.59 thf(fact_4716_length__takeWhile__le,axiom,
% 0.30/0.59 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_takeWhile_le
% 0.30/0.59 thf(fact_4717_sorted__takeWhile,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ A,P: A > $o] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.59 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_takeWhile
% 0.30/0.59 thf(fact_4718_takeWhile__nth,axiom,
% 0.30/0.59 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 0.30/0.59 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
% 0.30/0.59 => ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
% 0.30/0.59 = ( nth @ A @ Xs @ J ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % takeWhile_nth
% 0.30/0.59 thf(fact_4719_nth__length__takeWhile,axiom,
% 0.30/0.59 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.30/0.59 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 => ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nth_length_takeWhile
% 0.30/0.59 thf(fact_4720_sum__list_Oeq__foldr,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( monoid_add @ A )
% 0.30/0.59 => ( ( groups8242544230860333062m_list @ A )
% 0.30/0.59 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_list.eq_foldr
% 0.30/0.59 thf(fact_4721_length__takeWhile__less__P__nth,axiom,
% 0.30/0.59 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 0.30/0.59 ( ! [I4: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I4 @ J )
% 0.30/0.59 => ( P @ ( nth @ A @ Xs @ I4 ) ) )
% 0.30/0.59 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 => ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_takeWhile_less_P_nth
% 0.30/0.59 thf(fact_4722_takeWhile__eq__take__P__nth,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 0.30/0.59 ( ! [I4: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I4 @ N )
% 0.30/0.59 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) )
% 0.30/0.59 => ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 => ~ ( P @ ( nth @ A @ Xs @ N ) ) )
% 0.30/0.59 => ( ( takeWhile @ A @ P @ Xs )
% 0.30/0.59 = ( take @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % takeWhile_eq_take_P_nth
% 0.30/0.59 thf(fact_4723_nths__shift__lemma__Suc,axiom,
% 0.30/0.59 ! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
% 0.30/0.59 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.30/0.59 @ ( filter @ ( product_prod @ A @ nat )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
% 0.30/0.59 @ ( zip @ A @ nat @ Xs @ Is ) ) )
% 0.30/0.59 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.30/0.59 @ ( filter @ ( product_prod @ A @ nat )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
% 0.30/0.59 @ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_shift_lemma_Suc
% 0.30/0.59 thf(fact_4724_horner__sum__foldr,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_semiring_0 @ A )
% 0.30/0.59 => ( ( groups4207007520872428315er_sum @ B @ A )
% 0.30/0.59 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 0.30/0.59 ( foldr @ B @ A
% 0.30/0.59 @ ^ [X2: B,B3: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A5 @ B3 ) )
% 0.30/0.59 @ Xs3
% 0.30/0.59 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % horner_sum_foldr
% 0.30/0.59 thf(fact_4725_nths__shift__lemma,axiom,
% 0.30/0.59 ! [A: $tType,A6: set @ nat,Xs: list @ A,I: nat] :
% 0.30/0.59 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.30/0.59 @ ( filter @ ( product_prod @ A @ nat )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A6 )
% 0.30/0.59 @ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
% 0.30/0.59 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.30/0.59 @ ( filter @ ( product_prod @ A @ nat )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A6 )
% 0.30/0.59 @ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_shift_lemma
% 0.30/0.59 thf(fact_4726_filter__equals__takeWhile__sorted__rev,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,Xs: list @ B,T2: A] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 0.30/0.59 => ( ( filter @ B
% 0.30/0.59 @ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
% 0.30/0.59 @ Xs )
% 0.30/0.59 = ( takeWhile @ B
% 0.30/0.59 @ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
% 0.30/0.59 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % filter_equals_takeWhile_sorted_rev
% 0.30/0.59 thf(fact_4727_length__transpose,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.30/0.59 ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 0.30/0.59 = ( foldr @ ( list @ A ) @ nat
% 0.30/0.59 @ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.59 @ Xs
% 0.30/0.59 @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_transpose
% 0.30/0.59 thf(fact_4728_foldr__max__sorted,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ A,Y3: A] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 0.30/0.59 => ( ( ( Xs
% 0.30/0.59 = ( nil @ A ) )
% 0.30/0.59 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
% 0.30/0.59 = Y3 ) )
% 0.30/0.59 & ( ( Xs
% 0.30/0.59 != ( nil @ A ) )
% 0.30/0.59 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
% 0.30/0.59 = ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % foldr_max_sorted
% 0.30/0.59 thf(fact_4729_rec__nat__add__eq__if,axiom,
% 0.30/0.59 ! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
% 0.30/0.59 ( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.30/0.59 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % rec_nat_add_eq_if
% 0.30/0.59 thf(fact_4730_case__nat__add__eq__if,axiom,
% 0.30/0.59 ! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
% 0.30/0.59 ( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.30/0.59 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_nat_add_eq_if
% 0.30/0.59 thf(fact_4731_divmod__integer__eq__cases,axiom,
% 0.30/0.59 ( code_divmod_integer
% 0.30/0.59 = ( ^ [K3: code_integer,L3: code_integer] :
% 0.30/0.59 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.59 @ ( K3
% 0.30/0.59 = ( zero_zero @ code_integer ) )
% 0.30/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 0.30/0.59 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.59 @ ( L3
% 0.30/0.59 = ( zero_zero @ code_integer ) )
% 0.30/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
% 0.30/0.59 @ ( 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
% 0.30/0.59 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.59 @ ( ( sgn_sgn @ code_integer @ K3 )
% 0.30/0.59 = ( sgn_sgn @ code_integer @ L3 ) )
% 0.30/0.59 @ ( code_divmod_abs @ K3 @ L3 )
% 0.30/0.59 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.59 @ ^ [R5: code_integer,S6: code_integer] :
% 0.30/0.59 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.30/0.59 @ ( S6
% 0.30/0.59 = ( zero_zero @ code_integer ) )
% 0.30/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 0.30/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S6 ) ) )
% 0.30/0.59 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_integer_eq_cases
% 0.30/0.59 thf(fact_4732_old_Onat_Osimps_I7_J,axiom,
% 0.30/0.59 ! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
% 0.30/0.59 ( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
% 0.30/0.59 = ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.nat.simps(7)
% 0.30/0.59 thf(fact_4733_old_Onat_Osimps_I6_J,axiom,
% 0.30/0.59 ! [T: $tType,F1: T,F22: nat > T > T] :
% 0.30/0.59 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
% 0.30/0.59 = F1 ) ).
% 0.30/0.59
% 0.30/0.59 % old.nat.simps(6)
% 0.30/0.59 thf(fact_4734_case__nat__numeral,axiom,
% 0.30/0.59 ! [A: $tType,A3: A,F: nat > A,V: num] :
% 0.30/0.59 ( ( case_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
% 0.30/0.59 = ( F @ ( pred_numeral @ V ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_nat_numeral
% 0.30/0.59 thf(fact_4735_rec__nat__numeral,axiom,
% 0.30/0.59 ! [A: $tType,A3: A,F: nat > A > A,V: num] :
% 0.30/0.59 ( ( rec_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
% 0.30/0.59 = ( F @ ( pred_numeral @ V ) @ ( rec_nat @ A @ A3 @ F @ ( pred_numeral @ V ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % rec_nat_numeral
% 0.30/0.59 thf(fact_4736_comp__cong,axiom,
% 0.30/0.59 ! [C: $tType,B: $tType,D: $tType,A: $tType,E4: $tType,F: B > A,G: C > B,X: C,F5: D > A,G4: E4 > D,X5: E4] :
% 0.30/0.59 ( ( ( F @ ( G @ X ) )
% 0.30/0.59 = ( F5 @ ( G4 @ X5 ) ) )
% 0.30/0.59 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.59 = ( comp @ D @ A @ E4 @ F5 @ G4 @ X5 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % comp_cong
% 0.30/0.59 thf(fact_4737_nat_Ocase__distrib,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
% 0.30/0.59 ( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 0.30/0.59 = ( case_nat @ B @ ( H2 @ F1 )
% 0.30/0.59 @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
% 0.30/0.59 @ Nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.case_distrib
% 0.30/0.59 thf(fact_4738_old_Onat_Osimps_I5_J,axiom,
% 0.30/0.59 ! [A: $tType,F1: A,F22: nat > A,X22: nat] :
% 0.30/0.59 ( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
% 0.30/0.59 = ( F22 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.nat.simps(5)
% 0.30/0.59 thf(fact_4739_old_Onat_Osimps_I4_J,axiom,
% 0.30/0.59 ! [A: $tType,F1: A,F22: nat > A] :
% 0.30/0.59 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
% 0.30/0.59 = F1 ) ).
% 0.30/0.59
% 0.30/0.59 % old.nat.simps(4)
% 0.30/0.59 thf(fact_4740_nat_Odisc__eq__case_I2_J,axiom,
% 0.30/0.59 ! [Nat: nat] :
% 0.30/0.59 ( ( Nat
% 0.30/0.59 != ( zero_zero @ nat ) )
% 0.30/0.59 = ( case_nat @ $o @ $false
% 0.30/0.59 @ ^ [Uu: nat] : $true
% 0.30/0.59 @ Nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.disc_eq_case(2)
% 0.30/0.59 thf(fact_4741_nat_Odisc__eq__case_I1_J,axiom,
% 0.30/0.59 ! [Nat: nat] :
% 0.30/0.59 ( ( Nat
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 = ( case_nat @ $o @ $true
% 0.30/0.59 @ ^ [Uu: nat] : $false
% 0.30/0.59 @ Nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.disc_eq_case(1)
% 0.30/0.59 thf(fact_4742_rec__nat__Suc__imp,axiom,
% 0.30/0.59 ! [A: $tType,F: nat > A,F1: A,F22: nat > A > A,N: nat] :
% 0.30/0.59 ( ( F
% 0.30/0.59 = ( rec_nat @ A @ F1 @ F22 ) )
% 0.30/0.59 => ( ( F @ ( suc @ N ) )
% 0.30/0.59 = ( F22 @ N @ ( F @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % rec_nat_Suc_imp
% 0.30/0.59 thf(fact_4743_rec__nat__0__imp,axiom,
% 0.30/0.59 ! [A: $tType,F: nat > A,F1: A,F22: nat > A > A] :
% 0.30/0.59 ( ( F
% 0.30/0.59 = ( rec_nat @ A @ F1 @ F22 ) )
% 0.30/0.59 => ( ( F @ ( zero_zero @ nat ) )
% 0.30/0.59 = F1 ) ) ).
% 0.30/0.59
% 0.30/0.59 % rec_nat_0_imp
% 0.30/0.59 thf(fact_4744_sum__comp__morphism,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.59 ( ( ( comm_monoid_add @ B )
% 0.30/0.59 & ( comm_monoid_add @ A ) )
% 0.30/0.59 => ! [H2: B > A,G: C > B,A6: set @ C] :
% 0.30/0.59 ( ( ( H2 @ ( zero_zero @ B ) )
% 0.30/0.59 = ( zero_zero @ A ) )
% 0.30/0.59 => ( ! [X3: B,Y5: B] :
% 0.30/0.59 ( ( H2 @ ( plus_plus @ B @ X3 @ Y5 ) )
% 0.30/0.59 = ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.59 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A6 )
% 0.30/0.59 = ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_comp_morphism
% 0.30/0.59 thf(fact_4745_uminus__sum__list__map,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( ab_group_add @ A )
% 0.30/0.59 => ! [F: B > A,Xs: list @ B] :
% 0.30/0.59 ( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 0.30/0.59 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_sum_list_map
% 0.30/0.59 thf(fact_4746_less__eq__nat_Osimps_I2_J,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.30/0.59 = ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_nat.simps(2)
% 0.30/0.59 thf(fact_4747_max__Suc1,axiom,
% 0.30/0.59 ! [N: nat,M: nat] :
% 0.30/0.59 ( ( ord_max @ nat @ ( suc @ N ) @ M )
% 0.30/0.59 = ( case_nat @ nat @ ( suc @ N )
% 0.30/0.59 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N @ M4 ) )
% 0.30/0.59 @ M ) ) ).
% 0.30/0.59
% 0.30/0.59 % max_Suc1
% 0.30/0.59 thf(fact_4748_max__Suc2,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_max @ nat @ M @ ( suc @ N ) )
% 0.30/0.59 = ( case_nat @ nat @ ( suc @ N )
% 0.30/0.59 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N ) )
% 0.30/0.59 @ M ) ) ).
% 0.30/0.59
% 0.30/0.59 % max_Suc2
% 0.30/0.59 thf(fact_4749_sum_Oreindex__nontrivial,axiom,
% 0.30/0.59 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [A6: set @ B,H2: B > C,G: C > A] :
% 0.30/0.59 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.59 => ( ! [X3: B,Y5: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A6 )
% 0.30/0.59 => ( ( member @ B @ Y5 @ A6 )
% 0.30/0.59 => ( ( X3 != Y5 )
% 0.30/0.59 => ( ( ( H2 @ X3 )
% 0.30/0.59 = ( H2 @ Y5 ) )
% 0.30/0.59 => ( ( G @ ( H2 @ X3 ) )
% 0.30/0.59 = ( zero_zero @ A ) ) ) ) ) )
% 0.30/0.59 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.reindex_nontrivial
% 0.30/0.59 thf(fact_4750_prod_Oreindex__nontrivial,axiom,
% 0.30/0.59 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [A6: set @ B,H2: B > C,G: C > A] :
% 0.30/0.59 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.59 => ( ! [X3: B,Y5: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A6 )
% 0.30/0.59 => ( ( member @ B @ Y5 @ A6 )
% 0.30/0.59 => ( ( X3 != Y5 )
% 0.30/0.59 => ( ( ( H2 @ X3 )
% 0.30/0.59 = ( H2 @ Y5 ) )
% 0.30/0.59 => ( ( G @ ( H2 @ X3 ) )
% 0.30/0.59 = ( one_one @ A ) ) ) ) ) )
% 0.30/0.59 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.reindex_nontrivial
% 0.30/0.59 thf(fact_4751_sum__image__le,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.59 ( ( ordere6911136660526730532id_add @ B )
% 0.30/0.59 => ! [I6: set @ C,G: A > B,F: C > A] :
% 0.30/0.59 ( ( finite_finite2 @ C @ I6 )
% 0.30/0.59 => ( ! [I4: C] :
% 0.30/0.59 ( ( member @ C @ I4 @ I6 )
% 0.30/0.59 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I4 ) ) ) )
% 0.30/0.59 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum_image_le
% 0.30/0.59 thf(fact_4752_diff__Suc,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( minus_minus @ nat @ M @ ( suc @ N ) )
% 0.30/0.59 = ( case_nat @ nat @ ( zero_zero @ nat )
% 0.30/0.59 @ ^ [K3: nat] : K3
% 0.30/0.59 @ ( minus_minus @ nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % diff_Suc
% 0.30/0.59 thf(fact_4753_min__Suc1,axiom,
% 0.30/0.59 ! [N: nat,M: nat] :
% 0.30/0.59 ( ( ord_min @ nat @ ( suc @ N ) @ M )
% 0.30/0.59 = ( case_nat @ nat @ ( zero_zero @ nat )
% 0.30/0.59 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N @ M4 ) )
% 0.30/0.59 @ M ) ) ).
% 0.30/0.59
% 0.30/0.59 % min_Suc1
% 0.30/0.59 thf(fact_4754_min__Suc2,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_min @ nat @ M @ ( suc @ N ) )
% 0.30/0.59 = ( case_nat @ nat @ ( zero_zero @ nat )
% 0.30/0.59 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N ) )
% 0.30/0.59 @ M ) ) ).
% 0.30/0.59
% 0.30/0.59 % min_Suc2
% 0.30/0.59 thf(fact_4755_Nitpick_Ocase__nat__unfold,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( case_nat @ A )
% 0.30/0.59 = ( ^ [X2: A,F2: nat > A,N3: nat] :
% 0.30/0.59 ( if @ A
% 0.30/0.59 @ ( N3
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 @ X2
% 0.30/0.59 @ ( F2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nitpick.case_nat_unfold
% 0.30/0.59 thf(fact_4756_nths__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( nths @ A )
% 0.30/0.59 = ( ^ [Xs3: list @ A,A7: set @ nat] :
% 0.30/0.59 ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.30/0.59 @ ( filter @ ( product_prod @ A @ nat )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
% 0.30/0.59 @ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_def
% 0.30/0.59 thf(fact_4757_length__product__lists,axiom,
% 0.30/0.59 ! [B: $tType,Xss: list @ ( list @ B )] :
% 0.30/0.59 ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
% 0.30/0.59 = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_product_lists
% 0.30/0.59 thf(fact_4758_nat_Osplit__sels_I2_J,axiom,
% 0.30/0.59 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
% 0.30/0.59 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 0.30/0.59 = ( ~ ( ( ( Nat
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 & ~ ( P @ F1 ) )
% 0.30/0.59 | ( ( Nat
% 0.30/0.59 = ( suc @ ( pred @ Nat ) ) )
% 0.30/0.59 & ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.split_sels(2)
% 0.30/0.59 thf(fact_4759_nths__singleton,axiom,
% 0.30/0.59 ! [A: $tType,A6: set @ nat,X: A] :
% 0.30/0.59 ( ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 0.30/0.59 => ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
% 0.30/0.59 = ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.30/0.59 & ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 0.30/0.59 => ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
% 0.30/0.59 = ( nil @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_singleton
% 0.30/0.59 thf(fact_4760_card_Ocomp__fun__commute__on,axiom,
% 0.30/0.59 ( ( comp @ nat @ nat @ nat @ suc @ suc )
% 0.30/0.59 = ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% 0.30/0.59
% 0.30/0.59 % card.comp_fun_commute_on
% 0.30/0.59 thf(fact_4761_nths__all,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 0.30/0.59 ( ! [I4: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 => ( member @ nat @ I4 @ I6 ) )
% 0.30/0.59 => ( ( nths @ A @ Xs @ I6 )
% 0.30/0.59 = Xs ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_all
% 0.30/0.59 thf(fact_4762_sorted__nths,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ A,I6: set @ nat] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.59 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_nths
% 0.30/0.59 thf(fact_4763_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_Suc_atMost_Suc_shift
% 0.30/0.59 thf(fact_4764_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_Suc_lessThan_Suc_shift
% 0.30/0.59 thf(fact_4765_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_Suc_atMost_Suc_shift
% 0.30/0.59 thf(fact_4766_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_Suc_lessThan_Suc_shift
% 0.30/0.59 thf(fact_4767_lift__def,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( heap_Time_lift @ A @ B )
% 0.30/0.59 = ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % lift_def
% 0.30/0.59 thf(fact_4768_drop__eq__nths,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( drop @ A )
% 0.30/0.59 = ( ^ [N3: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % drop_eq_nths
% 0.30/0.59 thf(fact_4769_euclidean__size__int__def,axiom,
% 0.30/0.59 ( ( euclid6346220572633701492n_size @ int )
% 0.30/0.59 = ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % euclidean_size_int_def
% 0.30/0.59 thf(fact_4770_pred__def,axiom,
% 0.30/0.59 ( pred
% 0.30/0.59 = ( case_nat @ nat @ ( zero_zero @ nat )
% 0.30/0.59 @ ^ [X24: nat] : X24 ) ) ).
% 0.30/0.59
% 0.30/0.59 % pred_def
% 0.30/0.59 thf(fact_4771_length__nths,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 0.30/0.59 ( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
% 0.30/0.59 = ( finite_card @ nat
% 0.30/0.59 @ ( collect @ nat
% 0.30/0.59 @ ^ [I5: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 & ( member @ nat @ I5 @ I6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_nths
% 0.30/0.59 thf(fact_4772_sum_OatLeast0__atMost__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast0_atMost_Suc_shift
% 0.30/0.59 thf(fact_4773_sum_OatLeast0__lessThan__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast0_lessThan_Suc_shift
% 0.30/0.59 thf(fact_4774_prod_OatLeast0__atMost__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast0_atMost_Suc_shift
% 0.30/0.59 thf(fact_4775_prod_OatLeast0__lessThan__Suc__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast0_lessThan_Suc_shift
% 0.30/0.59 thf(fact_4776_sum_OatLeastLessThan__shift__0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeastLessThan_shift_0
% 0.30/0.59 thf(fact_4777_prod_OatLeastLessThan__shift__0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeastLessThan_shift_0
% 0.30/0.59 thf(fact_4778_sum_OatLeast__atMost__pred__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.59 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.59 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.59 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_atMost_pred_shift
% 0.30/0.59 thf(fact_4779_sum_OatLeast__lessThan__pred__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.59 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.59 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_lessThan_pred_shift
% 0.30/0.59 thf(fact_4780_prod_OatLeast__atMost__pred__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.59 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.59 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.59 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_atMost_pred_shift
% 0.30/0.59 thf(fact_4781_prod_OatLeast__lessThan__pred__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: nat > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.30/0.59 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.59 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_lessThan_pred_shift
% 0.30/0.59 thf(fact_4782_sum_OatLeast__int__atMost__int__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: int > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_int_atMost_int_shift
% 0.30/0.59 thf(fact_4783_filter__eq__nths,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( filter @ A )
% 0.30/0.59 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.30/0.59 ( nths @ A @ Xs3
% 0.30/0.59 @ ( collect @ nat
% 0.30/0.59 @ ^ [I5: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.59 & ( P4 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % filter_eq_nths
% 0.30/0.59 thf(fact_4784_prod_OatLeast__int__atMost__int__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: int > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_int_atMost_int_shift
% 0.30/0.59 thf(fact_4785_sum_OatLeast__int__lessThan__int__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [G: int > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeast_int_lessThan_int_shift
% 0.30/0.59 thf(fact_4786_sum_OatLeastAtMost__shift__0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.59 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.atLeastAtMost_shift_0
% 0.30/0.59 thf(fact_4787_prod_OatLeastAtMost__shift__0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [M: nat,N: nat,G: nat > A] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.59 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeastAtMost_shift_0
% 0.30/0.59 thf(fact_4788_prod_OatLeast__int__lessThan__int__shift,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: int > A,M: nat,N: nat] :
% 0.30/0.59 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.atLeast_int_lessThan_int_shift
% 0.30/0.59 thf(fact_4789_Gcd__int__def,axiom,
% 0.30/0.59 ( ( gcd_Gcd @ int )
% 0.30/0.59 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Gcd_int_def
% 0.30/0.59 thf(fact_4790_nths__Cons,axiom,
% 0.30/0.59 ! [A: $tType,X: A,L: list @ A,A6: set @ nat] :
% 0.30/0.59 ( ( nths @ A @ ( cons @ A @ X @ L ) @ A6 )
% 0.30/0.59 = ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
% 0.30/0.59 @ ( nths @ A @ L
% 0.30/0.59 @ ( collect @ nat
% 0.30/0.59 @ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nths_Cons
% 0.30/0.59 thf(fact_4791_nat_Osplit__sels_I1_J,axiom,
% 0.30/0.59 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
% 0.30/0.59 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 0.30/0.59 = ( ( ( Nat
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 => ( P @ F1 ) )
% 0.30/0.59 & ( ( Nat
% 0.30/0.59 = ( suc @ ( pred @ Nat ) ) )
% 0.30/0.59 => ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.split_sels(1)
% 0.30/0.59 thf(fact_4792_Code__Numeral_Onegative__def,axiom,
% 0.30/0.59 ( code_negative
% 0.30/0.59 = ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Code_Numeral.negative_def
% 0.30/0.59 thf(fact_4793_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
% 0.30/0.59 ! [A: $tType,F: nat > ( set @ A ),S3: set @ A] :
% 0.30/0.59 ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I4 ) @ S3 )
% 0.30/0.59 => ( ( finite_finite2 @ A @ S3 )
% 0.30/0.59 => ( ? [N9: nat] :
% 0.30/0.59 ( ! [N4: nat] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ N4 @ N9 )
% 0.30/0.59 => ! [M6: nat] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ M6 @ N9 )
% 0.30/0.59 => ( ( ord_less @ nat @ M6 @ N4 )
% 0.30/0.59 => ( ord_less @ ( set @ A ) @ ( F @ M6 ) @ ( F @ N4 ) ) ) ) )
% 0.30/0.59 & ! [N4: nat] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ N9 @ N4 )
% 0.30/0.59 => ( ( F @ N9 )
% 0.30/0.59 = ( F @ N4 ) ) ) )
% 0.30/0.59 => ( ( F @ ( finite_card @ A @ S3 ) )
% 0.30/0.59 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_mono_strict_prefix_implies_finite_fixpoint
% 0.30/0.59 thf(fact_4794_range__abs__Nats,axiom,
% 0.30/0.59 ( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
% 0.30/0.59 = ( semiring_1_Nats @ int ) ) ).
% 0.30/0.59
% 0.30/0.59 % range_abs_Nats
% 0.30/0.59 thf(fact_4795_Sup__atLeastAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_atLeastAtMost
% 0.30/0.59 thf(fact_4796_cSup__atLeastAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_atLeastAtMost
% 0.30/0.59 thf(fact_4797_cSup__atLeastLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_atLeastLessThan
% 0.30/0.59 thf(fact_4798_Sup__atLeastLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_atLeastLessThan
% 0.30/0.59 thf(fact_4799_cSup__greaterThanAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_greaterThanAtMost
% 0.30/0.59 thf(fact_4800_Sup__greaterThanAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_greaterThanAtMost
% 0.30/0.59 thf(fact_4801_Sup__greaterThanLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_greaterThanLessThan
% 0.30/0.59 thf(fact_4802_cSup__greaterThanLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_greaterThanLessThan
% 0.30/0.59 thf(fact_4803_Nats__0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_0
% 0.30/0.59 thf(fact_4804_cSup__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 & ( no_bot @ A ) )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ A3 @ Y5 ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ X8 )
% 0.30/0.59 = A3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_eq
% 0.30/0.59 thf(fact_4805_cSup__eq__maximum,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Z3: A,X8: set @ A] :
% 0.30/0.59 ( ( member @ A @ Z3 @ X8 )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ X8 )
% 0.30/0.59 = Z3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_eq_maximum
% 0.30/0.59 thf(fact_4806_Nats__1,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_1
% 0.30/0.59 thf(fact_4807_Nats__add,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [A3: A,B2: A] :
% 0.30/0.59 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_add
% 0.30/0.59 thf(fact_4808_Nats__mult,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [A3: A,B2: A] :
% 0.30/0.59 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_mult
% 0.30/0.59 thf(fact_4809_Nats__numeral,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_numeral
% 0.30/0.59 thf(fact_4810_of__nat__in__Nats,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_nat_in_Nats
% 0.30/0.59 thf(fact_4811_Nats__induct,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [X: A,P: A > $o] :
% 0.30/0.59 ( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ A @ N4 ) )
% 0.30/0.59 => ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_induct
% 0.30/0.59 thf(fact_4812_Nats__cases,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ! [X: A] :
% 0.30/0.59 ( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ~ ! [N4: nat] :
% 0.30/0.59 ( X
% 0.30/0.59 != ( semiring_1_of_nat @ A @ N4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_cases
% 0.30/0.59 thf(fact_4813_fact__in__Nats,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_char_0 @ A )
% 0.30/0.59 => ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % fact_in_Nats
% 0.30/0.59 thf(fact_4814_cSup__eq__non__empty,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ A3 @ Y5 ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ X8 )
% 0.30/0.59 = A3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_eq_non_empty
% 0.30/0.59 thf(fact_4815_cSup__least,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,Z3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_least
% 0.30/0.59 thf(fact_4816_le__cSup__finite,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,X: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( member @ A @ X @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_cSup_finite
% 0.30/0.59 thf(fact_4817_less__cSupD,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,Z3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 0.30/0.59 => ? [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 & ( ord_less @ A @ Z3 @ X3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_cSupD
% 0.30/0.59 thf(fact_4818_less__cSupE,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [Y3: A,X8: set @ A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 0.30/0.59 => ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ~ ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_cSupE
% 0.30/0.59 thf(fact_4819_finite__imp__Sup__less,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,X: A,A3: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( member @ A @ X @ X8 )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less @ A @ X3 @ A3 ) )
% 0.30/0.59 => ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_imp_Sup_less
% 0.30/0.59 thf(fact_4820_inf__Sup,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [A3: A,B7: set @ A] :
% 0.30/0.59 ( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B7 ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( inf_inf @ A @ A3 ) @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_Sup
% 0.30/0.59 thf(fact_4821_Sup__inf__eq__bot__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A3: A] :
% 0.30/0.59 ( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
% 0.30/0.59 = ( bot_bot @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ B7 )
% 0.30/0.59 => ( ( inf_inf @ A @ X2 @ A3 )
% 0.30/0.59 = ( bot_bot @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_inf_eq_bot_iff
% 0.30/0.59 thf(fact_4822_finite__subset__Union,axiom,
% 0.30/0.59 ! [A: $tType,A6: set @ A,B12: set @ ( set @ A )] :
% 0.30/0.59 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.59 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
% 0.30/0.59 => ~ ! [F6: set @ ( set @ A )] :
% 0.30/0.59 ( ( finite_finite2 @ ( set @ A ) @ F6 )
% 0.30/0.59 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F6 @ B12 )
% 0.30/0.59 => ~ ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ F6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_subset_Union
% 0.30/0.59 thf(fact_4823_card__Union__le__sum__card,axiom,
% 0.30/0.59 ! [A: $tType,U4: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_Union_le_sum_card
% 0.30/0.59 thf(fact_4824_SUP__inf,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [F: B > A,B7: set @ B,A3: A] :
% 0.30/0.59 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [B3: B] : ( inf_inf @ A @ ( F @ B3 ) @ A3 )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_inf
% 0.30/0.59 thf(fact_4825_Sup__inf,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A3: A] :
% 0.30/0.59 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ A @ A
% 0.30/0.59 @ ^ [B3: A] : ( inf_inf @ A @ B3 @ A3 )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_inf
% 0.30/0.59 thf(fact_4826_inf__SUP,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [A3: A,F: B > A,B7: set @ B] :
% 0.30/0.59 ( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [B3: B] : ( inf_inf @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_SUP
% 0.30/0.59 thf(fact_4827_SUP__inf__distrib2,axiom,
% 0.30/0.59 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
% 0.30/0.59 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [A5: B] :
% 0.30/0.59 ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ C @ A
% 0.30/0.59 @ ^ [B3: C] : ( inf_inf @ A @ ( F @ A5 ) @ ( G @ B3 ) )
% 0.30/0.59 @ B7 ) )
% 0.30/0.59 @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_inf_distrib2
% 0.30/0.59 thf(fact_4828_cSUP__least,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [A6: set @ B,F: B > A,M7: A] :
% 0.30/0.59 ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ M7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSUP_least
% 0.30/0.59 thf(fact_4829_finite__Sup__less__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ X8 )
% 0.30/0.59 => ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_Sup_less_iff
% 0.30/0.59 thf(fact_4830_cSup__abs__le,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( linordered_idom @ A ) )
% 0.30/0.59 => ! [S3: set @ A,A3: A] :
% 0.30/0.59 ( ( S3
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ S3 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_abs_le
% 0.30/0.59 thf(fact_4831_Nats__diff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linordered_idom @ A )
% 0.30/0.59 => ! [A3: A,B2: A] :
% 0.30/0.59 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 0.30/0.59 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.30/0.59 => ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_diff
% 0.30/0.59 thf(fact_4832_sum_OUnion__comp,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_add @ A )
% 0.30/0.59 => ! [B7: set @ ( set @ B ),G: B > A] :
% 0.30/0.59 ( ! [X3: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ X3 @ B7 )
% 0.30/0.59 => ( finite_finite2 @ B @ X3 ) )
% 0.30/0.59 => ( ! [A14: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ A14 @ B7 )
% 0.30/0.59 => ! [A25: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ A25 @ B7 )
% 0.30/0.59 => ( ( A14 != A25 )
% 0.30/0.59 => ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A14 )
% 0.30/0.59 => ( ( member @ B @ X3 @ A25 )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( zero_zero @ A ) ) ) ) ) ) )
% 0.30/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
% 0.30/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sum.Union_comp
% 0.30/0.59 thf(fact_4833_prod_OUnion__comp,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [B7: set @ ( set @ B ),G: B > A] :
% 0.30/0.59 ( ! [X3: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ X3 @ B7 )
% 0.30/0.59 => ( finite_finite2 @ B @ X3 ) )
% 0.30/0.59 => ( ! [A14: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ A14 @ B7 )
% 0.30/0.59 => ! [A25: set @ B] :
% 0.30/0.59 ( ( member @ ( set @ B ) @ A25 @ B7 )
% 0.30/0.59 => ( ( A14 != A25 )
% 0.30/0.59 => ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A14 )
% 0.30/0.59 => ( ( member @ B @ X3 @ A25 )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( one_one @ A ) ) ) ) ) ) )
% 0.30/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
% 0.30/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.Union_comp
% 0.30/0.59 thf(fact_4834_card__Union__le__sum__card__weak,axiom,
% 0.30/0.59 ! [A: $tType,U4: set @ ( set @ A )] :
% 0.30/0.59 ( ! [X3: set @ A] :
% 0.30/0.59 ( ( member @ ( set @ A ) @ X3 @ U4 )
% 0.30/0.59 => ( finite_finite2 @ A @ X3 ) )
% 0.30/0.59 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_Union_le_sum_card_weak
% 0.30/0.59 thf(fact_4835_Nats__subset__Ints,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_subset_Ints
% 0.30/0.59 thf(fact_4836_UN__UN__finite__eq,axiom,
% 0.30/0.59 ! [A: $tType,A6: nat > ( set @ A )] :
% 0.30/0.59 ( ( complete_Sup_Sup @ ( set @ A )
% 0.30/0.59 @ ( image @ nat @ ( set @ A )
% 0.30/0.59 @ ^ [N3: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 0.30/0.59 @ ( top_top @ ( set @ nat ) ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % UN_UN_finite_eq
% 0.30/0.59 thf(fact_4837_cSup__asclose,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( linordered_idom @ A ) )
% 0.30/0.59 => ! [S3: set @ A,L: A,E: A] :
% 0.30/0.59 ( ( S3
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ S3 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cSup_asclose
% 0.30/0.59 thf(fact_4838_card__UN__le,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,I6: set @ A,A6: A > ( set @ B )] :
% 0.30/0.59 ( ( finite_finite2 @ A @ I6 )
% 0.30/0.59 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A6 @ I6 ) ) )
% 0.30/0.59 @ ( groups7311177749621191930dd_sum @ A @ nat
% 0.30/0.59 @ ^ [I5: A] : ( finite_card @ B @ ( A6 @ I5 ) )
% 0.30/0.59 @ I6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_UN_le
% 0.30/0.59 thf(fact_4839_UN__finite__subset,axiom,
% 0.30/0.59 ! [A: $tType,A6: nat > ( set @ A ),C6: set @ A] :
% 0.30/0.59 ( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ C6 )
% 0.30/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % UN_finite_subset
% 0.30/0.59 thf(fact_4840_UN__finite2__eq,axiom,
% 0.30/0.59 ! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
% 0.30/0.59 ( ! [N4: nat] :
% 0.30/0.59 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % UN_finite2_eq
% 0.30/0.59 thf(fact_4841_Nats__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ( ( semiring_1_Nats @ A )
% 0.30/0.59 = ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_def
% 0.30/0.59 thf(fact_4842_card__partition,axiom,
% 0.30/0.59 ! [A: $tType,C6: set @ ( set @ A ),K: nat] :
% 0.30/0.59 ( ( finite_finite2 @ ( set @ A ) @ C6 )
% 0.30/0.59 => ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
% 0.30/0.59 => ( ! [C3: set @ A] :
% 0.30/0.59 ( ( member @ ( set @ A ) @ C3 @ C6 )
% 0.30/0.59 => ( ( finite_card @ A @ C3 )
% 0.30/0.59 = K ) )
% 0.30/0.59 => ( ! [C1: set @ A,C22: set @ A] :
% 0.30/0.59 ( ( member @ ( set @ A ) @ C1 @ C6 )
% 0.30/0.59 => ( ( member @ ( set @ A ) @ C22 @ C6 )
% 0.30/0.59 => ( ( C1 != C22 )
% 0.30/0.59 => ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
% 0.30/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.30/0.59 => ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C6 ) )
% 0.30/0.59 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_partition
% 0.30/0.59 thf(fact_4843_Nats__altdef2,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linordered_idom @ A )
% 0.30/0.59 => ( ( semiring_1_Nats @ A )
% 0.30/0.59 = ( collect @ A
% 0.30/0.59 @ ^ [N3: A] :
% 0.30/0.59 ( ( member @ A @ N3 @ ( ring_1_Ints @ A ) )
% 0.30/0.59 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_altdef2
% 0.30/0.59 thf(fact_4844_UN__finite2__subset,axiom,
% 0.30/0.59 ! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
% 0.30/0.59 ( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
% 0.30/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % UN_finite2_subset
% 0.30/0.59 thf(fact_4845_UN__le__eq__Un0,axiom,
% 0.30/0.59 ! [A: $tType,M7: nat > ( set @ A ),N: nat] :
% 0.30/0.59 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N ) ) )
% 0.30/0.59 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % UN_le_eq_Un0
% 0.30/0.59 thf(fact_4846_SUP__eq__top__iff,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B] :
% 0.30/0.59 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = ( top_top @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 0.30/0.59 => ? [Y6: B] :
% 0.30/0.59 ( ( member @ B @ Y6 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ X2 @ ( F @ Y6 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_eq_top_iff
% 0.30/0.59 thf(fact_4847_Sup__eq__top__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [A6: set @ A] :
% 0.30/0.59 ( ( ( complete_Sup_Sup @ A @ A6 )
% 0.30/0.59 = ( top_top @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 0.30/0.59 => ? [Y6: A] :
% 0.30/0.59 ( ( member @ A @ Y6 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_eq_top_iff
% 0.30/0.59 thf(fact_4848_mono__SUP,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( comple6319245703460814977attice @ B ) )
% 0.30/0.59 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 0.30/0.59 ( ( order_mono @ A @ B @ F )
% 0.30/0.59 => ( ord_less_eq @ B
% 0.30/0.59 @ ( complete_Sup_Sup @ B
% 0.30/0.59 @ ( image @ C @ B
% 0.30/0.59 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 0.30/0.59 @ I6 ) )
% 0.30/0.59 @ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_SUP
% 0.30/0.59 thf(fact_4849_Sup__nat__empty,axiom,
% 0.30/0.59 ( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.59 = ( zero_zero @ nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_nat_empty
% 0.30/0.59 thf(fact_4850_Sup__upper2,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [U: A,A6: set @ A,V: A] :
% 0.30/0.59 ( ( member @ A @ U @ A6 )
% 0.30/0.59 => ( ( ord_less_eq @ A @ V @ U )
% 0.30/0.59 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_upper2
% 0.30/0.59 thf(fact_4851_Sup__le__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,B2: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ B2 )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_le_iff
% 0.30/0.59 thf(fact_4852_Sup__upper,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,A6: set @ A] :
% 0.30/0.59 ( ( member @ A @ X @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_upper
% 0.30/0.59 thf(fact_4853_Sup__least,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,Z3: A] :
% 0.30/0.59 ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_least
% 0.30/0.59 thf(fact_4854_Sup__mono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.59 ( ! [A8: A] :
% 0.30/0.59 ( ( member @ A @ A8 @ A6 )
% 0.30/0.59 => ? [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ B7 )
% 0.30/0.59 & ( ord_less_eq @ A @ A8 @ X4 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_mono
% 0.30/0.59 thf(fact_4855_Sup__eqI,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,X: A] :
% 0.30/0.59 ( ! [Y5: A] :
% 0.30/0.59 ( ( member @ A @ Y5 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [Z5: A] :
% 0.30/0.59 ( ( member @ A @ Z5 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ Z5 @ Y5 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ Y5 ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ A6 )
% 0.30/0.59 = X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_eqI
% 0.30/0.59 thf(fact_4856_less__Sup__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [A3: A,S3: set @ A] :
% 0.30/0.59 ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) )
% 0.30/0.59 = ( ? [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ S3 )
% 0.30/0.59 & ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_Sup_iff
% 0.30/0.59 thf(fact_4857_le__Sup__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [X: A,A6: set @ A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) )
% 0.30/0.59 = ( ! [Y6: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y6 @ X )
% 0.30/0.59 => ? [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_Sup_iff
% 0.30/0.59 thf(fact_4858_SUP__eq,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ? [X4: C] :
% 0.30/0.59 ( ( member @ C @ X4 @ B7 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
% 0.30/0.59 => ( ! [J2: C] :
% 0.30/0.59 ( ( member @ C @ J2 @ B7 )
% 0.30/0.59 => ? [X4: B] :
% 0.30/0.59 ( ( member @ B @ X4 @ A6 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_eq
% 0.30/0.59 thf(fact_4859_less__eq__Sup,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,U: A] :
% 0.30/0.59 ( ! [V4: A] :
% 0.30/0.59 ( ( member @ A @ V4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ V4 ) )
% 0.30/0.59 => ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_Sup
% 0.30/0.59 thf(fact_4860_Sup__subset__mono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_subset_mono
% 0.30/0.59 thf(fact_4861_SUP__upper2,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I: B,A6: set @ B,U: A,F: B > A] :
% 0.30/0.59 ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ( ord_less_eq @ A @ U @ ( F @ I ) )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_upper2
% 0.30/0.59 thf(fact_4862_SUP__le__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,U: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
% 0.30/0.59 = ( ! [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_le_iff
% 0.30/0.59 thf(fact_4863_SUP__upper,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I: B,A6: set @ B,F: B > A] :
% 0.30/0.59 ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_upper
% 0.30/0.59 thf(fact_4864_SUP__mono_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [F: B > A,G: B > A,A6: set @ B] :
% 0.30/0.59 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_mono'
% 0.30/0.59 thf(fact_4865_SUP__least,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,F: B > A,U: A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ I4 ) @ U ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_least
% 0.30/0.59 thf(fact_4866_SUP__mono,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
% 0.30/0.59 ( ! [N4: B] :
% 0.30/0.59 ( ( member @ B @ N4 @ A6 )
% 0.30/0.59 => ? [X4: C] :
% 0.30/0.59 ( ( member @ C @ X4 @ B7 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_mono
% 0.30/0.59 thf(fact_4867_SUP__eqI,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,F: B > A,X: A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ I4 ) @ X ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [I3: B] :
% 0.30/0.59 ( ( member @ B @ I3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ I3 ) @ Y5 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ Y5 ) )
% 0.30/0.59 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_eqI
% 0.30/0.59 thf(fact_4868_SUP__lessD,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 0.30/0.59 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
% 0.30/0.59 => ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_lessD
% 0.30/0.59 thf(fact_4869_less__SUP__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [A3: A,F: B > A,A6: set @ B] :
% 0.30/0.59 ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.59 = ( ? [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_SUP_iff
% 0.30/0.59 thf(fact_4870_le__SUP__iff,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [X: A,F: B > A,A6: set @ B] :
% 0.30/0.59 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.59 = ( ! [Y6: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y6 @ X )
% 0.30/0.59 => ? [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ Y6 @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_SUP_iff
% 0.30/0.59 thf(fact_4871_SUP__eq__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I6: set @ B,C2: A,F: B > A] :
% 0.30/0.59 ( ( I6
% 0.30/0.59 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 => ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ I6 )
% 0.30/0.59 => ( ord_less_eq @ A @ C2 @ ( F @ I4 ) ) )
% 0.30/0.59 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I6 ) )
% 0.30/0.59 = C2 )
% 0.30/0.59 = ( ! [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 => ( ( F @ X2 )
% 0.30/0.59 = C2 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_eq_iff
% 0.30/0.59 thf(fact_4872_Sup__inter__less__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Sup_inter_less_eq
% 0.30/0.59 thf(fact_4873_SUP__subset__mono,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,B7: set @ B,F: B > A,G: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_subset_mono
% 0.30/0.59 thf(fact_4874_mono__Sup,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( comple6319245703460814977attice @ B ) )
% 0.30/0.59 => ! [F: A > B,A6: set @ A] :
% 0.30/0.59 ( ( order_mono @ A @ B @ F )
% 0.30/0.59 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_Sup
% 0.30/0.59 thf(fact_4875_card__UNION,axiom,
% 0.30/0.59 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.59 ( ( finite_finite2 @ ( set @ A ) @ A6 )
% 0.30/0.59 => ( ! [X3: set @ A] :
% 0.30/0.59 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 0.30/0.59 => ( finite_finite2 @ A @ X3 ) )
% 0.30/0.59 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
% 0.30/0.59 = ( nat2
% 0.30/0.59 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
% 0.30/0.59 @ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
% 0.30/0.59 @ ( collect @ ( set @ ( set @ A ) )
% 0.30/0.59 @ ^ [I8: set @ ( set @ A )] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A6 )
% 0.30/0.59 & ( I8
% 0.30/0.59 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_UNION
% 0.30/0.59 thf(fact_4876_surj__int__encode,axiom,
% 0.30/0.59 ( ( image @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
% 0.30/0.59 = ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % surj_int_encode
% 0.30/0.59 thf(fact_4877_horner__sum__eq__sum__funpow,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_semiring_0 @ A )
% 0.30/0.59 => ( ( groups4207007520872428315er_sum @ B @ A )
% 0.30/0.59 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 0.30/0.59 ( groups7311177749621191930dd_sum @ nat @ A
% 0.30/0.59 @ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A5 ) @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) )
% 0.30/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % horner_sum_eq_sum_funpow
% 0.30/0.59 thf(fact_4878_Suc__funpow,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( compow @ ( nat > nat ) @ N @ suc )
% 0.30/0.59 = ( plus_plus @ nat @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % Suc_funpow
% 0.30/0.59 thf(fact_4879_funpow__0,axiom,
% 0.30/0.59 ! [A: $tType,F: A > A,X: A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X )
% 0.30/0.59 = X ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_0
% 0.30/0.59 thf(fact_4880_Inf__eq__bot__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [A6: set @ A] :
% 0.30/0.59 ( ( ( complete_Inf_Inf @ A @ A6 )
% 0.30/0.59 = ( bot_bot @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
% 0.30/0.59 => ? [Y6: A] :
% 0.30/0.59 ( ( member @ A @ Y6 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_eq_bot_iff
% 0.30/0.59 thf(fact_4881_Inf__atLeastAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_atLeastAtMost
% 0.30/0.59 thf(fact_4882_cInf__atLeastAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_atLeastAtMost
% 0.30/0.59 thf(fact_4883_cInf__atLeastLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_atLeastLessThan
% 0.30/0.59 thf(fact_4884_Inf__atLeastLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_atLeastLessThan
% 0.30/0.59 thf(fact_4885_surj__fn,axiom,
% 0.30/0.59 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.59 ( ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.59 = ( top_top @ ( set @ A ) ) )
% 0.30/0.59 => ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.59 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % surj_fn
% 0.30/0.59 thf(fact_4886_cInf__greaterThanAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_greaterThanAtMost
% 0.30/0.59 thf(fact_4887_Inf__greaterThanAtMost,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_greaterThanAtMost
% 0.30/0.59 thf(fact_4888_Inf__greaterThanLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [X: A,Y3: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 0.30/0.59 = X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_greaterThanLessThan
% 0.30/0.59 thf(fact_4889_cInf__greaterThanLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( dense_linorder @ A ) )
% 0.30/0.59 => ! [Y3: A,X: A] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ X )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
% 0.30/0.59 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_greaterThanLessThan
% 0.30/0.59 thf(fact_4890_INF__eq__bot__iff,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B] :
% 0.30/0.59 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = ( bot_bot @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
% 0.30/0.59 => ? [Y6: B] :
% 0.30/0.59 ( ( member @ B @ Y6 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ ( F @ Y6 ) @ X2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_eq_bot_iff
% 0.30/0.59 thf(fact_4891_Compl__UN,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
% 0.30/0.59 ( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
% 0.30/0.59 = ( complete_Inf_Inf @ ( set @ A )
% 0.30/0.59 @ ( image @ B @ ( set @ A )
% 0.30/0.59 @ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
% 0.30/0.59 @ A6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Compl_UN
% 0.30/0.59 thf(fact_4892_Compl__INT,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
% 0.30/0.59 ( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ ( set @ A )
% 0.30/0.59 @ ( image @ B @ ( set @ A )
% 0.30/0.59 @ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
% 0.30/0.59 @ A6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Compl_INT
% 0.30/0.59 thf(fact_4893_Inf__atMostLessThan,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A] :
% 0.30/0.59 ( ( ord_less @ A @ ( top_top @ A ) @ X )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
% 0.30/0.59 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_atMostLessThan
% 0.30/0.59 thf(fact_4894_comp__funpow,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,N: nat,F: A > A] :
% 0.30/0.59 ( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
% 0.30/0.59 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % comp_funpow
% 0.30/0.59 thf(fact_4895_Inf__greatest,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,Z3: A] :
% 0.30/0.59 ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_greatest
% 0.30/0.59 thf(fact_4896_le__Inf__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [B2: A,A6: set @ A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A6 ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_Inf_iff
% 0.30/0.59 thf(fact_4897_Inf__lower2,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [U: A,A6: set @ A,V: A] :
% 0.30/0.59 ( ( member @ A @ U @ A6 )
% 0.30/0.59 => ( ( ord_less_eq @ A @ U @ V )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_lower2
% 0.30/0.59 thf(fact_4898_Inf__lower,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [X: A,A6: set @ A] :
% 0.30/0.59 ( ( member @ A @ X @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_lower
% 0.30/0.59 thf(fact_4899_Inf__mono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A6: set @ A] :
% 0.30/0.59 ( ! [B5: A] :
% 0.30/0.59 ( ( member @ A @ B5 @ B7 )
% 0.30/0.59 => ? [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ A6 )
% 0.30/0.59 & ( ord_less_eq @ A @ X4 @ B5 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_mono
% 0.30/0.59 thf(fact_4900_Inf__eqI,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,X: A] :
% 0.30/0.59 ( ! [I4: A] :
% 0.30/0.59 ( ( member @ A @ I4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ I4 ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [I3: A] :
% 0.30/0.59 ( ( member @ A @ I3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ I3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ A6 )
% 0.30/0.59 = X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_eqI
% 0.30/0.59 thf(fact_4901_cInf__eq__minimum,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [Z3: A,X8: set @ A] :
% 0.30/0.59 ( ( member @ A @ Z3 @ X8 )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ X8 )
% 0.30/0.59 = Z3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_eq_minimum
% 0.30/0.59 thf(fact_4902_cInf__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 & ( no_top @ A ) )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ A3 @ X3 ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ A3 ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ X8 )
% 0.30/0.59 = A3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_eq
% 0.30/0.59 thf(fact_4903_Inf__less__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [S3: set @ A,A3: A] :
% 0.30/0.59 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 )
% 0.30/0.59 = ( ? [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ S3 )
% 0.30/0.59 & ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_less_iff
% 0.30/0.59 thf(fact_4904_mono__pow,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [F: A > A,N: nat] :
% 0.30/0.59 ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_pow
% 0.30/0.59 thf(fact_4905_funpow__mult,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,M: nat,F: A > A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F ) )
% 0.30/0.59 = ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_mult
% 0.30/0.59 thf(fact_4906_int__encode__eq,axiom,
% 0.30/0.59 ! [X: int,Y3: int] :
% 0.30/0.59 ( ( ( nat_int_encode @ X )
% 0.30/0.59 = ( nat_int_encode @ Y3 ) )
% 0.30/0.59 = ( X = Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_encode_eq
% 0.30/0.59 thf(fact_4907_funpow__swap1,axiom,
% 0.30/0.59 ! [A: $tType,F: A > A,N: nat,X: A] :
% 0.30/0.59 ( ( F @ ( compow @ ( A > A ) @ N @ F @ X ) )
% 0.30/0.59 = ( compow @ ( A > A ) @ N @ F @ ( F @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_swap1
% 0.30/0.59 thf(fact_4908_funpow__mod__eq,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,F: A > A,X: A,M: nat] :
% 0.30/0.59 ( ( ( compow @ ( A > A ) @ N @ F @ X )
% 0.30/0.59 = X )
% 0.30/0.59 => ( ( compow @ ( A > A ) @ ( modulo_modulo @ nat @ M @ N ) @ F @ X )
% 0.30/0.59 = ( compow @ ( A > A ) @ M @ F @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_mod_eq
% 0.30/0.59 thf(fact_4909_Inf__le__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [A6: set @ A,X: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X )
% 0.30/0.59 = ( ! [Y6: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y6 )
% 0.30/0.59 => ? [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_le_iff
% 0.30/0.59 thf(fact_4910_INF__eq,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,B7: set @ C,G: C > A,F: B > A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ? [X4: C] :
% 0.30/0.59 ( ( member @ C @ X4 @ B7 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
% 0.30/0.59 => ( ! [J2: C] :
% 0.30/0.59 ( ( member @ C @ J2 @ B7 )
% 0.30/0.59 => ? [X4: B] :
% 0.30/0.59 ( ( member @ B @ X4 @ A6 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_eq
% 0.30/0.59 thf(fact_4911_cInf__eq__non__empty,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ A3 @ X3 ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [X4: A] :
% 0.30/0.59 ( ( member @ A @ X4 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ A3 ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ X8 )
% 0.30/0.59 = A3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_eq_non_empty
% 0.30/0.59 thf(fact_4912_cInf__greatest,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,Z3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_greatest
% 0.30/0.59 thf(fact_4913_Inf__less__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,U: A] :
% 0.30/0.59 ( ! [V4: A] :
% 0.30/0.59 ( ( member @ A @ V4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ V4 @ U ) )
% 0.30/0.59 => ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ U ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_less_eq
% 0.30/0.59 thf(fact_4914_cInf__le__finite,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [X8: set @ A,X: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( member @ A @ X @ X8 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_le_finite
% 0.30/0.59 thf(fact_4915_cInf__lessD,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,Z3: A] :
% 0.30/0.59 ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z3 )
% 0.30/0.59 => ? [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 & ( ord_less @ A @ X3 @ Z3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_lessD
% 0.30/0.59 thf(fact_4916_Inf__superset__mono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A6: set @ A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_superset_mono
% 0.30/0.59 thf(fact_4917_finite__imp__less__Inf,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,X: A,A3: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( member @ A @ X @ X8 )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ X8 )
% 0.30/0.59 => ( ord_less @ A @ A3 @ X3 ) )
% 0.30/0.59 => ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_imp_less_Inf
% 0.30/0.59 thf(fact_4918_funpow__times__power,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( monoid_mult @ A )
% 0.30/0.59 => ! [F: A > nat,X: A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ ( F @ X ) @ ( times_times @ A @ X ) )
% 0.30/0.59 = ( times_times @ A @ ( power_power @ A @ X @ ( F @ X ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_times_power
% 0.30/0.59 thf(fact_4919_funpow__Suc__right,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,F: A > A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 0.30/0.59 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_Suc_right
% 0.30/0.59 thf(fact_4920_funpow_Osimps_I2_J,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,F: A > A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 0.30/0.59 = ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow.simps(2)
% 0.30/0.59 thf(fact_4921_sup__Inf,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [A3: A,B7: set @ A] :
% 0.30/0.59 ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B7 ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( sup_sup @ A @ A3 ) @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sup_Inf
% 0.30/0.59 thf(fact_4922_Inf__sup__eq__top__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A3: A] :
% 0.30/0.59 ( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
% 0.30/0.59 = ( top_top @ A ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ B7 )
% 0.30/0.59 => ( ( sup_sup @ A @ X2 @ A3 )
% 0.30/0.59 = ( top_top @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_sup_eq_top_iff
% 0.30/0.59 thf(fact_4923_funpow__add,axiom,
% 0.30/0.59 ! [A: $tType,M: nat,N: nat,F: A > A] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F )
% 0.30/0.59 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_add
% 0.30/0.59 thf(fact_4924_funpow__mono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( order @ A )
% 0.30/0.59 => ! [F: A > A,A6: A,B7: A,N: nat] :
% 0.30/0.59 ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ( ord_less_eq @ A @ A6 @ B7 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ A6 ) @ ( compow @ ( A > A ) @ N @ F @ B7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_mono
% 0.30/0.59 thf(fact_4925_INF__greatest,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,U: A,F: B > A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ ( F @ I4 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_greatest
% 0.30/0.59 thf(fact_4926_le__INF__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [U: A,F: B > A,A6: set @ B] :
% 0.30/0.59 ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.59 = ( ! [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_INF_iff
% 0.30/0.59 thf(fact_4927_INF__lower2,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I: B,A6: set @ B,F: B > A,U: A] :
% 0.30/0.59 ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ( ord_less_eq @ A @ ( F @ I ) @ U )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_lower2
% 0.30/0.59 thf(fact_4928_INF__mono_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [F: B > A,G: B > A,A6: set @ B] :
% 0.30/0.59 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_mono'
% 0.30/0.59 thf(fact_4929_INF__lower,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I: B,A6: set @ B,F: B > A] :
% 0.30/0.59 ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ I ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_lower
% 0.30/0.59 thf(fact_4930_INF__mono,axiom,
% 0.30/0.59 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [B7: set @ B,A6: set @ C,F: C > A,G: B > A] :
% 0.30/0.59 ( ! [M6: B] :
% 0.30/0.59 ( ( member @ B @ M6 @ B7 )
% 0.30/0.59 => ? [X4: C] :
% 0.30/0.59 ( ( member @ C @ X4 @ A6 )
% 0.30/0.59 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_mono
% 0.30/0.59 thf(fact_4931_INF__eqI,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,X: A,F: B > A] :
% 0.30/0.59 ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ X @ ( F @ I4 ) ) )
% 0.30/0.59 => ( ! [Y5: A] :
% 0.30/0.59 ( ! [I3: B] :
% 0.30/0.59 ( ( member @ B @ I3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ ( F @ I3 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.59 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.59 = X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_eqI
% 0.30/0.59 thf(fact_4932_less__INF__D,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [Y3: A,F: B > A,A6: set @ B,I: B] :
% 0.30/0.59 ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.59 => ( ( member @ B @ I @ A6 )
% 0.30/0.59 => ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_INF_D
% 0.30/0.59 thf(fact_4933_INF__less__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,A3: A] :
% 0.30/0.59 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
% 0.30/0.59 = ( ? [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_less_iff
% 0.30/0.59 thf(fact_4934_INF__sup,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [F: B > A,B7: set @ B,A3: A] :
% 0.30/0.59 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [B3: B] : ( sup_sup @ A @ ( F @ B3 ) @ A3 )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_sup
% 0.30/0.59 thf(fact_4935_Inf__sup,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [B7: set @ A,A3: A] :
% 0.30/0.59 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ A @ A
% 0.30/0.59 @ ^ [B3: A] : ( sup_sup @ A @ B3 @ A3 )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_sup
% 0.30/0.59 thf(fact_4936_sup__INF,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [A3: A,F: B > A,B7: set @ B] :
% 0.30/0.59 ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [B3: B] : ( sup_sup @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.59 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sup_INF
% 0.30/0.59 thf(fact_4937_INF__sup__distrib2,axiom,
% 0.30/0.59 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
% 0.30/0.59 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [A5: B] :
% 0.30/0.59 ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ C @ A
% 0.30/0.59 @ ^ [B3: C] : ( sup_sup @ A @ ( F @ A5 ) @ ( G @ B3 ) )
% 0.30/0.59 @ B7 ) )
% 0.30/0.59 @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_sup_distrib2
% 0.30/0.59 thf(fact_4938_INF__le__iff,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comple5582772986160207858norder @ A )
% 0.30/0.59 => ! [F: B > A,A6: set @ B,X: A] :
% 0.30/0.59 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ X )
% 0.30/0.59 = ( ! [Y6: A] :
% 0.30/0.59 ( ( ord_less @ A @ X @ Y6 )
% 0.30/0.59 => ? [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ A6 )
% 0.30/0.59 & ( ord_less @ A @ ( F @ X2 ) @ Y6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_le_iff
% 0.30/0.59 thf(fact_4939_cINF__greatest,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( condit1219197933456340205attice @ A )
% 0.30/0.59 => ! [A6: set @ B,M: A,F: B > A] :
% 0.30/0.59 ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ A6 )
% 0.30/0.59 => ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cINF_greatest
% 0.30/0.59 thf(fact_4940_INF__eq__iff,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [I6: set @ B,F: B > A,C2: A] :
% 0.30/0.59 ( ( I6
% 0.30/0.59 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 => ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ I6 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ I4 ) @ C2 ) )
% 0.30/0.59 => ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ I6 ) )
% 0.30/0.59 = C2 )
% 0.30/0.59 = ( ! [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 => ( ( F @ X2 )
% 0.30/0.59 = C2 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_eq_iff
% 0.30/0.59 thf(fact_4941_finite__less__Inf__iff,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 => ! [X8: set @ A,A3: A] :
% 0.30/0.59 ( ( finite_finite2 @ A @ X8 )
% 0.30/0.59 => ( ( X8
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) )
% 0.30/0.59 = ( ! [X2: A] :
% 0.30/0.59 ( ( member @ A @ X2 @ X8 )
% 0.30/0.59 => ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_less_Inf_iff
% 0.30/0.59 thf(fact_4942_Inf__le__Sup,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A] :
% 0.30/0.59 ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_le_Sup
% 0.30/0.59 thf(fact_4943_cInf__abs__ge,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( linordered_idom @ A ) )
% 0.30/0.59 => ! [S3: set @ A,A3: A] :
% 0.30/0.59 ( ( S3
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ S3 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_abs_ge
% 0.30/0.59 thf(fact_4944_uminus__Sup,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple489889107523837845lgebra @ A )
% 0.30/0.59 => ! [A6: set @ A] :
% 0.30/0.59 ( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A6 ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_Sup
% 0.30/0.59 thf(fact_4945_uminus__Inf,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple489889107523837845lgebra @ A )
% 0.30/0.59 => ! [A6: set @ A] :
% 0.30/0.59 ( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A6 ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_Inf
% 0.30/0.59 thf(fact_4946_less__eq__Inf__inter,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_Inf_inter
% 0.30/0.59 thf(fact_4947_Kleene__iter__lpfp,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( order_bot @ A )
% 0.30/0.59 => ! [F: A > A,P5: A,K: nat] :
% 0.30/0.59 ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ( ord_less_eq @ A @ ( F @ P5 ) @ P5 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Kleene_iter_lpfp
% 0.30/0.59 thf(fact_4948_Kleene__iter__gpfp,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( order_top @ A )
% 0.30/0.59 => ! [F: A > A,P5: A,K: nat] :
% 0.30/0.59 ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ( ord_less_eq @ A @ P5 @ ( F @ P5 ) )
% 0.30/0.59 => ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Kleene_iter_gpfp
% 0.30/0.59 thf(fact_4949_funpow__mono2,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( order @ A )
% 0.30/0.59 => ! [F: A > A,I: nat,J: nat,X: A,Y3: A] :
% 0.30/0.59 ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.59 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq @ A @ X @ ( F @ X ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F @ X ) @ ( compow @ ( A > A ) @ J @ F @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_mono2
% 0.30/0.59 thf(fact_4950_INF__superset__mono,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [B7: set @ B,A6: set @ B,F: B > A,G: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ B7 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_superset_mono
% 0.30/0.59 thf(fact_4951_uminus__INF,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple489889107523837845lgebra @ A )
% 0.30/0.59 => ! [B7: B > A,A6: set @ B] :
% 0.30/0.59 ( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
% 0.30/0.59 @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_INF
% 0.30/0.59 thf(fact_4952_uminus__SUP,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple489889107523837845lgebra @ A )
% 0.30/0.59 => ! [B7: B > A,A6: set @ B] :
% 0.30/0.59 ( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
% 0.30/0.59 @ A6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_SUP
% 0.30/0.59 thf(fact_4953_mono__INF,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( comple6319245703460814977attice @ B ) )
% 0.30/0.59 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 0.30/0.59 ( ( order_mono @ A @ B @ F )
% 0.30/0.59 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
% 0.30/0.59 @ ( complete_Inf_Inf @ B
% 0.30/0.59 @ ( image @ C @ B
% 0.30/0.59 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 0.30/0.59 @ I6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_INF
% 0.30/0.59 thf(fact_4954_mono__Inf,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 & ( comple6319245703460814977attice @ B ) )
% 0.30/0.59 => ! [F: A > B,A6: set @ A] :
% 0.30/0.59 ( ( order_mono @ A @ B @ F )
% 0.30/0.59 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_Inf
% 0.30/0.59 thf(fact_4955_of__nat__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ( ( semiring_1_of_nat @ A )
% 0.30/0.59 = ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_nat_def
% 0.30/0.59 thf(fact_4956_numeral__add__unfold__funpow,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_numeral @ A )
% 0.30/0.59 => ! [K: num,A3: A] :
% 0.30/0.59 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
% 0.30/0.59 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % numeral_add_unfold_funpow
% 0.30/0.59 thf(fact_4957_INF__le__SUP,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comple6319245703460814977attice @ A )
% 0.30/0.59 => ! [A6: set @ B,F: B > A] :
% 0.30/0.59 ( ( A6
% 0.30/0.59 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_le_SUP
% 0.30/0.59 thf(fact_4958_mono__funpow,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( lattice @ A )
% 0.30/0.59 & ( order_bot @ A ) )
% 0.30/0.59 => ! [Q: A > A] :
% 0.30/0.59 ( ( order_mono @ A @ A @ Q )
% 0.30/0.59 => ( order_mono @ nat @ A
% 0.30/0.59 @ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_funpow
% 0.30/0.59 thf(fact_4959_cInf__asclose,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( condit6923001295902523014norder @ A )
% 0.30/0.59 & ( linordered_idom @ A ) )
% 0.30/0.59 => ! [S3: set @ A,L: A,E: A] :
% 0.30/0.59 ( ( S3
% 0.30/0.59 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ S3 )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % cInf_asclose
% 0.30/0.59 thf(fact_4960_funpow__decreasing,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( lattice @ A )
% 0.30/0.59 & ( order_bot @ A ) )
% 0.30/0.59 => ! [M: nat,N: nat,F: A > A] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.59 => ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_decreasing
% 0.30/0.59 thf(fact_4961_funpow__increasing,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ( lattice @ A )
% 0.30/0.59 & ( order_top @ A ) )
% 0.30/0.59 => ! [M: nat,N: nat,F: A > A] :
% 0.30/0.59 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.59 => ( ( order_mono @ A @ A @ F )
% 0.30/0.59 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % funpow_increasing
% 0.30/0.59 thf(fact_4962_numeral__unfold__funpow,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( semiring_1 @ A )
% 0.30/0.59 => ( ( numeral_numeral @ A )
% 0.30/0.59 = ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % numeral_unfold_funpow
% 0.30/0.59 thf(fact_4963_SUP__INF,axiom,
% 0.30/0.59 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [P: C > B > A] :
% 0.30/0.59 ( ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [Y6: B] :
% 0.30/0.59 ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ C @ A
% 0.30/0.59 @ ^ [X2: C] : ( P @ X2 @ Y6 )
% 0.30/0.59 @ ( top_top @ ( set @ C ) ) ) )
% 0.30/0.59 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.59 = ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ ( B > C ) @ A
% 0.30/0.59 @ ^ [X2: B > C] :
% 0.30/0.59 ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [Y6: B] : ( P @ ( X2 @ Y6 ) @ Y6 )
% 0.30/0.59 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.59 @ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % SUP_INF
% 0.30/0.59 thf(fact_4964_INF__SUP,axiom,
% 0.30/0.59 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.59 ( ( comple592849572758109894attice @ A )
% 0.30/0.59 => ! [P: C > B > A] :
% 0.30/0.59 ( ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [Y6: B] :
% 0.30/0.59 ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ C @ A
% 0.30/0.59 @ ^ [X2: C] : ( P @ X2 @ Y6 )
% 0.30/0.59 @ ( top_top @ ( set @ C ) ) ) )
% 0.30/0.59 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.59 = ( complete_Sup_Sup @ A
% 0.30/0.59 @ ( image @ ( B > C ) @ A
% 0.30/0.59 @ ^ [F2: B > C] :
% 0.30/0.59 ( complete_Inf_Inf @ A
% 0.30/0.59 @ ( image @ B @ A
% 0.30/0.59 @ ^ [X2: B] : ( P @ ( F2 @ X2 ) @ X2 )
% 0.30/0.59 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.59 @ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % INF_SUP
% 0.30/0.59 thf(fact_4965_relpowp__fun__conv,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) )
% 0.30/0.59 = ( ^ [N3: nat,P4: A > A > $o,X2: A,Y6: A] :
% 0.30/0.59 ? [F2: nat > A] :
% 0.30/0.59 ( ( ( F2 @ ( zero_zero @ nat ) )
% 0.30/0.59 = X2 )
% 0.30/0.59 & ( ( F2 @ N3 )
% 0.30/0.59 = Y6 )
% 0.30/0.59 & ! [I5: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ I5 @ N3 )
% 0.30/0.59 => ( P4 @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_fun_conv
% 0.30/0.59 thf(fact_4966_relpowp__1,axiom,
% 0.30/0.59 ! [A: $tType,P: A > A > $o] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_1
% 0.30/0.59 thf(fact_4967_Nat_Ofunpow__code__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( funpow @ A )
% 0.30/0.59 = ( compow @ ( A > A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nat.funpow_code_def
% 0.30/0.59 thf(fact_4968_Inf__int__def,axiom,
% 0.30/0.59 ( ( complete_Inf_Inf @ int )
% 0.30/0.59 = ( ^ [X7: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X7 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Inf_int_def
% 0.30/0.59 thf(fact_4969_relpowp__Suc__I2,axiom,
% 0.30/0.59 ! [A: $tType,P: A > A > $o,X: A,Y3: A,N: nat,Z3: A] :
% 0.30/0.59 ( ( P @ X @ Y3 )
% 0.30/0.59 => ( ( compow @ ( A > A > $o ) @ N @ P @ Y3 @ Z3 )
% 0.30/0.59 => ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_Suc_I2
% 0.30/0.59 thf(fact_4970_relpowp__Suc__E2,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 0.30/0.59 => ~ ! [Y5: A] :
% 0.30/0.59 ( ( P @ X @ Y5 )
% 0.30/0.59 => ~ ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_Suc_E2
% 0.30/0.59 thf(fact_4971_relpowp__Suc__D2,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 0.30/0.59 => ? [Y5: A] :
% 0.30/0.59 ( ( P @ X @ Y5 )
% 0.30/0.59 & ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_Suc_D2
% 0.30/0.59 thf(fact_4972_relpowp__Suc__I,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Y3: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y3 )
% 0.30/0.59 => ( ( P @ Y3 @ Z3 )
% 0.30/0.59 => ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_Suc_I
% 0.30/0.59 thf(fact_4973_relpowp__Suc__E,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 0.30/0.59 => ~ ! [Y5: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y5 )
% 0.30/0.59 => ~ ( P @ Y5 @ Z3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_Suc_E
% 0.30/0.59 thf(fact_4974_relpowp__0__I,axiom,
% 0.30/0.59 ! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_0_I
% 0.30/0.59 thf(fact_4975_relpowp__0__E,axiom,
% 0.30/0.59 ! [A: $tType,P: A > A > $o,X: A,Y3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y3 )
% 0.30/0.59 => ( X = Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_0_E
% 0.30/0.59 thf(fact_4976_relpowp_Osimps_I1_J,axiom,
% 0.30/0.59 ! [A: $tType,R2: A > A > $o] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
% 0.30/0.59 = ( ^ [Y: A,Z2: A] : ( Y = Z2 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp.simps(1)
% 0.30/0.59 thf(fact_4977_relpowp__E,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
% 0.30/0.59 => ( ( ( N
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 => ( X != Z3 ) )
% 0.30/0.59 => ~ ! [Y5: A,M6: nat] :
% 0.30/0.59 ( ( N
% 0.30/0.59 = ( suc @ M6 ) )
% 0.30/0.59 => ( ( compow @ ( A > A > $o ) @ M6 @ P @ X @ Y5 )
% 0.30/0.59 => ~ ( P @ Y5 @ Z3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_E
% 0.30/0.59 thf(fact_4978_relpowp__E2,axiom,
% 0.30/0.59 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 0.30/0.59 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
% 0.30/0.59 => ( ( ( N
% 0.30/0.59 = ( zero_zero @ nat ) )
% 0.30/0.59 => ( X != Z3 ) )
% 0.30/0.59 => ~ ! [Y5: A,M6: nat] :
% 0.30/0.59 ( ( N
% 0.30/0.59 = ( suc @ M6 ) )
% 0.30/0.59 => ( ( P @ X @ Y5 )
% 0.30/0.59 => ~ ( compow @ ( A > A > $o ) @ M6 @ P @ Y5 @ Z3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_E2
% 0.30/0.59 thf(fact_4979_relpowp__bot,axiom,
% 0.30/0.59 ! [A: $tType,N: nat] :
% 0.30/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.59 => ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
% 0.30/0.59 = ( bot_bot @ ( A > A > $o ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % relpowp_bot
% 0.30/0.59 thf(fact_4980_set__zip,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.30/0.59 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 0.30/0.59 = ( collect @ ( product_prod @ A @ B )
% 0.30/0.59 @ ^ [Uu: product_prod @ A @ B] :
% 0.30/0.59 ? [I5: nat] :
% 0.30/0.59 ( ( Uu
% 0.30/0.59 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I5 ) @ ( nth @ B @ Ys2 @ I5 ) ) )
% 0.30/0.59 & ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_zip
% 0.30/0.59 thf(fact_4981_measure__function__int,axiom,
% 0.30/0.59 fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% 0.30/0.59
% 0.30/0.59 % measure_function_int
% 0.30/0.59 thf(fact_4982_surj__int__decode,axiom,
% 0.30/0.59 ( ( image @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.59 = ( top_top @ ( set @ int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % surj_int_decode
% 0.30/0.59 thf(fact_4983_int__decode__inverse,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 0.30/0.59 = N ) ).
% 0.30/0.59
% 0.30/0.59 % int_decode_inverse
% 0.30/0.59 thf(fact_4984_int__encode__inverse,axiom,
% 0.30/0.59 ! [X: int] :
% 0.30/0.59 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 0.30/0.59 = X ) ).
% 0.30/0.59
% 0.30/0.59 % int_encode_inverse
% 0.30/0.59 thf(fact_4985_measure__snd,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,F: A > nat] :
% 0.30/0.59 ( ( fun_is_measure @ A @ F )
% 0.30/0.59 => ( fun_is_measure @ ( product_prod @ B @ A )
% 0.30/0.59 @ ^ [P6: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % measure_snd
% 0.30/0.59 thf(fact_4986_measure__fst,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType,F: A > nat] :
% 0.30/0.59 ( ( fun_is_measure @ A @ F )
% 0.30/0.59 => ( fun_is_measure @ ( product_prod @ A @ B )
% 0.30/0.59 @ ^ [P6: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % measure_fst
% 0.30/0.59 thf(fact_4987_lexordp_Omono,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ord @ A )
% 0.30/0.59 => ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
% 0.30/0.59 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
% 0.30/0.59 ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.59 ( ( X12
% 0.30/0.59 = ( nil @ A ) )
% 0.30/0.59 & ( X24
% 0.30/0.59 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 0.30/0.59 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.59 ( ( X12
% 0.30/0.59 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.59 & ( X24
% 0.30/0.59 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.59 & ( ord_less @ A @ X2 @ Y6 ) )
% 0.30/0.59 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.59 ( ( X12
% 0.30/0.59 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.59 & ( X24
% 0.30/0.59 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.59 & ~ ( ord_less @ A @ X2 @ Y6 )
% 0.30/0.59 & ~ ( ord_less @ A @ Y6 @ X2 )
% 0.30/0.59 & ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % lexordp.mono
% 0.30/0.59 thf(fact_4988_is__measure_Osimps,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( fun_is_measure @ A )
% 0.30/0.59 = ( ^ [A5: A > nat] :
% 0.30/0.59 ? [X7: A > nat] :
% 0.30/0.59 ( ^ [Y: A > nat,Z2: A > nat] : ( Y = Z2 )
% 0.30/0.59 @ A5
% 0.30/0.59 @ X7 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_measure.simps
% 0.30/0.59 thf(fact_4989_is__measure__trivial,axiom,
% 0.30/0.59 ! [A: $tType,F: A > nat] : ( fun_is_measure @ A @ F ) ).
% 0.30/0.59
% 0.30/0.59 % is_measure_trivial
% 0.30/0.59 thf(fact_4990_int__decode__eq,axiom,
% 0.30/0.59 ! [X: nat,Y3: nat] :
% 0.30/0.59 ( ( ( nat_int_decode @ X )
% 0.30/0.59 = ( nat_int_decode @ Y3 ) )
% 0.30/0.59 = ( X = Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_decode_eq
% 0.30/0.59 thf(fact_4991_bot2E,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 0.30/0.59 ~ ( bot_bot @ ( A > B > $o ) @ X @ Y3 ) ).
% 0.30/0.59
% 0.30/0.59 % bot2E
% 0.30/0.59 thf(fact_4992_measure__size,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( size @ A )
% 0.30/0.59 => ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % measure_size
% 0.30/0.59 thf(fact_4993_set__conv__nth,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( set2 @ A )
% 0.30/0.59 = ( ^ [Xs3: list @ A] :
% 0.30/0.59 ( collect @ A
% 0.30/0.59 @ ^ [Uu: A] :
% 0.30/0.59 ? [I5: nat] :
% 0.30/0.59 ( ( Uu
% 0.30/0.59 = ( nth @ A @ Xs3 @ I5 ) )
% 0.30/0.59 & ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_conv_nth
% 0.30/0.59 thf(fact_4994_set__nths,axiom,
% 0.30/0.59 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 0.30/0.59 ( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
% 0.30/0.59 = ( collect @ A
% 0.30/0.59 @ ^ [Uu: A] :
% 0.30/0.59 ? [I5: nat] :
% 0.30/0.59 ( ( Uu
% 0.30/0.59 = ( nth @ A @ Xs @ I5 ) )
% 0.30/0.59 & ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.59 & ( member @ nat @ I5 @ I6 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_nths
% 0.30/0.59 thf(fact_4995_sup__bot_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.59 => ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
% 0.30/0.59 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.59 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sup_bot.semilattice_neutr_order_axioms
% 0.30/0.59 thf(fact_4996_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.59 ( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % max_nat.semilattice_neutr_order_axioms
% 0.30/0.59 thf(fact_4997_times__int_Oabs__eq,axiom,
% 0.30/0.59 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( times_times @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( abs_Integ
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) )
% 0.30/0.59 @ Xa3
% 0.30/0.59 @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % times_int.abs_eq
% 0.30/0.59 thf(fact_4998_int_Oabs__induct,axiom,
% 0.30/0.59 ! [P: int > $o,X: int] :
% 0.30/0.59 ( ! [Y5: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y5 ) )
% 0.30/0.59 => ( P @ X ) ) ).
% 0.30/0.59
% 0.30/0.59 % int.abs_induct
% 0.30/0.59 thf(fact_4999_semilattice__neutr__order_Oneutr__eq__iff,axiom,
% 0.30/0.59 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.59 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.59 => ( ( Z3
% 0.30/0.59 = ( F @ A3 @ B2 ) )
% 0.30/0.59 = ( ( A3 = Z3 )
% 0.30/0.59 & ( B2 = Z3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % semilattice_neutr_order.neutr_eq_iff
% 0.30/0.59 thf(fact_5000_semilattice__neutr__order_Oeq__neutr__iff,axiom,
% 0.30/0.59 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.59 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.59 => ( ( ( F @ A3 @ B2 )
% 0.30/0.59 = Z3 )
% 0.30/0.59 = ( ( A3 = Z3 )
% 0.30/0.59 & ( B2 = Z3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % semilattice_neutr_order.eq_neutr_iff
% 0.30/0.59 thf(fact_5001_eq__Abs__Integ,axiom,
% 0.30/0.59 ! [Z3: int] :
% 0.30/0.59 ~ ! [X3: nat,Y5: nat] :
% 0.30/0.59 ( Z3
% 0.30/0.59 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y5 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % eq_Abs_Integ
% 0.30/0.59 thf(fact_5002_mono__compose,axiom,
% 0.30/0.59 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.59 ( ( ( order @ A )
% 0.30/0.59 & ( order @ C ) )
% 0.30/0.59 => ! [Q: A > B > C,F: D > B] :
% 0.30/0.59 ( ( order_mono @ A @ ( B > C ) @ Q )
% 0.30/0.59 => ( order_mono @ A @ ( D > C )
% 0.30/0.59 @ ^ [I5: A,X2: D] : ( Q @ I5 @ ( F @ X2 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mono_compose
% 0.30/0.59 thf(fact_5003_zero__int__def,axiom,
% 0.30/0.59 ( ( zero_zero @ int )
% 0.30/0.59 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_int_def
% 0.30/0.59 thf(fact_5004_int__def,axiom,
% 0.30/0.59 ( ( semiring_1_of_nat @ int )
% 0.30/0.59 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_def
% 0.30/0.59 thf(fact_5005_nat_Oabs__eq,axiom,
% 0.30/0.59 ! [X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( nat2 @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.abs_eq
% 0.30/0.59 thf(fact_5006_Nats__altdef1,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ( ( semiring_1_Nats @ A )
% 0.30/0.59 = ( collect @ A
% 0.30/0.59 @ ^ [Uu: A] :
% 0.30/0.59 ? [N3: int] :
% 0.30/0.59 ( ( Uu
% 0.30/0.59 = ( ring_1_of_int @ A @ N3 ) )
% 0.30/0.59 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Nats_altdef1
% 0.30/0.59 thf(fact_5007_uminus__int_Oabs__eq,axiom,
% 0.30/0.59 ! [X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( abs_Integ
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 )
% 0.30/0.59 @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_int.abs_eq
% 0.30/0.59 thf(fact_5008_one__int__def,axiom,
% 0.30/0.59 ( ( one_one @ int )
% 0.30/0.59 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_int_def
% 0.30/0.59 thf(fact_5009_of__int_Oabs__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ! [X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( product_case_prod @ nat @ nat @ A
% 0.30/0.59 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
% 0.30/0.59 @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_int.abs_eq
% 0.30/0.59 thf(fact_5010_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.59 ( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
% 0.30/0.59 @ ^ [M2: nat,N3: nat] :
% 0.30/0.59 ( ( dvd_dvd @ nat @ M2 @ N3 )
% 0.30/0.59 & ( M2 != N3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % gcd_nat.semilattice_neutr_order_axioms
% 0.30/0.59 thf(fact_5011_less__int_Oabs__eq,axiom,
% 0.30/0.59 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( ord_less @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
% 0.30/0.59 @ Xa3
% 0.30/0.59 @ X ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_int.abs_eq
% 0.30/0.59 thf(fact_5012_less__eq__int_Oabs__eq,axiom,
% 0.30/0.59 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
% 0.30/0.59 @ Xa3
% 0.30/0.59 @ X ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_int.abs_eq
% 0.30/0.59 thf(fact_5013_plus__int_Oabs__eq,axiom,
% 0.30/0.59 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( plus_plus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( abs_Integ
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) )
% 0.30/0.59 @ Xa3
% 0.30/0.59 @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % plus_int.abs_eq
% 0.30/0.59 thf(fact_5014_minus__int_Oabs__eq,axiom,
% 0.30/0.59 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.30/0.59 ( ( minus_minus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 0.30/0.59 = ( abs_Integ
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) )
% 0.30/0.59 @ Xa3
% 0.30/0.59 @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minus_int.abs_eq
% 0.30/0.59 thf(fact_5015_inf__top_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.59 => ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_top.semilattice_neutr_order_axioms
% 0.30/0.59 thf(fact_5016_take__bit__numeral__minus__numeral__int,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int )
% 0.30/0.59 @ ^ [Q3: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q3 ) ) )
% 0.30/0.59 @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % take_bit_numeral_minus_numeral_int
% 0.30/0.59 thf(fact_5017_prod_Oinsert_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [I6: set @ B,P5: B > A,I: B] :
% 0.30/0.59 ( ( finite_finite2 @ B
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 & ( ( P5 @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 => ( ( ( member @ B @ I @ I6 )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) )
% 0.30/0.59 & ( ~ ( member @ B @ I @ I6 )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 0.30/0.59 = ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.insert'
% 0.30/0.59 thf(fact_5018_sorted__insort__is__snoc,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ A,A3: A] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.59 => ( ! [X3: A] :
% 0.30/0.59 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.59 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 0.30/0.59 => ( ( linorder_insort_key @ A @ A
% 0.30/0.59 @ ^ [X2: A] : X2
% 0.30/0.59 @ A3
% 0.30/0.59 @ Xs )
% 0.30/0.59 = ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_insort_is_snoc
% 0.30/0.59 thf(fact_5019_length__insort,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,X: B,Xs: list @ B] :
% 0.30/0.59 ( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
% 0.30/0.59 = ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_insort
% 0.30/0.59 thf(fact_5020_prod_Oempty_H,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [P5: B > A] :
% 0.30/0.59 ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.59 = ( one_one @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.empty'
% 0.30/0.59 thf(fact_5021_take__bit__numeral__numeral,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.59 => ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.59 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % take_bit_numeral_numeral
% 0.30/0.59 thf(fact_5022_prod_Onon__neutral_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [G: B > A,I6: set @ B] :
% 0.30/0.59 ( ( groups1962203154675924110t_prod @ B @ A @ G
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 & ( ( G @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.non_neutral'
% 0.30/0.59 thf(fact_5023_insort__key_Osimps_I2_J,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,X: B,Y3: B,Ys2: list @ B] :
% 0.30/0.59 ( ( ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.59 => ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
% 0.30/0.59 = ( cons @ B @ X @ ( cons @ B @ Y3 @ Ys2 ) ) ) )
% 0.30/0.59 & ( ~ ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.59 => ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
% 0.30/0.59 = ( cons @ B @ Y3 @ ( linorder_insort_key @ B @ A @ F @ X @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % insort_key.simps(2)
% 0.30/0.59 thf(fact_5024_sorted__insort,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [X: A,Xs: list @ A] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 0.30/0.59 @ ( linorder_insort_key @ A @ A
% 0.30/0.59 @ ^ [X2: A] : X2
% 0.30/0.59 @ X
% 0.30/0.59 @ Xs ) )
% 0.30/0.59 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_insort
% 0.30/0.59 thf(fact_5025_prod_Odistrib__triv_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 0.30/0.59 ( ( finite_finite2 @ B @ I6 )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A
% 0.30/0.59 @ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 0.30/0.59 @ I6 )
% 0.30/0.59 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.distrib_triv'
% 0.30/0.59 thf(fact_5026_insort__is__Cons,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ B,F: B > A,A3: B] :
% 0.30/0.59 ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 0.30/0.59 => ( ord_less_eq @ A @ ( F @ A3 ) @ ( F @ X3 ) ) )
% 0.30/0.59 => ( ( linorder_insort_key @ B @ A @ F @ A3 @ Xs )
% 0.30/0.59 = ( cons @ B @ A3 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % insort_is_Cons
% 0.30/0.59 thf(fact_5027_sorted__insort__key,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,X: B,Xs: list @ B] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) )
% 0.30/0.59 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_insort_key
% 0.30/0.59 thf(fact_5028_prod_Omono__neutral__cong__right_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( one_one @ A ) ) )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ S3 )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( H2 @ X3 ) ) )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.mono_neutral_cong_right'
% 0.30/0.59 thf(fact_5029_prod_Omono__neutral__cong__left_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.59 => ( ! [I4: B] :
% 0.30/0.59 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.59 => ( ( H2 @ I4 )
% 0.30/0.59 = ( one_one @ A ) ) )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ S3 )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( H2 @ X3 ) ) )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.mono_neutral_cong_left'
% 0.30/0.59 thf(fact_5030_prod_Omono__neutral__right_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( one_one @ A ) ) )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.mono_neutral_right'
% 0.30/0.59 thf(fact_5031_prod_Omono__neutral__left_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 0.30/0.59 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 0.30/0.59 => ( ! [X3: B] :
% 0.30/0.59 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 0.30/0.59 => ( ( G @ X3 )
% 0.30/0.59 = ( one_one @ A ) ) )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
% 0.30/0.59 = ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.mono_neutral_left'
% 0.30/0.59 thf(fact_5032_prod_Odistrib_H,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 0.30/0.59 ( ( finite_finite2 @ B
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 & ( ( G @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 => ( ( finite_finite2 @ B
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I6 )
% 0.30/0.59 & ( ( H2 @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A
% 0.30/0.59 @ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 0.30/0.59 @ I6 )
% 0.30/0.59 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.distrib'
% 0.30/0.59 thf(fact_5033_prod_OG__def,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( comm_monoid_mult @ A )
% 0.30/0.59 => ( ( groups1962203154675924110t_prod @ B @ A )
% 0.30/0.59 = ( ^ [P6: B > A,I8: set @ B] :
% 0.30/0.59 ( if @ A
% 0.30/0.59 @ ( finite_finite2 @ B
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I8 )
% 0.30/0.59 & ( ( P6 @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 @ ( groups7121269368397514597t_prod @ B @ A @ P6
% 0.30/0.59 @ ( collect @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( member @ B @ X2 @ I8 )
% 0.30/0.59 & ( ( P6 @ X2 )
% 0.30/0.59 != ( one_one @ A ) ) ) ) )
% 0.30/0.59 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod.G_def
% 0.30/0.59 thf(fact_5034_filter__insort,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,Xs: list @ B,P: B > $o,X: B] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.59 => ( ( P @ X )
% 0.30/0.59 => ( ( filter @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
% 0.30/0.59 = ( linorder_insort_key @ B @ A @ F @ X @ ( filter @ B @ P @ Xs ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % filter_insort
% 0.30/0.59 thf(fact_5035_and__minus__numerals_I3_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % and_minus_numerals(3)
% 0.30/0.59 thf(fact_5036_and__minus__numerals_I7_J,axiom,
% 0.30/0.59 ! [N: num,M: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % and_minus_numerals(7)
% 0.30/0.59 thf(fact_5037_and__minus__numerals_I8_J,axiom,
% 0.30/0.59 ! [N: num,M: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % and_minus_numerals(8)
% 0.30/0.59 thf(fact_5038_and__minus__numerals_I4_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % and_minus_numerals(4)
% 0.30/0.59 thf(fact_5039_int__numeral__not__and__num,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_numeral_not_and_num
% 0.30/0.59 thf(fact_5040_int__numeral__and__not__num,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.59 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % int_numeral_and_not_num
% 0.30/0.59 thf(fact_5041_insort__key__remove1,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [A3: B,Xs: list @ B,F: B > A] :
% 0.30/0.59 ( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
% 0.30/0.59 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.59 => ( ( ( hd @ B
% 0.30/0.59 @ ( filter @ B
% 0.30/0.59 @ ^ [X2: B] :
% 0.30/0.59 ( ( F @ A3 )
% 0.30/0.59 = ( F @ X2 ) )
% 0.30/0.59 @ Xs ) )
% 0.30/0.59 = A3 )
% 0.30/0.59 => ( ( linorder_insort_key @ B @ A @ F @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
% 0.30/0.59 = Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % insort_key_remove1
% 0.30/0.59 thf(fact_5042_less__eq__int_Orep__eq,axiom,
% 0.30/0.59 ( ( ord_less_eq @ int )
% 0.30/0.59 = ( ^ [X2: int,Xa4: int] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.59 @ ^ [Y6: nat,Z7: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
% 0.30/0.59 @ ( rep_Integ @ X2 )
% 0.30/0.59 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_int.rep_eq
% 0.30/0.59 thf(fact_5043_less__int_Orep__eq,axiom,
% 0.30/0.59 ( ( ord_less @ int )
% 0.30/0.59 = ( ^ [X2: int,Xa4: int] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.59 @ ^ [Y6: nat,Z7: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
% 0.30/0.59 @ ( rep_Integ @ X2 )
% 0.30/0.59 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_int.rep_eq
% 0.30/0.59 thf(fact_5044_sorted__remove1,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [Xs: list @ A,A3: A] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.59 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_remove1
% 0.30/0.59 thf(fact_5045_sorted__map__remove1,axiom,
% 0.30/0.59 ! [A: $tType,B: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [F: B > A,Xs: list @ B,X: B] :
% 0.30/0.59 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.59 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % sorted_map_remove1
% 0.30/0.59 thf(fact_5046_length__remove1,axiom,
% 0.30/0.59 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.59 ( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.59 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 0.30/0.59 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
% 0.30/0.59 & ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.59 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 0.30/0.59 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % length_remove1
% 0.30/0.59 thf(fact_5047_nat_Orep__eq,axiom,
% 0.30/0.59 ( nat2
% 0.30/0.59 = ( ^ [X2: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat.rep_eq
% 0.30/0.59 thf(fact_5048_insort__remove1,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [A3: A,Xs: list @ A] :
% 0.30/0.59 ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
% 0.30/0.59 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.59 => ( ( linorder_insort_key @ A @ A
% 0.30/0.59 @ ^ [X2: A] : X2
% 0.30/0.59 @ A3
% 0.30/0.59 @ ( remove1 @ A @ A3 @ Xs ) )
% 0.30/0.59 = Xs ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % insort_remove1
% 0.30/0.59 thf(fact_5049_of__int_Orep__eq,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ( ( ring_1_of_int @ A )
% 0.30/0.59 = ( ^ [X2: int] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ A
% 0.30/0.59 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
% 0.30/0.59 @ ( rep_Integ @ X2 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_int.rep_eq
% 0.30/0.59 thf(fact_5050_uminus__int__def,axiom,
% 0.30/0.59 ( ( uminus_uminus @ int )
% 0.30/0.59 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % uminus_int_def
% 0.30/0.59 thf(fact_5051_numeral__xor__num,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.59 => ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.59 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % numeral_xor_num
% 0.30/0.59 thf(fact_5052_numeral__and__num,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.59 => ! [M: num,N: num] :
% 0.30/0.59 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.59 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % numeral_and_num
% 0.30/0.59 thf(fact_5053_times__int__def,axiom,
% 0.30/0.59 ( ( times_times @ int )
% 0.30/0.59 = ( 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 )
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % times_int_def
% 0.30/0.59 thf(fact_5054_minus__int__def,axiom,
% 0.30/0.59 ( ( minus_minus @ int )
% 0.30/0.59 = ( 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 )
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minus_int_def
% 0.30/0.59 thf(fact_5055_plus__int__def,axiom,
% 0.30/0.59 ( ( plus_plus @ int )
% 0.30/0.59 = ( 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 )
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.59 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.59 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.59 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % plus_int_def
% 0.30/0.59 thf(fact_5056_rp__inv__image__def,axiom,
% 0.30/0.59 ! [B: $tType,A: $tType] :
% 0.30/0.59 ( ( fun_rp_inv_image @ A @ B )
% 0.30/0.59 = ( 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 ) ) ) )
% 0.30/0.59 @ ^ [R6: set @ ( product_prod @ A @ A ),S7: 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 @ S7 @ F2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % rp_inv_image_def
% 0.30/0.59 thf(fact_5057_ring__1__class_Oof__int__def,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( ring_1 @ A )
% 0.30/0.59 => ( ( ring_1_of_int @ A )
% 0.30/0.59 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
% 0.30/0.59 @ ( product_case_prod @ nat @ nat @ A
% 0.30/0.59 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ring_1_class.of_int_def
% 0.30/0.59 thf(fact_5058_card__Min__le__sum,axiom,
% 0.30/0.59 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.59 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.59 => ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % card_Min_le_sum
% 0.30/0.59 thf(fact_5059_of__nat__eq__id,axiom,
% 0.30/0.59 ( ( semiring_1_of_nat @ nat )
% 0.30/0.59 = ( id @ nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_nat_eq_id
% 0.30/0.59 thf(fact_5060_id__funpow,axiom,
% 0.30/0.59 ! [A: $tType,N: nat] :
% 0.30/0.59 ( ( compow @ ( A > A ) @ N @ ( id @ A ) )
% 0.30/0.59 = ( id @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % id_funpow
% 0.30/0.59 thf(fact_5061_rotate0,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( rotate @ A @ ( zero_zero @ nat ) )
% 0.30/0.59 = ( id @ ( list @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % rotate0
% 0.30/0.59 thf(fact_5062_group__add__class_Ominus__comp__minus,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( group_add @ A )
% 0.30/0.59 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 0.30/0.59 = ( id @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % group_add_class.minus_comp_minus
% 0.30/0.59 thf(fact_5063_boolean__algebra__class_Ominus__comp__minus,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( boolea8198339166811842893lgebra @ A )
% 0.30/0.59 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 0.30/0.59 = ( id @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % boolean_algebra_class.minus_comp_minus
% 0.30/0.59 thf(fact_5064_Min__singleton,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( linorder @ A )
% 0.30/0.59 => ! [X: A] :
% 0.30/0.59 ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.59 = X ) ) ).
% 0.30/0.59
% 0.30/0.59 % Min_singleton
% 0.30/0.59 thf(fact_5065_push__bit__0__id,axiom,
% 0.30/0.59 ! [A: $tType] :
% 0.30/0.59 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.59 => ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
% 0.30/0.60 = ( id @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % push_bit_0_id
% 0.30/0.60 thf(fact_5066_drop__bit__0,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.60 => ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
% 0.30/0.60 = ( id @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % drop_bit_0
% 0.30/0.60 thf(fact_5067_Min_Obounded__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.bounded_iff
% 0.30/0.60 thf(fact_5068_Min__gr__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_gr_iff
% 0.30/0.60 thf(fact_5069_Min__const,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ B,C2: A] :
% 0.30/0.60 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A
% 0.30/0.60 @ ( image @ B @ A
% 0.30/0.60 @ ^ [Uu: B] : C2
% 0.30/0.60 @ A6 ) )
% 0.30/0.60 = C2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_const
% 0.30/0.60 thf(fact_5070_Min__insert,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_insert
% 0.30/0.60 thf(fact_5071_funpow__simps__right_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,F: A > A] :
% 0.30/0.60 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F )
% 0.30/0.60 = ( id @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % funpow_simps_right(1)
% 0.30/0.60 thf(fact_5072_Min_OcoboundedI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,A3: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A3 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.coboundedI
% 0.30/0.60 thf(fact_5073_Min__eqI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ! [Y5: A] :
% 0.30/0.60 ( ( member @ A @ Y5 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ Y5 ) )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.60 = X ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_eqI
% 0.30/0.60 thf(fact_5074_Min__le,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_le
% 0.30/0.60 thf(fact_5075_Min__in,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_in
% 0.30/0.60 thf(fact_5076_Min_Oin__idem,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.in_idem
% 0.30/0.60 thf(fact_5077_less__int__def,axiom,
% 0.30/0.60 ( ( ord_less @ int )
% 0.30/0.60 = ( 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 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_int_def
% 0.30/0.60 thf(fact_5078_less__eq__int__def,axiom,
% 0.30/0.60 ( ( ord_less_eq @ int )
% 0.30/0.60 = ( 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 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_int_def
% 0.30/0.60 thf(fact_5079_nat__def,axiom,
% 0.30/0.60 ( nat2
% 0.30/0.60 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nat_def
% 0.30/0.60 thf(fact_5080_Min_OboundedI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ! [A8: A] :
% 0.30/0.60 ( ( member @ A @ A8 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ A8 ) )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.boundedI
% 0.30/0.60 thf(fact_5081_Min_OboundedE,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 => ! [A10: A] :
% 0.30/0.60 ( ( member @ A @ A10 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.boundedE
% 0.30/0.60 thf(fact_5082_eq__Min__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,M: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( M
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( ( member @ A @ M @ A6 )
% 0.30/0.60 & ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_Min_iff
% 0.30/0.60 thf(fact_5083_Min__le__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
% 0.30/0.60 = ( ? [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 & ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_le_iff
% 0.30/0.60 thf(fact_5084_Min__eq__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,M: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.60 = M )
% 0.30/0.60 = ( ( member @ A @ M @ A6 )
% 0.30/0.60 & ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_eq_iff
% 0.30/0.60 thf(fact_5085_Min__less__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
% 0.30/0.60 = ( ? [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 & ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_less_iff
% 0.30/0.60 thf(fact_5086_Min__insert2,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,A3: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ! [B5: A] :
% 0.30/0.60 ( ( member @ A @ B5 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ A3 @ B5 ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A3 @ A6 ) )
% 0.30/0.60 = A3 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_insert2
% 0.30/0.60 thf(fact_5087_Min__Inf,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( comple5582772986160207858norder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.60 = ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_Inf
% 0.30/0.60 thf(fact_5088_Min_Osubset__imp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.subset_imp
% 0.30/0.60 thf(fact_5089_Min__antimono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [M7: set @ A,N6: set @ A] :
% 0.30/0.60 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
% 0.30/0.60 => ( ( M7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_antimono
% 0.30/0.60 thf(fact_5090_hom__Min__commute,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [H2: A > A,N6: set @ A] :
% 0.30/0.60 ( ! [X3: A,Y5: A] :
% 0.30/0.60 ( ( H2 @ ( ord_min @ A @ X3 @ Y5 ) )
% 0.30/0.60 = ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.60 => ( ( N6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N6 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % hom_Min_commute
% 0.30/0.60 thf(fact_5091_Min_Osubset,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( B7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.60 => ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.subset
% 0.30/0.60 thf(fact_5092_Min_Oclosed,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.60 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.closed
% 0.30/0.60 thf(fact_5093_Min_Oinsert__not__elem,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.insert_not_elem
% 0.30/0.60 thf(fact_5094_mono__Min__commute,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( linorder @ A )
% 0.30/0.60 & ( linorder @ B ) )
% 0.30/0.60 => ! [F: A > B,A6: set @ A] :
% 0.30/0.60 ( ( order_mono @ A @ B @ F )
% 0.30/0.60 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( F @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mono_Min_commute
% 0.30/0.60 thf(fact_5095_Min_Ounion,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.60 => ( ( B7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.60 = ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( lattic643756798350308766er_Min @ A @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.union
% 0.30/0.60 thf(fact_5096_Min__add__commute,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( linord4140545234300271783up_add @ A )
% 0.30/0.60 => ! [S3: set @ B,F: B > A,K: A] :
% 0.30/0.60 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.60 => ( ( S3
% 0.30/0.60 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A
% 0.30/0.60 @ ( image @ B @ A
% 0.30/0.60 @ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
% 0.30/0.60 @ S3 ) )
% 0.30/0.60 = ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min_add_commute
% 0.30/0.60 thf(fact_5097_Min_Oinsert__remove,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = X ) )
% 0.30/0.60 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.insert_remove
% 0.30/0.60 thf(fact_5098_Min_Oremove,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.60 = X ) )
% 0.30/0.60 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.60 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Min.remove
% 0.30/0.60 thf(fact_5099_positive__def,axiom,
% 0.30/0.60 ( positive
% 0.30/0.60 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % positive_def
% 0.30/0.60 thf(fact_5100_dual__Max,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ( ( lattices_Max @ A
% 0.30/0.60 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dual_Max
% 0.30/0.60 thf(fact_5101_dual__max,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ( ( max @ A
% 0.30/0.60 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 0.30/0.60 = ( ord_min @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dual_max
% 0.30/0.60 thf(fact_5102_euclidean__size__nat__def,axiom,
% 0.30/0.60 ( ( euclid6346220572633701492n_size @ nat )
% 0.30/0.60 = ( id @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % euclidean_size_nat_def
% 0.30/0.60 thf(fact_5103_linorder_OMax_Ocong,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( lattices_Max @ A )
% 0.30/0.60 = ( lattices_Max @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % linorder.Max.cong
% 0.30/0.60 thf(fact_5104_ord_Omax__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( max @ A )
% 0.30/0.60 = ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ B3 @ A5 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ord.max_def
% 0.30/0.60 thf(fact_5105_ord_Omax_Ocong,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( max @ A )
% 0.30/0.60 = ( max @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % ord.max.cong
% 0.30/0.60 thf(fact_5106_dual__min,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ( ( min @ A
% 0.30/0.60 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 0.30/0.60 = ( ord_max @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dual_min
% 0.30/0.60 thf(fact_5107_sum__le__card__Max,axiom,
% 0.30/0.60 ! [A: $tType,A6: set @ A,F: A > nat] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % sum_le_card_Max
% 0.30/0.60 thf(fact_5108_euclidean__size__integer__def,axiom,
% 0.30/0.60 ( ( euclid6346220572633701492n_size @ code_integer )
% 0.30/0.60 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % euclidean_size_integer_def
% 0.30/0.60 thf(fact_5109_Max__singleton,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [X: A] :
% 0.30/0.60 ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 = X ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_singleton
% 0.30/0.60 thf(fact_5110_Max__divisors__self__nat,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( N
% 0.30/0.60 != ( zero_zero @ nat ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ nat
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
% 0.30/0.60 = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_divisors_self_nat
% 0.30/0.60 thf(fact_5111_Max_Obounded__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 0.30/0.60 = ( ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.bounded_iff
% 0.30/0.60 thf(fact_5112_Max__less__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 0.30/0.60 = ( ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_less_iff
% 0.30/0.60 thf(fact_5113_Max__const,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ B,C2: A] :
% 0.30/0.60 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A
% 0.30/0.60 @ ( image @ B @ A
% 0.30/0.60 @ ^ [Uu: B] : C2
% 0.30/0.60 @ A6 ) )
% 0.30/0.60 = C2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_const
% 0.30/0.60 thf(fact_5114_Max__insert,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_insert
% 0.30/0.60 thf(fact_5115_minus__Max__eq__Min,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.60 => ! [S3: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.60 => ( ( S3
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
% 0.30/0.60 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_Max_eq_Min
% 0.30/0.60 thf(fact_5116_minus__Min__eq__Max,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linord5086331880401160121up_add @ A )
% 0.30/0.60 => ! [S3: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.60 => ( ( S3
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_Min_eq_Max
% 0.30/0.60 thf(fact_5117_ord_Omin_Ocong,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( min @ A )
% 0.30/0.60 = ( min @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % ord.min.cong
% 0.30/0.60 thf(fact_5118_ord_Omin__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( min @ A )
% 0.30/0.60 = ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ A5 @ B3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ord.min_def
% 0.30/0.60 thf(fact_5119_Max_OcoboundedI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,A3: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.coboundedI
% 0.30/0.60 thf(fact_5120_Max__eq__if,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.60 => ( ! [X3: A] :
% 0.30/0.60 ( ( member @ A @ X3 @ A6 )
% 0.30/0.60 => ? [Xa: A] :
% 0.30/0.60 ( ( member @ A @ Xa @ B7 )
% 0.30/0.60 & ( ord_less_eq @ A @ X3 @ Xa ) ) )
% 0.30/0.60 => ( ! [X3: A] :
% 0.30/0.60 ( ( member @ A @ X3 @ B7 )
% 0.30/0.60 => ? [Xa: A] :
% 0.30/0.60 ( ( member @ A @ Xa @ A6 )
% 0.30/0.60 & ( ord_less_eq @ A @ X3 @ Xa ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_eq_if
% 0.30/0.60 thf(fact_5121_Max__eqI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ! [Y5: A] :
% 0.30/0.60 ( ( member @ A @ Y5 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = X ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_eqI
% 0.30/0.60 thf(fact_5122_Max__ge,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_ge
% 0.30/0.60 thf(fact_5123_Max__in,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_in
% 0.30/0.60 thf(fact_5124_Max_Oin__idem,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.in_idem
% 0.30/0.60 thf(fact_5125_Max_OboundedI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ! [A8: A] :
% 0.30/0.60 ( ( member @ A @ A8 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ A8 @ X ) )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.boundedI
% 0.30/0.60 thf(fact_5126_Max_OboundedE,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 0.30/0.60 => ! [A10: A] :
% 0.30/0.60 ( ( member @ A @ A10 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.boundedE
% 0.30/0.60 thf(fact_5127_eq__Max__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,M: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( M
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( ( member @ A @ M @ A6 )
% 0.30/0.60 & ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_Max_iff
% 0.30/0.60 thf(fact_5128_Max__ge__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( ? [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 & ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_ge_iff
% 0.30/0.60 thf(fact_5129_Max__eq__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,M: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = M )
% 0.30/0.60 = ( ( member @ A @ M @ A6 )
% 0.30/0.60 & ! [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_eq_iff
% 0.30/0.60 thf(fact_5130_Max__gr__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( ? [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ A6 )
% 0.30/0.60 & ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_gr_iff
% 0.30/0.60 thf(fact_5131_Max__insert2,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,A3: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ! [B5: A] :
% 0.30/0.60 ( ( member @ A @ B5 @ A6 )
% 0.30/0.60 => ( ord_less_eq @ A @ B5 @ A3 ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A3 @ A6 ) )
% 0.30/0.60 = A3 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_insert2
% 0.30/0.60 thf(fact_5132_Max__Sup,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( comple5582772986160207858norder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_Sup
% 0.30/0.60 thf(fact_5133_Max_Osubset__imp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.subset_imp
% 0.30/0.60 thf(fact_5134_Max__mono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [M7: set @ A,N6: set @ A] :
% 0.30/0.60 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
% 0.30/0.60 => ( ( M7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_mono
% 0.30/0.60 thf(fact_5135_hom__Max__commute,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [H2: A > A,N6: set @ A] :
% 0.30/0.60 ( ! [X3: A,Y5: A] :
% 0.30/0.60 ( ( H2 @ ( ord_max @ A @ X3 @ Y5 ) )
% 0.30/0.60 = ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.60 => ( ( N6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N6 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % hom_Max_commute
% 0.30/0.60 thf(fact_5136_Max_Osubset,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( B7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.60 => ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B7 ) @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.subset
% 0.30/0.60 thf(fact_5137_Max_Oclosed,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.60 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.closed
% 0.30/0.60 thf(fact_5138_Max_Oinsert__not__elem,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.insert_not_elem
% 0.30/0.60 thf(fact_5139_mono__Max__commute,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( linorder @ A )
% 0.30/0.60 & ( linorder @ B ) )
% 0.30/0.60 => ! [F: A > B,A6: set @ A] :
% 0.30/0.60 ( ( order_mono @ A @ B @ F )
% 0.30/0.60 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( F @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mono_Max_commute
% 0.30/0.60 thf(fact_5140_Max_Ounion,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( A6
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.60 => ( ( B7
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.60 = ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.union
% 0.30/0.60 thf(fact_5141_Sup__nat__def,axiom,
% 0.30/0.60 ( ( complete_Sup_Sup @ nat )
% 0.30/0.60 = ( ^ [X7: set @ nat] :
% 0.30/0.60 ( if @ nat
% 0.30/0.60 @ ( X7
% 0.30/0.60 = ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.60 @ ( zero_zero @ nat )
% 0.30/0.60 @ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Sup_nat_def
% 0.30/0.60 thf(fact_5142_card__le__Suc__Max,axiom,
% 0.30/0.60 ! [S3: set @ nat] :
% 0.30/0.60 ( ( finite_finite2 @ nat @ S3 )
% 0.30/0.60 => ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % card_le_Suc_Max
% 0.30/0.60 thf(fact_5143_Max__add__commute,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( linord4140545234300271783up_add @ A )
% 0.30/0.60 => ! [S3: set @ B,F: B > A,K: A] :
% 0.30/0.60 ( ( finite_finite2 @ B @ S3 )
% 0.30/0.60 => ( ( S3
% 0.30/0.60 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A
% 0.30/0.60 @ ( image @ B @ A
% 0.30/0.60 @ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
% 0.30/0.60 @ S3 ) )
% 0.30/0.60 = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_add_commute
% 0.30/0.60 thf(fact_5144_divide__nat__def,axiom,
% 0.30/0.60 ( ( divide_divide @ nat )
% 0.30/0.60 = ( ^ [M2: nat,N3: nat] :
% 0.30/0.60 ( if @ nat
% 0.30/0.60 @ ( N3
% 0.30/0.60 = ( zero_zero @ nat ) )
% 0.30/0.60 @ ( zero_zero @ nat )
% 0.30/0.60 @ ( lattic643756798349783984er_Max @ nat
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M2 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_nat_def
% 0.30/0.60 thf(fact_5145_gcd__is__Max__divisors__nat,axiom,
% 0.30/0.60 ! [N: nat,M: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.60 => ( ( gcd_gcd @ nat @ M @ N )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ nat
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [D5: nat] :
% 0.30/0.60 ( ( dvd_dvd @ nat @ D5 @ M )
% 0.30/0.60 & ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % gcd_is_Max_divisors_nat
% 0.30/0.60 thf(fact_5146_Max_Oinsert__remove,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = X ) )
% 0.30/0.60 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.60 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.insert_remove
% 0.30/0.60 thf(fact_5147_Max_Oremove,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [A6: set @ A,X: A] :
% 0.30/0.60 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.60 => ( ( member @ A @ X @ A6 )
% 0.30/0.60 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = X ) )
% 0.30/0.60 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.60 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.60 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max.remove
% 0.30/0.60 thf(fact_5148_nat__of__integer__def,axiom,
% 0.30/0.60 ( code_nat_of_integer
% 0.30/0.60 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % nat_of_integer_def
% 0.30/0.60 thf(fact_5149_Gcd__eq__Max,axiom,
% 0.30/0.60 ! [M7: set @ nat] :
% 0.30/0.60 ( ( finite_finite2 @ nat @ M7 )
% 0.30/0.60 => ( ( M7
% 0.30/0.60 != ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.60 => ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.60 => ( ( gcd_Gcd @ nat @ M7 )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ nat
% 0.30/0.60 @ ( complete_Inf_Inf @ ( set @ nat )
% 0.30/0.60 @ ( image @ nat @ ( set @ nat )
% 0.30/0.60 @ ^ [M2: nat] :
% 0.30/0.60 ( collect @ nat
% 0.30/0.60 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
% 0.30/0.60 @ M7 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Gcd_eq_Max
% 0.30/0.60 thf(fact_5150_dual__Min,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ( ( lattices_Min @ A
% 0.30/0.60 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dual_Min
% 0.30/0.60 thf(fact_5151_mask__integer__def,axiom,
% 0.30/0.60 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 0.30/0.60 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mask_integer_def
% 0.30/0.60 thf(fact_5152_positive_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % positive.rsp
% 0.30/0.60 thf(fact_5153_Max__divisors__self__int,axiom,
% 0.30/0.60 ! [N: int] :
% 0.30/0.60 ( ( N
% 0.30/0.60 != ( zero_zero @ int ) )
% 0.30/0.60 => ( ( lattic643756798349783984er_Max @ int
% 0.30/0.60 @ ( collect @ int
% 0.30/0.60 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
% 0.30/0.60 = ( abs_abs @ int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Max_divisors_self_int
% 0.30/0.60 thf(fact_5154_power__transfer,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType] :
% 0.30/0.60 ( ( ( power @ B )
% 0.30/0.60 & ( power @ A ) )
% 0.30/0.60 => ! [R2: A > B > $o] :
% 0.30/0.60 ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.60 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.30/0.60 => ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ R2 )
% 0.30/0.60 @ ( power_power @ A )
% 0.30/0.60 @ ( power_power @ B ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_transfer
% 0.30/0.60 thf(fact_5155_linorder_OMin_Ocong,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( lattices_Min @ A )
% 0.30/0.60 = ( lattices_Min @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % linorder.Min.cong
% 0.30/0.60 thf(fact_5156_transfer__rule__numeral,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType] :
% 0.30/0.60 ( ( ( monoid_add @ B )
% 0.30/0.60 & ( semiring_numeral @ B )
% 0.30/0.60 & ( monoid_add @ A )
% 0.30/0.60 & ( semiring_numeral @ A ) )
% 0.30/0.60 => ! [R2: A > B > $o] :
% 0.30/0.60 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.60 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.60 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.30/0.60 => ( bNF_rel_fun @ num @ num @ A @ B
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ R2
% 0.30/0.60 @ ( numeral_numeral @ A )
% 0.30/0.60 @ ( numeral_numeral @ B ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % transfer_rule_numeral
% 0.30/0.60 thf(fact_5157_transfer__rule__of__int,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType] :
% 0.30/0.60 ( ( ( ring_1 @ B )
% 0.30/0.60 & ( ring_1 @ A ) )
% 0.30/0.60 => ! [R2: A > B > $o] :
% 0.30/0.60 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.60 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.60 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.30/0.60 => ( ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
% 0.30/0.60 => ( bNF_rel_fun @ int @ int @ A @ B
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ R2
% 0.30/0.60 @ ( ring_1_of_int @ A )
% 0.30/0.60 @ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % transfer_rule_of_int
% 0.30/0.60 thf(fact_5158_gcd__is__Max__divisors__int,axiom,
% 0.30/0.60 ! [N: int,M: int] :
% 0.30/0.60 ( ( N
% 0.30/0.60 != ( zero_zero @ int ) )
% 0.30/0.60 => ( ( gcd_gcd @ int @ M @ N )
% 0.30/0.60 = ( lattic643756798349783984er_Max @ int
% 0.30/0.60 @ ( collect @ int
% 0.30/0.60 @ ^ [D5: int] :
% 0.30/0.60 ( ( dvd_dvd @ int @ D5 @ M )
% 0.30/0.60 & ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % gcd_is_Max_divisors_int
% 0.30/0.60 thf(fact_5159_transfer__rule__of__nat,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType] :
% 0.30/0.60 ( ( ( semiring_1 @ B )
% 0.30/0.60 & ( semiring_1 @ A ) )
% 0.30/0.60 => ! [R2: A > B > $o] :
% 0.30/0.60 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.60 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.60 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.30/0.60 => ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ R2
% 0.30/0.60 @ ( semiring_1_of_nat @ A )
% 0.30/0.60 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % transfer_rule_of_nat
% 0.30/0.60 thf(fact_5160_integer__of__nat__def,axiom,
% 0.30/0.60 ( code_integer_of_nat
% 0.30/0.60 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat_def
% 0.30/0.60 thf(fact_5161_antimono__funpow,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( lattice @ A )
% 0.30/0.60 & ( order_top @ A ) )
% 0.30/0.60 => ! [Q: A > A] :
% 0.30/0.60 ( ( order_mono @ A @ A @ Q )
% 0.30/0.60 => ( order_antimono @ nat @ A
% 0.30/0.60 @ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( top_top @ A ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimono_funpow
% 0.30/0.60 thf(fact_5162_nat__of__integer__integer__of__nat,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
% 0.30/0.60 = N ) ).
% 0.30/0.60
% 0.30/0.60 % nat_of_integer_integer_of_nat
% 0.30/0.60 thf(fact_5163_integer__of__nat_Orep__eq,axiom,
% 0.30/0.60 ! [X: nat] :
% 0.30/0.60 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 0.30/0.60 = ( semiring_1_of_nat @ int @ X ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat.rep_eq
% 0.30/0.60 thf(fact_5164_int__of__integer__integer__of__nat,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 0.30/0.60 = ( semiring_1_of_nat @ int @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % int_of_integer_integer_of_nat
% 0.30/0.60 thf(fact_5165_euclidean__size__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ nat @ nat
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( euclid6346220572633701492n_size @ int )
% 0.30/0.60 @ ( euclid6346220572633701492n_size @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % euclidean_size_integer.rsp
% 0.30/0.60 thf(fact_5166_euclidean__size__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( euclid6346220572633701492n_size @ nat )
% 0.30/0.60 @ ( euclid6346220572633701492n_size @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % euclidean_size_natural.rsp
% 0.30/0.60 thf(fact_5167_dup_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
% 0.30/0.60 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dup.rsp
% 0.30/0.60 thf(fact_5168_sub_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ int @ int
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
% 0.30/0.60 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % sub.rsp
% 0.30/0.60 thf(fact_5169_less__eq__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( ord_less_eq @ int )
% 0.30/0.60 @ ( ord_less_eq @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_integer.rsp
% 0.30/0.60 thf(fact_5170_less__eq__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( ord_less_eq @ nat )
% 0.30/0.60 @ ( ord_less_eq @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_natural.rsp
% 0.30/0.60 thf(fact_5171_less__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( ord_less @ int )
% 0.30/0.60 @ ( ord_less @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_integer.rsp
% 0.30/0.60 thf(fact_5172_less__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( ord_less @ nat )
% 0.30/0.60 @ ( ord_less @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_natural.rsp
% 0.30/0.60 thf(fact_5173_uminus__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( uminus_uminus @ int )
% 0.30/0.60 @ ( uminus_uminus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_integer.rsp
% 0.30/0.60 thf(fact_5174_times__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( times_times @ nat )
% 0.30/0.60 @ ( times_times @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_natural.rsp
% 0.30/0.60 thf(fact_5175_times__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( times_times @ int )
% 0.30/0.60 @ ( times_times @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_integer.rsp
% 0.30/0.60 thf(fact_5176_natural__of__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ nat @ nat
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ nat2
% 0.30/0.60 @ nat2 ) ).
% 0.30/0.60
% 0.30/0.60 % natural_of_integer.rsp
% 0.30/0.60 thf(fact_5177_Suc_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ suc
% 0.30/0.60 @ suc ) ).
% 0.30/0.60
% 0.30/0.60 % Suc.rsp
% 0.30/0.60 thf(fact_5178_modulo__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( modulo_modulo @ int )
% 0.30/0.60 @ ( modulo_modulo @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % modulo_integer.rsp
% 0.30/0.60 thf(fact_5179_modulo__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( modulo_modulo @ nat )
% 0.30/0.60 @ ( modulo_modulo @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % modulo_natural.rsp
% 0.30/0.60 thf(fact_5180_sgn__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( sgn_sgn @ int )
% 0.30/0.60 @ ( sgn_sgn @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % sgn_integer.rsp
% 0.30/0.60 thf(fact_5181_plus__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( plus_plus @ nat )
% 0.30/0.60 @ ( plus_plus @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_natural.rsp
% 0.30/0.60 thf(fact_5182_plus__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( plus_plus @ int )
% 0.30/0.60 @ ( plus_plus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_integer.rsp
% 0.30/0.60 thf(fact_5183_divide__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( divide_divide @ nat )
% 0.30/0.60 @ ( divide_divide @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_natural.rsp
% 0.30/0.60 thf(fact_5184_divide__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( divide_divide @ int )
% 0.30/0.60 @ ( divide_divide @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_integer.rsp
% 0.30/0.60 thf(fact_5185_abs__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( abs_abs @ int )
% 0.30/0.60 @ ( abs_abs @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % abs_integer.rsp
% 0.30/0.60 thf(fact_5186_minus__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( minus_minus @ int )
% 0.30/0.60 @ ( minus_minus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_integer.rsp
% 0.30/0.60 thf(fact_5187_minus__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( minus_minus @ nat )
% 0.30/0.60 @ ( minus_minus @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_natural.rsp
% 0.30/0.60 thf(fact_5188_integer__of__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ int @ int
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( semiring_1_of_nat @ int )
% 0.30/0.60 @ ( semiring_1_of_nat @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_natural.rsp
% 0.30/0.60 thf(fact_5189_push__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se4730199178511100633sh_bit @ nat )
% 0.30/0.60 @ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % push_bit_natural.rsp
% 0.30/0.60 thf(fact_5190_push__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se4730199178511100633sh_bit @ int )
% 0.30/0.60 @ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % push_bit_integer.rsp
% 0.30/0.60 thf(fact_5191_flip__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se8732182000553998342ip_bit @ int )
% 0.30/0.60 @ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % flip_bit_integer.rsp
% 0.30/0.60 thf(fact_5192_flip__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se8732182000553998342ip_bit @ nat )
% 0.30/0.60 @ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % flip_bit_natural.rsp
% 0.30/0.60 thf(fact_5193_take__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se2584673776208193580ke_bit @ nat )
% 0.30/0.60 @ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_natural.rsp
% 0.30/0.60 thf(fact_5194_take__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se2584673776208193580ke_bit @ int )
% 0.30/0.60 @ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_integer.rsp
% 0.30/0.60 thf(fact_5195_and__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5824344872417868541ns_and @ nat )
% 0.30/0.60 @ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % and_natural.rsp
% 0.30/0.60 thf(fact_5196_and__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5824344872417868541ns_and @ int )
% 0.30/0.60 @ ( bit_se5824344872417868541ns_and @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % and_integer.rsp
% 0.30/0.60 thf(fact_5197_unset__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se2638667681897837118et_bit @ nat )
% 0.30/0.60 @ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % unset_bit_natural.rsp
% 0.30/0.60 thf(fact_5198_unset__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se2638667681897837118et_bit @ int )
% 0.30/0.60 @ ( bit_se2638667681897837118et_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % unset_bit_integer.rsp
% 0.30/0.60 thf(fact_5199_xor__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5824344971392196577ns_xor @ int )
% 0.30/0.60 @ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % xor_integer.rsp
% 0.30/0.60 thf(fact_5200_xor__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5824344971392196577ns_xor @ nat )
% 0.30/0.60 @ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % xor_natural.rsp
% 0.30/0.60 thf(fact_5201_set__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5668285175392031749et_bit @ int )
% 0.30/0.60 @ ( bit_se5668285175392031749et_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % set_bit_integer.rsp
% 0.30/0.60 thf(fact_5202_set__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se5668285175392031749et_bit @ nat )
% 0.30/0.60 @ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % set_bit_natural.rsp
% 0.30/0.60 thf(fact_5203_drop__bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se4197421643247451524op_bit @ nat )
% 0.30/0.60 @ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % drop_bit_natural.rsp
% 0.30/0.60 thf(fact_5204_drop__bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se4197421643247451524op_bit @ int )
% 0.30/0.60 @ ( bit_se4197421643247451524op_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % drop_bit_integer.rsp
% 0.30/0.60 thf(fact_5205_or__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se1065995026697491101ons_or @ nat )
% 0.30/0.60 @ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % or_natural.rsp
% 0.30/0.60 thf(fact_5206_or__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.60 @ ( bit_se1065995026697491101ons_or @ int )
% 0.30/0.60 @ ( bit_se1065995026697491101ons_or @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % or_integer.rsp
% 0.30/0.60 thf(fact_5207_bit__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 0.30/0.60 @ ( bit_se5641148757651400278ts_bit @ int )
% 0.30/0.60 @ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % bit_integer.rsp
% 0.30/0.60 thf(fact_5208_not__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bit_ri4277139882892585799ns_not @ int )
% 0.30/0.60 @ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_integer.rsp
% 0.30/0.60 thf(fact_5209_bit__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 0.30/0.60 @ ( bit_se5641148757651400278ts_bit @ nat )
% 0.30/0.60 @ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % bit_natural.rsp
% 0.30/0.60 thf(fact_5210_mask__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ int @ int
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bit_se2239418461657761734s_mask @ int )
% 0.30/0.60 @ ( bit_se2239418461657761734s_mask @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % mask_integer.rsp
% 0.30/0.60 thf(fact_5211_mask__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( bit_se2239418461657761734s_mask @ nat )
% 0.30/0.60 @ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % mask_natural.rsp
% 0.30/0.60 thf(fact_5212_division__segment__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ int @ int
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( euclid7384307370059645450egment @ int )
% 0.30/0.60 @ ( euclid7384307370059645450egment @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % division_segment_integer.rsp
% 0.30/0.60 thf(fact_5213_division__segment__natural_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( euclid7384307370059645450egment @ nat )
% 0.30/0.60 @ ( euclid7384307370059645450egment @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % division_segment_natural.rsp
% 0.30/0.60 thf(fact_5214_Fract_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ratrel )
% 0.30/0.60 @ ^ [A5: int,B3: int] :
% 0.30/0.60 ( if @ ( product_prod @ int @ int )
% 0.30/0.60 @ ( B3
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ A5 @ B3 ) )
% 0.30/0.60 @ ^ [A5: int,B3: int] :
% 0.30/0.60 ( if @ ( product_prod @ int @ int )
% 0.30/0.60 @ ( B3
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Fract.rsp
% 0.30/0.60 thf(fact_5215_num__of__integer_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ int @ int @ num @ num
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 0.30/0.60 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % num_of_integer.rsp
% 0.30/0.60 thf(fact_5216_antimonoD,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( order @ A )
% 0.30/0.60 & ( order @ B ) )
% 0.30/0.60 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.60 ( ( order_antimono @ A @ B @ F )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.60 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimonoD
% 0.30/0.60 thf(fact_5217_antimonoE,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( order @ A )
% 0.30/0.60 & ( order @ B ) )
% 0.30/0.60 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.60 ( ( order_antimono @ A @ B @ F )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.60 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimonoE
% 0.30/0.60 thf(fact_5218_antimonoI,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( order @ A )
% 0.30/0.60 & ( order @ B ) )
% 0.30/0.60 => ! [F: A > B] :
% 0.30/0.60 ( ! [X3: A,Y5: A] :
% 0.30/0.60 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.60 => ( ord_less_eq @ B @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.30/0.60 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimonoI
% 0.30/0.60 thf(fact_5219_antimono__def,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( order @ A )
% 0.30/0.60 & ( order @ B ) )
% 0.30/0.60 => ( ( order_antimono @ A @ B )
% 0.30/0.60 = ( ^ [F2: A > B] :
% 0.30/0.60 ! [X2: A,Y6: A] :
% 0.30/0.60 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.30/0.60 => ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimono_def
% 0.30/0.60 thf(fact_5220_transfer__rule__of__bool,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType] :
% 0.30/0.60 ( ( ( zero_neq_one @ B )
% 0.30/0.60 & ( zero_neq_one @ A ) )
% 0.30/0.60 => ! [R2: A > B > $o] :
% 0.30/0.60 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.60 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.60 => ( bNF_rel_fun @ $o @ $o @ A @ B
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 0.30/0.60 @ R2
% 0.30/0.60 @ ( zero_neq_one_of_bool @ A )
% 0.30/0.60 @ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % transfer_rule_of_bool
% 0.30/0.60 thf(fact_5221_times__rat_Orsp,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_rat.rsp
% 0.30/0.60 thf(fact_5222_plus__rat_Orsp,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_rat.rsp
% 0.30/0.60 thf(fact_5223_integer__of__nat__eq__of__nat,axiom,
% 0.30/0.60 ( code_integer_of_nat
% 0.30/0.60 = ( semiring_1_of_nat @ code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat_eq_of_nat
% 0.30/0.60 thf(fact_5224_antimono__iff__le__Suc,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( order @ A )
% 0.30/0.60 => ( ( order_antimono @ nat @ A )
% 0.30/0.60 = ( ^ [F2: nat > A] :
% 0.30/0.60 ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % antimono_iff_le_Suc
% 0.30/0.60 thf(fact_5225_integer__of__nat__0,axiom,
% 0.30/0.60 ( ( code_integer_of_nat @ ( zero_zero @ nat ) )
% 0.30/0.60 = ( zero_zero @ code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat_0
% 0.30/0.60 thf(fact_5226_integer__of__nat_Oabs__eq,axiom,
% 0.30/0.60 ( code_integer_of_nat
% 0.30/0.60 = ( ^ [X2: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat.abs_eq
% 0.30/0.60 thf(fact_5227_integer__of__nat__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
% 0.30/0.60 = ( numeral_numeral @ code_integer @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat_numeral
% 0.30/0.60 thf(fact_5228_min__of__antimono,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( linorder @ A )
% 0.30/0.60 & ( linorder @ B ) )
% 0.30/0.60 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.60 ( ( order_antimono @ A @ B @ F )
% 0.30/0.60 => ( ( ord_min @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.60 = ( F @ ( ord_max @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % min_of_antimono
% 0.30/0.60 thf(fact_5229_max__of__antimono,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( linorder @ A )
% 0.30/0.60 & ( linorder @ B ) )
% 0.30/0.60 => ! [F: A > B,X: A,Y3: A] :
% 0.30/0.60 ( ( order_antimono @ A @ B @ F )
% 0.30/0.60 => ( ( ord_max @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.60 = ( F @ ( ord_min @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_of_antimono
% 0.30/0.60 thf(fact_5230_integer__of__nat__1,axiom,
% 0.30/0.60 ( ( code_integer_of_nat @ ( one_one @ nat ) )
% 0.30/0.60 = ( one_one @ code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % integer_of_nat_1
% 0.30/0.60 thf(fact_5231_uminus__rat_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_rat.rsp
% 0.30/0.60 thf(fact_5232_inverse__rat_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] :
% 0.30/0.60 ( if @ ( product_prod @ int @ int )
% 0.30/0.60 @ ( ( product_fst @ int @ int @ X2 )
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] :
% 0.30/0.60 ( if @ ( product_prod @ int @ int )
% 0.30/0.60 @ ( ( product_fst @ int @ int @ X2 )
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % inverse_rat.rsp
% 0.30/0.60 thf(fact_5233_plus__rat_Otransfer,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 0.30/0.60 @ ( plus_plus @ rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_rat.transfer
% 0.30/0.60 thf(fact_5234_inverse__rat_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] :
% 0.30/0.60 ( if @ ( product_prod @ int @ int )
% 0.30/0.60 @ ( ( product_fst @ int @ int @ X2 )
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.60 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
% 0.30/0.60 @ ( inverse_inverse @ rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % inverse_rat.transfer
% 0.30/0.60 thf(fact_5235_positive_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
% 0.30/0.60 @ positive ) ).
% 0.30/0.60
% 0.30/0.60 % positive.transfer
% 0.30/0.60 thf(fact_5236_one__rat_Otransfer,axiom,
% 0.30/0.60 pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% 0.30/0.60
% 0.30/0.60 % one_rat.transfer
% 0.30/0.60 thf(fact_5237_zero__rat_Otransfer,axiom,
% 0.30/0.60 pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% 0.30/0.60
% 0.30/0.60 % zero_rat.transfer
% 0.30/0.60 thf(fact_5238_uminus__rat_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
% 0.30/0.60 @ ( uminus_uminus @ rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_rat.transfer
% 0.30/0.60 thf(fact_5239_times__rat_Otransfer,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 0.30/0.60 @ ( times_times @ rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_rat.transfer
% 0.30/0.60 thf(fact_5240_times__int_Otransfer,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
% 0.30/0.60 @ ( times_times @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_int.transfer
% 0.30/0.60 thf(fact_5241_num_Orec__transfer,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,S3: A > B > $o] :
% 0.30/0.60 ( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S3
% 0.30/0.60 @ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
% 0.30/0.60 @ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ A @ B
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ S3 ) ) )
% 0.30/0.60 @ ( rec_num @ A )
% 0.30/0.60 @ ( rec_num @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % num.rec_transfer
% 0.30/0.60 thf(fact_5242_num__of__integer__def,axiom,
% 0.30/0.60 ( code_num_of_integer
% 0.30/0.60 = ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % num_of_integer_def
% 0.30/0.60 thf(fact_5243_verit__eq__simplify_I20_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X22: num] :
% 0.30/0.60 ( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
% 0.30/0.60 = ( F22 @ X22 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X22 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(20)
% 0.30/0.60 thf(fact_5244_verit__eq__simplify_I19_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
% 0.30/0.60 ( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
% 0.30/0.60 = F1 ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(19)
% 0.30/0.60 thf(fact_5245_verit__eq__simplify_I21_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
% 0.30/0.60 ( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
% 0.30/0.60 = ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(21)
% 0.30/0.60 thf(fact_5246_zero__int_Otransfer,axiom,
% 0.30/0.60 pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% 0.30/0.60
% 0.30/0.60 % zero_int.transfer
% 0.30/0.60 thf(fact_5247_int__transfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ pcr_int
% 0.30/0.60 @ ^ [N3: nat] : ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) )
% 0.30/0.60 @ ( semiring_1_of_nat @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % int_transfer
% 0.30/0.60 thf(fact_5248_uminus__int_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
% 0.30/0.60 @ ( uminus_uminus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_int.transfer
% 0.30/0.60 thf(fact_5249_nat_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
% 0.30/0.60 @ nat2 ) ).
% 0.30/0.60
% 0.30/0.60 % nat.transfer
% 0.30/0.60 thf(fact_5250_one__int_Otransfer,axiom,
% 0.30/0.60 pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% 0.30/0.60
% 0.30/0.60 % one_int.transfer
% 0.30/0.60 thf(fact_5251_of__int_Otransfer,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ring_1 @ A )
% 0.30/0.60 => ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
% 0.30/0.60 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ A
% 0.30/0.60 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
% 0.30/0.60 @ ( ring_1_of_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % of_int.transfer
% 0.30/0.60 thf(fact_5252_less__int_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 0.30/0.60 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 0.30/0.60 @ ( ord_less @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_int.transfer
% 0.30/0.60 thf(fact_5253_less__eq__int_Otransfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 0.30/0.60 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 0.30/0.60 @ ( ord_less_eq @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_int.transfer
% 0.30/0.60 thf(fact_5254_plus__int_Otransfer,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
% 0.30/0.60 @ ( plus_plus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_int.transfer
% 0.30/0.60 thf(fact_5255_minus__int_Otransfer,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
% 0.30/0.60 @ ( minus_minus @ int ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_int.transfer
% 0.30/0.60 thf(fact_5256_num_Ocase__transfer,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,S3: A > B > $o] :
% 0.30/0.60 ( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S3
% 0.30/0.60 @ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ A @ B
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ S3 )
% 0.30/0.60 @ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ A @ B
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ S3 )
% 0.30/0.60 @ ( bNF_rel_fun @ num @ num @ A @ B
% 0.30/0.60 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.60 @ S3 ) ) )
% 0.30/0.60 @ ( case_num @ A )
% 0.30/0.60 @ ( case_num @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % num.case_transfer
% 0.30/0.60 thf(fact_5257_times__int_Orsp,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % times_int.rsp
% 0.30/0.60 thf(fact_5258_list_Osize__gen_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
% 0.30/0.60 ( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
% 0.30/0.60 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % list.size_gen(2)
% 0.30/0.60 thf(fact_5259_intrel__iff,axiom,
% 0.30/0.60 ! [X: nat,Y3: nat,U: nat,V: nat] :
% 0.30/0.60 ( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ ( product_Pair @ nat @ nat @ U @ V ) )
% 0.30/0.60 = ( ( plus_plus @ nat @ X @ V )
% 0.30/0.60 = ( plus_plus @ nat @ U @ Y3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % intrel_iff
% 0.30/0.60 thf(fact_5260_int_Orel__eq__transfer,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 0.30/0.60 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ intrel
% 0.30/0.60 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % int.rel_eq_transfer
% 0.30/0.60 thf(fact_5261_int_Oabs__eq__iff,axiom,
% 0.30/0.60 ! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
% 0.30/0.60 ( ( ( abs_Integ @ X )
% 0.30/0.60 = ( abs_Integ @ Y3 ) )
% 0.30/0.60 = ( intrel @ X @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % int.abs_eq_iff
% 0.30/0.60 thf(fact_5262_num_Ocase__distrib,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
% 0.30/0.60 ( ( H2 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
% 0.30/0.60 = ( case_num @ B @ ( H2 @ F1 )
% 0.30/0.60 @ ^ [X2: num] : ( H2 @ ( F22 @ X2 ) )
% 0.30/0.60 @ ^ [X2: num] : ( H2 @ ( F32 @ X2 ) )
% 0.30/0.60 @ Num ) ) ).
% 0.30/0.60
% 0.30/0.60 % num.case_distrib
% 0.30/0.60 thf(fact_5263_verit__eq__simplify_I17_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A,F32: num > A,X22: num] :
% 0.30/0.60 ( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
% 0.30/0.60 = ( F22 @ X22 ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(17)
% 0.30/0.60 thf(fact_5264_verit__eq__simplify_I16_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A,F32: num > A] :
% 0.30/0.60 ( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
% 0.30/0.60 = F1 ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(16)
% 0.30/0.60 thf(fact_5265_zero__int_Orsp,axiom,
% 0.30/0.60 intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_int.rsp
% 0.30/0.60 thf(fact_5266_verit__eq__simplify_I18_J,axiom,
% 0.30/0.60 ! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
% 0.30/0.60 ( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
% 0.30/0.60 = ( F32 @ X32 ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_eq_simplify(18)
% 0.30/0.60 thf(fact_5267_list_Osize__gen_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,X: A > nat] :
% 0.30/0.60 ( ( size_list @ A @ X @ ( nil @ A ) )
% 0.30/0.60 = ( zero_zero @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % list.size_gen(1)
% 0.30/0.60 thf(fact_5268_size__list__estimation,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
% 0.30/0.60 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.60 => ( ( ord_less @ nat @ Y3 @ ( F @ X ) )
% 0.30/0.60 => ( ord_less @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % size_list_estimation
% 0.30/0.60 thf(fact_5269_size__list__pointwise,axiom,
% 0.30/0.60 ! [A: $tType,Xs: list @ A,F: A > nat,G: A > nat] :
% 0.30/0.60 ( ! [X3: A] :
% 0.30/0.60 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.30/0.60 => ( ord_less_eq @ nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.60 => ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % size_list_pointwise
% 0.30/0.60 thf(fact_5270_size__list__estimation_H,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
% 0.30/0.60 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.60 => ( ( ord_less_eq @ nat @ Y3 @ ( F @ X ) )
% 0.30/0.60 => ( ord_less_eq @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % size_list_estimation'
% 0.30/0.60 thf(fact_5271_uminus__int_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_int.rsp
% 0.30/0.60 thf(fact_5272_nat_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
% 0.30/0.60 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nat.rsp
% 0.30/0.60 thf(fact_5273_one__int_Orsp,axiom,
% 0.30/0.60 intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_int.rsp
% 0.30/0.60 thf(fact_5274_of__int_Orsp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ring_1 @ A )
% 0.30/0.60 => ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
% 0.30/0.60 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ A
% 0.30/0.60 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ A
% 0.30/0.60 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % of_int.rsp
% 0.30/0.60 thf(fact_5275_intrel__def,axiom,
% 0.30/0.60 ( intrel
% 0.30/0.60 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] :
% 0.30/0.60 ( ( plus_plus @ nat @ X2 @ V5 )
% 0.30/0.60 = ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % intrel_def
% 0.30/0.60 thf(fact_5276_less__int_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 0.30/0.60 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_int.rsp
% 0.30/0.60 thf(fact_5277_less__eq__int_Orsp,axiom,
% 0.30/0.60 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 0.30/0.60 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 0.30/0.60 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ $o
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_int.rsp
% 0.30/0.60 thf(fact_5278_plus__int_Orsp,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % plus_int.rsp
% 0.30/0.60 thf(fact_5279_minus__int_Orsp,axiom,
% 0.30/0.60 ( 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 )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
% 0.30/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.30/0.60 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.30/0.60 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_int.rsp
% 0.30/0.60 thf(fact_5280_Nitpick_Osize__list__simp_I1_J,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( size_list @ A )
% 0.30/0.60 = ( ^ [F2: A > nat,Xs3: list @ A] :
% 0.30/0.60 ( if @ nat
% 0.30/0.60 @ ( Xs3
% 0.30/0.60 = ( nil @ A ) )
% 0.30/0.60 @ ( zero_zero @ nat )
% 0.30/0.60 @ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Nitpick.size_list_simp(1)
% 0.30/0.60 thf(fact_5281_execute__make,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 0.30/0.60 @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_make
% 0.30/0.60 thf(fact_5282_relpow__finite__bounded1,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
% 0.30/0.60 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [N3: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.60 & ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_finite_bounded1
% 0.30/0.60 thf(fact_5283_tl__upt,axiom,
% 0.30/0.60 ! [M: nat,N: nat] :
% 0.30/0.60 ( ( tl @ nat @ ( upt @ M @ N ) )
% 0.30/0.60 = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % tl_upt
% 0.30/0.60 thf(fact_5284_relpow__1,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R2 )
% 0.30/0.60 = R2 ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_1
% 0.30/0.60 thf(fact_5285_length__tl,axiom,
% 0.30/0.60 ! [A: $tType,Xs: list @ A] :
% 0.30/0.60 ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
% 0.30/0.60 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % length_tl
% 0.30/0.60 thf(fact_5286_finite__relpow,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.60 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % finite_relpow
% 0.30/0.60 thf(fact_5287_tl__replicate,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,X: A] :
% 0.30/0.60 ( ( tl @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.60 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% 0.30/0.60
% 0.30/0.60 % tl_replicate
% 0.30/0.60 thf(fact_5288_execute__heap,axiom,
% 0.30/0.60 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
% 0.30/0.60 = ( 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 ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_heap
% 0.30/0.60 thf(fact_5289_execute__guard_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ( P @ H2 )
% 0.30/0.60 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_guard(2)
% 0.30/0.60 thf(fact_5290_Heap__eqI,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
% 0.30/0.60 ( ! [H6: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ F @ H6 )
% 0.30/0.60 = ( heap_Time_execute @ A @ G @ H6 ) )
% 0.30/0.60 => ( F = G ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap_eqI
% 0.30/0.60 thf(fact_5291_effectI,axiom,
% 0.30/0.60 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R: A,H: heap_ext @ product_unit,N: nat] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ C2 @ H2 )
% 0.30/0.60 = ( 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 @ H @ N ) ) ) )
% 0.30/0.60 => ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % effectI
% 0.30/0.60 thf(fact_5292_effect__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap_Time_effect @ A )
% 0.30/0.60 = ( ^ [C4: heap_Time_Heap @ A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: A,N3: nat] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ C4 @ H4 )
% 0.30/0.60 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % effect_def
% 0.30/0.60 thf(fact_5293_sorted__tl,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [Xs: list @ A] :
% 0.30/0.60 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.60 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % sorted_tl
% 0.30/0.60 thf(fact_5294_relpow__Suc__E,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 0.30/0.60 => ~ ! [Y5: A] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_Suc_E
% 0.30/0.60 thf(fact_5295_relpow__Suc__I,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Y3: A,N: nat,R2: set @ ( product_prod @ A @ A ),Z3: A] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 0.30/0.60 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_Suc_I
% 0.30/0.60 thf(fact_5296_relpow__Suc__D2,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 0.30/0.60 => ? [Y5: A] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 0.30/0.60 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_Suc_D2
% 0.30/0.60 thf(fact_5297_relpow__Suc__E2,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 0.30/0.60 => ~ ! [Y5: A] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 0.30/0.60 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_Suc_E2
% 0.30/0.60 thf(fact_5298_relpow__Suc__I2,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Y3: A,R2: set @ ( product_prod @ A @ A ),Z3: A,N: nat] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
% 0.30/0.60 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_Suc_I2
% 0.30/0.60 thf(fact_5299_relpow__0__I,axiom,
% 0.30/0.60 ! [A: $tType,X: A,R2: 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 ) @ R2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_0_I
% 0.30/0.60 thf(fact_5300_relpow__0__E,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) )
% 0.30/0.60 => ( X = Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_0_E
% 0.30/0.60 thf(fact_5301_take__tl,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.60 ( ( take @ A @ N @ ( tl @ A @ Xs ) )
% 0.30/0.60 = ( tl @ A @ ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_tl
% 0.30/0.60 thf(fact_5302_drop__Suc,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.60 ( ( drop @ A @ ( suc @ N ) @ Xs )
% 0.30/0.60 = ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % drop_Suc
% 0.30/0.60 thf(fact_5303_execute__assert_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( P @ X )
% 0.30/0.60 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
% 0.30/0.60 = ( 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 @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_assert(1)
% 0.30/0.60 thf(fact_5304_execute__tap,axiom,
% 0.30/0.60 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_tap
% 0.30/0.60 thf(fact_5305_execute__return,axiom,
% 0.30/0.60 ! [A: $tType,X: A] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
% 0.30/0.60 = ( 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 ) ) )
% 0.30/0.60 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_return
% 0.30/0.60 thf(fact_5306_execute__ureturn,axiom,
% 0.30/0.60 ! [A: $tType,X: A] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
% 0.30/0.60 = ( 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 ) ) )
% 0.30/0.60 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_ureturn
% 0.30/0.60 thf(fact_5307_relpow__E,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 => ( ( ( N
% 0.30/0.60 = ( zero_zero @ nat ) )
% 0.30/0.60 => ( X != Z3 ) )
% 0.30/0.60 => ~ ! [Y5: A,M6: nat] :
% 0.30/0.60 ( ( N
% 0.30/0.60 = ( suc @ M6 ) )
% 0.30/0.60 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) )
% 0.30/0.60 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_E
% 0.30/0.60 thf(fact_5308_relpow__E2,axiom,
% 0.30/0.60 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 => ( ( ( N
% 0.30/0.60 = ( zero_zero @ nat ) )
% 0.30/0.60 => ( X != Z3 ) )
% 0.30/0.60 => ~ ! [Y5: A,M6: nat] :
% 0.30/0.60 ( ( N
% 0.30/0.60 = ( suc @ M6 ) )
% 0.30/0.60 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 0.30/0.60 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_E2
% 0.30/0.60 thf(fact_5309_tl__take,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.60 ( ( tl @ A @ ( take @ A @ N @ Xs ) )
% 0.30/0.60 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % tl_take
% 0.30/0.60 thf(fact_5310_relpow__empty,axiom,
% 0.30/0.60 ! [A: $tType,N: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.60 => ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.60 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_empty
% 0.30/0.60 thf(fact_5311_relpow__finite__bounded,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
% 0.30/0.60 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_finite_bounded
% 0.30/0.60 thf(fact_5312_Nitpick_Osize__list__simp_I2_J,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( size_size @ ( list @ A ) )
% 0.30/0.60 = ( ^ [Xs3: list @ A] :
% 0.30/0.60 ( if @ nat
% 0.30/0.60 @ ( Xs3
% 0.30/0.60 = ( nil @ A ) )
% 0.30/0.60 @ ( zero_zero @ nat )
% 0.30/0.60 @ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Nitpick.size_list_simp(2)
% 0.30/0.60 thf(fact_5313_nth__tl,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.60 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
% 0.30/0.60 => ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
% 0.30/0.60 = ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nth_tl
% 0.30/0.60 thf(fact_5314_relpow__fun__conv,axiom,
% 0.30/0.60 ! [A: $tType,A3: A,B2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 0.30/0.60 = ( ? [F2: nat > A] :
% 0.30/0.60 ( ( ( F2 @ ( zero_zero @ nat ) )
% 0.30/0.60 = A3 )
% 0.30/0.60 & ( ( F2 @ N )
% 0.30/0.60 = B2 )
% 0.30/0.60 & ! [I5: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ I5 @ N )
% 0.30/0.60 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) @ R2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % relpow_fun_conv
% 0.30/0.60 thf(fact_5315_execute__of__list,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [Xs: list @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
% 0.30/0.60 @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_of_list
% 0.30/0.60 thf(fact_5316_execute__new,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [N: nat,X: A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.30/0.60 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 0.30/0.60 @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_new
% 0.30/0.60 thf(fact_5317_take__Suc,axiom,
% 0.30/0.60 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.30/0.60 ( ( Xs
% 0.30/0.60 != ( nil @ A ) )
% 0.30/0.60 => ( ( take @ A @ ( suc @ N ) @ Xs )
% 0.30/0.60 = ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N @ ( tl @ A @ Xs ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_Suc
% 0.30/0.60 thf(fact_5318_ntrancl__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( transitive_ntrancl @ A )
% 0.30/0.60 = ( ^ [N3: nat,R6: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ^ [I5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I5 @ R6 )
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [I5: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I5 )
% 0.30/0.60 & ( ord_less_eq @ nat @ I5 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ntrancl_def
% 0.30/0.60 thf(fact_5319_option_Osize_I4_J,axiom,
% 0.30/0.60 ! [A: $tType,X22: A] :
% 0.30/0.60 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
% 0.30/0.60 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % option.size(4)
% 0.30/0.60 thf(fact_5320_trancl__finite__eq__relpow,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ( transitive_trancl @ A @ R2 )
% 0.30/0.60 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [N3: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.60 & ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % trancl_finite_eq_relpow
% 0.30/0.60 thf(fact_5321_ntrancl__Zero,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R2 )
% 0.30/0.60 = R2 ) ).
% 0.30/0.60
% 0.30/0.60 % ntrancl_Zero
% 0.30/0.60 thf(fact_5322_take__bit__num__simps_I2_J,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( bit_take_bit_num @ ( suc @ N ) @ one2 )
% 0.30/0.60 = ( some @ num @ one2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_simps(2)
% 0.30/0.60 thf(fact_5323_take__bit__num__simps_I4_J,axiom,
% 0.30/0.60 ! [N: nat,M: num] :
% 0.30/0.60 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.30/0.60 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_simps(4)
% 0.30/0.60 thf(fact_5324_execute__len,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [A3: array @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H2 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_len
% 0.30/0.60 thf(fact_5325_finite__trancl__ntranl,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ( transitive_trancl @ A @ R2 )
% 0.30/0.60 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) @ ( one_one @ nat ) ) @ R2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % finite_trancl_ntranl
% 0.30/0.60 thf(fact_5326_trancl__set__ntrancl,axiom,
% 0.30/0.60 ! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
% 0.30/0.60 = ( 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 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % trancl_set_ntrancl
% 0.30/0.60 thf(fact_5327_trancl__power,axiom,
% 0.30/0.60 ! [A: $tType,P5: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R2 ) )
% 0.30/0.60 = ( ? [N3: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.60 & ( member @ ( product_prod @ A @ A ) @ P5 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % trancl_power
% 0.30/0.60 thf(fact_5328_option_Osize__neq,axiom,
% 0.30/0.60 ! [A: $tType,X: option @ A] :
% 0.30/0.60 ( ( size_size @ ( option @ A ) @ X )
% 0.30/0.60 != ( zero_zero @ nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % option.size_neq
% 0.30/0.60 thf(fact_5329_option_Osize__gen_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,X: A > nat,X22: A] :
% 0.30/0.60 ( ( size_option @ A @ X @ ( some @ A @ X22 ) )
% 0.30/0.60 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % option.size_gen(2)
% 0.30/0.60 thf(fact_5330_sorted__find__Min,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linorder @ A )
% 0.30/0.60 => ! [Xs: list @ A,P: A > $o] :
% 0.30/0.60 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.60 => ( ? [X4: A] :
% 0.30/0.60 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 0.30/0.60 & ( P @ X4 ) )
% 0.30/0.60 => ( ( find @ A @ P @ Xs )
% 0.30/0.60 = ( some @ A
% 0.30/0.60 @ ( lattic643756798350308766er_Min @ A
% 0.30/0.60 @ ( collect @ A
% 0.30/0.60 @ ^ [X2: A] :
% 0.30/0.60 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.30/0.60 & ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % sorted_find_Min
% 0.30/0.60 thf(fact_5331_timeFrame_Osimps_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,R: A,H2: heap_ext @ product_unit,N2: nat] :
% 0.30/0.60 ( ( 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 @ H2 @ N2 ) ) ) )
% 0.30/0.60 = ( 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 @ H2 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % timeFrame.simps(1)
% 0.30/0.60 thf(fact_5332_timeFrame__zero,axiom,
% 0.30/0.60 ! [A: $tType,H2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H2 )
% 0.30/0.60 = H2 ) ).
% 0.30/0.60
% 0.30/0.60 % timeFrame_zero
% 0.30/0.60 thf(fact_5333_timeFrame__assoc,axiom,
% 0.30/0.60 ! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
% 0.30/0.60 = ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% 0.30/0.60
% 0.30/0.60 % timeFrame_assoc
% 0.30/0.60 thf(fact_5334_find__Some__iff2,axiom,
% 0.30/0.60 ! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
% 0.30/0.60 ( ( ( some @ A @ X )
% 0.30/0.60 = ( find @ A @ P @ Xs ) )
% 0.30/0.60 = ( ? [I5: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.60 & ( P @ ( nth @ A @ Xs @ I5 ) )
% 0.30/0.60 & ( X
% 0.30/0.60 = ( nth @ A @ Xs @ I5 ) )
% 0.30/0.60 & ! [J3: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ J3 @ I5 )
% 0.30/0.60 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % find_Some_iff2
% 0.30/0.60 thf(fact_5335_find__Some__iff,axiom,
% 0.30/0.60 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 0.30/0.60 ( ( ( find @ A @ P @ Xs )
% 0.30/0.60 = ( some @ A @ X ) )
% 0.30/0.60 = ( ? [I5: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.60 & ( P @ ( nth @ A @ Xs @ I5 ) )
% 0.30/0.60 & ( X
% 0.30/0.60 = ( nth @ A @ Xs @ I5 ) )
% 0.30/0.60 & ! [J3: nat] :
% 0.30/0.60 ( ( ord_less @ nat @ J3 @ I5 )
% 0.30/0.60 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % find_Some_iff
% 0.30/0.60 thf(fact_5336_tap__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap_Time_tap @ A )
% 0.30/0.60 = ( ^ [F2: ( heap_ext @ product_unit ) > A] :
% 0.30/0.60 ( heap_Time_Heap2 @ A
% 0.30/0.60 @ ^ [H4: 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 @ H4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % tap_def
% 0.30/0.60 thf(fact_5337_execute__bind_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 = ( 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 @ H @ N ) ) ) )
% 0.30/0.60 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 0.30/0.60 = ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_bind(1)
% 0.30/0.60 thf(fact_5338_rtrancl__finite__eq__relpow,axiom,
% 0.30/0.60 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.60 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.30/0.60 => ( ( transitive_rtrancl @ A @ R2 )
% 0.30/0.60 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.60 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 0.30/0.60 @ ( collect @ nat
% 0.30/0.60 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % rtrancl_finite_eq_relpow
% 0.30/0.60 thf(fact_5339_Heap_Oinject,axiom,
% 0.30/0.60 ! [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 ) ) )] :
% 0.30/0.60 ( ( ( heap_Time_Heap2 @ A @ X )
% 0.30/0.60 = ( heap_Time_Heap2 @ A @ Ya ) )
% 0.30/0.60 = ( X = Ya ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap.inject
% 0.30/0.60 thf(fact_5340_Heap__Time__Monad_Obind__bind,axiom,
% 0.30/0.60 ! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
% 0.30/0.60 = ( heap_Time_bind @ A @ B @ F
% 0.30/0.60 @ ^ [X2: A] : ( heap_Time_bind @ C @ B @ ( G @ X2 ) @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap_Time_Monad.bind_bind
% 0.30/0.60 thf(fact_5341_Heap__execute,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.30/0.60 ( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
% 0.30/0.60 = F ) ).
% 0.30/0.60
% 0.30/0.60 % Heap_execute
% 0.30/0.60 thf(fact_5342_execute_Osimps,axiom,
% 0.30/0.60 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F ) )
% 0.30/0.60 = F ) ).
% 0.30/0.60
% 0.30/0.60 % execute.simps
% 0.30/0.60 thf(fact_5343_distrib__if__bind,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
% 0.30/0.60 ( ( B2
% 0.30/0.60 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
% 0.30/0.60 = ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
% 0.30/0.60 & ( ~ B2
% 0.30/0.60 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
% 0.30/0.60 = ( heap_Time_bind @ B @ A @ D2 @ F ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % distrib_if_bind
% 0.30/0.60 thf(fact_5344_Heap_Oexhaust,axiom,
% 0.30/0.60 ! [A: $tType,Y3: heap_Time_Heap @ A] :
% 0.30/0.60 ~ ! [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.30/0.60 ( Y3
% 0.30/0.60 != ( heap_Time_Heap2 @ A @ X3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap.exhaust
% 0.30/0.60 thf(fact_5345_assert__cong,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F5: A > ( heap_Time_Heap @ B ),X: A] :
% 0.30/0.60 ( ( P = P2 )
% 0.30/0.60 => ( ! [X3: A] :
% 0.30/0.60 ( ( P2 @ X3 )
% 0.30/0.60 => ( ( F @ X3 )
% 0.30/0.60 = ( F5 @ X3 ) ) )
% 0.30/0.60 => ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
% 0.30/0.60 = ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F5 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % assert_cong
% 0.30/0.60 thf(fact_5346_bind__ureturn,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.30/0.60 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
% 0.30/0.60 = F ) ).
% 0.30/0.60
% 0.30/0.60 % bind_ureturn
% 0.30/0.60 thf(fact_5347_ureturn__bind,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 0.30/0.60 ( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
% 0.30/0.60 = ( F @ X ) ) ).
% 0.30/0.60
% 0.30/0.60 % ureturn_bind
% 0.30/0.60 thf(fact_5348_effect__bindE,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,R4: A,N: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 @ H3 @ R4 @ N )
% 0.30/0.60 => ~ ! [H7: heap_ext @ product_unit,R3: B,N1: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ B @ F @ H2 @ H7 @ R3 @ N1 )
% 0.30/0.60 => ! [N22: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ ( G @ R3 ) @ H7 @ H3 @ R4 @ N22 )
% 0.30/0.60 => ( N
% 0.30/0.60 != ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % effect_bindE
% 0.30/0.60 thf(fact_5349_effect__bindI,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,R4: B,N2: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ F @ H2 @ H @ R @ N )
% 0.30/0.60 => ( ( heap_Time_effect @ B @ ( G @ R ) @ H @ H3 @ R4 @ N2 )
% 0.30/0.60 => ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 @ H3 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % effect_bindI
% 0.30/0.60 thf(fact_5350_bind__lift,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
% 0.30/0.60 ( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
% 0.30/0.60 = ( heap_Time_bind @ B @ A @ F
% 0.30/0.60 @ ^ [X2: B] : ( heap_Time_return @ A @ ( G @ X2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % bind_lift
% 0.30/0.60 thf(fact_5351_Heap_Osize_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.30/0.60 ( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X ) )
% 0.30/0.60 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap.size(2)
% 0.30/0.60 thf(fact_5352_heap__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap_Time_heap @ A )
% 0.30/0.60 = ( ^ [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 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % heap_def
% 0.30/0.60 thf(fact_5353_execute__bind__eq__SomeI,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y3: B,H3: heap_ext @ product_unit,N2: nat] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 = ( 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 @ H @ N ) ) ) )
% 0.30/0.60 => ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H )
% 0.30/0.60 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
% 0.30/0.60 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 0.30/0.60 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_bind_eq_SomeI
% 0.30/0.60 thf(fact_5354_bind__return,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.30/0.60 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
% 0.30/0.60 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 0.30/0.60 @ ^ [Uu: product_unit] : F ) ) ).
% 0.30/0.60
% 0.30/0.60 % bind_return
% 0.30/0.60 thf(fact_5355_return__bind,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 0.30/0.60 ( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
% 0.30/0.60 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 0.30/0.60 @ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % return_bind
% 0.30/0.60 thf(fact_5356_Heap_Osize__gen,axiom,
% 0.30/0.60 ! [A: $tType,Xa3: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.30/0.60 ( ( heap_Time_size_Heap @ A @ Xa3 @ ( heap_Time_Heap2 @ A @ X ) )
% 0.30/0.60 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap.size_gen
% 0.30/0.60 thf(fact_5357_success__bind__executeI,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 = ( 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 @ H @ N ) ) ) )
% 0.30/0.60 => ( ( heap_Time_success @ B @ ( G @ X ) @ H )
% 0.30/0.60 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_bind_executeI
% 0.30/0.60 thf(fact_5358_take__bit__num__def,axiom,
% 0.30/0.60 ( bit_take_bit_num
% 0.30/0.60 = ( ^ [N3: nat,M2: num] :
% 0.30/0.60 ( if @ ( option @ num )
% 0.30/0.60 @ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) )
% 0.30/0.60 = ( zero_zero @ nat ) )
% 0.30/0.60 @ ( none @ num )
% 0.30/0.60 @ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_def
% 0.30/0.60 thf(fact_5359_power__int__numeral__neg__numeral,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [M: num,N: num] :
% 0.30/0.60 ( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.60 = ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_numeral_neg_numeral
% 0.30/0.60 thf(fact_5360_power__int__1__left,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [N: int] :
% 0.30/0.60 ( ( power_int @ A @ ( one_one @ A ) @ N )
% 0.30/0.60 = ( one_one @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_1_left
% 0.30/0.60 thf(fact_5361_power__int__1__right,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( inverse @ A )
% 0.30/0.60 & ( monoid_mult @ A ) )
% 0.30/0.60 => ! [Y3: A] :
% 0.30/0.60 ( ( power_int @ A @ Y3 @ ( one_one @ int ) )
% 0.30/0.60 = Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_1_right
% 0.30/0.60 thf(fact_5362_power__int__sgn,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,N: int] :
% 0.30/0.60 ( ( sgn_sgn @ A @ ( power_int @ A @ A3 @ N ) )
% 0.30/0.60 = ( power_int @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_sgn
% 0.30/0.60 thf(fact_5363_power__int__mult__distrib__numeral2,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [X: A,W2: num,M: int] :
% 0.30/0.60 ( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mult_distrib_numeral2
% 0.30/0.60 thf(fact_5364_power__int__mult__distrib__numeral1,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [W2: num,Y3: A,M: int] :
% 0.30/0.60 ( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) @ M )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mult_distrib_numeral1
% 0.30/0.60 thf(fact_5365_power__int__eq__0__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ( power_int @ A @ X @ N )
% 0.30/0.60 = ( zero_zero @ A ) )
% 0.30/0.60 = ( ( X
% 0.30/0.60 = ( zero_zero @ A ) )
% 0.30/0.60 & ( N
% 0.30/0.60 != ( zero_zero @ int ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_eq_0_iff
% 0.30/0.60 thf(fact_5366_power__int__0__left,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [M: int] :
% 0.30/0.60 ( ( M
% 0.30/0.60 != ( zero_zero @ int ) )
% 0.30/0.60 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.60 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_0_left
% 0.30/0.60 thf(fact_5367_power__int__0__right,axiom,
% 0.30/0.60 ! [B: $tType] :
% 0.30/0.60 ( ( ( inverse @ B )
% 0.30/0.60 & ( power @ B ) )
% 0.30/0.60 => ! [X: B] :
% 0.30/0.60 ( ( power_int @ B @ X @ ( zero_zero @ int ) )
% 0.30/0.60 = ( one_one @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_0_right
% 0.30/0.60 thf(fact_5368_abs__power__int__minus,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,N: int] :
% 0.30/0.60 ( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 0.30/0.60 = ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % abs_power_int_minus
% 0.30/0.60 thf(fact_5369_power__int__of__nat,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( inverse @ A )
% 0.30/0.60 & ( power @ A ) )
% 0.30/0.60 => ! [X: A,N: nat] :
% 0.30/0.60 ( ( power_int @ A @ X @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.60 = ( power_power @ A @ X @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_of_nat
% 0.30/0.60 thf(fact_5370_power__int__mult__numeral,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: num,N: num] :
% 0.30/0.60 ( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 0.30/0.60 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mult_numeral
% 0.30/0.60 thf(fact_5371_take__bit__num__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
% 0.30/0.60 = ( none @ num ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_simps(1)
% 0.30/0.60 thf(fact_5372_power__int__minus__one__mult__self,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [M: int] :
% 0.30/0.60 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) )
% 0.30/0.60 = ( one_one @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_one_mult_self
% 0.30/0.60 thf(fact_5373_power__int__minus__one__mult__self_H,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [M: int,B2: A] :
% 0.30/0.60 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ B2 ) )
% 0.30/0.60 = B2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_one_mult_self'
% 0.30/0.60 thf(fact_5374_power__int__numeral,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( inverse @ A )
% 0.30/0.60 & ( power @ A ) )
% 0.30/0.60 => ! [X: A,N: num] :
% 0.30/0.60 ( ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) )
% 0.30/0.60 = ( power_power @ A @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_numeral
% 0.30/0.60 thf(fact_5375_power__int__add__numeral2,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: num,N: num,B2: A] :
% 0.30/0.60 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_add_numeral2
% 0.30/0.60 thf(fact_5376_power__int__add__numeral,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: num,N: num] :
% 0.30/0.60 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.60 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_add_numeral
% 0.30/0.60 thf(fact_5377_power__int__minus1__right,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( inverse @ A )
% 0.30/0.60 & ( monoid_mult @ A ) )
% 0.30/0.60 => ! [Y3: A] :
% 0.30/0.60 ( ( power_int @ A @ Y3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.30/0.60 = ( inverse_inverse @ A @ Y3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus1_right
% 0.30/0.60 thf(fact_5378_power__int__mono__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,B2: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.30/0.60 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) )
% 0.30/0.60 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mono_iff
% 0.30/0.60 thf(fact_5379_power__int__minus__left__even,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [N: int,A3: A] :
% 0.30/0.60 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.60 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.60 = ( power_int @ A @ A3 @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_left_even
% 0.30/0.60 thf(fact_5380_power__int__minus__left__odd,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [N: int,A3: A] :
% 0.30/0.60 ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.60 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.60 = ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_left_odd
% 0.30/0.60 thf(fact_5381_take__bit__num__simps_I3_J,axiom,
% 0.30/0.60 ! [N: nat,M: num] :
% 0.30/0.60 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.30/0.60 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 0.30/0.60 @ ^ [Q3: num] : ( some @ num @ ( bit0 @ Q3 ) )
% 0.30/0.60 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_simps(3)
% 0.30/0.60 thf(fact_5382_power__int__not__zero,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( N
% 0.30/0.60 = ( zero_zero @ int ) ) )
% 0.30/0.60 => ( ( power_int @ A @ X @ N )
% 0.30/0.60 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_not_zero
% 0.30/0.60 thf(fact_5383_zero__le__power__int,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.60 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_le_power_int
% 0.30/0.60 thf(fact_5384_success__LetI,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( X = T2 )
% 0.30/0.60 => ( ( heap_Time_success @ B @ ( F @ X ) @ H2 )
% 0.30/0.60 => ( heap_Time_success @ B @ ( F @ T2 ) @ H2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_LetI
% 0.30/0.60 thf(fact_5385_power__int__commutes,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
% 0.30/0.60 = ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_commutes
% 0.30/0.60 thf(fact_5386_power__int__mult__distrib,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [X: A,Y3: A,M: int] :
% 0.30/0.60 ( ( power_int @ A @ ( times_times @ A @ X @ Y3 ) @ M )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mult_distrib
% 0.30/0.60 thf(fact_5387_power__int__mult,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: int,N: int] :
% 0.30/0.60 ( ( power_int @ A @ X @ ( times_times @ int @ M @ N ) )
% 0.30/0.60 = ( power_int @ A @ ( power_int @ A @ X @ M ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mult
% 0.30/0.60 thf(fact_5388_power__int__inverse,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( power_int @ A @ ( inverse_inverse @ A @ X ) @ N )
% 0.30/0.60 = ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_inverse
% 0.30/0.60 thf(fact_5389_success__ifI,axiom,
% 0.30/0.60 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,E: heap_Time_Heap @ A] :
% 0.30/0.60 ( ( C2
% 0.30/0.60 => ( heap_Time_success @ A @ T2 @ H2 ) )
% 0.30/0.60 => ( ( ~ C2
% 0.30/0.60 => ( heap_Time_success @ A @ E @ H2 ) )
% 0.30/0.60 => ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_ifI
% 0.30/0.60 thf(fact_5390_power__int__divide__distrib,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [X: A,Y3: A,M: int] :
% 0.30/0.60 ( ( power_int @ A @ ( divide_divide @ A @ X @ Y3 ) @ M )
% 0.30/0.60 = ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_divide_distrib
% 0.30/0.60 thf(fact_5391_power__int__abs,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,N: int] :
% 0.30/0.60 ( ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) )
% 0.30/0.60 = ( power_int @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_abs
% 0.30/0.60 thf(fact_5392_effect__success,axiom,
% 0.30/0.60 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N )
% 0.30/0.60 => ( heap_Time_success @ A @ C2 @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % effect_success
% 0.30/0.60 thf(fact_5393_success__effectE,axiom,
% 0.30/0.60 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_success @ A @ C2 @ H2 )
% 0.30/0.60 => ~ ! [R3: A,H7: heap_ext @ product_unit,N4: nat] :
% 0.30/0.60 ~ ( heap_Time_effect @ A @ C2 @ H2 @ H7 @ R3 @ N4 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_effectE
% 0.30/0.60 thf(fact_5394_zero__less__power__int,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.60 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_less_power_int
% 0.30/0.60 thf(fact_5395_power__int__one__over,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
% 0.30/0.60 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_one_over
% 0.30/0.60 thf(fact_5396_power__int__minus,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( power_int @ A @ X @ ( uminus_uminus @ int @ N ) )
% 0.30/0.60 = ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus
% 0.30/0.60 thf(fact_5397_success__returnI,axiom,
% 0.30/0.60 ! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H2 ) ).
% 0.30/0.60
% 0.30/0.60 % success_returnI
% 0.30/0.60 thf(fact_5398_success__heapI,axiom,
% 0.30/0.60 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H2 ) ).
% 0.30/0.60
% 0.30/0.60 % success_heapI
% 0.30/0.60 thf(fact_5399_success__guardI,axiom,
% 0.30/0.60 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ( P @ H2 )
% 0.30/0.60 => ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_guardI
% 0.30/0.60 thf(fact_5400_success__assertI,axiom,
% 0.30/0.60 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( P @ X )
% 0.30/0.60 => ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_assertI
% 0.30/0.60 thf(fact_5401_success__ureturnI,axiom,
% 0.30/0.60 ! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 ) ).
% 0.30/0.60
% 0.30/0.60 % success_ureturnI
% 0.30/0.60 thf(fact_5402_success__makeI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_makeI
% 0.30/0.60 thf(fact_5403_success__of__listI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_of_listI
% 0.30/0.60 thf(fact_5404_success__newI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [N: nat,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_newI
% 0.30/0.60 thf(fact_5405_success__tapI,axiom,
% 0.30/0.60 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H2 ) ).
% 0.30/0.60
% 0.30/0.60 % success_tapI
% 0.30/0.60 thf(fact_5406_success__lenI,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap @ A )
% 0.30/0.60 => ! [A3: array @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_lenI
% 0.30/0.60 thf(fact_5407_power__int__0__left__If,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [M: int] :
% 0.30/0.60 ( ( ( M
% 0.30/0.60 = ( zero_zero @ int ) )
% 0.30/0.60 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.60 = ( one_one @ A ) ) )
% 0.30/0.60 & ( ( M
% 0.30/0.60 != ( zero_zero @ int ) )
% 0.30/0.60 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 0.30/0.60 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_0_left_If
% 0.30/0.60 thf(fact_5408_power__int__increasing,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [N: int,N6: int,A3: A] :
% 0.30/0.60 ( ( ord_less_eq @ int @ N @ N6 )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.60 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_increasing
% 0.30/0.60 thf(fact_5409_power__int__strict__increasing,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [N: int,N6: int,A3: A] :
% 0.30/0.60 ( ( ord_less @ int @ N @ N6 )
% 0.30/0.60 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.60 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_strict_increasing
% 0.30/0.60 thf(fact_5410_power__int__minus__one__minus,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [N: int] :
% 0.30/0.60 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
% 0.30/0.60 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_one_minus
% 0.30/0.60 thf(fact_5411_power__int__diff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [X: A,M: int,N: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( M != N ) )
% 0.30/0.60 => ( ( power_int @ A @ X @ ( minus_minus @ int @ M @ N ) )
% 0.30/0.60 = ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_diff
% 0.30/0.60 thf(fact_5412_power__int__minus__one__diff__commute,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [A3: int,B2: int] :
% 0.30/0.60 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
% 0.30/0.60 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_one_diff_commute
% 0.30/0.60 thf(fact_5413_success__bind__effectI,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ F @ H2 @ H @ X @ N )
% 0.30/0.60 => ( ( heap_Time_success @ B @ ( G @ X ) @ H )
% 0.30/0.60 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_bind_effectI
% 0.30/0.60 thf(fact_5414_power__int__strict__decreasing,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [N: int,N6: int,A3: A] :
% 0.30/0.60 ( ( ord_less @ int @ N @ N6 )
% 0.30/0.60 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.60 => ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_strict_decreasing
% 0.30/0.60 thf(fact_5415_power__int__mono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,Y3: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.60 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.60 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y3 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_mono
% 0.30/0.60 thf(fact_5416_power__int__strict__antimono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,B2: A,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.60 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
% 0.30/0.60 => ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_strict_antimono
% 0.30/0.60 thf(fact_5417_one__le__power__int,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
% 0.30/0.60 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_le_power_int
% 0.30/0.60 thf(fact_5418_one__less__power__int,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_less_power_int
% 0.30/0.60 thf(fact_5419_power__int__add,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: int,N: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( ( plus_plus @ int @ M @ N )
% 0.30/0.60 != ( zero_zero @ int ) ) )
% 0.30/0.60 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N ) )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_add
% 0.30/0.60 thf(fact_5420_option_Osize__gen_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,X: A > nat] :
% 0.30/0.60 ( ( size_option @ A @ X @ ( none @ A ) )
% 0.30/0.60 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % option.size_gen(1)
% 0.30/0.60 thf(fact_5421_option_Osize_I3_J,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 0.30/0.60 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % option.size(3)
% 0.30/0.60 thf(fact_5422_power__int__minus__left__distrib,axiom,
% 0.30/0.60 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.60 ( ( ( division_ring @ A )
% 0.30/0.60 & ( one @ B )
% 0.30/0.60 & ( uminus @ B ) )
% 0.30/0.60 => ! [X: C,A3: A,N: int] :
% 0.30/0.60 ( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
% 0.30/0.60 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_left_distrib
% 0.30/0.60 thf(fact_5423_take__bit__num__eq__None__imp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.60 => ! [M: nat,N: num] :
% 0.30/0.60 ( ( ( bit_take_bit_num @ M @ N )
% 0.30/0.60 = ( none @ num ) )
% 0.30/0.60 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
% 0.30/0.60 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % take_bit_num_eq_None_imp
% 0.30/0.60 thf(fact_5424_and__num__eq__None__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.60 => ! [M: num,N: num] :
% 0.30/0.60 ( ( ( bit_un7362597486090784418nd_num @ M @ N )
% 0.30/0.60 = ( none @ num ) )
% 0.30/0.60 = ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.60 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % and_num_eq_None_iff
% 0.30/0.60 thf(fact_5425_xor__num__eq__None__iff,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.60 => ! [M: num,N: num] :
% 0.30/0.60 ( ( ( bit_un2480387367778600638or_num @ M @ N )
% 0.30/0.60 = ( none @ num ) )
% 0.30/0.60 = ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.30/0.60 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % xor_num_eq_None_iff
% 0.30/0.60 thf(fact_5426_power__int__antimono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,B2: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.30/0.60 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
% 0.30/0.60 => ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_antimono
% 0.30/0.60 thf(fact_5427_power__int__strict__mono,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [A3: A,B2: A,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ A3 @ B2 )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_strict_mono
% 0.30/0.60 thf(fact_5428_power__int__le__one,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.60 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 0.30/0.60 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_le_one
% 0.30/0.60 thf(fact_5429_power__int__decreasing,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [N: int,N6: int,A3: A] :
% 0.30/0.60 ( ( ord_less_eq @ int @ N @ N6 )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.60 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.60 => ( ( ( A3
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( N6
% 0.30/0.60 != ( zero_zero @ int ) )
% 0.30/0.60 | ( N
% 0.30/0.60 = ( zero_zero @ int ) ) )
% 0.30/0.60 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_decreasing
% 0.30/0.60 thf(fact_5430_power__int__le__imp__le__exp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,M: int,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 0.30/0.60 => ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
% 0.30/0.60 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_le_imp_le_exp
% 0.30/0.60 thf(fact_5431_power__int__le__imp__less__exp,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( linordered_field @ A )
% 0.30/0.60 => ! [X: A,M: int,N: int] :
% 0.30/0.60 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 0.30/0.60 => ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
% 0.30/0.60 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.30/0.60 => ( ord_less @ int @ M @ N ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_le_imp_less_exp
% 0.30/0.60 thf(fact_5432_power__int__minus__mult,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( field @ A )
% 0.30/0.60 => ! [X: A,N: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( N
% 0.30/0.60 != ( zero_zero @ int ) ) )
% 0.30/0.60 => ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
% 0.30/0.60 = ( power_int @ A @ X @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_mult
% 0.30/0.60 thf(fact_5433_power__int__add__1_H,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( M
% 0.30/0.60 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.30/0.60 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
% 0.30/0.60 = ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_add_1'
% 0.30/0.60 thf(fact_5434_power__int__add__1,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [X: A,M: int] :
% 0.30/0.60 ( ( ( X
% 0.30/0.60 != ( zero_zero @ A ) )
% 0.30/0.60 | ( M
% 0.30/0.60 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.30/0.60 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
% 0.30/0.60 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_add_1
% 0.30/0.60 thf(fact_5435_power__int__minus__left,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( division_ring @ A )
% 0.30/0.60 => ! [N: int,A3: A] :
% 0.30/0.60 ( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.60 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.60 = ( power_int @ A @ A3 @ N ) ) )
% 0.30/0.60 & ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 0.30/0.60 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.30/0.60 = ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_minus_left
% 0.30/0.60 thf(fact_5436_successE,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_success @ A @ F @ H2 )
% 0.30/0.60 => ~ ! [R3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ H7 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % successE
% 0.30/0.60 thf(fact_5437_and__not__num__eq__None__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( bit_and_not_num @ M @ N )
% 0.30/0.60 = ( none @ num ) )
% 0.30/0.60 = ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.30/0.60 = ( zero_zero @ int ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % and_not_num_eq_None_iff
% 0.30/0.60 thf(fact_5438_power__int__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( ( inverse @ A )
% 0.30/0.60 & ( power @ A ) )
% 0.30/0.60 => ( ( power_int @ A )
% 0.30/0.60 = ( ^ [X2: A,N3: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) @ ( power_power @ A @ X2 @ ( nat2 @ N3 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_int_def
% 0.30/0.60 thf(fact_5439_execute__bind__success,axiom,
% 0.30/0.60 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( heap_Time_success @ A @ F @ H2 )
% 0.30/0.60 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 0.30/0.60 = ( 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 @ H2 ) ) ) ) @ ( 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 @ H2 ) ) ) ) @ ( 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 @ H2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_bind_success
% 0.30/0.60 thf(fact_5440_effectE,axiom,
% 0.30/0.60 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.30/0.60 ( ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N )
% 0.30/0.60 => ~ ( ( R
% 0.30/0.60 = ( 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 @ H2 ) ) ) )
% 0.30/0.60 => ( ( H
% 0.30/0.60 = ( 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 @ H2 ) ) ) ) )
% 0.30/0.60 => ( ( N
% 0.30/0.60 = ( 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 @ H2 ) ) ) ) )
% 0.30/0.60 => ~ ( heap_Time_success @ A @ C2 @ H2 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % effectE
% 0.30/0.60 thf(fact_5441_in__finite__psubset,axiom,
% 0.30/0.60 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 0.30/0.60 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A6 @ B7 ) @ ( finite_psubset @ A ) )
% 0.30/0.60 = ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.60 & ( finite_finite2 @ A @ B7 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % in_finite_psubset
% 0.30/0.60 thf(fact_5442_timeFrame_Osimps_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,N: nat] :
% 0.30/0.60 ( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.60 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % timeFrame.simps(2)
% 0.30/0.60 thf(fact_5443_execute__bind_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.60 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 0.30/0.60 = ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_bind(2)
% 0.30/0.60 thf(fact_5444_successI,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.60 => ( heap_Time_success @ A @ F @ H2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % successI
% 0.30/0.60 thf(fact_5445_success__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( heap_Time_success @ A )
% 0.30/0.60 = ( ^ [F2: heap_Time_Heap @ A,H4: heap_ext @ product_unit] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ F2 @ H4 )
% 0.30/0.60 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % success_def
% 0.30/0.60 thf(fact_5446_execute__guard_I1_J,axiom,
% 0.30/0.60 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.60 ( ~ ( P @ H2 )
% 0.30/0.60 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
% 0.30/0.60 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_guard(1)
% 0.30/0.60 thf(fact_5447_execute__assert_I2_J,axiom,
% 0.30/0.60 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ~ ( P @ X )
% 0.30/0.60 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
% 0.30/0.60 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % execute_assert(2)
% 0.30/0.60 thf(fact_5448_finite__psubset__def,axiom,
% 0.30/0.60 ! [A: $tType] :
% 0.30/0.60 ( ( finite_psubset @ A )
% 0.30/0.60 = ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.30/0.60 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.30/0.60 @ ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.60 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 0.30/0.60 & ( finite_finite2 @ A @ B6 ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % finite_psubset_def
% 0.30/0.60 thf(fact_5449_Heap__cases,axiom,
% 0.30/0.60 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 0.30/0.60 ( ! [X3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 0.30/0.60 ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H7 ) ) )
% 0.30/0.60 => ( ( heap_Time_execute @ A @ F @ H2 )
% 0.30/0.60 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Heap_cases
% 0.30/0.60 thf(fact_5450_timeFrame_Ocases,axiom,
% 0.30/0.60 ! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.30/0.60 ( ! [N4: nat,R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 0.30/0.60 ( X
% 0.30/0.60 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( 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 @ H6 @ N10 ) ) ) ) )
% 0.30/0.61 => ~ ! [N4: nat] :
% 0.30/0.61 ( X
% 0.30/0.61 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % timeFrame.cases
% 0.30/0.61 thf(fact_5451_Max_Oinfinite,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 0.30/0.61 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Max.infinite
% 0.30/0.61 thf(fact_5452_Min_Oinfinite,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 0.30/0.61 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Min.infinite
% 0.30/0.61 thf(fact_5453_guard__def,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( heap_Time_guard @ A )
% 0.30/0.61 = ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.61 ( heap_Time_Heap2 @ A
% 0.30/0.61 @ ^ [H4: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H4 ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H4 ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % guard_def
% 0.30/0.61 thf(fact_5454_timeFrame_Oelims,axiom,
% 0.30/0.61 ! [A: $tType,X: nat,Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.61 ( ( ( heap_Time_timeFrame @ A @ X @ Xa3 )
% 0.30/0.61 = Y3 )
% 0.30/0.61 => ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 0.30/0.61 ( ( Xa3
% 0.30/0.61 = ( 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 @ H6 @ N10 ) ) ) )
% 0.30/0.61 => ( Y3
% 0.30/0.61 != ( 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 @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) ) )
% 0.30/0.61 => ~ ( ( Xa3
% 0.30/0.61 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.61 => ( Y3
% 0.30/0.61 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % timeFrame.elims
% 0.30/0.61 thf(fact_5455_in__measure,axiom,
% 0.30/0.61 ! [A: $tType,X: A,Y3: A,F: A > nat] :
% 0.30/0.61 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measure @ A @ F ) )
% 0.30/0.61 = ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % in_measure
% 0.30/0.61 thf(fact_5456_pred__nat__def,axiom,
% 0.30/0.61 ( pred_nat
% 0.30/0.61 = ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.61 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.61 @ ^ [M2: nat,N3: nat] :
% 0.30/0.61 ( N3
% 0.30/0.61 = ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % pred_nat_def
% 0.30/0.61 thf(fact_5457_Min_Oeq__fold_H,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ( ( lattic643756798350308766er_Min @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] :
% 0.30/0.61 ( the2 @ A
% 0.30/0.61 @ ( finite_fold @ A @ ( option @ A )
% 0.30/0.61 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y6 ) )
% 0.30/0.61 @ ( none @ A )
% 0.30/0.61 @ A7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Min.eq_fold'
% 0.30/0.61 thf(fact_5458_sum_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( comm_monoid_add @ A )
% 0.30/0.61 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 0.30/0.61 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.eq_fold
% 0.30/0.61 thf(fact_5459_prod_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( comm_monoid_mult @ A )
% 0.30/0.61 => ( ( groups7121269368397514597t_prod @ B @ A )
% 0.30/0.61 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod.eq_fold
% 0.30/0.61 thf(fact_5460_Max_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Max.eq_fold
% 0.30/0.61 thf(fact_5461_Min_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Min.eq_fold
% 0.30/0.61 thf(fact_5462_execute__bind__case,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit] :
% 0.30/0.61 ( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 )
% 0.30/0.61 = ( 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 ) ) )
% 0.30/0.61 @ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.30/0.61 @ ^ [X2: B] :
% 0.30/0.61 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.30/0.61 @ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ A @ N3 @ ( heap_Time_execute @ A @ ( G @ X2 ) @ H5 ) ) ) )
% 0.30/0.61 @ ( heap_Time_execute @ B @ F @ H2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % execute_bind_case
% 0.30/0.61 thf(fact_5463_less__eq,axiom,
% 0.30/0.61 ! [M: nat,N: nat] :
% 0.30/0.61 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
% 0.30/0.61 = ( ord_less @ nat @ M @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % less_eq
% 0.30/0.61 thf(fact_5464_pred__nat__trancl__eq__le,axiom,
% 0.30/0.61 ! [M: nat,N: nat] :
% 0.30/0.61 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
% 0.30/0.61 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % pred_nat_trancl_eq_le
% 0.30/0.61 thf(fact_5465_Heap__Time__Monad_Obind__def,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( heap_Time_bind @ A @ B )
% 0.30/0.61 = ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
% 0.30/0.61 ( heap_Time_Heap2 @ B
% 0.30/0.61 @ ^ [H4: heap_ext @ product_unit] :
% 0.30/0.61 ( 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 ) ) )
% 0.30/0.61 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.30/0.61 @ ^ [R5: A] :
% 0.30/0.61 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.30/0.61 @ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ B @ N3 @ ( heap_Time_execute @ B @ ( G2 @ R5 ) @ H5 ) ) ) )
% 0.30/0.61 @ ( heap_Time_execute @ A @ F2 @ H4 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Heap_Time_Monad.bind_def
% 0.30/0.61 thf(fact_5466_Max_Oeq__fold_H,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ( ( lattic643756798349783984er_Max @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] :
% 0.30/0.61 ( the2 @ A
% 0.30/0.61 @ ( finite_fold @ A @ ( option @ A )
% 0.30/0.61 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y6 ) )
% 0.30/0.61 @ ( none @ A )
% 0.30/0.61 @ A7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Max.eq_fold'
% 0.30/0.61 thf(fact_5467_Sup__fin_Oeq__fold_H,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] :
% 0.30/0.61 ( the2 @ A
% 0.30/0.61 @ ( finite_fold @ A @ ( option @ A )
% 0.30/0.61 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y6 ) )
% 0.30/0.61 @ ( none @ A )
% 0.30/0.61 @ A7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.eq_fold'
% 0.30/0.61 thf(fact_5468_Inf__fin_Oeq__fold_H,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] :
% 0.30/0.61 ( the2 @ A
% 0.30/0.61 @ ( finite_fold @ A @ ( option @ A )
% 0.30/0.61 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y6 ) )
% 0.30/0.61 @ ( none @ A )
% 0.30/0.61 @ A7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.eq_fold'
% 0.30/0.61 thf(fact_5469_timeFrame_Opelims,axiom,
% 0.30/0.61 ! [A: $tType,X: nat,Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.30/0.61 ( ( ( heap_Time_timeFrame @ A @ X @ Xa3 )
% 0.30/0.61 = Y3 )
% 0.30/0.61 => ( ( 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 @ Xa3 ) )
% 0.30/0.61 => ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 0.30/0.61 ( ( Xa3
% 0.30/0.61 = ( 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 @ H6 @ N10 ) ) ) )
% 0.30/0.61 => ( ( Y3
% 0.30/0.61 = ( 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 @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) )
% 0.30/0.61 => ~ ( 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 ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) ) ) ) )
% 0.30/0.61 => ~ ( ( Xa3
% 0.30/0.61 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.61 => ( ( Y3
% 0.30/0.61 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.61 => ~ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % timeFrame.pelims
% 0.30/0.61 thf(fact_5470_Inf__fin_Osingleton,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = X ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.singleton
% 0.30/0.61 thf(fact_5471_Sup__fin_Osingleton,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = X ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.singleton
% 0.30/0.61 thf(fact_5472_inf__Sup__absorb,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,A3: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.61 => ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 0.30/0.61 = A3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inf_Sup_absorb
% 0.30/0.61 thf(fact_5473_sup__Inf__absorb,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,A3: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.61 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 )
% 0.30/0.61 = A3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sup_Inf_absorb
% 0.30/0.61 thf(fact_5474_Inf__fin_Oinsert,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.insert
% 0.30/0.61 thf(fact_5475_Sup__fin_Oinsert,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.insert
% 0.30/0.61 thf(fact_5476_Inf__fin__Min,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ( semilattice_inf @ A )
% 0.30/0.61 & ( linorder @ A ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A )
% 0.30/0.61 = ( lattic643756798350308766er_Min @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin_Min
% 0.30/0.61 thf(fact_5477_Sup__fin__Max,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ( semilattice_sup @ A )
% 0.30/0.61 & ( linorder @ A ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A )
% 0.30/0.61 = ( lattic643756798349783984er_Max @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin_Max
% 0.30/0.61 thf(fact_5478_Inf__fin__le__Sup__fin,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( lattice @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin_le_Sup_fin
% 0.30/0.61 thf(fact_5479_Sup__fin_OcoboundedI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,A3: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.coboundedI
% 0.30/0.61 thf(fact_5480_Inf__fin_OcoboundedI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,A3: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.coboundedI
% 0.30/0.61 thf(fact_5481_Inf__fin_Oin__idem,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.in_idem
% 0.30/0.61 thf(fact_5482_Sup__fin_Oin__idem,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.in_idem
% 0.30/0.61 thf(fact_5483_Sup__fin_Obounded__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
% 0.30/0.61 = ( ! [X2: A] :
% 0.30/0.61 ( ( member @ A @ X2 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.bounded_iff
% 0.30/0.61 thf(fact_5484_Inf__fin_Obounded__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 0.30/0.61 = ( ! [X2: A] :
% 0.30/0.61 ( ( member @ A @ X2 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.bounded_iff
% 0.30/0.61 thf(fact_5485_Sup__fin_OboundedI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ! [A8: A] :
% 0.30/0.61 ( ( member @ A @ A8 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ A8 @ X ) )
% 0.30/0.61 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.boundedI
% 0.30/0.61 thf(fact_5486_Sup__fin_OboundedE,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
% 0.30/0.61 => ! [A10: A] :
% 0.30/0.61 ( ( member @ A @ A10 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.boundedE
% 0.30/0.61 thf(fact_5487_Inf__fin_OboundedI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ! [A8: A] :
% 0.30/0.61 ( ( member @ A @ A8 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X @ A8 ) )
% 0.30/0.61 => ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.boundedI
% 0.30/0.61 thf(fact_5488_Inf__fin_OboundedE,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 0.30/0.61 => ! [A10: A] :
% 0.30/0.61 ( ( member @ A @ A10 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.boundedE
% 0.30/0.61 thf(fact_5489_Sup__fin__Sup,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( comple6319245703460814977attice @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 0.30/0.61 = ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin_Sup
% 0.30/0.61 thf(fact_5490_card_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( finite_card @ A )
% 0.30/0.61 = ( finite_fold @ A @ nat
% 0.30/0.61 @ ^ [Uu: A] : suc
% 0.30/0.61 @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % card.eq_fold
% 0.30/0.61 thf(fact_5491_Inf__fin__Inf,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( comple6319245703460814977attice @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 0.30/0.61 = ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin_Inf
% 0.30/0.61 thf(fact_5492_Inf__fin_Oinfinite,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 0.30/0.61 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.infinite
% 0.30/0.61 thf(fact_5493_Sup__fin_Oinfinite,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 0.30/0.61 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.infinite
% 0.30/0.61 thf(fact_5494_Sup__fin_Osubset__imp,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.subset_imp
% 0.30/0.61 thf(fact_5495_Inf__fin_Osubset__imp,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.subset_imp
% 0.30/0.61 thf(fact_5496_Inf__fin_Ohom__commute,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [H2: A > A,N6: set @ A] :
% 0.30/0.61 ( ! [X3: A,Y5: A] :
% 0.30/0.61 ( ( H2 @ ( inf_inf @ A @ X3 @ Y5 ) )
% 0.30/0.61 = ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.61 => ( ( N6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
% 0.30/0.61 = ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.hom_commute
% 0.30/0.61 thf(fact_5497_Sup__fin_Ohom__commute,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [H2: A > A,N6: set @ A] :
% 0.30/0.61 ( ! [X3: A,Y5: A] :
% 0.30/0.61 ( ( H2 @ ( sup_sup @ A @ X3 @ Y5 ) )
% 0.30/0.61 = ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.61 => ( ( N6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
% 0.30/0.61 = ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.hom_commute
% 0.30/0.61 thf(fact_5498_Inf__fin_Osubset,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.61 => ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.subset
% 0.30/0.61 thf(fact_5499_Sup__fin_Osubset,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.61 => ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B7 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.subset
% 0.30/0.61 thf(fact_5500_Inf__fin_Oinsert__not__elem,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.insert_not_elem
% 0.30/0.61 thf(fact_5501_Inf__fin_Oclosed,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ! [X3: A,Y5: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.61 => ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.closed
% 0.30/0.61 thf(fact_5502_Sup__fin_Oclosed,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ! [X3: A,Y5: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.61 => ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.closed
% 0.30/0.61 thf(fact_5503_Sup__fin_Oinsert__not__elem,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.insert_not_elem
% 0.30/0.61 thf(fact_5504_Inf__fin_Ounion,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.61 = ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.union
% 0.30/0.61 thf(fact_5505_Sup__fin_Ounion,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.61 = ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.union
% 0.30/0.61 thf(fact_5506_Inf__fin_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.eq_fold
% 0.30/0.61 thf(fact_5507_Sup__fin_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.eq_fold
% 0.30/0.61 thf(fact_5508_inf__Sup1__distrib,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( distrib_lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic5882676163264333800up_fin @ A
% 0.30/0.61 @ ( collect @ A
% 0.30/0.61 @ ^ [Uu: A] :
% 0.30/0.61 ? [A5: A] :
% 0.30/0.61 ( ( Uu
% 0.30/0.61 = ( inf_inf @ A @ X @ A5 ) )
% 0.30/0.61 & ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inf_Sup1_distrib
% 0.30/0.61 thf(fact_5509_inf__Sup2__distrib,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( distrib_lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) )
% 0.30/0.61 = ( lattic5882676163264333800up_fin @ A
% 0.30/0.61 @ ( collect @ A
% 0.30/0.61 @ ^ [Uu: A] :
% 0.30/0.61 ? [A5: A,B3: A] :
% 0.30/0.61 ( ( Uu
% 0.30/0.61 = ( inf_inf @ A @ A5 @ B3 ) )
% 0.30/0.61 & ( member @ A @ A5 @ A6 )
% 0.30/0.61 & ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inf_Sup2_distrib
% 0.30/0.61 thf(fact_5510_sup__Inf2__distrib,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( distrib_lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.61 => ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) )
% 0.30/0.61 = ( lattic7752659483105999362nf_fin @ A
% 0.30/0.61 @ ( collect @ A
% 0.30/0.61 @ ^ [Uu: A] :
% 0.30/0.61 ? [A5: A,B3: A] :
% 0.30/0.61 ( ( Uu
% 0.30/0.61 = ( sup_sup @ A @ A5 @ B3 ) )
% 0.30/0.61 & ( member @ A @ A5 @ A6 )
% 0.30/0.61 & ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sup_Inf2_distrib
% 0.30/0.61 thf(fact_5511_sup__Inf1__distrib,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( distrib_lattice @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 0.30/0.61 = ( lattic7752659483105999362nf_fin @ A
% 0.30/0.61 @ ( collect @ A
% 0.30/0.61 @ ^ [Uu: A] :
% 0.30/0.61 ? [A5: A] :
% 0.30/0.61 ( ( Uu
% 0.30/0.61 = ( sup_sup @ A @ X @ A5 ) )
% 0.30/0.61 & ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sup_Inf1_distrib
% 0.30/0.61 thf(fact_5512_Inf__fin_Oinsert__remove,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = X ) )
% 0.30/0.61 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.insert_remove
% 0.30/0.61 thf(fact_5513_Inf__fin_Oremove,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_inf @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 0.30/0.61 = X ) )
% 0.30/0.61 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 0.30/0.61 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Inf_fin.remove
% 0.30/0.61 thf(fact_5514_Sup__fin_Oinsert__remove,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = X ) )
% 0.30/0.61 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 0.30/0.61 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.insert_remove
% 0.30/0.61 thf(fact_5515_Sup__fin_Oremove,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semilattice_sup @ A )
% 0.30/0.61 => ! [A6: set @ A,X: A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( member @ A @ X @ A6 )
% 0.30/0.61 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 0.30/0.61 = X ) )
% 0.30/0.61 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 0.30/0.61 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_fin.remove
% 0.30/0.61 thf(fact_5516_raise__def,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( heap_Time_raise @ A )
% 0.30/0.61 = ( ^ [S6: list @ char] :
% 0.30/0.61 ( heap_Time_Heap2 @ A
% 0.30/0.61 @ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % raise_def
% 0.30/0.61 thf(fact_5517_Gcd__fin_Oeq__fold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ( ( semiring_gcd_Gcd_fin @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Gcd_fin.eq_fold
% 0.30/0.61 thf(fact_5518_funpow__inj__finite,axiom,
% 0.30/0.61 ! [A: $tType,P5: A > A,X: A] :
% 0.30/0.61 ( ( inj_on @ A @ A @ P5 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.61 => ( ( finite_finite2 @ A
% 0.30/0.61 @ ( collect @ A
% 0.30/0.61 @ ^ [Y6: A] :
% 0.30/0.61 ? [N3: nat] :
% 0.30/0.61 ( Y6
% 0.30/0.61 = ( compow @ ( A > A ) @ N3 @ P5 @ X ) ) ) )
% 0.30/0.61 => ~ ! [N4: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 0.30/0.61 => ( ( compow @ ( A > A ) @ N4 @ P5 @ X )
% 0.30/0.61 != X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % funpow_inj_finite
% 0.30/0.61 thf(fact_5519_inj__uminus,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ordered_ab_group_add @ A )
% 0.30/0.61 => ! [A6: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_uminus
% 0.30/0.61 thf(fact_5520_raise__bind,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,E: list @ char,F: B > ( heap_Time_Heap @ A )] :
% 0.30/0.61 ( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E ) @ F )
% 0.30/0.61 = ( heap_Time_raise @ A @ E ) ) ).
% 0.30/0.61
% 0.30/0.61 % raise_bind
% 0.30/0.61 thf(fact_5521_Gcd__fin_Oempty,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 = ( zero_zero @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Gcd_fin.empty
% 0.30/0.61 thf(fact_5522_Gcd__fin_Oinfinite,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 0.30/0.61 = ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Gcd_fin.infinite
% 0.30/0.61 thf(fact_5523_is__unit__Gcd__fin__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) @ ( one_one @ A ) )
% 0.30/0.61 = ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 0.30/0.61 = ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_unit_Gcd_fin_iff
% 0.30/0.61 thf(fact_5524_inj__mult__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( idom @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.61 = ( A3
% 0.30/0.61 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_mult_left
% 0.30/0.61 thf(fact_5525_inj__divide__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( inj_on @ A @ A
% 0.30/0.61 @ ^ [B3: A] : ( divide_divide @ A @ B3 @ A3 )
% 0.30/0.61 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.61 = ( A3
% 0.30/0.61 != ( zero_zero @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_divide_right
% 0.30/0.61 thf(fact_5526_inj__fn,axiom,
% 0.30/0.61 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.61 ( ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.61 => ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_fn
% 0.30/0.61 thf(fact_5527_subset__image__inj,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,S3: set @ A,F: B > A,T6: set @ B] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F @ T6 ) )
% 0.30/0.61 = ( ? [U5: set @ B] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ B ) @ U5 @ T6 )
% 0.30/0.61 & ( inj_on @ B @ A @ F @ U5 )
% 0.30/0.61 & ( S3
% 0.30/0.61 = ( image @ B @ A @ F @ U5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % subset_image_inj
% 0.30/0.61 thf(fact_5528_inj__list__encode,axiom,
% 0.30/0.61 ! [A6: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_list_encode
% 0.30/0.61 thf(fact_5529_inj__prod__encode,axiom,
% 0.30/0.61 ! [A6: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_prod_encode
% 0.30/0.61 thf(fact_5530_inj__Suc,axiom,
% 0.30/0.61 ! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_Suc
% 0.30/0.61 thf(fact_5531_inj__on__of__nat,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_char_0 @ A )
% 0.30/0.61 => ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_of_nat
% 0.30/0.61 thf(fact_5532_inj__int__encode,axiom,
% 0.30/0.61 ! [A6: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_int_encode
% 0.30/0.61 thf(fact_5533_inj__int__decode,axiom,
% 0.30/0.61 ! [A6: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_int_decode
% 0.30/0.61 thf(fact_5534_linorder__inj__onI,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( order @ A )
% 0.30/0.61 => ! [A6: set @ A,F: A > B] :
% 0.30/0.61 ( ! [X3: A,Y5: A] :
% 0.30/0.61 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.61 => ( ( member @ A @ X3 @ A6 )
% 0.30/0.61 => ( ( member @ A @ Y5 @ A6 )
% 0.30/0.61 => ( ( F @ X3 )
% 0.30/0.61 != ( F @ Y5 ) ) ) ) )
% 0.30/0.61 => ( ! [X3: A,Y5: A] :
% 0.30/0.61 ( ( member @ A @ X3 @ A6 )
% 0.30/0.61 => ( ( member @ A @ Y5 @ A6 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.61 | ( ord_less_eq @ A @ Y5 @ X3 ) ) ) )
% 0.30/0.61 => ( inj_on @ A @ B @ F @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % linorder_inj_onI
% 0.30/0.61 thf(fact_5535_inj__on__mult,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semidom_divide @ A )
% 0.30/0.61 => ! [A3: A,A6: set @ A] :
% 0.30/0.61 ( ( A3
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 => ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_mult
% 0.30/0.61 thf(fact_5536_linorder__injI,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [F: A > B] :
% 0.30/0.61 ( ! [X3: A,Y5: A] :
% 0.30/0.61 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.61 => ( ( F @ X3 )
% 0.30/0.61 != ( F @ Y5 ) ) )
% 0.30/0.61 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % linorder_injI
% 0.30/0.61 thf(fact_5537_inj__on__strict__subset,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,F: A > B,B7: set @ A,A6: set @ A] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F @ B7 )
% 0.30/0.61 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.61 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ ( image @ A @ B @ F @ B7 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_strict_subset
% 0.30/0.61 thf(fact_5538_inj__of__nat,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_char_0 @ A )
% 0.30/0.61 => ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_of_nat
% 0.30/0.61 thf(fact_5539_inj__on__diff__nat,axiom,
% 0.30/0.61 ! [N6: set @ nat,K: nat] :
% 0.30/0.61 ( ! [N4: nat] :
% 0.30/0.61 ( ( member @ nat @ N4 @ N6 )
% 0.30/0.61 => ( ord_less_eq @ nat @ K @ N4 ) )
% 0.30/0.61 => ( inj_on @ nat @ nat
% 0.30/0.61 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
% 0.30/0.61 @ N6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_diff_nat
% 0.30/0.61 thf(fact_5540_inj__on__set__encode,axiom,
% 0.30/0.61 inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_set_encode
% 0.30/0.61 thf(fact_5541_inj__on__iff__surj,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,A6: set @ A,A11: set @ B] :
% 0.30/0.61 ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( ? [F2: A > B] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F2 @ A6 )
% 0.30/0.61 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ A11 ) ) )
% 0.30/0.61 = ( ? [G2: B > A] :
% 0.30/0.61 ( ( image @ B @ A @ G2 @ A11 )
% 0.30/0.61 = A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_iff_surj
% 0.30/0.61 thf(fact_5542_pigeonhole,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
% 0.30/0.61 ( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( finite_card @ B @ A6 ) )
% 0.30/0.61 => ~ ( inj_on @ B @ A @ F @ A6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % pigeonhole
% 0.30/0.61 thf(fact_5543_effect__raiseE,axiom,
% 0.30/0.61 ! [A: $tType,X: list @ char,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 0.30/0.61 ~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H2 @ H @ R @ N ) ).
% 0.30/0.61
% 0.30/0.61 % effect_raiseE
% 0.30/0.61 thf(fact_5544_finite__imp__nat__seg__image__inj__on,axiom,
% 0.30/0.61 ! [A: $tType,A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ? [N4: nat,F3: nat > A] :
% 0.30/0.61 ( ( A6
% 0.30/0.61 = ( image @ nat @ A @ F3
% 0.30/0.61 @ ( collect @ nat
% 0.30/0.61 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) )
% 0.30/0.61 & ( inj_on @ nat @ A @ F3
% 0.30/0.61 @ ( collect @ nat
% 0.30/0.61 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % finite_imp_nat_seg_image_inj_on
% 0.30/0.61 thf(fact_5545_finite__imp__inj__to__nat__seg,axiom,
% 0.30/0.61 ! [A: $tType,A6: set @ A] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ? [F3: A > nat,N4: nat] :
% 0.30/0.61 ( ( ( image @ A @ nat @ F3 @ A6 )
% 0.30/0.61 = ( collect @ nat
% 0.30/0.61 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) )
% 0.30/0.61 & ( inj_on @ A @ nat @ F3 @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % finite_imp_inj_to_nat_seg
% 0.30/0.61 thf(fact_5546_inj__image__Compl__subset,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.61 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_image_Compl_subset
% 0.30/0.61 thf(fact_5547_inj__on__nth,axiom,
% 0.30/0.61 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 0.30/0.61 ( ( distinct @ A @ Xs )
% 0.30/0.61 => ( ! [X3: nat] :
% 0.30/0.61 ( ( member @ nat @ X3 @ I6 )
% 0.30/0.61 => ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 0.30/0.61 => ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_nth
% 0.30/0.61 thf(fact_5548_infinite__iff__countable__subset,axiom,
% 0.30/0.61 ! [A: $tType,S3: set @ A] :
% 0.30/0.61 ( ( ~ ( finite_finite2 @ A @ S3 ) )
% 0.30/0.61 = ( ? [F2: nat > A] :
% 0.30/0.61 ( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.61 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % infinite_iff_countable_subset
% 0.30/0.61 thf(fact_5549_infinite__countable__subset,axiom,
% 0.30/0.61 ! [A: $tType,S3: set @ A] :
% 0.30/0.61 ( ~ ( finite_finite2 @ A @ S3 )
% 0.30/0.61 => ? [F3: nat > A] :
% 0.30/0.61 ( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.61 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % infinite_countable_subset
% 0.30/0.61 thf(fact_5550_inj__on__funpow__least,axiom,
% 0.30/0.61 ! [A: $tType,N: nat,F: A > A,S: A] :
% 0.30/0.61 ( ( ( compow @ ( A > A ) @ N @ F @ S )
% 0.30/0.61 = S )
% 0.30/0.61 => ( ! [M6: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M6 )
% 0.30/0.61 => ( ( ord_less @ nat @ M6 @ N )
% 0.30/0.61 => ( ( compow @ ( A > A ) @ M6 @ F @ S )
% 0.30/0.61 != S ) ) )
% 0.30/0.61 => ( inj_on @ nat @ A
% 0.30/0.61 @ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F @ S )
% 0.30/0.61 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_funpow_least
% 0.30/0.61 thf(fact_5551_inj__on__iff__card__le,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.61 => ( ( ? [F2: A > B] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F2 @ A6 )
% 0.30/0.61 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ B7 ) ) )
% 0.30/0.61 = ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % inj_on_iff_card_le
% 0.30/0.61 thf(fact_5552_card__inj__on__le,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,F: A > B,A6: set @ A,B7: set @ B] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F @ A6 )
% 0.30/0.61 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ B7 )
% 0.30/0.61 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.61 => ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % card_inj_on_le
% 0.30/0.61 thf(fact_5553_card__le__inj,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
% 0.30/0.61 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.61 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.61 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) )
% 0.30/0.61 => ? [F3: A > B] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A6 ) @ B7 )
% 0.30/0.61 & ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % card_le_inj
% 0.30/0.61 thf(fact_5554_execute__raise,axiom,
% 0.30/0.61 ! [A: $tType,S: list @ char] :
% 0.30/0.61 ( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
% 0.30/0.61 = ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % execute_raise
% 0.30/0.61 thf(fact_5555_map__sorted__distinct__set__unique,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [F: B > A,Xs: list @ B,Ys2: list @ B] :
% 0.30/0.61 ( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys2 ) ) )
% 0.30/0.61 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.61 => ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.61 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys2 ) )
% 0.30/0.61 => ( ( distinct @ A @ ( map @ B @ A @ F @ Ys2 ) )
% 0.30/0.61 => ( ( ( set2 @ B @ Xs )
% 0.30/0.61 = ( set2 @ B @ Ys2 ) )
% 0.30/0.61 => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % map_sorted_distinct_set_unique
% 0.30/0.61 thf(fact_5556_Gcd__fin__0__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 0.30/0.61 = ( zero_zero @ A ) )
% 0.30/0.61 = ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.61 & ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Gcd_fin_0_iff
% 0.30/0.61 thf(fact_5557_greaterThan__Suc,axiom,
% 0.30/0.61 ! [K: nat] :
% 0.30/0.61 ( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
% 0.30/0.61 = ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % greaterThan_Suc
% 0.30/0.61 thf(fact_5558_last__list__update,axiom,
% 0.30/0.61 ! [A: $tType,Xs: list @ A,K: nat,X: A] :
% 0.30/0.61 ( ( Xs
% 0.30/0.61 != ( nil @ A ) )
% 0.30/0.61 => ( ( ( K
% 0.30/0.61 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.30/0.61 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.30/0.61 = X ) )
% 0.30/0.61 & ( ( K
% 0.30/0.61 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.30/0.61 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.30/0.61 = ( last @ A @ Xs ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % last_list_update
% 0.30/0.61 thf(fact_5559_last__upt,axiom,
% 0.30/0.61 ! [I: nat,J: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ I @ J )
% 0.30/0.61 => ( ( last @ nat @ ( upt @ I @ J ) )
% 0.30/0.61 = ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % last_upt
% 0.30/0.61 thf(fact_5560_greaterThan__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ! [I: A,K: A] :
% 0.30/0.61 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 0.30/0.61 = ( ord_less @ A @ K @ I ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % greaterThan_iff
% 0.30/0.61 thf(fact_5561_greaterThan__subset__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [X: A,Y3: A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y3 ) )
% 0.30/0.61 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % greaterThan_subset_iff
% 0.30/0.61 thf(fact_5562_last__replicate,axiom,
% 0.30/0.61 ! [A: $tType,N: nat,X: A] :
% 0.30/0.61 ( ( N
% 0.30/0.61 != ( zero_zero @ nat ) )
% 0.30/0.61 => ( ( last @ A @ ( replicate @ A @ N @ X ) )
% 0.30/0.61 = X ) ) ).
% 0.30/0.61
% 0.30/0.61 % last_replicate
% 0.30/0.61 thf(fact_5563_Compl__greaterThan,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [K: A] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 0.30/0.61 = ( set_ord_atMost @ A @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_greaterThan
% 0.30/0.61 thf(fact_5564_Compl__atMost,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [K: A] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 0.30/0.61 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_atMost
% 0.30/0.61 thf(fact_5565_Sup__greaterThanAtLeast,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( comple6319245703460814977attice @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.30/0.61 => ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
% 0.30/0.61 = ( top_top @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sup_greaterThanAtLeast
% 0.30/0.61 thf(fact_5566_image__uminus__greaterThan,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ordered_ab_group_add @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
% 0.30/0.61 = ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % image_uminus_greaterThan
% 0.30/0.61 thf(fact_5567_image__uminus__lessThan,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ordered_ab_group_add @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
% 0.30/0.61 = ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % image_uminus_lessThan
% 0.30/0.61 thf(fact_5568_last__drop,axiom,
% 0.30/0.61 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.61 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.61 => ( ( last @ A @ ( drop @ A @ N @ Xs ) )
% 0.30/0.61 = ( last @ A @ Xs ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % last_drop
% 0.30/0.61 thf(fact_5569_greaterThan__def,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ( ( set_ord_greaterThan @ A )
% 0.30/0.61 = ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % greaterThan_def
% 0.30/0.61 thf(fact_5570_ivl__disj__un__one_I5_J,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [L: A,U: A] :
% 0.30/0.61 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.61 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.30/0.61 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ivl_disj_un_one(5)
% 0.30/0.61 thf(fact_5571_greaterThan__0,axiom,
% 0.30/0.61 ( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
% 0.30/0.61 = ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % greaterThan_0
% 0.30/0.61 thf(fact_5572_last__conv__nth,axiom,
% 0.30/0.61 ! [A: $tType,Xs: list @ A] :
% 0.30/0.61 ( ( Xs
% 0.30/0.61 != ( nil @ A ) )
% 0.30/0.61 => ( ( last @ A @ Xs )
% 0.30/0.61 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % last_conv_nth
% 0.30/0.61 thf(fact_5573_interval__cases,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit6923001295902523014norder @ A )
% 0.30/0.61 => ! [S3: set @ A] :
% 0.30/0.61 ( ! [A8: A,B5: A,X3: A] :
% 0.30/0.61 ( ( member @ A @ A8 @ S3 )
% 0.30/0.61 => ( ( member @ A @ B5 @ S3 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ A8 @ X3 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ X3 @ B5 )
% 0.30/0.61 => ( member @ A @ X3 @ S3 ) ) ) ) )
% 0.30/0.61 => ? [A8: A,B5: A] :
% 0.30/0.61 ( ( S3
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( top_top @ ( set @ A ) ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_ord_lessThan @ A @ B5 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_ord_atMost @ A @ B5 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_ord_greaterThan @ A @ A8 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_ord_atLeast @ A @ A8 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_or5935395276787703475ssThan @ A @ A8 @ B5 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_or3652927894154168847AtMost @ A @ A8 @ B5 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_or7035219750837199246ssThan @ A @ A8 @ B5 ) )
% 0.30/0.61 | ( S3
% 0.30/0.61 = ( set_or1337092689740270186AtMost @ A @ A8 @ B5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % interval_cases
% 0.30/0.61 thf(fact_5574_sum__mult__sum__if__inj,axiom,
% 0.30/0.61 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.61 ( ( semiring_0 @ B )
% 0.30/0.61 => ! [F: A > B,G: C > B,A6: set @ A,B7: set @ C] :
% 0.30/0.61 ( ( inj_on @ ( product_prod @ A @ C ) @ B
% 0.30/0.61 @ ( product_case_prod @ A @ C @ B
% 0.30/0.61 @ ^ [A5: A,B3: C] : ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 0.30/0.61 @ ( product_Sigma @ A @ C @ A6
% 0.30/0.61 @ ^ [Uu: A] : B7 ) )
% 0.30/0.61 => ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
% 0.30/0.61 = ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
% 0.30/0.61 @ ( collect @ B
% 0.30/0.61 @ ^ [Uu: B] :
% 0.30/0.61 ? [A5: A,B3: C] :
% 0.30/0.61 ( ( Uu
% 0.30/0.61 = ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 0.30/0.61 & ( member @ A @ A5 @ A6 )
% 0.30/0.61 & ( member @ C @ B3 @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mult_sum_if_inj
% 0.30/0.61 thf(fact_5575_arg__min__inj__eq,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( order @ B )
% 0.30/0.61 => ! [F: A > B,P: A > $o,A3: A] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F @ ( collect @ A @ P ) )
% 0.30/0.61 => ( ( P @ A3 )
% 0.30/0.61 => ( ! [Y5: A] :
% 0.30/0.61 ( ( P @ Y5 )
% 0.30/0.61 => ( ord_less_eq @ B @ ( F @ A3 ) @ ( F @ Y5 ) ) )
% 0.30/0.61 => ( ( lattices_ord_arg_min @ A @ B @ F @ P )
% 0.30/0.61 = A3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_inj_eq
% 0.30/0.61 thf(fact_5576_atLeast__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ! [I: A,K: A] :
% 0.30/0.61 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 0.30/0.61 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_iff
% 0.30/0.61 thf(fact_5577_atLeast__0,axiom,
% 0.30/0.61 ( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
% 0.30/0.61 = ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_0
% 0.30/0.61 thf(fact_5578_atLeast__subset__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ! [X: A,Y3: A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y3 ) )
% 0.30/0.61 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_subset_iff
% 0.30/0.61 thf(fact_5579_Compl__atLeast,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [K: A] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 0.30/0.61 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_atLeast
% 0.30/0.61 thf(fact_5580_Compl__lessThan,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [K: A] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 0.30/0.61 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_lessThan
% 0.30/0.61 thf(fact_5581_Icc__subset__Ici__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ! [L: A,H2: A,L4: A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) )
% 0.30/0.61 = ( ~ ( ord_less_eq @ A @ L @ H2 )
% 0.30/0.61 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Icc_subset_Ici_iff
% 0.30/0.61 thf(fact_5582_image__uminus__atMost,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ordered_ab_group_add @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
% 0.30/0.61 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % image_uminus_atMost
% 0.30/0.61 thf(fact_5583_image__uminus__atLeast,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ordered_ab_group_add @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
% 0.30/0.61 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % image_uminus_atLeast
% 0.30/0.61 thf(fact_5584_arg__min__nat__lemma,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.30/0.61 ( ( P @ K )
% 0.30/0.61 => ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
% 0.30/0.61 & ! [Y4: A] :
% 0.30/0.61 ( ( P @ Y4 )
% 0.30/0.61 => ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y4 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_nat_lemma
% 0.30/0.61 thf(fact_5585_arg__min__nat__le,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 0.30/0.61 ( ( P @ X )
% 0.30/0.61 => ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_nat_le
% 0.30/0.61 thf(fact_5586_atLeast__def,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ( ( set_ord_atLeast @ A )
% 0.30/0.61 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_def
% 0.30/0.61 thf(fact_5587_arg__min__equality,axiom,
% 0.30/0.61 ! [A: $tType,C: $tType] :
% 0.30/0.61 ( ( order @ A )
% 0.30/0.61 => ! [P: C > $o,K: C,F: C > A] :
% 0.30/0.61 ( ( P @ K )
% 0.30/0.61 => ( ! [X3: C] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 => ( ord_less_eq @ A @ ( F @ K ) @ ( F @ X3 ) ) )
% 0.30/0.61 => ( ( F @ ( lattices_ord_arg_min @ C @ A @ F @ P ) )
% 0.30/0.61 = ( F @ K ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_equality
% 0.30/0.61 thf(fact_5588_arg__minI,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( ord @ B )
% 0.30/0.61 => ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
% 0.30/0.61 ( ( P @ X )
% 0.30/0.61 => ( ! [Y5: A] :
% 0.30/0.61 ( ( P @ Y5 )
% 0.30/0.61 => ~ ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X ) ) )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 => ( ! [Y4: A] :
% 0.30/0.61 ( ( P @ Y4 )
% 0.30/0.61 => ~ ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 0.30/0.61 => ( Q @ X3 ) ) )
% 0.30/0.61 => ( Q @ ( lattices_ord_arg_min @ A @ B @ F @ P ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_minI
% 0.30/0.61 thf(fact_5589_atLeast__Suc__greaterThan,axiom,
% 0.30/0.61 ! [K: nat] :
% 0.30/0.61 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 0.30/0.61 = ( set_ord_greaterThan @ nat @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_Suc_greaterThan
% 0.30/0.61 thf(fact_5590_Ex__inj__on__UNION__Sigma,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,A6: B > ( set @ A ),I6: set @ B] :
% 0.30/0.61 ? [F3: A > ( product_prod @ B @ A )] :
% 0.30/0.61 ( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) )
% 0.30/0.61 & ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) ) @ ( product_Sigma @ B @ A @ I6 @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Ex_inj_on_UNION_Sigma
% 0.30/0.61 thf(fact_5591_ivl__disj__un__one_I8_J,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [L: A,U: A] :
% 0.30/0.61 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.61 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.30/0.61 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ivl_disj_un_one(8)
% 0.30/0.61 thf(fact_5592_Ici__subset__Ioi__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 0.30/0.61 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Ici_subset_Ioi_iff
% 0.30/0.61 thf(fact_5593_card__cartesian__product,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
% 0.30/0.61 ( ( finite_card @ ( product_prod @ A @ B )
% 0.30/0.61 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.61 @ ^ [Uu: A] : B7 ) )
% 0.30/0.61 = ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % card_cartesian_product
% 0.30/0.61 thf(fact_5594_ivl__disj__un__one_I7_J,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [L: A,U: A] :
% 0.30/0.61 ( ( ord_less_eq @ A @ L @ U )
% 0.30/0.61 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.30/0.61 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ivl_disj_un_one(7)
% 0.30/0.61 thf(fact_5595_ivl__disj__un__one_I6_J,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [L: A,U: A] :
% 0.30/0.61 ( ( ord_less @ A @ L @ U )
% 0.30/0.61 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.30/0.61 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ivl_disj_un_one(6)
% 0.30/0.61 thf(fact_5596_lists__length__Suc__eq,axiom,
% 0.30/0.61 ! [A: $tType,A6: set @ A,N: nat] :
% 0.30/0.61 ( ( collect @ ( list @ A )
% 0.30/0.61 @ ^ [Xs3: list @ A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 0.30/0.61 & ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.61 = ( suc @ N ) ) ) )
% 0.30/0.61 = ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
% 0.30/0.61 @ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
% 0.30/0.61 @ ^ [Xs3: list @ A,N3: A] : ( cons @ A @ N3 @ Xs3 ) )
% 0.30/0.61 @ ( product_Sigma @ ( list @ A ) @ A
% 0.30/0.61 @ ( collect @ ( list @ A )
% 0.30/0.61 @ ^ [Xs3: list @ A] :
% 0.30/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 0.30/0.61 & ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.61 = N ) ) )
% 0.30/0.61 @ ^ [Uu: list @ A] : A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % lists_length_Suc_eq
% 0.30/0.61 thf(fact_5597_arg__min__on__def,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ( ( lattic7623131987881927897min_on @ B @ A )
% 0.30/0.61 = ( ^ [F2: B > A,S7: set @ B] :
% 0.30/0.61 ( lattices_ord_arg_min @ B @ A @ F2
% 0.30/0.61 @ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_on_def
% 0.30/0.61 thf(fact_5598_atLeast__Suc,axiom,
% 0.30/0.61 ! [K: nat] :
% 0.30/0.61 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 0.30/0.61 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % atLeast_Suc
% 0.30/0.61 thf(fact_5599_Compl__Times__UNIV1,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,A6: set @ B] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 0.30/0.61 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.30/0.61 @ ^ [Uu: A] : A6 ) )
% 0.30/0.61 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.30/0.61 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_Times_UNIV1
% 0.30/0.61 thf(fact_5600_Compl__Times__UNIV2,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,A6: set @ A] :
% 0.30/0.61 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 0.30/0.61 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.61 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 0.30/0.61 = ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A6 )
% 0.30/0.61 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Compl_Times_UNIV2
% 0.30/0.61 thf(fact_5601_mult__inj__if__coprime__nat,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,F: A > nat,A6: set @ A,G: B > nat,B7: set @ B] :
% 0.30/0.61 ( ( inj_on @ A @ nat @ F @ A6 )
% 0.30/0.61 => ( ( inj_on @ B @ nat @ G @ B7 )
% 0.30/0.61 => ( ! [A8: A,B5: B] :
% 0.30/0.61 ( ( member @ A @ A8 @ A6 )
% 0.30/0.61 => ( ( member @ B @ B5 @ B7 )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ nat @ ( F @ A8 ) @ ( G @ B5 ) ) ) )
% 0.30/0.61 => ( inj_on @ ( product_prod @ A @ B ) @ nat
% 0.30/0.61 @ ( product_case_prod @ A @ B @ nat
% 0.30/0.61 @ ^ [A5: A,B3: B] : ( times_times @ nat @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 0.30/0.61 @ ( product_Sigma @ A @ B @ A6
% 0.30/0.61 @ ^ [Uu: A] : B7 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_inj_if_coprime_nat
% 0.30/0.61 thf(fact_5602_coprime__mult__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.61 = ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mult_left_iff
% 0.30/0.61 thf(fact_5603_coprime__mult__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.61 = ( ( algebr8660921524188924756oprime @ A @ C2 @ A3 )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mult_right_iff
% 0.30/0.61 thf(fact_5604_coprime__minus__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_minus_left_iff
% 0.30/0.61 thf(fact_5605_coprime__minus__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_minus_right_iff
% 0.30/0.61 thf(fact_5606_coprime__self,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
% 0.30/0.61 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_self
% 0.30/0.61 thf(fact_5607_coprime__mod__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( A3
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B2 @ A3 ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mod_right_iff
% 0.30/0.61 thf(fact_5608_coprime__mod__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( euclid3725896446679973847miring @ A )
% 0.30/0.61 => ! [B2: A,A3: A] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mod_left_iff
% 0.30/0.61 thf(fact_5609_coprime__power__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A,N: nat] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B2 @ N ) )
% 0.30/0.61 = ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 | ( N
% 0.30/0.61 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_power_right_iff
% 0.30/0.61 thf(fact_5610_coprime__power__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,N: nat,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
% 0.30/0.61 = ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 | ( N
% 0.30/0.61 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_power_left_iff
% 0.30/0.61 thf(fact_5611_coprime__imp__gcd__eq__1,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.61 = ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_imp_gcd_eq_1
% 0.30/0.61 thf(fact_5612_coprime__0__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.61 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_0_right_iff
% 0.30/0.61 thf(fact_5613_coprime__0__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.61 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_0_left_iff
% 0.30/0.61 thf(fact_5614_coprime__mult__self__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.61 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mult_self_right_iff
% 0.30/0.61 thf(fact_5615_coprime__mult__self__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.61 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_mult_self_left_iff
% 0.30/0.61 thf(fact_5616_is__unit__gcd,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_unit_gcd
% 0.30/0.61 thf(fact_5617_arg__min__natI,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 0.30/0.61 ( ( P @ K )
% 0.30/0.61 => ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_natI
% 0.30/0.61 thf(fact_5618_coprime__add__one__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_add_one_left
% 0.30/0.61 thf(fact_5619_coprime__add__one__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_add_one_right
% 0.30/0.61 thf(fact_5620_coprime__diff__one__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ring_gcd @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_diff_one_left
% 0.30/0.61 thf(fact_5621_coprime__doff__one__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ring_gcd @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_doff_one_right
% 0.30/0.61 thf(fact_5622_coprime__dvd__mult__right__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.61 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_dvd_mult_right_iff
% 0.30/0.61 thf(fact_5623_coprime__dvd__mult__left__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.61 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_dvd_mult_left_iff
% 0.30/0.61 thf(fact_5624_divides__mult,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ B2 @ C2 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divides_mult
% 0.30/0.61 thf(fact_5625_coprime__Suc__0__left,axiom,
% 0.30/0.61 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_Suc_0_left
% 0.30/0.61 thf(fact_5626_coprime__Suc__0__right,axiom,
% 0.30/0.61 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_Suc_0_right
% 0.30/0.61 thf(fact_5627_coprime__1__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_1_left
% 0.30/0.61 thf(fact_5628_coprime__1__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_1_right
% 0.30/0.61 thf(fact_5629_coprime__Suc__left__nat,axiom,
% 0.30/0.61 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_Suc_left_nat
% 0.30/0.61 thf(fact_5630_coprime__Suc__right__nat,axiom,
% 0.30/0.61 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_Suc_right_nat
% 0.30/0.61 thf(fact_5631_coprime__commute,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A )
% 0.30/0.61 = ( ^ [B3: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_commute
% 0.30/0.61 thf(fact_5632_coprime__divisors,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ B2 @ D2 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_divisors
% 0.30/0.61 thf(fact_5633_coprime__crossproduct__nat,axiom,
% 0.30/0.61 ! [A3: nat,D2: nat,B2: nat,C2: nat] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ nat @ A3 @ D2 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ nat @ B2 @ C2 )
% 0.30/0.61 => ( ( ( times_times @ nat @ A3 @ C2 )
% 0.30/0.61 = ( times_times @ nat @ B2 @ D2 ) )
% 0.30/0.61 = ( ( A3 = B2 )
% 0.30/0.61 & ( C2 = D2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_crossproduct_nat
% 0.30/0.61 thf(fact_5634_is__unit__right__imp__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [B2: A,A3: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_unit_right_imp_coprime
% 0.30/0.61 thf(fact_5635_is__unit__left__imp__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_unit_left_imp_coprime
% 0.30/0.61 thf(fact_5636_coprime__common__divisor,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_common_divisor
% 0.30/0.61 thf(fact_5637_coprime__absorb__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [Y3: A,X: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ Y3 @ X )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
% 0.30/0.61 = ( dvd_dvd @ A @ Y3 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_absorb_right
% 0.30/0.61 thf(fact_5638_coprime__imp__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [C2: A,D2: A,A3: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 0.30/0.61 => ( ! [E2: A] :
% 0.30/0.61 ( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
% 0.30/0.61 => ( ( dvd_dvd @ A @ E2 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ E2 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ E2 @ C2 ) ) ) )
% 0.30/0.61 => ( ! [E2: A] :
% 0.30/0.61 ( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
% 0.30/0.61 => ( ( dvd_dvd @ A @ E2 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ E2 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ E2 @ D2 ) ) ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_imp_coprime
% 0.30/0.61 thf(fact_5639_coprime__absorb__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [X: A,Y3: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ X @ Y3 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
% 0.30/0.61 = ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_absorb_left
% 0.30/0.61 thf(fact_5640_not__coprimeI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.30/0.61 => ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.61 => ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % not_coprimeI
% 0.30/0.61 thf(fact_5641_not__coprimeE,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.61 => ~ ! [C3: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ C3 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C3 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % not_coprimeE
% 0.30/0.61 thf(fact_5642_coprime__def,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A )
% 0.30/0.61 = ( ^ [A5: A,B3: A] :
% 0.30/0.61 ! [C4: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ C4 @ A5 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C4 @ B3 )
% 0.30/0.61 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_def
% 0.30/0.61 thf(fact_5643_coprimeI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( algebraic_semidom @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ! [C3: A] :
% 0.30/0.61 ( ( dvd_dvd @ A @ C3 @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ A @ C3 @ B2 )
% 0.30/0.61 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprimeI
% 0.30/0.61 thf(fact_5644_coprime__common__divisor__nat,axiom,
% 0.30/0.61 ! [A3: nat,B2: nat,X: nat] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
% 0.30/0.61 => ( ( dvd_dvd @ nat @ X @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ nat @ X @ B2 )
% 0.30/0.61 => ( X
% 0.30/0.61 = ( one_one @ nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_common_divisor_nat
% 0.30/0.61 thf(fact_5645_gcd__eq__1__imp__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.61 = ( one_one @ A ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_eq_1_imp_coprime
% 0.30/0.61 thf(fact_5646_coprime__iff__gcd__eq__1,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A )
% 0.30/0.61 = ( ^ [A5: A,B3: A] :
% 0.30/0.61 ( ( gcd_gcd @ A @ A5 @ B3 )
% 0.30/0.61 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_iff_gcd_eq_1
% 0.30/0.61 thf(fact_5647_gcd__mult__right__right__cancel,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.61 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_mult_right_right_cancel
% 0.30/0.61 thf(fact_5648_gcd__mult__right__left__cancel,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 0.30/0.61 => ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.61 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_mult_right_left_cancel
% 0.30/0.61 thf(fact_5649_gcd__mult__left__right__cancel,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 0.30/0.61 => ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
% 0.30/0.61 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_mult_left_right_cancel
% 0.30/0.61 thf(fact_5650_gcd__mult__left__left__cancel,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [B2: A,C2: A,A3: A] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 0.30/0.61 => ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 )
% 0.30/0.61 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_mult_left_left_cancel
% 0.30/0.61 thf(fact_5651_mult__mod__cancel__right,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.61 & ( semiring_gcd @ A ) )
% 0.30/0.61 => ! [A3: A,N: A,M: A,B2: A] :
% 0.30/0.61 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M )
% 0.30/0.61 = ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M ) )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ M @ N )
% 0.30/0.61 => ( ( modulo_modulo @ A @ A3 @ M )
% 0.30/0.61 = ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_mod_cancel_right
% 0.30/0.61 thf(fact_5652_mult__mod__cancel__left,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.61 & ( semiring_gcd @ A ) )
% 0.30/0.61 => ! [N: A,A3: A,M: A,B2: A] :
% 0.30/0.61 ( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M )
% 0.30/0.61 = ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M ) )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ A @ M @ N )
% 0.30/0.61 => ( ( modulo_modulo @ A @ A3 @ M )
% 0.30/0.61 = ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_mod_cancel_left
% 0.30/0.61 thf(fact_5653_invertible__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( euclid8851590272496341667cancel @ A )
% 0.30/0.61 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.61 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.61 = ( one_one @ A ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % invertible_coprime
% 0.30/0.61 thf(fact_5654_gcd__coprime__exists,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 => ? [A15: A,B8: A] :
% 0.30/0.61 ( ( A3
% 0.30/0.61 = ( times_times @ A @ A15 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 0.30/0.61 & ( B2
% 0.30/0.61 = ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ A @ A15 @ B8 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_coprime_exists
% 0.30/0.61 thf(fact_5655_gcd__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A,A4: A,B4: A] :
% 0.30/0.61 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 => ( ( A3
% 0.30/0.61 = ( times_times @ A @ A4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 0.30/0.61 => ( ( B2
% 0.30/0.61 = ( times_times @ A @ B4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_coprime
% 0.30/0.61 thf(fact_5656_div__gcd__coprime,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( semiring_gcd @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( ( A3
% 0.30/0.61 != ( zero_zero @ A ) )
% 0.30/0.61 | ( B2
% 0.30/0.61 != ( zero_zero @ A ) ) )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % div_gcd_coprime
% 0.30/0.61 thf(fact_5657_coprime__diff__one__right__nat,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_diff_one_right_nat
% 0.30/0.61 thf(fact_5658_coprime__diff__one__left__nat,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.61 => ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_diff_one_left_nat
% 0.30/0.61 thf(fact_5659_set__encode__vimage__Suc,axiom,
% 0.30/0.61 ! [A6: set @ nat] :
% 0.30/0.61 ( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A6 ) )
% 0.30/0.61 = ( divide_divide @ nat @ ( nat_set_encode @ A6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % set_encode_vimage_Suc
% 0.30/0.61 thf(fact_5660_floor__rat__def,axiom,
% 0.30/0.61 ( ( archim6421214686448440834_floor @ rat )
% 0.30/0.61 = ( ^ [X2: rat] :
% 0.30/0.61 ( the @ int
% 0.30/0.61 @ ^ [Z7: int] :
% 0.30/0.61 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z7 ) @ X2 )
% 0.30/0.61 & ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % floor_rat_def
% 0.30/0.61 thf(fact_5661_sorted__insort__insert__key,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [F: B > A,Xs: list @ B,X: B] :
% 0.30/0.61 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 0.30/0.61 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sorted_insort_insert_key
% 0.30/0.61 thf(fact_5662_coprime__abs__left__iff,axiom,
% 0.30/0.61 ! [K: int,L: int] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ int @ ( abs_abs @ int @ K ) @ L )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_abs_left_iff
% 0.30/0.61 thf(fact_5663_coprime__abs__right__iff,axiom,
% 0.30/0.61 ! [K: int,L: int] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ int @ K @ ( abs_abs @ int @ L ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_abs_right_iff
% 0.30/0.61 thf(fact_5664_coprime__int__iff,axiom,
% 0.30/0.61 ! [M: nat,N: nat] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ nat @ M @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_int_iff
% 0.30/0.61 thf(fact_5665_normalize__stable,axiom,
% 0.30/0.61 ! [Q4: int,P5: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ int @ P5 @ Q4 )
% 0.30/0.61 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 0.30/0.61 = ( product_Pair @ int @ int @ P5 @ Q4 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % normalize_stable
% 0.30/0.61 thf(fact_5666_coprime__nat__abs__left__iff,axiom,
% 0.30/0.61 ! [K: int,N: nat] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_nat_abs_left_iff
% 0.30/0.61 thf(fact_5667_coprime__nat__abs__right__iff,axiom,
% 0.30/0.61 ! [N: nat,K: int] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 0.30/0.61 = ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_nat_abs_right_iff
% 0.30/0.61 thf(fact_5668_vimage__if,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,C2: B,A6: set @ B,D2: B,B7: set @ A] :
% 0.30/0.61 ( ( ( member @ B @ C2 @ A6 )
% 0.30/0.61 => ( ( ( member @ B @ D2 @ A6 )
% 0.30/0.61 => ( ( vimage @ A @ B
% 0.30/0.61 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 0.30/0.61 @ A6 )
% 0.30/0.61 = ( top_top @ ( set @ A ) ) ) )
% 0.30/0.61 & ( ~ ( member @ B @ D2 @ A6 )
% 0.30/0.61 => ( ( vimage @ A @ B
% 0.30/0.61 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 0.30/0.61 @ A6 )
% 0.30/0.61 = B7 ) ) ) )
% 0.30/0.61 & ( ~ ( member @ B @ C2 @ A6 )
% 0.30/0.61 => ( ( ( member @ B @ D2 @ A6 )
% 0.30/0.61 => ( ( vimage @ A @ B
% 0.30/0.61 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 0.30/0.61 @ A6 )
% 0.30/0.61 = ( uminus_uminus @ ( set @ A ) @ B7 ) ) )
% 0.30/0.61 & ( ~ ( member @ B @ D2 @ A6 )
% 0.30/0.61 => ( ( vimage @ A @ B
% 0.30/0.61 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 0.30/0.61 @ A6 )
% 0.30/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % vimage_if
% 0.30/0.61 thf(fact_5669_vimage__Compl,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 0.30/0.61 ( ( vimage @ A @ B @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) )
% 0.30/0.61 = ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % vimage_Compl
% 0.30/0.61 thf(fact_5670_coprime__crossproduct__int,axiom,
% 0.30/0.61 ! [A3: int,D2: int,B2: int,C2: int] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ int @ A3 @ D2 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ int @ B2 @ C2 )
% 0.30/0.61 => ( ( ( times_times @ int @ ( abs_abs @ int @ A3 ) @ ( abs_abs @ int @ C2 ) )
% 0.30/0.61 = ( times_times @ int @ ( abs_abs @ int @ B2 ) @ ( abs_abs @ int @ D2 ) ) )
% 0.30/0.61 = ( ( ( abs_abs @ int @ A3 )
% 0.30/0.61 = ( abs_abs @ int @ B2 ) )
% 0.30/0.61 & ( ( abs_abs @ int @ C2 )
% 0.30/0.61 = ( abs_abs @ int @ D2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_crossproduct_int
% 0.30/0.61 thf(fact_5671_finite__vimage__Suc__iff,axiom,
% 0.30/0.61 ! [F4: set @ nat] :
% 0.30/0.61 ( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F4 ) )
% 0.30/0.61 = ( finite_finite2 @ nat @ F4 ) ) ).
% 0.30/0.61
% 0.30/0.61 % finite_vimage_Suc_iff
% 0.30/0.61 thf(fact_5672_Nitpick_OThe__psimp,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,X: A] :
% 0.30/0.61 ( ( P
% 0.30/0.61 = ( ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.30/0.61 @ X ) )
% 0.30/0.61 => ( ( the @ A @ P )
% 0.30/0.61 = X ) ) ).
% 0.30/0.61
% 0.30/0.61 % Nitpick.The_psimp
% 0.30/0.61 thf(fact_5673_vimage__Suc__insert__Suc,axiom,
% 0.30/0.61 ! [N: nat,A6: set @ nat] :
% 0.30/0.61 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A6 ) )
% 0.30/0.61 = ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % vimage_Suc_insert_Suc
% 0.30/0.61 thf(fact_5674_vimage__Suc__insert__0,axiom,
% 0.30/0.61 ! [A6: set @ nat] :
% 0.30/0.61 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 0.30/0.61 = ( vimage @ nat @ nat @ suc @ A6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % vimage_Suc_insert_0
% 0.30/0.61 thf(fact_5675_coprime__common__divisor__int,axiom,
% 0.30/0.61 ! [A3: int,B2: int,X: int] :
% 0.30/0.61 ( ( algebr8660921524188924756oprime @ int @ A3 @ B2 )
% 0.30/0.61 => ( ( dvd_dvd @ int @ X @ A3 )
% 0.30/0.61 => ( ( dvd_dvd @ int @ X @ B2 )
% 0.30/0.61 => ( ( abs_abs @ int @ X )
% 0.30/0.61 = ( one_one @ int ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % coprime_common_divisor_int
% 0.30/0.61 thf(fact_5676_sorted__insort__insert,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [Xs: list @ A,X: A] :
% 0.30/0.61 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.61 => ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 0.30/0.61 @ ( linord329482645794927042rt_key @ A @ A
% 0.30/0.61 @ ^ [X2: A] : X2
% 0.30/0.61 @ X
% 0.30/0.61 @ Xs ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sorted_insort_insert
% 0.30/0.61 thf(fact_5677_card__vimage__inj__on__le,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,F: A > B,D4: set @ A,A6: set @ B] :
% 0.30/0.61 ( ( inj_on @ A @ B @ F @ D4 )
% 0.30/0.61 => ( ( finite_finite2 @ B @ A6 )
% 0.30/0.61 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ D4 ) ) @ ( finite_card @ B @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % card_vimage_inj_on_le
% 0.30/0.61 thf(fact_5678_set__decode__div__2,axiom,
% 0.30/0.61 ! [X: nat] :
% 0.30/0.61 ( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.30/0.61 = ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % set_decode_div_2
% 0.30/0.61 thf(fact_5679_old_Orec__nat__def,axiom,
% 0.30/0.61 ! [T: $tType] :
% 0.30/0.61 ( ( rec_nat @ T )
% 0.30/0.61 = ( ^ [F12: T,F23: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % old.rec_nat_def
% 0.30/0.61 thf(fact_5680_rat__floor__lemma,axiom,
% 0.30/0.61 ! [A3: int,B2: int] :
% 0.30/0.61 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 & ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_floor_lemma
% 0.30/0.61 thf(fact_5681_sgn__rat,axiom,
% 0.30/0.61 ! [A3: int,B2: int] :
% 0.30/0.61 ( ( sgn_sgn @ rat @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ring_1_of_int @ rat @ ( times_times @ int @ ( sgn_sgn @ int @ A3 ) @ ( sgn_sgn @ int @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sgn_rat
% 0.30/0.61 thf(fact_5682_minus__rat__cancel,axiom,
% 0.30/0.61 ! [A3: int,B2: int] :
% 0.30/0.61 ( ( fract @ ( uminus_uminus @ int @ A3 ) @ ( uminus_uminus @ int @ B2 ) )
% 0.30/0.61 = ( fract @ A3 @ B2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_rat_cancel
% 0.30/0.61 thf(fact_5683_minus__rat,axiom,
% 0.30/0.61 ! [A3: int,B2: int] :
% 0.30/0.61 ( ( uminus_uminus @ rat @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( fract @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_rat
% 0.30/0.61 thf(fact_5684_divide__rat,axiom,
% 0.30/0.61 ! [A3: int,B2: int,C2: int,D2: int] :
% 0.30/0.61 ( ( divide_divide @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( fract @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ C2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divide_rat
% 0.30/0.61 thf(fact_5685_mult__rat,axiom,
% 0.30/0.61 ! [A3: int,B2: int,C2: int,D2: int] :
% 0.30/0.61 ( ( times_times @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( fract @ ( times_times @ int @ A3 @ C2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_rat
% 0.30/0.61 thf(fact_5686_less__rat,axiom,
% 0.30/0.61 ! [B2: int,D2: int,A3: int,C2: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( D2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % less_rat
% 0.30/0.61 thf(fact_5687_le__rat,axiom,
% 0.30/0.61 ! [B2: int,D2: int,A3: int,C2: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( D2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % le_rat
% 0.30/0.61 thf(fact_5688_add__rat,axiom,
% 0.30/0.61 ! [B2: int,D2: int,A3: int,C2: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( D2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_rat
% 0.30/0.61 thf(fact_5689_diff__rat,axiom,
% 0.30/0.61 ! [B2: int,D2: int,A3: int,C2: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( D2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( minus_minus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( fract @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % diff_rat
% 0.30/0.61 thf(fact_5690_Rat__cases,axiom,
% 0.30/0.61 ! [Q4: rat] :
% 0.30/0.61 ~ ! [A8: int,B5: int] :
% 0.30/0.61 ( ( Q4
% 0.30/0.61 = ( fract @ A8 @ B5 ) )
% 0.30/0.61 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 0.30/0.61 => ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rat_cases
% 0.30/0.61 thf(fact_5691_Rat__induct,axiom,
% 0.30/0.61 ! [P: rat > $o,Q4: rat] :
% 0.30/0.61 ( ! [A8: int,B5: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
% 0.30/0.61 => ( P @ ( fract @ A8 @ B5 ) ) ) )
% 0.30/0.61 => ( P @ Q4 ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rat_induct
% 0.30/0.61 thf(fact_5692_Fract__of__nat__eq,axiom,
% 0.30/0.61 ! [K: nat] :
% 0.30/0.61 ( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
% 0.30/0.61 = ( semiring_1_of_nat @ rat @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_of_nat_eq
% 0.30/0.61 thf(fact_5693_eq__rat_I3_J,axiom,
% 0.30/0.61 ! [A3: int,C2: int] :
% 0.30/0.61 ( ( fract @ ( zero_zero @ int ) @ A3 )
% 0.30/0.61 = ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % eq_rat(3)
% 0.30/0.61 thf(fact_5694_eq__rat_I1_J,axiom,
% 0.30/0.61 ! [B2: int,D2: int,A3: int,C2: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( D2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( ( fract @ A3 @ B2 )
% 0.30/0.61 = ( fract @ C2 @ D2 ) )
% 0.30/0.61 = ( ( times_times @ int @ A3 @ D2 )
% 0.30/0.61 = ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % eq_rat(1)
% 0.30/0.61 thf(fact_5695_mult__rat__cancel,axiom,
% 0.30/0.61 ! [C2: int,A3: int,B2: int] :
% 0.30/0.61 ( ( C2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( fract @ ( times_times @ int @ C2 @ A3 ) @ ( times_times @ int @ C2 @ B2 ) )
% 0.30/0.61 = ( fract @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_rat_cancel
% 0.30/0.61 thf(fact_5696_eq__rat_I2_J,axiom,
% 0.30/0.61 ! [A3: int] :
% 0.30/0.61 ( ( fract @ A3 @ ( zero_zero @ int ) )
% 0.30/0.61 = ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % eq_rat(2)
% 0.30/0.61 thf(fact_5697_Rat__induct__pos,axiom,
% 0.30/0.61 ! [P: rat > $o,Q4: rat] :
% 0.30/0.61 ( ! [A8: int,B5: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 0.30/0.61 => ( P @ ( fract @ A8 @ B5 ) ) )
% 0.30/0.61 => ( P @ Q4 ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rat_induct_pos
% 0.30/0.61 thf(fact_5698_rat__number__collapse_I6_J,axiom,
% 0.30/0.61 ! [K: int] :
% 0.30/0.61 ( ( fract @ K @ ( zero_zero @ int ) )
% 0.30/0.61 = ( zero_zero @ rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_collapse(6)
% 0.30/0.61 thf(fact_5699_rat__number__collapse_I1_J,axiom,
% 0.30/0.61 ! [K: int] :
% 0.30/0.61 ( ( fract @ ( zero_zero @ int ) @ K )
% 0.30/0.61 = ( zero_zero @ rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_collapse(1)
% 0.30/0.61 thf(fact_5700_One__rat__def,axiom,
% 0.30/0.61 ( ( one_one @ rat )
% 0.30/0.61 = ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % One_rat_def
% 0.30/0.61 thf(fact_5701_Fract__of__int__eq,axiom,
% 0.30/0.61 ! [K: int] :
% 0.30/0.61 ( ( fract @ K @ ( one_one @ int ) )
% 0.30/0.61 = ( ring_1_of_int @ rat @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_of_int_eq
% 0.30/0.61 thf(fact_5702_quotient__of__def,axiom,
% 0.30/0.61 ( quotient_of
% 0.30/0.61 = ( ^ [X2: rat] :
% 0.30/0.61 ( the @ ( product_prod @ int @ int )
% 0.30/0.61 @ ^ [Pair: product_prod @ int @ int] :
% 0.30/0.61 ( ( X2
% 0.30/0.61 = ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
% 0.30/0.61 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % quotient_of_def
% 0.30/0.61 thf(fact_5703_Rat__cases__nonzero,axiom,
% 0.30/0.61 ! [Q4: rat] :
% 0.30/0.61 ( ! [A8: int,B5: int] :
% 0.30/0.61 ( ( Q4
% 0.30/0.61 = ( fract @ A8 @ B5 ) )
% 0.30/0.61 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 0.30/0.61 => ( ( A8
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) )
% 0.30/0.61 => ( Q4
% 0.30/0.61 = ( zero_zero @ rat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rat_cases_nonzero
% 0.30/0.61 thf(fact_5704_Zero__rat__def,axiom,
% 0.30/0.61 ( ( zero_zero @ rat )
% 0.30/0.61 = ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Zero_rat_def
% 0.30/0.61 thf(fact_5705_rat__number__expand_I3_J,axiom,
% 0.30/0.61 ( ( numeral_numeral @ rat )
% 0.30/0.61 = ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_expand(3)
% 0.30/0.61 thf(fact_5706_rat__number__collapse_I3_J,axiom,
% 0.30/0.61 ! [W2: num] :
% 0.30/0.61 ( ( fract @ ( numeral_numeral @ int @ W2 ) @ ( one_one @ int ) )
% 0.30/0.61 = ( numeral_numeral @ rat @ W2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_collapse(3)
% 0.30/0.61 thf(fact_5707_quotient__of__unique,axiom,
% 0.30/0.61 ! [R: rat] :
% 0.30/0.61 ? [X3: product_prod @ int @ int] :
% 0.30/0.61 ( ( R
% 0.30/0.61 = ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
% 0.30/0.61 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
% 0.30/0.61 & ! [Y4: product_prod @ int @ int] :
% 0.30/0.61 ( ( ( R
% 0.30/0.61 = ( fract @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
% 0.30/0.61 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y4 ) )
% 0.30/0.61 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
% 0.30/0.61 => ( Y4 = X3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % quotient_of_unique
% 0.30/0.61 thf(fact_5708_Fract_Oabs__eq,axiom,
% 0.30/0.61 ( fract
% 0.30/0.61 = ( ^ [Xa4: int,X2: int] :
% 0.30/0.61 ( abs_Rat
% 0.30/0.61 @ ( if @ ( product_prod @ int @ int )
% 0.30/0.61 @ ( X2
% 0.30/0.61 = ( zero_zero @ int ) )
% 0.30/0.61 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.61 @ ( product_Pair @ int @ int @ Xa4 @ X2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract.abs_eq
% 0.30/0.61 thf(fact_5709_positive__rat,axiom,
% 0.30/0.61 ! [A3: int,B2: int] :
% 0.30/0.61 ( ( positive @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % positive_rat
% 0.30/0.61 thf(fact_5710_Fract__less__zero__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
% 0.30/0.61 = ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_less_zero_iff
% 0.30/0.61 thf(fact_5711_zero__less__Fract__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zero_less_Fract_iff
% 0.30/0.61 thf(fact_5712_one__less__Fract__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ord_less @ int @ B2 @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_less_Fract_iff
% 0.30/0.61 thf(fact_5713_Fract__less__one__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
% 0.30/0.61 = ( ord_less @ int @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_less_one_iff
% 0.30/0.61 thf(fact_5714_rat__number__collapse_I5_J,axiom,
% 0.30/0.61 ( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
% 0.30/0.61 = ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_collapse(5)
% 0.30/0.61 thf(fact_5715_Fract_Otransfer,axiom,
% 0.30/0.61 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
% 0.30/0.61 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.61 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
% 0.30/0.61 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.61 @ pcr_rat )
% 0.30/0.61 @ ^ [A5: int,B3: int] :
% 0.30/0.61 ( if @ ( product_prod @ int @ int )
% 0.30/0.61 @ ( B3
% 0.30/0.61 = ( zero_zero @ int ) )
% 0.30/0.61 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.61 @ ( product_Pair @ int @ int @ A5 @ B3 ) )
% 0.30/0.61 @ fract ) ).
% 0.30/0.61
% 0.30/0.61 % Fract.transfer
% 0.30/0.61 thf(fact_5716_Fract__add__one,axiom,
% 0.30/0.61 ! [N: int,M: int] :
% 0.30/0.61 ( ( N
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
% 0.30/0.61 = ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_add_one
% 0.30/0.61 thf(fact_5717_Fract__le__zero__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
% 0.30/0.61 = ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_le_zero_iff
% 0.30/0.61 thf(fact_5718_zero__le__Fract__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zero_le_Fract_iff
% 0.30/0.61 thf(fact_5719_one__le__Fract__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_le_Fract_iff
% 0.30/0.61 thf(fact_5720_Fract__le__one__iff,axiom,
% 0.30/0.61 ! [B2: int,A3: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.30/0.61 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
% 0.30/0.61 = ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Fract_le_one_iff
% 0.30/0.61 thf(fact_5721_rat__number__expand_I5_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
% 0.30/0.61 = ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_expand(5)
% 0.30/0.61 thf(fact_5722_rat__number__collapse_I4_J,axiom,
% 0.30/0.61 ! [W2: num] :
% 0.30/0.61 ( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( one_one @ int ) )
% 0.30/0.61 = ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % rat_number_collapse(4)
% 0.30/0.61 thf(fact_5723_Rats__cases_H,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [X: A] :
% 0.30/0.61 ( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
% 0.30/0.61 => ~ ! [A8: int,B5: int] :
% 0.30/0.61 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 0.30/0.61 => ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
% 0.30/0.61 => ( X
% 0.30/0.61 != ( divide_divide @ A @ ( ring_1_of_int @ A @ A8 ) @ ( ring_1_of_int @ A @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_cases'
% 0.30/0.61 thf(fact_5724_horner__sum__transfer,axiom,
% 0.30/0.61 ! [C: $tType,A: $tType,B: $tType,D: $tType] :
% 0.30/0.61 ( ( ( comm_semiring_0 @ B )
% 0.30/0.61 & ( comm_semiring_0 @ A ) )
% 0.30/0.61 => ! [A6: A > B > $o,B7: C > D > $o] :
% 0.30/0.61 ( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.61 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.30/0.61 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.30/0.61 => ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B7 @ A6 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A6 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B7 ) @ A6 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % horner_sum_transfer
% 0.30/0.61 thf(fact_5725_of__rat__neg__numeral__eq,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [W2: num] :
% 0.30/0.61 ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) )
% 0.30/0.61 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_neg_numeral_eq
% 0.30/0.61 thf(fact_5726_Rats__minus__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: A] :
% 0.30/0.61 ( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( field_char_0_Rats @ A ) )
% 0.30/0.61 = ( member @ A @ A3 @ ( field_char_0_Rats @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_minus_iff
% 0.30/0.61 thf(fact_5727_of__rat__0,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
% 0.30/0.61 = ( zero_zero @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_0
% 0.30/0.61 thf(fact_5728_of__rat__eq__0__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat] :
% 0.30/0.61 ( ( ( field_char_0_of_rat @ A @ A3 )
% 0.30/0.61 = ( zero_zero @ A ) )
% 0.30/0.61 = ( A3
% 0.30/0.61 = ( zero_zero @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_eq_0_iff
% 0.30/0.61 thf(fact_5729_zero__eq__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat] :
% 0.30/0.61 ( ( ( zero_zero @ A )
% 0.30/0.61 = ( field_char_0_of_rat @ A @ A3 ) )
% 0.30/0.61 = ( ( zero_zero @ rat )
% 0.30/0.61 = A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zero_eq_of_rat_iff
% 0.30/0.61 thf(fact_5730_one__eq__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat] :
% 0.30/0.61 ( ( ( one_one @ A )
% 0.30/0.61 = ( field_char_0_of_rat @ A @ A3 ) )
% 0.30/0.61 = ( ( one_one @ rat )
% 0.30/0.61 = A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_eq_of_rat_iff
% 0.30/0.61 thf(fact_5731_of__rat__eq__1__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat] :
% 0.30/0.61 ( ( ( field_char_0_of_rat @ A @ A3 )
% 0.30/0.61 = ( one_one @ A ) )
% 0.30/0.61 = ( A3
% 0.30/0.61 = ( one_one @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_eq_1_iff
% 0.30/0.61 thf(fact_5732_of__rat__1,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
% 0.30/0.61 = ( one_one @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_1
% 0.30/0.61 thf(fact_5733_of__rat__of__nat__eq,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [N: nat] :
% 0.30/0.61 ( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
% 0.30/0.61 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_of_nat_eq
% 0.30/0.61 thf(fact_5734_of__rat__less__0__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
% 0.30/0.61 = ( ord_less @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_less_0_iff
% 0.30/0.61 thf(fact_5735_zero__less__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.30/0.61 = ( ord_less @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zero_less_of_rat_iff
% 0.30/0.61 thf(fact_5736_of__rat__le__0__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
% 0.30/0.61 = ( ord_less_eq @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_le_0_iff
% 0.30/0.61 thf(fact_5737_zero__le__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.30/0.61 = ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zero_le_of_rat_iff
% 0.30/0.61 thf(fact_5738_of__rat__neg__one,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 0.30/0.61 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_neg_one
% 0.30/0.61 thf(fact_5739_of__rat__less__1__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 0.30/0.61 = ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_less_1_iff
% 0.30/0.61 thf(fact_5740_one__less__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.30/0.61 = ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_less_of_rat_iff
% 0.30/0.61 thf(fact_5741_of__rat__le__1__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 0.30/0.61 = ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_le_1_iff
% 0.30/0.61 thf(fact_5742_one__le__of__rat__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat] :
% 0.30/0.61 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.30/0.61 = ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_le_of_rat_iff
% 0.30/0.61 thf(fact_5743_Rats__mult,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: A,B2: A] :
% 0.30/0.61 ( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
% 0.30/0.61 => ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
% 0.30/0.61 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_mult
% 0.30/0.61 thf(fact_5744_Rats__0,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_0
% 0.30/0.61 thf(fact_5745_Rats__1,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_1
% 0.30/0.61 thf(fact_5746_Rats__of__nat,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Rats_of_nat
% 0.30/0.61 thf(fact_5747_of__rat__minus,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat] :
% 0.30/0.61 ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ A3 ) )
% 0.30/0.61 = ( uminus_uminus @ A @ ( field_char_0_of_rat @ A @ A3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_minus
% 0.30/0.61 thf(fact_5748_of__rat__less,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat,S: rat] :
% 0.30/0.61 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
% 0.30/0.61 = ( ord_less @ rat @ R @ S ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_less
% 0.30/0.61 thf(fact_5749_of__rat__less__eq,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linordered_field @ A )
% 0.30/0.61 => ! [R: rat,S: rat] :
% 0.30/0.61 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
% 0.30/0.61 = ( ord_less_eq @ rat @ R @ S ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_less_eq
% 0.30/0.61 thf(fact_5750_of__rat__mult,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [A3: rat,B2: rat] :
% 0.30/0.61 ( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
% 0.30/0.61 = ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_mult
% 0.30/0.61 thf(fact_5751_list__all2__conv__all__nth,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( list_all2 @ A @ B )
% 0.30/0.61 = ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
% 0.30/0.61 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.30/0.61 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 0.30/0.61 & ! [I5: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.61 => ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ B @ Ys3 @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_all2_conv_all_nth
% 0.30/0.61 thf(fact_5752_list__all2__all__nthI,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
% 0.30/0.61 ( ( ( size_size @ ( list @ A ) @ A3 )
% 0.30/0.61 = ( size_size @ ( list @ B ) @ B2 ) )
% 0.30/0.61 => ( ! [N4: nat] :
% 0.30/0.61 ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ A3 ) )
% 0.30/0.61 => ( P @ ( nth @ A @ A3 @ N4 ) @ ( nth @ B @ B2 @ N4 ) ) )
% 0.30/0.61 => ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_all2_all_nthI
% 0.30/0.61 thf(fact_5753_list__all2__nthD2,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
% 0.30/0.61 ( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
% 0.30/0.61 => ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.61 => ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_all2_nthD2
% 0.30/0.61 thf(fact_5754_list__all2__nthD,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
% 0.30/0.61 ( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
% 0.30/0.61 => ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.61 => ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_all2_nthD
% 0.30/0.61 thf(fact_5755_of__rat__rat,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( field_char_0 @ A )
% 0.30/0.61 => ! [B2: int,A3: int] :
% 0.30/0.61 ( ( B2
% 0.30/0.61 != ( zero_zero @ int ) )
% 0.30/0.61 => ( ( field_char_0_of_rat @ A @ ( fract @ A3 @ B2 ) )
% 0.30/0.61 = ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_rat_rat
% 0.30/0.61 thf(fact_5756_sum__list__transfer,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( ( monoid_add @ B )
% 0.30/0.61 & ( monoid_add @ A ) )
% 0.30/0.61 => ! [A6: A > B > $o] :
% 0.30/0.61 ( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.30/0.61 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.30/0.61 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_list_transfer
% 0.30/0.61 thf(fact_5757_prod__list__transfer,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( ( monoid_mult @ B )
% 0.30/0.61 & ( monoid_mult @ A ) )
% 0.30/0.61 => ! [A6: A > B > $o] :
% 0.30/0.61 ( ( A6 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.30/0.61 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.30/0.61 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list_transfer
% 0.30/0.61 thf(fact_5758_bot_Oordering__top__axioms,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( order_bot @ A )
% 0.30/0.61 => ( ordering_top @ A
% 0.30/0.61 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.61 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 )
% 0.30/0.61 @ ( bot_bot @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % bot.ordering_top_axioms
% 0.30/0.61 thf(fact_5759_dropWhile__nth,axiom,
% 0.30/0.61 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 0.30/0.61 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 0.30/0.61 => ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
% 0.30/0.61 = ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dropWhile_nth
% 0.30/0.61 thf(fact_5760_prod__list_OCons,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( monoid_mult @ A )
% 0.30/0.61 => ! [X: A,Xs: list @ A] :
% 0.30/0.61 ( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs ) )
% 0.30/0.61 = ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list.Cons
% 0.30/0.61 thf(fact_5761_prod__list_ONil,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( monoid_mult @ A )
% 0.30/0.61 => ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
% 0.30/0.61 = ( one_one @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list.Nil
% 0.30/0.61 thf(fact_5762_prod__list_Oappend,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( monoid_mult @ A )
% 0.30/0.61 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.30/0.61 ( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.30/0.61 = ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ ( groups5270119922927024881d_list @ A @ Ys2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list.append
% 0.30/0.61 thf(fact_5763_ordering__top_Oextremum__uniqueI,axiom,
% 0.30/0.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/0.61 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.61 => ( ( Less_eq @ Top @ A3 )
% 0.30/0.61 => ( A3 = Top ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ordering_top.extremum_uniqueI
% 0.30/0.61 thf(fact_5764_ordering__top_Onot__eq__extremum,axiom,
% 0.30/0.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/0.61 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.61 => ( ( A3 != Top )
% 0.30/0.61 = ( Less @ A3 @ Top ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ordering_top.not_eq_extremum
% 0.30/0.61 thf(fact_5765_ordering__top_Oextremum__unique,axiom,
% 0.30/0.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/0.61 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.61 => ( ( Less_eq @ Top @ A3 )
% 0.30/0.61 = ( A3 = Top ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ordering_top.extremum_unique
% 0.30/0.61 thf(fact_5766_ordering__top_Oextremum__strict,axiom,
% 0.30/0.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/0.61 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.61 => ~ ( Less @ Top @ A3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % ordering_top.extremum_strict
% 0.30/0.61 thf(fact_5767_ordering__top_Oextremum,axiom,
% 0.30/0.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 0.30/0.61 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.61 => ( Less_eq @ A3 @ Top ) ) ).
% 0.30/0.61
% 0.30/0.61 % ordering_top.extremum
% 0.30/0.61 thf(fact_5768_length__dropWhile__le,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.61
% 0.30/0.61 % length_dropWhile_le
% 0.30/0.61 thf(fact_5769_sorted__dropWhile,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( linorder @ A )
% 0.30/0.61 => ! [Xs: list @ A,P: A > $o] :
% 0.30/0.61 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.61 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sorted_dropWhile
% 0.30/0.61 thf(fact_5770_prod__list__zero__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ( semiring_1 @ A )
% 0.30/0.61 & ( semiri3467727345109120633visors @ A ) )
% 0.30/0.61 => ! [Xs: list @ A] :
% 0.30/0.61 ( ( ( groups5270119922927024881d_list @ A @ Xs )
% 0.30/0.61 = ( zero_zero @ A ) )
% 0.30/0.61 = ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list_zero_iff
% 0.30/0.61 thf(fact_5771_gcd__nat_Oordering__top__axioms,axiom,
% 0.30/0.61 ( ordering_top @ nat @ ( dvd_dvd @ nat )
% 0.30/0.61 @ ^ [M2: nat,N3: nat] :
% 0.30/0.61 ( ( dvd_dvd @ nat @ M2 @ N3 )
% 0.30/0.61 & ( M2 != N3 ) )
% 0.30/0.61 @ ( zero_zero @ nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % gcd_nat.ordering_top_axioms
% 0.30/0.61 thf(fact_5772_prod__list_Oeq__foldr,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( monoid_mult @ A )
% 0.30/0.61 => ( ( groups5270119922927024881d_list @ A )
% 0.30/0.61 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_list.eq_foldr
% 0.30/0.61 thf(fact_5773_top_Oordering__top__axioms,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( order_top @ A )
% 0.30/0.61 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % top.ordering_top_axioms
% 0.30/0.61 thf(fact_5774_bot__nat__0_Oordering__top__axioms,axiom,
% 0.30/0.61 ( ordering_top @ nat
% 0.30/0.61 @ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
% 0.30/0.61 @ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 )
% 0.30/0.61 @ ( zero_zero @ nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % bot_nat_0.ordering_top_axioms
% 0.30/0.61 thf(fact_5775_mlex__eq,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( mlex_prod @ A )
% 0.30/0.61 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 0.30/0.61 ( collect @ ( product_prod @ A @ A )
% 0.30/0.61 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.61 @ ^ [X2: A,Y6: A] :
% 0.30/0.61 ( ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
% 0.30/0.61 | ( ( ord_less_eq @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
% 0.30/0.61 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y6 ) @ R6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mlex_eq
% 0.30/0.61 thf(fact_5776_arg__min__SOME__Min,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( linorder @ B )
% 0.30/0.61 => ! [S3: set @ A,F: A > B] :
% 0.30/0.61 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.61 => ( ( lattic7623131987881927897min_on @ A @ B @ F @ S3 )
% 0.30/0.61 = ( fChoice @ A
% 0.30/0.61 @ ^ [Y6: A] :
% 0.30/0.61 ( ( member @ A @ Y6 @ S3 )
% 0.30/0.61 & ( ( F @ Y6 )
% 0.30/0.61 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ S3 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_SOME_Min
% 0.30/0.61 thf(fact_5777_prod__decode__triangle__add,axiom,
% 0.30/0.61 ! [K: nat,M: nat] :
% 0.30/0.61 ( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) )
% 0.30/0.61 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_decode_triangle_add
% 0.30/0.61 thf(fact_5778_prod__decode__eq,axiom,
% 0.30/0.61 ! [X: nat,Y3: nat] :
% 0.30/0.61 ( ( ( nat_prod_decode @ X )
% 0.30/0.61 = ( nat_prod_decode @ Y3 ) )
% 0.30/0.61 = ( X = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_decode_eq
% 0.30/0.61 thf(fact_5779_some__equality,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A3: A] :
% 0.30/0.61 ( ( P @ A3 )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 => ( X3 = A3 ) )
% 0.30/0.61 => ( ( fChoice @ A @ P )
% 0.30/0.61 = A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % some_equality
% 0.30/0.61 thf(fact_5780_some__eq__trivial,axiom,
% 0.30/0.61 ! [A: $tType,X: A] :
% 0.30/0.61 ( ( fChoice @ A
% 0.30/0.61 @ ^ [Y6: A] : ( Y6 = X ) )
% 0.30/0.61 = X ) ).
% 0.30/0.61
% 0.30/0.61 % some_eq_trivial
% 0.30/0.61 thf(fact_5781_some__sym__eq__trivial,axiom,
% 0.30/0.61 ! [A: $tType,X: A] :
% 0.30/0.61 ( ( fChoice @ A
% 0.30/0.61 @ ( ^ [Y: A,Z2: A] : ( Y = Z2 )
% 0.30/0.61 @ X ) )
% 0.30/0.61 = X ) ).
% 0.30/0.61
% 0.30/0.61 % some_sym_eq_trivial
% 0.30/0.61 thf(fact_5782_Eps__case__prod__eq,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 0.30/0.61 ( ( fChoice @ ( product_prod @ A @ B )
% 0.30/0.61 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.61 @ ^ [X9: A,Y8: B] :
% 0.30/0.61 ( ( X = X9 )
% 0.30/0.61 & ( Y3 = Y8 ) ) ) )
% 0.30/0.61 = ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % Eps_case_prod_eq
% 0.30/0.61 thf(fact_5783_prod__decode__inverse,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
% 0.30/0.61 = N ) ).
% 0.30/0.61
% 0.30/0.61 % prod_decode_inverse
% 0.30/0.61 thf(fact_5784_prod__encode__inverse,axiom,
% 0.30/0.61 ! [X: product_prod @ nat @ nat] :
% 0.30/0.61 ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
% 0.30/0.61 = X ) ).
% 0.30/0.61
% 0.30/0.61 % prod_encode_inverse
% 0.30/0.61 thf(fact_5785_inj__prod__decode,axiom,
% 0.30/0.61 ! [A6: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_prod_decode
% 0.30/0.61 thf(fact_5786_someI2,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
% 0.30/0.61 ( ( P @ A3 )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 => ( Q @ X3 ) )
% 0.30/0.61 => ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % someI2
% 0.30/0.61 thf(fact_5787_someI__ex,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o] :
% 0.30/0.61 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.61 => ( P @ ( fChoice @ A @ P ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % someI_ex
% 0.30/0.61 thf(fact_5788_someI2__ex,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.61 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 => ( Q @ X3 ) )
% 0.30/0.61 => ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % someI2_ex
% 0.30/0.61 thf(fact_5789_someI2__bex,axiom,
% 0.30/0.61 ! [A: $tType,A6: set @ A,P: A > $o,Q: A > $o] :
% 0.30/0.61 ( ? [X4: A] :
% 0.30/0.61 ( ( member @ A @ X4 @ A6 )
% 0.30/0.61 & ( P @ X4 ) )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( ( member @ A @ X3 @ A6 )
% 0.30/0.61 & ( P @ X3 ) )
% 0.30/0.61 => ( Q @ X3 ) )
% 0.30/0.61 => ( Q
% 0.30/0.61 @ ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] :
% 0.30/0.61 ( ( member @ A @ X2 @ A6 )
% 0.30/0.61 & ( P @ X2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % someI2_bex
% 0.30/0.61 thf(fact_5790_some__eq__ex,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o] :
% 0.30/0.61 ( ( P @ ( fChoice @ A @ P ) )
% 0.30/0.61 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % some_eq_ex
% 0.30/0.61 thf(fact_5791_some1__equality,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A3: A] :
% 0.30/0.61 ( ? [X4: A] :
% 0.30/0.61 ( ( P @ X4 )
% 0.30/0.61 & ! [Y5: A] :
% 0.30/0.61 ( ( P @ Y5 )
% 0.30/0.61 => ( Y5 = X4 ) ) )
% 0.30/0.61 => ( ( P @ A3 )
% 0.30/0.61 => ( ( fChoice @ A @ P )
% 0.30/0.61 = A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % some1_equality
% 0.30/0.61 thf(fact_5792_verit__sko__ex_H,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A6: $o] :
% 0.30/0.61 ( ( ( P @ ( fChoice @ A @ P ) )
% 0.30/0.61 = A6 )
% 0.30/0.61 => ( ( ? [X7: A] : ( P @ X7 ) )
% 0.30/0.61 = A6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_ex'
% 0.30/0.61 thf(fact_5793_verit__sko__forall,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( ^ [P3: A > $o] :
% 0.30/0.61 ! [X6: A] : ( P3 @ X6 ) )
% 0.30/0.61 = ( ^ [P4: A > $o] :
% 0.30/0.61 ( P4
% 0.30/0.61 @ ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] :
% 0.30/0.61 ~ ( P4 @ X2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_forall
% 0.30/0.61 thf(fact_5794_verit__sko__forall_H,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A6: $o] :
% 0.30/0.61 ( ( ( P
% 0.30/0.61 @ ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] :
% 0.30/0.61 ~ ( P @ X2 ) ) )
% 0.30/0.61 = A6 )
% 0.30/0.61 => ( ( ! [X7: A] : ( P @ X7 ) )
% 0.30/0.61 = A6 ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_forall'
% 0.30/0.61 thf(fact_5795_verit__sko__forall_H_H,axiom,
% 0.30/0.61 ! [A: $tType,B7: A,A6: A,P: A > $o] :
% 0.30/0.61 ( ( B7 = A6 )
% 0.30/0.61 => ( ( ( fChoice @ A @ P )
% 0.30/0.61 = A6 )
% 0.30/0.61 = ( ( fChoice @ A @ P )
% 0.30/0.61 = B7 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_forall''
% 0.30/0.61 thf(fact_5796_verit__sko__ex__indirect,axiom,
% 0.30/0.61 ! [A: $tType,X: A,P: A > $o] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( fChoice @ A @ P ) )
% 0.30/0.61 => ( ( ? [X7: A] : ( P @ X7 ) )
% 0.30/0.61 = ( P @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_ex_indirect
% 0.30/0.61 thf(fact_5797_verit__sko__ex__indirect2,axiom,
% 0.30/0.61 ! [A: $tType,X: A,P: A > $o,P2: A > $o] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( fChoice @ A @ P ) )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 = ( P2 @ X3 ) )
% 0.30/0.61 => ( ( ? [X7: A] : ( P2 @ X7 ) )
% 0.30/0.61 = ( P @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_ex_indirect2
% 0.30/0.61 thf(fact_5798_verit__sko__forall__indirect,axiom,
% 0.30/0.61 ! [A: $tType,X: A,P: A > $o] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] :
% 0.30/0.61 ~ ( P @ X2 ) ) )
% 0.30/0.61 => ( ( ! [X7: A] : ( P @ X7 ) )
% 0.30/0.61 = ( P @ X ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_forall_indirect
% 0.30/0.61 thf(fact_5799_verit__sko__forall__indirect2,axiom,
% 0.30/0.61 ! [A: $tType,X: A,P: A > $o,P2: A > $o] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] :
% 0.30/0.61 ~ ( P @ X2 ) ) )
% 0.30/0.61 => ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 = ( P2 @ X3 ) )
% 0.30/0.61 => ( ( ! [X7: A] : ( P2 @ X7 ) )
% 0.30/0.61 = ( P @ X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_sko_forall_indirect2
% 0.30/0.61 thf(fact_5800_Nitpick_OEps__psimp,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,X: A,Y3: A] :
% 0.30/0.61 ( ( P @ X )
% 0.30/0.61 => ( ~ ( P @ Y3 )
% 0.30/0.61 => ( ( ( fChoice @ A @ P )
% 0.30/0.61 = Y3 )
% 0.30/0.61 => ( ( fChoice @ A @ P )
% 0.30/0.61 = X ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Nitpick.Eps_psimp
% 0.30/0.61 thf(fact_5801_someI,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,X: A] :
% 0.30/0.61 ( ( P @ X )
% 0.30/0.61 => ( P @ ( fChoice @ A @ P ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % someI
% 0.30/0.61 thf(fact_5802_Eps__cong,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.61 ( ! [X3: A] :
% 0.30/0.61 ( ( P @ X3 )
% 0.30/0.61 = ( Q @ X3 ) )
% 0.30/0.61 => ( ( fChoice @ A @ P )
% 0.30/0.61 = ( fChoice @ A @ Q ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Eps_cong
% 0.30/0.61 thf(fact_5803_tfl__some,axiom,
% 0.30/0.61 ! [A: $tType,P7: A > $o,X4: A] :
% 0.30/0.61 ( ( P7 @ X4 )
% 0.30/0.61 => ( P7 @ ( fChoice @ A @ P7 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % tfl_some
% 0.30/0.61 thf(fact_5804_some__eq__imp,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,A3: A,B2: A] :
% 0.30/0.61 ( ( ( fChoice @ A @ P )
% 0.30/0.61 = A3 )
% 0.30/0.61 => ( ( P @ B2 )
% 0.30/0.61 => ( P @ A3 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % some_eq_imp
% 0.30/0.61 thf(fact_5805_some__in__eq,axiom,
% 0.30/0.61 ! [A: $tType,A6: set @ A] :
% 0.30/0.61 ( ( member @ A
% 0.30/0.61 @ ( fChoice @ A
% 0.30/0.61 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 0.30/0.61 @ A6 )
% 0.30/0.61 = ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % some_in_eq
% 0.30/0.61 thf(fact_5806_exE__some,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,C2: A] :
% 0.30/0.61 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.61 => ( ( C2
% 0.30/0.61 = ( fChoice @ A @ P ) )
% 0.30/0.61 => ( P @ C2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % exE_some
% 0.30/0.61 thf(fact_5807_split__paired__Eps,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( fChoice @ ( product_prod @ A @ B ) )
% 0.30/0.61 = ( ^ [P4: ( product_prod @ A @ B ) > $o] :
% 0.30/0.61 ( fChoice @ ( product_prod @ A @ B )
% 0.30/0.61 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.61 @ ^ [A5: A,B3: B] : ( P4 @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % split_paired_Eps
% 0.30/0.61 thf(fact_5808_prod__decode__def,axiom,
% 0.30/0.61 ( nat_prod_decode
% 0.30/0.61 = ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % prod_decode_def
% 0.30/0.61 thf(fact_5809_Eps__case__prod,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.30/0.61 ( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
% 0.30/0.61 = ( fChoice @ ( product_prod @ A @ B )
% 0.30/0.61 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Eps_case_prod
% 0.30/0.61 thf(fact_5810_surj__prod__decode,axiom,
% 0.30/0.61 ( ( image @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.61 = ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % surj_prod_decode
% 0.30/0.61 thf(fact_5811_mlex__less,axiom,
% 0.30/0.61 ! [A: $tType,F: A > nat,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.61 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.61 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mlex_less
% 0.30/0.61 thf(fact_5812_mlex__iff,axiom,
% 0.30/0.61 ! [A: $tType,X: A,Y3: A,F: A > nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.61 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) )
% 0.30/0.61 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.61 | ( ( ( F @ X )
% 0.30/0.61 = ( F @ Y3 ) )
% 0.30/0.61 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mlex_iff
% 0.30/0.61 thf(fact_5813_mlex__leq,axiom,
% 0.30/0.61 ! [A: $tType,F: A > nat,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.61 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 0.30/0.61 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
% 0.30/0.61 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mlex_leq
% 0.30/0.61 thf(fact_5814_list__decode_Oelims,axiom,
% 0.30/0.61 ! [X: nat,Y3: list @ nat] :
% 0.30/0.61 ( ( ( nat_list_decode @ X )
% 0.30/0.61 = Y3 )
% 0.30/0.61 => ( ( ( X
% 0.30/0.61 = ( zero_zero @ nat ) )
% 0.30/0.61 => ( Y3
% 0.30/0.61 != ( nil @ nat ) ) )
% 0.30/0.61 => ~ ! [N4: nat] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( suc @ N4 ) )
% 0.30/0.61 => ( Y3
% 0.30/0.61 != ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.30/0.61 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 0.30/0.61 @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.elims
% 0.30/0.61 thf(fact_5815_arg__min__def,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ( ( lattices_ord_arg_min @ B @ A )
% 0.30/0.61 = ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % arg_min_def
% 0.30/0.61 thf(fact_5816_list__decode_Osimps_I2_J,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( nat_list_decode @ ( suc @ N ) )
% 0.30/0.61 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.30/0.61 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 0.30/0.61 @ ( nat_prod_decode @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.simps(2)
% 0.30/0.61 thf(fact_5817_list__encode__inverse,axiom,
% 0.30/0.61 ! [X: list @ nat] :
% 0.30/0.61 ( ( nat_list_decode @ ( nat_list_encode @ X ) )
% 0.30/0.61 = X ) ).
% 0.30/0.61
% 0.30/0.61 % list_encode_inverse
% 0.30/0.61 thf(fact_5818_list__decode__inverse,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( nat_list_encode @ ( nat_list_decode @ N ) )
% 0.30/0.61 = N ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode_inverse
% 0.30/0.61 thf(fact_5819_list__decode__eq,axiom,
% 0.30/0.61 ! [X: nat,Y3: nat] :
% 0.30/0.61 ( ( ( nat_list_decode @ X )
% 0.30/0.61 = ( nat_list_decode @ Y3 ) )
% 0.30/0.61 = ( X = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode_eq
% 0.30/0.61 thf(fact_5820_inj__list__decode,axiom,
% 0.30/0.61 ! [A6: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A6 ) ).
% 0.30/0.61
% 0.30/0.61 % inj_list_decode
% 0.30/0.61 thf(fact_5821_is__arg__min__antimono,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( order @ B )
% 0.30/0.61 => ! [F: A > B,P: A > $o,X: A,Y3: A] :
% 0.30/0.61 ( ( lattic501386751177426532rg_min @ A @ B @ F @ P @ X )
% 0.30/0.61 => ( ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) )
% 0.30/0.61 => ( ( P @ Y3 )
% 0.30/0.61 => ( lattic501386751177426532rg_min @ A @ B @ F @ P @ Y3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_arg_min_antimono
% 0.30/0.61 thf(fact_5822_is__arg__min__linorder,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( linorder @ B )
% 0.30/0.61 => ( ( lattic501386751177426532rg_min @ A @ B )
% 0.30/0.61 = ( ^ [F2: A > B,P4: A > $o,X2: A] :
% 0.30/0.61 ( ( P4 @ X2 )
% 0.30/0.61 & ! [Y6: A] :
% 0.30/0.61 ( ( P4 @ Y6 )
% 0.30/0.61 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_arg_min_linorder
% 0.30/0.61 thf(fact_5823_is__arg__min__def,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( ord @ A )
% 0.30/0.61 => ( ( lattic501386751177426532rg_min @ B @ A )
% 0.30/0.61 = ( ^ [F2: B > A,P4: B > $o,X2: B] :
% 0.30/0.61 ( ( P4 @ X2 )
% 0.30/0.61 & ~ ? [Y6: B] :
% 0.30/0.61 ( ( P4 @ Y6 )
% 0.30/0.61 & ( ord_less @ A @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_arg_min_def
% 0.30/0.61 thf(fact_5824_list__decode_Osimps_I1_J,axiom,
% 0.30/0.61 ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 0.30/0.61 = ( nil @ nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.simps(1)
% 0.30/0.61 thf(fact_5825_is__arg__min__arg__min__nat,axiom,
% 0.30/0.61 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 0.30/0.61 ( ( P @ X )
% 0.30/0.61 => ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % is_arg_min_arg_min_nat
% 0.30/0.61 thf(fact_5826_surj__list__decode,axiom,
% 0.30/0.61 ( ( image @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.61 = ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % surj_list_decode
% 0.30/0.61 thf(fact_5827_ex__is__arg__min__if__finite,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( order @ B )
% 0.30/0.61 => ! [S3: set @ A,F: A > B] :
% 0.30/0.61 ( ( finite_finite2 @ A @ S3 )
% 0.30/0.61 => ( ( S3
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ? [X_12: A] :
% 0.30/0.61 ( lattic501386751177426532rg_min @ A @ B @ F
% 0.30/0.61 @ ^ [X2: A] : ( member @ A @ X2 @ S3 )
% 0.30/0.61 @ X_12 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ex_is_arg_min_if_finite
% 0.30/0.61 thf(fact_5828_list__decode_Opelims,axiom,
% 0.30/0.61 ! [X: nat,Y3: list @ nat] :
% 0.30/0.61 ( ( ( nat_list_decode @ X )
% 0.30/0.61 = Y3 )
% 0.30/0.61 => ( ( accp @ nat @ nat_list_decode_rel @ X )
% 0.30/0.61 => ( ( ( X
% 0.30/0.61 = ( zero_zero @ nat ) )
% 0.30/0.61 => ( ( Y3
% 0.30/0.61 = ( nil @ nat ) )
% 0.30/0.61 => ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
% 0.30/0.61 => ~ ! [N4: nat] :
% 0.30/0.61 ( ( X
% 0.30/0.61 = ( suc @ N4 ) )
% 0.30/0.61 => ( ( Y3
% 0.30/0.61 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.30/0.61 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 0.30/0.61 @ ( nat_prod_decode @ N4 ) ) )
% 0.30/0.61 => ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.pelims
% 0.30/0.61 thf(fact_5829_mono__cSUP,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.61 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 & ( condit1219197933456340205attice @ B ) )
% 0.30/0.61 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 0.30/0.61 ( ( order_mono @ A @ B @ F )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A6 @ I6 ) )
% 0.30/0.61 => ( ( I6
% 0.30/0.61 != ( bot_bot @ ( set @ C ) ) )
% 0.30/0.61 => ( ord_less_eq @ B
% 0.30/0.61 @ ( complete_Sup_Sup @ B
% 0.30/0.61 @ ( image @ C @ B
% 0.30/0.61 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 0.30/0.61 @ I6 ) )
% 0.30/0.61 @ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mono_cSUP
% 0.30/0.61 thf(fact_5830_mono__cSup,axiom,
% 0.30/0.61 ! [B: $tType,A: $tType] :
% 0.30/0.61 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 & ( condit1219197933456340205attice @ B ) )
% 0.30/0.61 => ! [F: A > B,A6: set @ A] :
% 0.30/0.61 ( ( order_mono @ A @ B @ F )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ A6 )
% 0.30/0.61 => ( ( A6
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mono_cSup
% 0.30/0.61 thf(fact_5831_bdd__above_OI,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ! [A6: set @ A,M7: A] :
% 0.30/0.61 ( ! [X3: A] :
% 0.30/0.61 ( ( member @ A @ X3 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X3 @ M7 ) )
% 0.30/0.61 => ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % bdd_above.I
% 0.30/0.61 thf(fact_5832_bdd__above_OI2,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ! [A6: set @ B,F: B > A,M7: A] :
% 0.30/0.61 ( ! [X3: B] :
% 0.30/0.61 ( ( member @ B @ X3 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
% 0.30/0.61 => ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % bdd_above.I2
% 0.30/0.61 thf(fact_5833_bdd__above_OE,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ! [A6: set @ A] :
% 0.30/0.61 ( ( condit941137186595557371_above @ A @ A6 )
% 0.30/0.61 => ~ ! [M8: A] :
% 0.30/0.61 ~ ! [X4: A] :
% 0.30/0.61 ( ( member @ A @ X4 @ A6 )
% 0.30/0.61 => ( ord_less_eq @ A @ X4 @ M8 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % bdd_above.E
% 0.30/0.61 thf(fact_5834_bdd__above_Ounfold,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( preorder @ A )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A )
% 0.30/0.61 = ( ^ [A7: set @ A] :
% 0.30/0.61 ? [M9: A] :
% 0.30/0.61 ! [X2: A] :
% 0.30/0.61 ( ( member @ A @ X2 @ A7 )
% 0.30/0.61 => ( ord_less_eq @ A @ X2 @ M9 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % bdd_above.unfold
% 0.30/0.61 thf(fact_5835_cSup__upper2,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [X: A,X8: set @ A,Y3: A] :
% 0.30/0.61 ( ( member @ A @ X @ X8 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ Y3 @ X )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ X8 )
% 0.30/0.61 => ( ord_less_eq @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSup_upper2
% 0.30/0.61 thf(fact_5836_cSup__upper,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [X: A,X8: set @ A] :
% 0.30/0.61 ( ( member @ A @ X @ X8 )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ X8 )
% 0.30/0.61 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSup_upper
% 0.30/0.61 thf(fact_5837_list__decode_Opsimps_I1_J,axiom,
% 0.30/0.61 ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 0.30/0.61 => ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 0.30/0.61 = ( nil @ nat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.psimps(1)
% 0.30/0.61 thf(fact_5838_list__decode_Opinduct,axiom,
% 0.30/0.61 ! [A0: nat,P: nat > $o] :
% 0.30/0.61 ( ( accp @ nat @ nat_list_decode_rel @ A0 )
% 0.30/0.61 => ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 0.30/0.61 => ( P @ ( zero_zero @ nat ) ) )
% 0.30/0.61 => ( ! [N4: nat] :
% 0.30/0.61 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) )
% 0.30/0.61 => ( ! [X4: nat,Y4: nat] :
% 0.30/0.61 ( ( ( product_Pair @ nat @ nat @ X4 @ Y4 )
% 0.30/0.61 = ( nat_prod_decode @ N4 ) )
% 0.30/0.61 => ( P @ Y4 ) )
% 0.30/0.61 => ( P @ ( suc @ N4 ) ) ) )
% 0.30/0.61 => ( P @ A0 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % list_decode.pinduct
% 0.30/0.61 thf(fact_5839_cSUP__upper,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [X: B,A6: set @ B,F: B > A] :
% 0.30/0.61 ( ( member @ B @ X @ A6 )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.61 => ( ord_less_eq @ A @ ( F @ X ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSUP_upper
% 0.30/0.61 thf(fact_5840_cSUP__upper2,axiom,
% 0.30/0.61 ! [A: $tType,B: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [F: B > A,A6: set @ B,X: B,U: A] :
% 0.30/0.61 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.61 => ( ( member @ B @ X @ A6 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ U @ ( F @ X ) )
% 0.30/0.61 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSUP_upper2
% 0.30/0.61 thf(fact_5841_cSup__le__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [S3: set @ A,A3: A] :
% 0.30/0.61 ( ( S3
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ S3 )
% 0.30/0.61 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A3 )
% 0.30/0.61 = ( ! [X2: A] :
% 0.30/0.61 ( ( member @ A @ X2 @ S3 )
% 0.30/0.61 => ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSup_le_iff
% 0.30/0.61 thf(fact_5842_cSup__mono,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit1219197933456340205attice @ A )
% 0.30/0.61 => ! [B7: set @ A,A6: set @ A] :
% 0.30/0.61 ( ( B7
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ A6 )
% 0.30/0.61 => ( ! [B5: A] :
% 0.30/0.61 ( ( member @ A @ B5 @ B7 )
% 0.30/0.61 => ? [X4: A] :
% 0.30/0.61 ( ( member @ A @ X4 @ A6 )
% 0.30/0.61 & ( ord_less_eq @ A @ B5 @ X4 ) ) )
% 0.30/0.61 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B7 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % cSup_mono
% 0.30/0.61 thf(fact_5843_less__cSup__iff,axiom,
% 0.30/0.61 ! [A: $tType] :
% 0.30/0.61 ( ( condit6923001295902523014norder @ A )
% 0.30/0.61 => ! [X8: set @ A,Y3: A] :
% 0.30/0.61 ( ( X8
% 0.30/0.61 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.61 => ( ( condit941137186595557371_above @ A @ X8 )
% 0.30/0.62 => ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 0.30/0.62 = ( ? [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ X8 )
% 0.30/0.62 & ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_cSup_iff
% 0.30/0.62 thf(fact_5844_cSUP__lessD,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 0.30/0.62 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
% 0.30/0.62 => ( ( member @ B @ I @ A6 )
% 0.30/0.62 => ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSUP_lessD
% 0.30/0.62 thf(fact_5845_cSUP__le__iff,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ B,F: B > A,U: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
% 0.30/0.62 = ( ! [X2: B] :
% 0.30/0.62 ( ( member @ B @ X2 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSUP_le_iff
% 0.30/0.62 thf(fact_5846_cSUP__mono,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ B,G: C > A,B7: set @ C,F: B > A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B7 ) )
% 0.30/0.62 => ( ! [N4: B] :
% 0.30/0.62 ( ( member @ B @ N4 @ A6 )
% 0.30/0.62 => ? [X4: C] :
% 0.30/0.62 ( ( member @ C @ X4 @ B7 )
% 0.30/0.62 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSUP_mono
% 0.30/0.62 thf(fact_5847_cSup__subset__mono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ B7 )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSup_subset_mono
% 0.30/0.62 thf(fact_5848_less__cSUP__iff,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit6923001295902523014norder @ A )
% 0.30/0.62 => ! [A6: set @ B,F: B > A,A3: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.62 = ( ? [X2: B] :
% 0.30/0.62 ( ( member @ B @ X2 @ A6 )
% 0.30/0.62 & ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_cSUP_iff
% 0.30/0.62 thf(fact_5849_list__decode_Opsimps_I2_J,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
% 0.30/0.62 => ( ( nat_list_decode @ ( suc @ N ) )
% 0.30/0.62 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.30/0.62 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 0.30/0.62 @ ( nat_prod_decode @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % list_decode.psimps(2)
% 0.30/0.62 thf(fact_5850_cSUP__subset__mono,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B7 ) )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 0.30/0.62 => ( ! [X3: B] :
% 0.30/0.62 ( ( member @ B @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSUP_subset_mono
% 0.30/0.62 thf(fact_5851_cSup__inter__less__eq,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( condit941137186595557371_above @ A @ A6 )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ B7 )
% 0.30/0.62 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSup_inter_less_eq
% 0.30/0.62 thf(fact_5852_division__segment__integer__def,axiom,
% 0.30/0.62 ( ( euclid7384307370059645450egment @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % division_segment_integer_def
% 0.30/0.62 thf(fact_5853_mono__cInf,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 & ( condit1219197933456340205attice @ B ) )
% 0.30/0.62 => ! [F: A > B,A6: set @ A] :
% 0.30/0.62 ( ( order_mono @ A @ B @ F )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mono_cInf
% 0.30/0.62 thf(fact_5854_mono__cINF,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.62 ( ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 & ( condit1219197933456340205attice @ B ) )
% 0.30/0.62 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 0.30/0.62 ( ( order_mono @ A @ B @ F )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A6 @ I6 ) )
% 0.30/0.62 => ( ( I6
% 0.30/0.62 != ( bot_bot @ ( set @ C ) ) )
% 0.30/0.62 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
% 0.30/0.62 @ ( complete_Inf_Inf @ B
% 0.30/0.62 @ ( image @ C @ B
% 0.30/0.62 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 0.30/0.62 @ I6 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mono_cINF
% 0.30/0.62 thf(fact_5855_bdd__belowI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ! [A6: set @ A,M: A] :
% 0.30/0.62 ( ! [X3: A] :
% 0.30/0.62 ( ( member @ A @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ M @ X3 ) )
% 0.30/0.62 => ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_belowI
% 0.30/0.62 thf(fact_5856_bdd__below_OI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ! [A6: set @ A,M7: A] :
% 0.30/0.62 ( ! [X3: A] :
% 0.30/0.62 ( ( member @ A @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ M7 @ X3 ) )
% 0.30/0.62 => ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_below.I
% 0.30/0.62 thf(fact_5857_bdd__below__uminus,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ordered_ab_group_add @ A )
% 0.30/0.62 => ! [X8: set @ A] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
% 0.30/0.62 = ( condit941137186595557371_above @ A @ X8 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_below_uminus
% 0.30/0.62 thf(fact_5858_bdd__above__uminus,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ordered_ab_group_add @ A )
% 0.30/0.62 => ! [X8: set @ A] :
% 0.30/0.62 ( ( condit941137186595557371_above @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
% 0.30/0.62 = ( condit1013018076250108175_below @ A @ X8 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_above_uminus
% 0.30/0.62 thf(fact_5859_bdd__below_OI2,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ! [A6: set @ B,M7: A,F: B > A] :
% 0.30/0.62 ( ! [X3: B] :
% 0.30/0.62 ( ( member @ B @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ M7 @ ( F @ X3 ) ) )
% 0.30/0.62 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_below.I2
% 0.30/0.62 thf(fact_5860_bdd__belowI2,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ! [A6: set @ B,M: A,F: B > A] :
% 0.30/0.62 ( ! [X3: B] :
% 0.30/0.62 ( ( member @ B @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
% 0.30/0.62 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_belowI2
% 0.30/0.62 thf(fact_5861_cInf__lower2,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [X: A,X8: set @ A,Y3: A] :
% 0.30/0.62 ( ( member @ A @ X @ X8 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ X8 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_lower2
% 0.30/0.62 thf(fact_5862_cInf__lower,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [X: A,X8: set @ A] :
% 0.30/0.62 ( ( member @ A @ X @ X8 )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ X8 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_lower
% 0.30/0.62 thf(fact_5863_bdd__below_Ounfold,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A )
% 0.30/0.62 = ( ^ [A7: set @ A] :
% 0.30/0.62 ? [M9: A] :
% 0.30/0.62 ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ A7 )
% 0.30/0.62 => ( ord_less_eq @ A @ M9 @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_below.unfold
% 0.30/0.62 thf(fact_5864_bdd__below_OE,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ! [A6: set @ A] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ A6 )
% 0.30/0.62 => ~ ! [M8: A] :
% 0.30/0.62 ~ ! [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ M8 @ X4 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bdd_below.E
% 0.30/0.62 thf(fact_5865_cINF__lower2,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [F: B > A,A6: set @ B,X: B,U: A] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( member @ B @ X @ A6 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ ( F @ X ) @ U )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cINF_lower2
% 0.30/0.62 thf(fact_5866_cINF__lower,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [F: B > A,A6: set @ B,X: B] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( member @ B @ X @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cINF_lower
% 0.30/0.62 thf(fact_5867_le__cInf__iff,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [S3: set @ A,A3: A] :
% 0.30/0.62 ( ( S3
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ S3 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S3 ) )
% 0.30/0.62 = ( ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ S3 )
% 0.30/0.62 => ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % le_cInf_iff
% 0.30/0.62 thf(fact_5868_cInf__mono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [B7: set @ A,A6: set @ A] :
% 0.30/0.62 ( ( B7
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ A6 )
% 0.30/0.62 => ( ! [B5: A] :
% 0.30/0.62 ( ( member @ A @ B5 @ B7 )
% 0.30/0.62 => ? [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ A6 )
% 0.30/0.62 & ( ord_less_eq @ A @ X4 @ B5 ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_mono
% 0.30/0.62 thf(fact_5869_cInf__less__iff,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit6923001295902523014norder @ A )
% 0.30/0.62 => ! [X8: set @ A,Y3: A] :
% 0.30/0.62 ( ( X8
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ X8 )
% 0.30/0.62 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 )
% 0.30/0.62 = ( ? [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ X8 )
% 0.30/0.62 & ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_less_iff
% 0.30/0.62 thf(fact_5870_less__cINF__D,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.62 => ( ( member @ B @ I @ A6 )
% 0.30/0.62 => ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_cINF_D
% 0.30/0.62 thf(fact_5871_cINF__mono,axiom,
% 0.30/0.62 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [B7: set @ B,F: C > A,A6: set @ C,G: B > A] :
% 0.30/0.62 ( ( B7
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F @ A6 ) )
% 0.30/0.62 => ( ! [M6: B] :
% 0.30/0.62 ( ( member @ B @ M6 @ B7 )
% 0.30/0.62 => ? [X4: C] :
% 0.30/0.62 ( ( member @ C @ X4 @ A6 )
% 0.30/0.62 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cINF_mono
% 0.30/0.62 thf(fact_5872_le__cINF__iff,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ B,F: B > A,U: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 0.30/0.62 = ( ! [X2: B] :
% 0.30/0.62 ( ( member @ B @ X2 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % le_cINF_iff
% 0.30/0.62 thf(fact_5873_cInf__superset__mono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ B7 )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B7 ) @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_superset_mono
% 0.30/0.62 thf(fact_5874_cINF__less__iff,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit6923001295902523014norder @ A )
% 0.30/0.62 => ! [A6: set @ B,F: B > A,A3: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
% 0.30/0.62 = ( ? [X2: B] :
% 0.30/0.62 ( ( member @ B @ X2 @ A6 )
% 0.30/0.62 & ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cINF_less_iff
% 0.30/0.62 thf(fact_5875_uminus__integer__def,axiom,
% 0.30/0.62 ( ( uminus_uminus @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % uminus_integer_def
% 0.30/0.62 thf(fact_5876_abs__integer__def,axiom,
% 0.30/0.62 ( ( abs_abs @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_integer_def
% 0.30/0.62 thf(fact_5877_sgn__integer__def,axiom,
% 0.30/0.62 ( ( sgn_sgn @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sgn_integer_def
% 0.30/0.62 thf(fact_5878_cINF__superset__mono,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ B ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B7 ) )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 0.30/0.62 => ( ! [X3: B] :
% 0.30/0.62 ( ( member @ B @ X3 @ B7 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cINF_superset_mono
% 0.30/0.62 thf(fact_5879_less__eq__cInf__inter,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( condit1013018076250108175_below @ A @ A6 )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ B7 )
% 0.30/0.62 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_cInf_inter
% 0.30/0.62 thf(fact_5880_cInf__le__cSup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [A6: set @ A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ A6 )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_le_cSup
% 0.30/0.62 thf(fact_5881_not__integer__def,axiom,
% 0.30/0.62 ( ( bit_ri4277139882892585799ns_not @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % not_integer_def
% 0.30/0.62 thf(fact_5882_Code__Numeral_Odup__def,axiom,
% 0.30/0.62 ( code_dup
% 0.30/0.62 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
% 0.30/0.62 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.dup_def
% 0.30/0.62 thf(fact_5883_flip__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se8732182000553998342ip_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % flip_bit_integer_def
% 0.30/0.62 thf(fact_5884_set__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se5668285175392031749et_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % set_bit_integer_def
% 0.30/0.62 thf(fact_5885_Code__Numeral_Odup__code_I1_J,axiom,
% 0.30/0.62 ( ( code_dup @ ( zero_zero @ code_integer ) )
% 0.30/0.62 = ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.dup_code(1)
% 0.30/0.62 thf(fact_5886_dup_Orep__eq,axiom,
% 0.30/0.62 ! [X: code_integer] :
% 0.30/0.62 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 0.30/0.62 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % dup.rep_eq
% 0.30/0.62 thf(fact_5887_dup_Oabs__eq,axiom,
% 0.30/0.62 ! [X: int] :
% 0.30/0.62 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 0.30/0.62 = ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % dup.abs_eq
% 0.30/0.62 thf(fact_5888_take__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se2584673776208193580ke_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % take_bit_integer_def
% 0.30/0.62 thf(fact_5889_drop__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se4197421643247451524op_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % drop_bit_integer_def
% 0.30/0.62 thf(fact_5890_push__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se4730199178511100633sh_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % push_bit_integer_def
% 0.30/0.62 thf(fact_5891_unset__bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se2638667681897837118et_bit @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % unset_bit_integer_def
% 0.30/0.62 thf(fact_5892_Code__Numeral_Osub__code_I9_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62 = ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(9)
% 0.30/0.62 thf(fact_5893_Code__Numeral_Osub__code_I8_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62 = ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(8)
% 0.30/0.62 thf(fact_5894_xor__integer__def,axiom,
% 0.30/0.62 ( ( bit_se5824344971392196577ns_xor @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % xor_integer_def
% 0.30/0.62 thf(fact_5895_Code__Numeral_Osub__code_I1_J,axiom,
% 0.30/0.62 ( ( code_sub @ one2 @ one2 )
% 0.30/0.62 = ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(1)
% 0.30/0.62 thf(fact_5896_Code__Numeral_Osub__code_I6_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(6)
% 0.30/0.62 thf(fact_5897_Code__Numeral_Osub__code_I7_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(7)
% 0.30/0.62 thf(fact_5898_sub_Orep__eq,axiom,
% 0.30/0.62 ! [X: num,Xa3: num] :
% 0.30/0.62 ( ( code_int_of_integer @ ( code_sub @ X @ Xa3 ) )
% 0.30/0.62 = ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sub.rep_eq
% 0.30/0.62 thf(fact_5899_sub_Oabs__eq,axiom,
% 0.30/0.62 ( code_sub
% 0.30/0.62 = ( ^ [Xa4: num,X2: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sub.abs_eq
% 0.30/0.62 thf(fact_5900_plus__integer__def,axiom,
% 0.30/0.62 ( ( plus_plus @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer_def
% 0.30/0.62 thf(fact_5901_times__integer__def,axiom,
% 0.30/0.62 ( ( times_times @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer_def
% 0.30/0.62 thf(fact_5902_minus__integer__def,axiom,
% 0.30/0.62 ( ( minus_minus @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer_def
% 0.30/0.62 thf(fact_5903_divide__integer__def,axiom,
% 0.30/0.62 ( ( divide_divide @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % divide_integer_def
% 0.30/0.62 thf(fact_5904_modulo__integer__def,axiom,
% 0.30/0.62 ( ( modulo_modulo @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % modulo_integer_def
% 0.30/0.62 thf(fact_5905_and__integer__def,axiom,
% 0.30/0.62 ( ( bit_se5824344872417868541ns_and @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % and_integer_def
% 0.30/0.62 thf(fact_5906_or__integer__def,axiom,
% 0.30/0.62 ( ( bit_se1065995026697491101ons_or @ code_integer )
% 0.30/0.62 = ( 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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % or_integer_def
% 0.30/0.62 thf(fact_5907_Sup__int__def,axiom,
% 0.30/0.62 ( ( complete_Sup_Sup @ int )
% 0.30/0.62 = ( ^ [X7: set @ int] :
% 0.30/0.62 ( the @ int
% 0.30/0.62 @ ^ [X2: int] :
% 0.30/0.62 ( ( member @ int @ X2 @ X7 )
% 0.30/0.62 & ! [Y6: int] :
% 0.30/0.62 ( ( member @ int @ Y6 @ X7 )
% 0.30/0.62 => ( ord_less_eq @ int @ Y6 @ X2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_int_def
% 0.30/0.62 thf(fact_5908_lexordp__conv__lexord,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( ( ord_lexordp @ A )
% 0.30/0.62 = ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_conv_lexord
% 0.30/0.62 thf(fact_5909_sorted__wrt__iff__nth__Suc__transp,axiom,
% 0.30/0.62 ! [A: $tType,P: A > A > $o,Xs: list @ A] :
% 0.30/0.62 ( ( transp @ A @ P )
% 0.30/0.62 => ( ( sorted_wrt @ A @ P @ Xs )
% 0.30/0.62 = ( ! [I5: nat] :
% 0.30/0.62 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.62 => ( P @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sorted_wrt_iff_nth_Suc_transp
% 0.30/0.62 thf(fact_5910_lexordp__simps_I3_J,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
% 0.30/0.62 ( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
% 0.30/0.62 = ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.62 | ( ~ ( ord_less @ A @ Y3 @ X )
% 0.30/0.62 & ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_simps(3)
% 0.30/0.62 thf(fact_5911_lexordp__irreflexive,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [Xs: list @ A] :
% 0.30/0.62 ( ! [X3: A] :
% 0.30/0.62 ~ ( ord_less @ A @ X3 @ X3 )
% 0.30/0.62 => ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_irreflexive
% 0.30/0.62 thf(fact_5912_lexordp_OCons,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.62 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.62 => ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp.Cons
% 0.30/0.62 thf(fact_5913_lexordp_OCons__eq,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.62 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.62 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.30/0.62 => ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.30/0.62 => ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp.Cons_eq
% 0.30/0.62 thf(fact_5914_lexordp__append__leftD,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
% 0.30/0.62 ( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
% 0.30/0.62 => ( ! [A8: A] :
% 0.30/0.62 ~ ( ord_less @ A @ A8 @ A8 )
% 0.30/0.62 => ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_append_leftD
% 0.30/0.62 thf(fact_5915_Sup__eq__Inf,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ( ( complete_Sup_Sup @ A )
% 0.30/0.62 = ( ^ [A7: set @ A] :
% 0.30/0.62 ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [B3: A] :
% 0.30/0.62 ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ A7 )
% 0.30/0.62 => ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_eq_Inf
% 0.30/0.62 thf(fact_5916_Inf__eq__Sup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ( ( complete_Inf_Inf @ A )
% 0.30/0.62 = ( ^ [A7: set @ A] :
% 0.30/0.62 ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [B3: A] :
% 0.30/0.62 ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ A7 )
% 0.30/0.62 => ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Inf_eq_Sup
% 0.30/0.62 thf(fact_5917_lexordp__induct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.30/0.62 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.30/0.62 => ( ! [Y5: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y5 @ Ys4 ) )
% 0.30/0.62 => ( ! [X3: A,Xs2: list @ A,Y5: A,Ys4: list @ A] :
% 0.30/0.62 ( ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.62 => ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y5 @ Ys4 ) ) )
% 0.30/0.62 => ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
% 0.30/0.62 ( ( ord_lexordp @ A @ Xs2 @ Ys4 )
% 0.30/0.62 => ( ( P @ Xs2 @ Ys4 )
% 0.30/0.62 => ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys4 ) ) ) )
% 0.30/0.62 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_induct
% 0.30/0.62 thf(fact_5918_lexordp__cases,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.30/0.62 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 0.30/0.62 => ( ( ( Xs
% 0.30/0.62 = ( nil @ A ) )
% 0.30/0.62 => ! [Y5: A,Ys5: list @ A] :
% 0.30/0.62 ( Ys2
% 0.30/0.62 != ( cons @ A @ Y5 @ Ys5 ) ) )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ? [Xs4: list @ A] :
% 0.30/0.62 ( Xs
% 0.30/0.62 = ( cons @ A @ X3 @ Xs4 ) )
% 0.30/0.62 => ! [Y5: A] :
% 0.30/0.62 ( ? [Ys5: list @ A] :
% 0.30/0.62 ( Ys2
% 0.30/0.62 = ( cons @ A @ Y5 @ Ys5 ) )
% 0.30/0.62 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 0.30/0.62 => ~ ! [X3: A,Xs4: list @ A] :
% 0.30/0.62 ( ( Xs
% 0.30/0.62 = ( cons @ A @ X3 @ Xs4 ) )
% 0.30/0.62 => ! [Ys5: list @ A] :
% 0.30/0.62 ( ( Ys2
% 0.30/0.62 = ( cons @ A @ X3 @ Ys5 ) )
% 0.30/0.62 => ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_cases
% 0.30/0.62 thf(fact_5919_lexordp_Osimps,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ( ( ord_lexordp @ A )
% 0.30/0.62 = ( ^ [A12: list @ A,A23: list @ A] :
% 0.30/0.62 ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.62 ( ( A12
% 0.30/0.62 = ( nil @ A ) )
% 0.30/0.62 & ( A23
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 0.30/0.62 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.62 ( ( A12
% 0.30/0.62 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.62 & ( A23
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.62 & ( ord_less @ A @ X2 @ Y6 ) )
% 0.30/0.62 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.62 ( ( A12
% 0.30/0.62 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.62 & ( A23
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.62 & ~ ( ord_less @ A @ X2 @ Y6 )
% 0.30/0.62 & ~ ( ord_less @ A @ Y6 @ X2 )
% 0.30/0.62 & ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp.simps
% 0.30/0.62 thf(fact_5920_lexordp_Ocases,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [A1: list @ A,A22: list @ A] :
% 0.30/0.62 ( ( ord_lexordp @ A @ A1 @ A22 )
% 0.30/0.62 => ( ( ( A1
% 0.30/0.62 = ( nil @ A ) )
% 0.30/0.62 => ! [Y5: A,Ys4: list @ A] :
% 0.30/0.62 ( A22
% 0.30/0.62 != ( cons @ A @ Y5 @ Ys4 ) ) )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ? [Xs2: list @ A] :
% 0.30/0.62 ( A1
% 0.30/0.62 = ( cons @ A @ X3 @ Xs2 ) )
% 0.30/0.62 => ! [Y5: A] :
% 0.30/0.62 ( ? [Ys4: list @ A] :
% 0.30/0.62 ( A22
% 0.30/0.62 = ( cons @ A @ Y5 @ Ys4 ) )
% 0.30/0.62 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 0.30/0.62 => ~ ! [X3: A,Y5: A,Xs2: list @ A] :
% 0.30/0.62 ( ( A1
% 0.30/0.62 = ( cons @ A @ X3 @ Xs2 ) )
% 0.30/0.62 => ! [Ys4: list @ A] :
% 0.30/0.62 ( ( A22
% 0.30/0.62 = ( cons @ A @ Y5 @ Ys4 ) )
% 0.30/0.62 => ( ~ ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.62 => ( ~ ( ord_less @ A @ Y5 @ X3 )
% 0.30/0.62 => ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp.cases
% 0.30/0.62 thf(fact_5921_lexordp__append__left__rightI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ! [X: A,Y3: A,Us: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.62 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.62 => ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_append_left_rightI
% 0.30/0.62 thf(fact_5922_lexordp__iff,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( ( ord_lexordp @ A )
% 0.30/0.62 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.62 ( ? [X2: A,Vs2: list @ A] :
% 0.30/0.62 ( Ys3
% 0.30/0.62 = ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Vs2 ) ) )
% 0.30/0.62 | ? [Us2: list @ A,A5: A,B3: A,Vs2: list @ A,Ws: list @ A] :
% 0.30/0.62 ( ( ord_less @ A @ A5 @ B3 )
% 0.30/0.62 & ( Xs3
% 0.30/0.62 = ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
% 0.30/0.62 & ( Ys3
% 0.30/0.62 = ( append @ A @ Us2 @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lexordp_iff
% 0.30/0.62 thf(fact_5923_cInf__cSup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [S3: set @ A] :
% 0.30/0.62 ( ( S3
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit1013018076250108175_below @ A @ S3 )
% 0.30/0.62 => ( ( complete_Inf_Inf @ A @ S3 )
% 0.30/0.62 = ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [X2: A] :
% 0.30/0.62 ! [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ S3 )
% 0.30/0.62 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cInf_cSup
% 0.30/0.62 thf(fact_5924_cSup__cInf,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( condit1219197933456340205attice @ A )
% 0.30/0.62 => ! [S3: set @ A] :
% 0.30/0.62 ( ( S3
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( condit941137186595557371_above @ A @ S3 )
% 0.30/0.62 => ( ( complete_Sup_Sup @ A @ S3 )
% 0.30/0.62 = ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [X2: A] :
% 0.30/0.62 ! [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ S3 )
% 0.30/0.62 => ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cSup_cInf
% 0.30/0.62 thf(fact_5925_transp__gr,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ( transp @ A
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % transp_gr
% 0.30/0.62 thf(fact_5926_transp__ge,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ( transp @ A
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % transp_ge
% 0.30/0.62 thf(fact_5927_ord__class_Olexordp__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ( ( ord_lexordp @ A )
% 0.30/0.62 = ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
% 0.30/0.62 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
% 0.30/0.62 ( ? [Y6: A,Ys3: list @ A] :
% 0.30/0.62 ( ( X12
% 0.30/0.62 = ( nil @ A ) )
% 0.30/0.62 & ( X24
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 0.30/0.62 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.62 ( ( X12
% 0.30/0.62 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.62 & ( X24
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.62 & ( ord_less @ A @ X2 @ Y6 ) )
% 0.30/0.62 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.62 ( ( X12
% 0.30/0.62 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.62 & ( X24
% 0.30/0.62 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.62 & ~ ( ord_less @ A @ X2 @ Y6 )
% 0.30/0.62 & ~ ( ord_less @ A @ Y6 @ X2 )
% 0.30/0.62 & ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ord_class.lexordp_def
% 0.30/0.62 thf(fact_5928_Sup__Inf,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple592849572758109894attice @ A )
% 0.30/0.62 => ! [A6: set @ ( set @ A )] :
% 0.30/0.62 ( ( complete_Sup_Sup @ A @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A6 ) )
% 0.30/0.62 = ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
% 0.30/0.62 @ ( collect @ ( set @ A )
% 0.30/0.62 @ ^ [Uu: set @ A] :
% 0.30/0.62 ? [F2: ( set @ A ) > A] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ A] :
% 0.30/0.62 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_Inf
% 0.30/0.62 thf(fact_5929_Inf__Sup__le,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple592849572758109894attice @ A )
% 0.30/0.62 => ! [A6: set @ ( set @ A )] :
% 0.30/0.62 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
% 0.30/0.62 @ ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 0.30/0.62 @ ( collect @ ( set @ A )
% 0.30/0.62 @ ^ [Uu: set @ A] :
% 0.30/0.62 ? [F2: ( set @ A ) > A] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ A] :
% 0.30/0.62 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Inf_Sup_le
% 0.30/0.62 thf(fact_5930_Sup__Inf__le,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [A6: set @ ( set @ A )] :
% 0.30/0.62 ( ord_less_eq @ A
% 0.30/0.62 @ ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 0.30/0.62 @ ( collect @ ( set @ A )
% 0.30/0.62 @ ^ [Uu: set @ A] :
% 0.30/0.62 ? [F2: ( set @ A ) > A] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ A] :
% 0.30/0.62 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) )
% 0.30/0.62 @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_Inf_le
% 0.30/0.62 thf(fact_5931_finite__Inf__Sup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( finite8700451911770168679attice @ A )
% 0.30/0.62 => ! [A6: set @ ( set @ A )] :
% 0.30/0.62 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
% 0.30/0.62 @ ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 0.30/0.62 @ ( collect @ ( set @ A )
% 0.30/0.62 @ ^ [Uu: set @ A] :
% 0.30/0.62 ? [F2: ( set @ A ) > A] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ A] :
% 0.30/0.62 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % finite_Inf_Sup
% 0.30/0.62 thf(fact_5932_SUP__INF__set,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( comple592849572758109894attice @ A )
% 0.30/0.62 => ! [G: B > A,A6: set @ ( set @ B )] :
% 0.30/0.62 ( ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( image @ ( set @ B ) @ A
% 0.30/0.62 @ ^ [X2: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ X2 ) )
% 0.30/0.62 @ A6 ) )
% 0.30/0.62 = ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( image @ ( set @ B ) @ A
% 0.30/0.62 @ ^ [X2: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ X2 ) )
% 0.30/0.62 @ ( collect @ ( set @ B )
% 0.30/0.62 @ ^ [Uu: set @ B] :
% 0.30/0.62 ? [F2: ( set @ B ) > B] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ B] :
% 0.30/0.62 ( ( member @ ( set @ B ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % SUP_INF_set
% 0.30/0.62 thf(fact_5933_INF__SUP__set,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType] :
% 0.30/0.62 ( ( comple592849572758109894attice @ A )
% 0.30/0.62 => ! [G: B > A,A6: set @ ( set @ B )] :
% 0.30/0.62 ( ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( image @ ( set @ B ) @ A
% 0.30/0.62 @ ^ [B6: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) )
% 0.30/0.62 @ A6 ) )
% 0.30/0.62 = ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( image @ ( set @ B ) @ A
% 0.30/0.62 @ ^ [B6: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) )
% 0.30/0.62 @ ( collect @ ( set @ B )
% 0.30/0.62 @ ^ [Uu: set @ B] :
% 0.30/0.62 ? [F2: ( set @ B ) > B] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
% 0.30/0.62 & ! [X2: set @ B] :
% 0.30/0.62 ( ( member @ ( set @ B ) @ X2 @ A6 )
% 0.30/0.62 => ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % INF_SUP_set
% 0.30/0.62 thf(fact_5934_lfp__funpow,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,N: nat] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
% 0.30/0.62 = ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_funpow
% 0.30/0.62 thf(fact_5935_lfp__Kleene__iter,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,K: nat] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( bot_bot @ A ) )
% 0.30/0.62 = ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) )
% 0.30/0.62 => ( ( complete_lattice_lfp @ A @ F )
% 0.30/0.62 = ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_Kleene_iter
% 0.30/0.62 thf(fact_5936_transp__le,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % transp_le
% 0.30/0.62 thf(fact_5937_transp__less,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( preorder @ A )
% 0.30/0.62 => ( transp @ A @ ( ord_less @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % transp_less
% 0.30/0.62 thf(fact_5938_lfp__ordinal__induct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,P: A > $o] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ! [S4: A] :
% 0.30/0.62 ( ( P @ S4 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F ) )
% 0.30/0.62 => ( P @ ( F @ S4 ) ) ) )
% 0.30/0.62 => ( ! [M8: set @ A] :
% 0.30/0.62 ( ! [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ M8 )
% 0.30/0.62 => ( P @ X4 ) )
% 0.30/0.62 => ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
% 0.30/0.62 => ( P @ ( complete_lattice_lfp @ A @ F ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_ordinal_induct
% 0.30/0.62 thf(fact_5939_def__lfp__induct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [A6: A,F: A > A,P: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 = ( complete_lattice_lfp @ A @ F ) )
% 0.30/0.62 => ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ A6 @ P ) ) @ P )
% 0.30/0.62 => ( ord_less_eq @ A @ A6 @ P ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % def_lfp_induct
% 0.30/0.62 thf(fact_5940_lfp__induct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,P: A] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) @ P )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_induct
% 0.30/0.62 thf(fact_5941_le__rel__bool__arg__iff,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( ord @ A )
% 0.30/0.62 => ( ( ord_less_eq @ ( $o > A ) )
% 0.30/0.62 = ( ^ [X7: $o > A,Y9: $o > A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y9 @ $false ) )
% 0.30/0.62 & ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % le_rel_bool_arg_iff
% 0.30/0.62 thf(fact_5942_lfp__mono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,G: A > A] :
% 0.30/0.62 ( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_mono
% 0.30/0.62 thf(fact_5943_lfp__lowerbound,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,A6: A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ ( F @ A6 ) @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ A6 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_lowerbound
% 0.30/0.62 thf(fact_5944_lfp__greatest,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,A6: A] :
% 0.30/0.62 ( ! [U2: A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ ( F @ U2 ) @ U2 )
% 0.30/0.62 => ( ord_less_eq @ A @ A6 @ U2 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ A6 @ ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_greatest
% 0.30/0.62 thf(fact_5945_lfp__lfp,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A > A] :
% 0.30/0.62 ( ! [X3: A,Y5: A,W: A,Z4: A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ W @ Z4 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
% 0.30/0.62 => ( ( complete_lattice_lfp @ A
% 0.30/0.62 @ ^ [X2: A] : ( complete_lattice_lfp @ A @ ( F @ X2 ) ) )
% 0.30/0.62 = ( complete_lattice_lfp @ A
% 0.30/0.62 @ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_lfp
% 0.30/0.62 thf(fact_5946_lfp__eqI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F4: A > A,X: A] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F4 )
% 0.30/0.62 => ( ( ( F4 @ X )
% 0.30/0.62 = X )
% 0.30/0.62 => ( ! [Z4: A] :
% 0.30/0.62 ( ( ( F4 @ Z4 )
% 0.30/0.62 = Z4 )
% 0.30/0.62 => ( ord_less_eq @ A @ X @ Z4 ) )
% 0.30/0.62 => ( ( complete_lattice_lfp @ A @ F4 )
% 0.30/0.62 = X ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_eqI
% 0.30/0.62 thf(fact_5947_lfp__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ( ( complete_lattice_lfp @ A )
% 0.30/0.62 = ( ^ [F2: A > A] :
% 0.30/0.62 ( complete_Inf_Inf @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [U3: A] : ( ord_less_eq @ A @ ( F2 @ U3 ) @ U3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_def
% 0.30/0.62 thf(fact_5948_char__of__take__bit__eq,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.62 => ! [N: nat,M: A] :
% 0.30/0.62 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
% 0.30/0.62 => ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
% 0.30/0.62 = ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % char_of_take_bit_eq
% 0.30/0.62 thf(fact_5949_num__of__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.62 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 0.30/0.62 @ code_num_of_integer ) ).
% 0.30/0.62
% 0.30/0.62 % num_of_integer.transfer
% 0.30/0.62 thf(fact_5950_Max_Osemilattice__order__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Max.semilattice_order_set_axioms
% 0.30/0.62 thf(fact_5951_char__of__nat,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.62 => ! [N: nat] :
% 0.30/0.62 ( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.30/0.62 = ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % char_of_nat
% 0.30/0.62 thf(fact_5952_integer_Oid__abs__transfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ int @ int @ code_integer
% 0.30/0.62 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.62 @ code_pcr_integer
% 0.30/0.62 @ ^ [X2: int] : X2
% 0.30/0.62 @ code_integer_of_int ) ).
% 0.30/0.62
% 0.30/0.62 % integer.id_abs_transfer
% 0.30/0.62 thf(fact_5953_integer_Orep__transfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.62 @ ^ [X2: int] : X2
% 0.30/0.62 @ code_int_of_integer ) ).
% 0.30/0.62
% 0.30/0.62 % integer.rep_transfer
% 0.30/0.62 thf(fact_5954_zero__integer_Otransfer,axiom,
% 0.30/0.62 code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % zero_integer.transfer
% 0.30/0.62 thf(fact_5955_less__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.62 @ ( ord_less @ int )
% 0.30/0.62 @ ( ord_less @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer.transfer
% 0.30/0.62 thf(fact_5956_plus__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer.transfer
% 0.30/0.62 thf(fact_5957_uminus__integer_Otransfer,axiom,
% 0.30/0.62 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % uminus_integer.transfer
% 0.30/0.62 thf(fact_5958_times__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer.transfer
% 0.30/0.62 thf(fact_5959_one__integer_Otransfer,axiom,
% 0.30/0.62 code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % one_integer.transfer
% 0.30/0.62 thf(fact_5960_less__eq__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.62 @ ( ord_less_eq @ int )
% 0.30/0.62 @ ( ord_less_eq @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer.transfer
% 0.30/0.62 thf(fact_5961_minus__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer.transfer
% 0.30/0.62 thf(fact_5962_abs__integer_Otransfer,axiom,
% 0.30/0.62 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % abs_integer.transfer
% 0.30/0.62 thf(fact_5963_integer__of__nat_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ code_pcr_integer
% 0.30/0.62 @ ( semiring_1_of_nat @ int )
% 0.30/0.62 @ code_integer_of_nat ) ).
% 0.30/0.62
% 0.30/0.62 % integer_of_nat.transfer
% 0.30/0.62 thf(fact_5964_divide__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % divide_integer.transfer
% 0.30/0.62 thf(fact_5965_nat__of__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ nat2
% 0.30/0.62 @ code_nat_of_integer ) ).
% 0.30/0.62
% 0.30/0.62 % nat_of_integer.transfer
% 0.30/0.62 thf(fact_5966_modulo__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % modulo_integer.transfer
% 0.30/0.62 thf(fact_5967_sgn__integer_Otransfer,axiom,
% 0.30/0.62 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % sgn_integer.transfer
% 0.30/0.62 thf(fact_5968_dup_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
% 0.30/0.62 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
% 0.30/0.62 @ code_dup ) ).
% 0.30/0.62
% 0.30/0.62 % dup.transfer
% 0.30/0.62 thf(fact_5969_take__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se2584673776208193580ke_bit @ int )
% 0.30/0.62 @ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % take_bit_integer.transfer
% 0.30/0.62 thf(fact_5970_not__integer_Otransfer,axiom,
% 0.30/0.62 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % not_integer.transfer
% 0.30/0.62 thf(fact_5971_and__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % and_integer.transfer
% 0.30/0.62 thf(fact_5972_bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ int )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % bit_integer.transfer
% 0.30/0.62 thf(fact_5973_or__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % or_integer.transfer
% 0.30/0.62 thf(fact_5974_xor__integer_Otransfer,axiom,
% 0.30/0.62 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 ) ).
% 0.30/0.62
% 0.30/0.62 % xor_integer.transfer
% 0.30/0.62 thf(fact_5975_push__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se4730199178511100633sh_bit @ int )
% 0.30/0.62 @ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % push_bit_integer.transfer
% 0.30/0.62 thf(fact_5976_drop__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se4197421643247451524op_bit @ int )
% 0.30/0.62 @ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % drop_bit_integer.transfer
% 0.30/0.62 thf(fact_5977_mask__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ code_pcr_integer
% 0.30/0.62 @ ( bit_se2239418461657761734s_mask @ int )
% 0.30/0.62 @ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % mask_integer.transfer
% 0.30/0.62 thf(fact_5978_unset__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se2638667681897837118et_bit @ int )
% 0.30/0.62 @ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % unset_bit_integer.transfer
% 0.30/0.62 thf(fact_5979_set__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se5668285175392031749et_bit @ int )
% 0.30/0.62 @ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % set_bit_integer.transfer
% 0.30/0.62 thf(fact_5980_flip__bit__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.30/0.62 @ ( bit_se8732182000553998342ip_bit @ int )
% 0.30/0.62 @ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % flip_bit_integer.transfer
% 0.30/0.62 thf(fact_5981_division__segment__integer_Otransfer,axiom,
% 0.30/0.62 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % division_segment_integer.transfer
% 0.30/0.62 thf(fact_5982_euclidean__size__integer_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 0.30/0.62 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.62 @ ( euclid6346220572633701492n_size @ int )
% 0.30/0.62 @ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % euclidean_size_integer.transfer
% 0.30/0.62 thf(fact_5983_sub_Otransfer,axiom,
% 0.30/0.62 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
% 0.30/0.62 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.62 @ ( bNF_rel_fun @ num @ num @ int @ code_integer
% 0.30/0.62 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 0.30/0.62 @ code_pcr_integer )
% 0.30/0.62 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
% 0.30/0.62 @ code_sub ) ).
% 0.30/0.62
% 0.30/0.62 % sub.transfer
% 0.30/0.62 thf(fact_5984_Inf__fin_Osemilattice__order__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_inf @ A )
% 0.30/0.62 => ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Inf_fin.semilattice_order_set_axioms
% 0.30/0.62 thf(fact_5985_Min_Osemilattice__order__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Min.semilattice_order_set_axioms
% 0.30/0.62 thf(fact_5986_UNIV__char__of__nat,axiom,
% 0.30/0.62 ( ( top_top @ ( set @ char ) )
% 0.30/0.62 = ( image @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % UNIV_char_of_nat
% 0.30/0.62 thf(fact_5987_inj__on__char__of__nat,axiom,
% 0.30/0.62 inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_on_char_of_nat
% 0.30/0.62 thf(fact_5988_Sup__fin_Osemilattice__order__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_sup @ A )
% 0.30/0.62 => ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.62 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_fin.semilattice_order_set_axioms
% 0.30/0.62 thf(fact_5989_char__of__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_un5681908812861735899ations @ A )
% 0.30/0.62 => ( ( unique5772411509450598832har_of @ A )
% 0.30/0.62 = ( ^ [N3: A] :
% 0.30/0.62 ( char2
% 0.30/0.62 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
% 0.30/0.62 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % char_of_def
% 0.30/0.62 thf(fact_5990_range__nat__of__char,axiom,
% 0.30/0.62 ( ( image @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
% 0.30/0.62 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % range_nat_of_char
% 0.30/0.62 thf(fact_5991_Gcd__fin__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semiring_gcd @ A )
% 0.30/0.62 => ( ( semiring_gcd_Gcd_fin @ A )
% 0.30/0.62 = ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Gcd_fin_def
% 0.30/0.62 thf(fact_5992_char_Osize_I2_J,axiom,
% 0.30/0.62 ! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.30/0.62 ( ( size_size @ char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.30/0.62 = ( zero_zero @ nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % char.size(2)
% 0.30/0.62 thf(fact_5993_nat__of__char__less__256,axiom,
% 0.30/0.62 ! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nat_of_char_less_256
% 0.30/0.62 thf(fact_5994_assert__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( heap_Time_assert @ A )
% 0.30/0.62 = ( ^ [P4: A > $o,X2: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X2 ) @ ( heap_Time_return @ A @ X2 ) @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % assert_def
% 0.30/0.62 thf(fact_5995_char_Osize__gen,axiom,
% 0.30/0.62 ! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 0.30/0.62 ( ( size_char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 0.30/0.62 = ( zero_zero @ nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % char.size_gen
% 0.30/0.62 thf(fact_5996_n__lists_Osimps_I2_J,axiom,
% 0.30/0.62 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.62 ( ( n_lists @ A @ ( suc @ N ) @ Xs )
% 0.30/0.62 = ( concat @ ( list @ A )
% 0.30/0.62 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 0.30/0.62 @ ^ [Ys3: list @ A] :
% 0.30/0.62 ( map @ A @ ( list @ A )
% 0.30/0.62 @ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
% 0.30/0.62 @ Xs )
% 0.30/0.62 @ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % n_lists.simps(2)
% 0.30/0.62 thf(fact_5997_comm__append__is__replicate,axiom,
% 0.30/0.62 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.62 ( ( Xs
% 0.30/0.62 != ( nil @ A ) )
% 0.30/0.62 => ( ( Ys2
% 0.30/0.62 != ( nil @ A ) )
% 0.30/0.62 => ( ( ( append @ A @ Xs @ Ys2 )
% 0.30/0.62 = ( append @ A @ Ys2 @ Xs ) )
% 0.30/0.62 => ? [N4: nat,Zs2: list @ A] :
% 0.30/0.62 ( ( ord_less @ nat @ ( one_one @ nat ) @ N4 )
% 0.30/0.62 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N4 @ Zs2 ) )
% 0.30/0.62 = ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % comm_append_is_replicate
% 0.30/0.62 thf(fact_5998_strict__sorted__equal__Uniq,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ! [A6: set @ A] :
% 0.30/0.62 ( uniq @ ( list @ A )
% 0.30/0.62 @ ^ [Xs3: list @ A] :
% 0.30/0.62 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
% 0.30/0.62 & ( ( set2 @ A @ Xs3 )
% 0.30/0.62 = A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % strict_sorted_equal_Uniq
% 0.30/0.62 thf(fact_5999_integer__of__natural__of__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 0.30/0.62 = ( semiring_1_of_nat @ code_integer @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % integer_of_natural_of_nat
% 0.30/0.62 thf(fact_6000_integer__of__natural__of__integer,axiom,
% 0.30/0.62 ! [K: code_integer] :
% 0.30/0.62 ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
% 0.30/0.62 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % integer_of_natural_of_integer
% 0.30/0.62 thf(fact_6001_natural__of__integer__of__natural,axiom,
% 0.30/0.62 ! [N: code_natural] :
% 0.30/0.62 ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
% 0.30/0.62 = N ) ).
% 0.30/0.62
% 0.30/0.62 % natural_of_integer_of_natural
% 0.30/0.62 thf(fact_6002_card__def,axiom,
% 0.30/0.62 ! [B: $tType] :
% 0.30/0.62 ( ( finite_card @ B )
% 0.30/0.62 = ( finite_folding_F @ B @ nat
% 0.30/0.62 @ ^ [Uu: B] : suc
% 0.30/0.62 @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % card_def
% 0.30/0.62 thf(fact_6003_length__code,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( size_size @ ( list @ A ) )
% 0.30/0.62 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % length_code
% 0.30/0.62 thf(fact_6004_gen__length__code_I2_J,axiom,
% 0.30/0.62 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 0.30/0.62 ( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
% 0.30/0.62 = ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% 0.30/0.62
% 0.30/0.62 % gen_length_code(2)
% 0.30/0.62 thf(fact_6005_less__than__iff,axiom,
% 0.30/0.62 ! [X: nat,Y3: nat] :
% 0.30/0.62 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ less_than )
% 0.30/0.62 = ( ord_less @ nat @ X @ Y3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_than_iff
% 0.30/0.62 thf(fact_6006_Code__Numeral_Osub__code_I4_J,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( code_sub @ one2 @ ( bit0 @ N ) )
% 0.30/0.62 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(4)
% 0.30/0.62 thf(fact_6007_Code__Numeral_Odup__code_I3_J,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( code_dup @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.dup_code(3)
% 0.30/0.62 thf(fact_6008_less__integer__code_I7_J,axiom,
% 0.30/0.62 ! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(7)
% 0.30/0.62 thf(fact_6009_less__integer__code_I3_J,axiom,
% 0.30/0.62 ! [L: num] :
% 0.30/0.62 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(3)
% 0.30/0.62 thf(fact_6010_plus__integer__code_I6_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer_code(6)
% 0.30/0.62 thf(fact_6011_less__eq__integer__code_I7_J,axiom,
% 0.30/0.62 ! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(7)
% 0.30/0.62 thf(fact_6012_less__eq__integer__code_I3_J,axiom,
% 0.30/0.62 ! [L: num] :
% 0.30/0.62 ~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(3)
% 0.30/0.62 thf(fact_6013_less__integer__code_I9_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.30/0.62 = ( ord_less @ num @ L @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(9)
% 0.30/0.62 thf(fact_6014_less__eq__integer__code_I9_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.30/0.62 = ( ord_less_eq @ num @ L @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(9)
% 0.30/0.62 thf(fact_6015_minus__integer__code_I6_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_sub @ N @ M ) ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer_code(6)
% 0.30/0.62 thf(fact_6016_divmod__abs__code_I2_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.30/0.62 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % divmod_abs_code(2)
% 0.30/0.62 thf(fact_6017_Code__Numeral_Osub__code_I5_J,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( code_sub @ one2 @ ( bit1 @ N ) )
% 0.30/0.62 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(5)
% 0.30/0.62 thf(fact_6018_pair__less__def,axiom,
% 0.30/0.62 ( fun_pair_less
% 0.30/0.62 = ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% 0.30/0.62
% 0.30/0.62 % pair_less_def
% 0.30/0.62 thf(fact_6019_Gcd__int__set__eq__fold,axiom,
% 0.30/0.62 ! [Xs: list @ int] :
% 0.30/0.62 ( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
% 0.30/0.62 = ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Gcd_int_set_eq_fold
% 0.30/0.62 thf(fact_6020_Gcd__set__eq__fold,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semiring_Gcd @ A )
% 0.30/0.62 => ! [Xs: list @ A] :
% 0.30/0.62 ( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
% 0.30/0.62 = ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Gcd_set_eq_fold
% 0.30/0.62 thf(fact_6021_Gcd__fin_Oset__eq__fold,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semiring_gcd @ A )
% 0.30/0.62 => ! [Xs: list @ A] :
% 0.30/0.62 ( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
% 0.30/0.62 = ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Gcd_fin.set_eq_fold
% 0.30/0.62 thf(fact_6022_mono__transfer,axiom,
% 0.30/0.62 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 0.30/0.62 ( ( ( order @ B )
% 0.30/0.62 & ( order @ D )
% 0.30/0.62 & ( order @ C )
% 0.30/0.62 & ( order @ A ) )
% 0.30/0.62 => ! [A6: A > B > $o,B7: C > D > $o] :
% 0.30/0.62 ( ( bi_total @ A @ B @ A6 )
% 0.30/0.62 => ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A6
% 0.30/0.62 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
% 0.30/0.62 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.62 @ ( ord_less_eq @ A )
% 0.30/0.62 @ ( ord_less_eq @ B ) )
% 0.30/0.62 => ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B7
% 0.30/0.62 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B7
% 0.30/0.62 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.62 @ ( ord_less_eq @ C )
% 0.30/0.62 @ ( ord_less_eq @ D ) )
% 0.30/0.62 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B7 )
% 0.30/0.62 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 0.30/0.62 @ ( order_mono @ A @ C )
% 0.30/0.62 @ ( order_mono @ B @ D ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mono_transfer
% 0.30/0.62 thf(fact_6023_binomial__def,axiom,
% 0.30/0.62 ( binomial
% 0.30/0.62 = ( ^ [N3: nat,K3: nat] :
% 0.30/0.62 ( finite_card @ ( set @ nat )
% 0.30/0.62 @ ( collect @ ( set @ nat )
% 0.30/0.62 @ ^ [K6: set @ nat] :
% 0.30/0.62 ( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 0.30/0.62 & ( ( finite_card @ nat @ K6 )
% 0.30/0.62 = K3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % binomial_def
% 0.30/0.62 thf(fact_6024_Pow__Compl,axiom,
% 0.30/0.62 ! [A: $tType,A6: set @ A] :
% 0.30/0.62 ( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.62 = ( collect @ ( set @ A )
% 0.30/0.62 @ ^ [Uu: set @ A] :
% 0.30/0.62 ? [B6: set @ A] :
% 0.30/0.62 ( ( Uu
% 0.30/0.62 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.30/0.62 & ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Pow_Compl
% 0.30/0.62 thf(fact_6025_int_Obi__total,axiom,
% 0.30/0.62 bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% 0.30/0.62
% 0.30/0.62 % int.bi_total
% 0.30/0.62 thf(fact_6026_integer_Obi__total,axiom,
% 0.30/0.62 bi_total @ int @ code_integer @ code_pcr_integer ).
% 0.30/0.62
% 0.30/0.62 % integer.bi_total
% 0.30/0.62 thf(fact_6027_Gcd__nat__set__eq__fold,axiom,
% 0.30/0.62 ! [Xs: list @ nat] :
% 0.30/0.62 ( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
% 0.30/0.62 = ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Gcd_nat_set_eq_fold
% 0.30/0.62 thf(fact_6028_Heap__lub__empty,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
% 0.30/0.62 = ( heap_Time_Heap2 @ A
% 0.30/0.62 @ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Heap_lub_empty
% 0.30/0.62 thf(fact_6029_gfp__Kleene__iter,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,K: nat] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( top_top @ A ) )
% 0.30/0.62 = ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) )
% 0.30/0.62 => ( ( complete_lattice_gfp @ A @ F )
% 0.30/0.62 = ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_Kleene_iter
% 0.30/0.62 thf(fact_6030_gfp__gfp,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A > A] :
% 0.30/0.62 ( ! [X3: A,Y5: A,W: A,Z4: A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ W @ Z4 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
% 0.30/0.62 => ( ( complete_lattice_gfp @ A
% 0.30/0.62 @ ^ [X2: A] : ( complete_lattice_gfp @ A @ ( F @ X2 ) ) )
% 0.30/0.62 = ( complete_lattice_gfp @ A
% 0.30/0.62 @ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_gfp
% 0.30/0.62 thf(fact_6031_gfp__mono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,G: A > A] :
% 0.30/0.62 ( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ ( complete_lattice_gfp @ A @ G ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_mono
% 0.30/0.62 thf(fact_6032_gfp__upperbound,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [X8: A,F: A > A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ X8 @ ( F @ X8 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_upperbound
% 0.30/0.62 thf(fact_6033_gfp__least,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,X8: A] :
% 0.30/0.62 ( ! [U2: A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ U2 @ ( F @ U2 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ U2 @ X8 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ X8 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_least
% 0.30/0.62 thf(fact_6034_gfp__eqI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F4: A > A,X: A] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F4 )
% 0.30/0.62 => ( ( ( F4 @ X )
% 0.30/0.62 = X )
% 0.30/0.62 => ( ! [Z4: A] :
% 0.30/0.62 ( ( ( F4 @ Z4 )
% 0.30/0.62 = Z4 )
% 0.30/0.62 => ( ord_less_eq @ A @ Z4 @ X ) )
% 0.30/0.62 => ( ( complete_lattice_gfp @ A @ F4 )
% 0.30/0.62 = X ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_eqI
% 0.30/0.62 thf(fact_6035_gfp__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ( ( complete_lattice_gfp @ A )
% 0.30/0.62 = ( ^ [F2: A > A] :
% 0.30/0.62 ( complete_Sup_Sup @ A
% 0.30/0.62 @ ( collect @ A
% 0.30/0.62 @ ^ [U3: A] : ( ord_less_eq @ A @ U3 @ ( F2 @ U3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_def
% 0.30/0.62 thf(fact_6036_coinduct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,X8: A] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % coinduct
% 0.30/0.62 thf(fact_6037_def__coinduct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [A6: A,F: A > A,X8: A] :
% 0.30/0.62 ( ( A6
% 0.30/0.62 = ( complete_lattice_gfp @ A @ F ) )
% 0.30/0.62 => ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ A6 ) ) )
% 0.30/0.62 => ( ord_less_eq @ A @ X8 @ A6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % def_coinduct
% 0.30/0.62 thf(fact_6038_coinduct__lemma,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [X8: A,F: A > A] :
% 0.30/0.62 ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
% 0.30/0.62 => ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % coinduct_lemma
% 0.30/0.62 thf(fact_6039_gfp__ordinal__induct,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,P: A > $o] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ! [S4: A] :
% 0.30/0.62 ( ( P @ S4 )
% 0.30/0.62 => ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ S4 )
% 0.30/0.62 => ( P @ ( F @ S4 ) ) ) )
% 0.30/0.62 => ( ! [M8: set @ A] :
% 0.30/0.62 ( ! [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ M8 )
% 0.30/0.62 => ( P @ X4 ) )
% 0.30/0.62 => ( P @ ( complete_Inf_Inf @ A @ M8 ) ) )
% 0.30/0.62 => ( P @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_ordinal_induct
% 0.30/0.62 thf(fact_6040_gfp__funpow,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A,N: nat] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
% 0.30/0.62 = ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gfp_funpow
% 0.30/0.62 thf(fact_6041_lfp__le__gfp,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple6319245703460814977attice @ A )
% 0.30/0.62 => ! [F: A > A] :
% 0.30/0.62 ( ( order_mono @ A @ A @ F )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lfp_le_gfp
% 0.30/0.62 thf(fact_6042_Code__Numeral_Osub__code_I2_J,axiom,
% 0.30/0.62 ! [M: num] :
% 0.30/0.62 ( ( code_sub @ ( bit0 @ M ) @ one2 )
% 0.30/0.62 = ( code_Pos @ ( bitM @ M ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(2)
% 0.30/0.62 thf(fact_6043_is__singleton__altdef,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( is_singleton @ A )
% 0.30/0.62 = ( ^ [A7: set @ A] :
% 0.30/0.62 ( ( finite_card @ A @ A7 )
% 0.30/0.62 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % is_singleton_altdef
% 0.30/0.62 thf(fact_6044_Code__Numeral_OPos__def,axiom,
% 0.30/0.62 ( code_Pos
% 0.30/0.62 = ( numeral_numeral @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.Pos_def
% 0.30/0.62 thf(fact_6045_Pos__fold_I2_J,axiom,
% 0.30/0.62 ! [K: num] :
% 0.30/0.62 ( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
% 0.30/0.62 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Pos_fold(2)
% 0.30/0.62 thf(fact_6046_Pos__fold_I1_J,axiom,
% 0.30/0.62 ( ( numeral_numeral @ code_integer @ one2 )
% 0.30/0.62 = ( code_Pos @ one2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % Pos_fold(1)
% 0.30/0.62 thf(fact_6047_Pos__fold_I3_J,axiom,
% 0.30/0.62 ! [K: num] :
% 0.30/0.62 ( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
% 0.30/0.62 = ( code_Pos @ ( bit1 @ K ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Pos_fold(3)
% 0.30/0.62 thf(fact_6048_one__integer__code,axiom,
% 0.30/0.62 ( ( one_one @ code_integer )
% 0.30/0.62 = ( code_Pos @ one2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % one_integer_code
% 0.30/0.62 thf(fact_6049_Code__Numeral_Odup__code_I2_J,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( code_dup @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.dup_code(2)
% 0.30/0.62 thf(fact_6050_less__integer__code_I4_J,axiom,
% 0.30/0.62 ! [K: num] :
% 0.30/0.62 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(4)
% 0.30/0.62 thf(fact_6051_less__integer__code_I2_J,axiom,
% 0.30/0.62 ! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(2)
% 0.30/0.62 thf(fact_6052_plus__integer__code_I3_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer_code(3)
% 0.30/0.62 thf(fact_6053_less__eq__integer__code_I4_J,axiom,
% 0.30/0.62 ! [K: num] :
% 0.30/0.62 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(4)
% 0.30/0.62 thf(fact_6054_less__eq__integer__code_I2_J,axiom,
% 0.30/0.62 ! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(2)
% 0.30/0.62 thf(fact_6055_Code__Numeral_ONeg__def,axiom,
% 0.30/0.62 ( code_Neg
% 0.30/0.62 = ( ^ [N3: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.Neg_def
% 0.30/0.62 thf(fact_6056_uminus__integer__code_I2_J,axiom,
% 0.30/0.62 ! [M: num] :
% 0.30/0.62 ( ( uminus_uminus @ code_integer @ ( code_Pos @ M ) )
% 0.30/0.62 = ( code_Neg @ M ) ) ).
% 0.30/0.62
% 0.30/0.62 % uminus_integer_code(2)
% 0.30/0.62 thf(fact_6057_uminus__integer__code_I3_J,axiom,
% 0.30/0.62 ! [M: num] :
% 0.30/0.62 ( ( uminus_uminus @ code_integer @ ( code_Neg @ M ) )
% 0.30/0.62 = ( code_Pos @ M ) ) ).
% 0.30/0.62
% 0.30/0.62 % uminus_integer_code(3)
% 0.30/0.62 thf(fact_6058_less__integer__code_I8_J,axiom,
% 0.30/0.62 ! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(8)
% 0.30/0.62 thf(fact_6059_less__integer__code_I6_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(6)
% 0.30/0.62 thf(fact_6060_times__integer__code_I3_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer_code(3)
% 0.30/0.62 thf(fact_6061_less__eq__integer__code_I8_J,axiom,
% 0.30/0.62 ! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(8)
% 0.30/0.62 thf(fact_6062_less__eq__integer__code_I6_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(6)
% 0.30/0.62 thf(fact_6063_less__integer__code_I5_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.30/0.62 = ( ord_less @ num @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_integer_code(5)
% 0.30/0.62 thf(fact_6064_less__eq__integer__code_I5_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.30/0.62 = ( ord_less_eq @ num @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % less_eq_integer_code(5)
% 0.30/0.62 thf(fact_6065_minus__integer__code_I3_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_sub @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer_code(3)
% 0.30/0.62 thf(fact_6066_divmod__abs__code_I1_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.30/0.62 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % divmod_abs_code(1)
% 0.30/0.62 thf(fact_6067_minus__integer__code_I5_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer_code(5)
% 0.30/0.62 thf(fact_6068_minus__integer__code_I4_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % minus_integer_code(4)
% 0.30/0.62 thf(fact_6069_times__integer__code_I6_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer_code(6)
% 0.30/0.62 thf(fact_6070_times__integer__code_I5_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer_code(5)
% 0.30/0.62 thf(fact_6071_times__integer__code_I4_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % times_integer_code(4)
% 0.30/0.62 thf(fact_6072_plus__integer__code_I4_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 0.30/0.62 = ( code_sub @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer_code(4)
% 0.30/0.62 thf(fact_6073_plus__integer__code_I5_J,axiom,
% 0.30/0.62 ! [M: num,N: num] :
% 0.30/0.62 ( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 0.30/0.62 = ( code_sub @ N @ M ) ) ).
% 0.30/0.62
% 0.30/0.62 % plus_integer_code(5)
% 0.30/0.62 thf(fact_6074_divmod__abs__code_I3_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
% 0.30/0.62 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % divmod_abs_code(3)
% 0.30/0.62 thf(fact_6075_divmod__abs__code_I4_J,axiom,
% 0.30/0.62 ! [K: num,L: num] :
% 0.30/0.62 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
% 0.30/0.62 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.30/0.62
% 0.30/0.62 % divmod_abs_code(4)
% 0.30/0.62 thf(fact_6076_Code__Numeral_Osub__code_I3_J,axiom,
% 0.30/0.62 ! [M: num] :
% 0.30/0.62 ( ( code_sub @ ( bit1 @ M ) @ one2 )
% 0.30/0.62 = ( code_Pos @ ( bit0 @ M ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Code_Numeral.sub_code(3)
% 0.30/0.62 thf(fact_6077_prod__list__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( monoid_mult @ A )
% 0.30/0.62 => ( ( groups5270119922927024881d_list @ A )
% 0.30/0.62 = ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % prod_list_def
% 0.30/0.62 thf(fact_6078_semilattice__order__set_Osubset__imp,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.62 => ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.subset_imp
% 0.30/0.62 thf(fact_6079_semilattice__set_OF_Ocong,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( lattic1715443433743089157tice_F @ A )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.F.cong
% 0.30/0.62 thf(fact_6080_Inf__fin__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_inf @ A )
% 0.30/0.62 => ( ( lattic7752659483105999362nf_fin @ A )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Inf_fin_def
% 0.30/0.62 thf(fact_6081_Sup__fin__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_sup @ A )
% 0.30/0.62 => ( ( lattic5882676163264333800up_fin @ A )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_fin_def
% 0.30/0.62 thf(fact_6082_Max__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( ( lattic643756798349783984er_Max @ A )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ ( ord_max @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Max_def
% 0.30/0.62 thf(fact_6083_Min__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( ( lattic643756798350308766er_Min @ A )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ ( ord_min @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Min_def
% 0.30/0.62 thf(fact_6084_semilattice__order__set_OcoboundedI,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.62 => ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.coboundedI
% 0.30/0.62 thf(fact_6085_semilattice__order__set_OboundedE,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 0.30/0.62 => ! [A10: A] :
% 0.30/0.62 ( ( member @ A @ A10 @ A6 )
% 0.30/0.62 => ( Less_eq @ X @ A10 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.boundedE
% 0.30/0.62 thf(fact_6086_semilattice__order__set_OboundedI,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ! [A8: A] :
% 0.30/0.62 ( ( member @ A @ A8 @ A6 )
% 0.30/0.62 => ( Less_eq @ X @ A8 ) )
% 0.30/0.62 => ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.boundedI
% 0.30/0.62 thf(fact_6087_semilattice__order__set_Obounded__iff,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 0.30/0.62 = ( ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ A6 )
% 0.30/0.62 => ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.bounded_iff
% 0.30/0.62 thf(fact_6088_sum__list__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( monoid_add @ A )
% 0.30/0.62 => ( ( groups8242544230860333062m_list @ A )
% 0.30/0.62 = ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_list_def
% 0.30/0.62 thf(fact_6089_semilattice__set_Oeq__fold_H,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 0.30/0.62 = ( the2 @ A
% 0.30/0.62 @ ( finite_fold @ A @ ( option @ A )
% 0.30/0.62 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F @ X2 ) @ Y6 ) )
% 0.30/0.62 @ ( none @ A )
% 0.30/0.62 @ A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.eq_fold'
% 0.30/0.62 thf(fact_6090_semilattice__set_Oinsert__remove,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.62 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 0.30/0.62 = X ) )
% 0.30/0.62 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 0.30/0.62 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.insert_remove
% 0.30/0.62 thf(fact_6091_Min_Osemilattice__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( lattic149705377957585745ce_set @ A @ ( ord_min @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Min.semilattice_set_axioms
% 0.30/0.62 thf(fact_6092_Max_Osemilattice__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( lattic149705377957585745ce_set @ A @ ( ord_max @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Max.semilattice_set_axioms
% 0.30/0.62 thf(fact_6093_Inf__fin_Osemilattice__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_inf @ A )
% 0.30/0.62 => ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Inf_fin.semilattice_set_axioms
% 0.30/0.62 thf(fact_6094_Sup__fin_Osemilattice__set__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( semilattice_sup @ A )
% 0.30/0.62 => ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sup_fin.semilattice_set_axioms
% 0.30/0.62 thf(fact_6095_semilattice__order__set_Oaxioms_I2_J,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.62 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.62 => ( lattic149705377957585745ce_set @ A @ F ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_order_set.axioms(2)
% 0.30/0.62 thf(fact_6096_semilattice__set_Oin__idem,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.in_idem
% 0.30/0.62 thf(fact_6097_semilattice__set_Osingleton,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.62 = X ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.singleton
% 0.30/0.62 thf(fact_6098_semilattice__set_Ohom__commute,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,H2: A > A,N6: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ! [X3: A,Y5: A] :
% 0.30/0.62 ( ( H2 @ ( F @ X3 @ Y5 ) )
% 0.30/0.62 = ( F @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 0.30/0.62 => ( ( finite_finite2 @ A @ N6 )
% 0.30/0.62 => ( ( N6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F @ N6 ) )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ F @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.hom_commute
% 0.30/0.62 thf(fact_6099_semilattice__set_Osubset,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( B7
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.62 => ( ( F @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 0.30/0.62 = ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.subset
% 0.30/0.62 thf(fact_6100_semilattice__set_Oinsert__not__elem,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ~ ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 0.30/0.62 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.insert_not_elem
% 0.30/0.62 thf(fact_6101_semilattice__set_Oinsert,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 0.30/0.62 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.insert
% 0.30/0.62 thf(fact_6102_semilattice__set_Oclosed,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.62 => ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.closed
% 0.30/0.62 thf(fact_6103_semilattice__set_Ounion,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.62 => ( ( B7
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.62 = ( F @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.union
% 0.30/0.62 thf(fact_6104_semilattice__set_Oeq__fold,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 0.30/0.62 = ( finite_fold @ A @ A @ F @ X @ A6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.eq_fold
% 0.30/0.62 thf(fact_6105_semilattice__set_Oinfinite,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 0.30/0.62 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.infinite
% 0.30/0.62 thf(fact_6106_semilattice__set_Oremove,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 0.30/0.62 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.62 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 0.30/0.62 = X ) )
% 0.30/0.62 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 0.30/0.62 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_set.remove
% 0.30/0.62 thf(fact_6107_Least__mono,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( ( order @ A )
% 0.30/0.62 & ( order @ B ) )
% 0.30/0.62 => ! [F: A > B,S3: set @ A] :
% 0.30/0.62 ( ( order_mono @ A @ B @ F )
% 0.30/0.62 => ( ? [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ S3 )
% 0.30/0.62 & ! [Xa2: A] :
% 0.30/0.62 ( ( member @ A @ Xa2 @ S3 )
% 0.30/0.62 => ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
% 0.30/0.62 => ( ( ord_Least @ B
% 0.30/0.62 @ ^ [Y6: B] : ( member @ B @ Y6 @ ( image @ A @ B @ F @ S3 ) ) )
% 0.30/0.62 = ( F
% 0.30/0.62 @ ( ord_Least @ A
% 0.30/0.62 @ ^ [X2: A] : ( member @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_mono
% 0.30/0.62 thf(fact_6108_cr__int__def,axiom,
% 0.30/0.62 ( cr_int
% 0.30/0.62 = ( ^ [X2: product_prod @ nat @ nat] :
% 0.30/0.62 ( ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.62 @ ( abs_Integ @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cr_int_def
% 0.30/0.62 thf(fact_6109_Least__eq__0,axiom,
% 0.30/0.62 ! [P: nat > $o] :
% 0.30/0.62 ( ( P @ ( zero_zero @ nat ) )
% 0.30/0.62 => ( ( ord_Least @ nat @ P )
% 0.30/0.62 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_eq_0
% 0.30/0.62 thf(fact_6110_Least__le,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,K: A] :
% 0.30/0.62 ( ( P @ K )
% 0.30/0.62 => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_le
% 0.30/0.62 thf(fact_6111_Least1I,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( order @ A )
% 0.30/0.62 => ! [P: A > $o] :
% 0.30/0.62 ( ? [X4: A] :
% 0.30/0.62 ( ( P @ X4 )
% 0.30/0.62 & ! [Y5: A] :
% 0.30/0.62 ( ( P @ Y5 )
% 0.30/0.62 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 0.30/0.62 & ! [Y5: A] :
% 0.30/0.62 ( ( ( P @ Y5 )
% 0.30/0.62 & ! [Ya2: A] :
% 0.30/0.62 ( ( P @ Ya2 )
% 0.30/0.62 => ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
% 0.30/0.62 => ( Y5 = X4 ) ) )
% 0.30/0.62 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least1I
% 0.30/0.62 thf(fact_6112_Least1__le,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( order @ A )
% 0.30/0.62 => ! [P: A > $o,Z3: A] :
% 0.30/0.62 ( ? [X4: A] :
% 0.30/0.62 ( ( P @ X4 )
% 0.30/0.62 & ! [Y5: A] :
% 0.30/0.62 ( ( P @ Y5 )
% 0.30/0.62 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 0.30/0.62 & ! [Y5: A] :
% 0.30/0.62 ( ( ( P @ Y5 )
% 0.30/0.62 & ! [Ya2: A] :
% 0.30/0.62 ( ( P @ Ya2 )
% 0.30/0.62 => ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
% 0.30/0.62 => ( Y5 = X4 ) ) )
% 0.30/0.62 => ( ( P @ Z3 )
% 0.30/0.62 => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least1_le
% 0.30/0.62 thf(fact_6113_LeastI2__order,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( order @ A )
% 0.30/0.62 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.30/0.62 ( ( P @ X )
% 0.30/0.62 => ( ! [Y5: A] :
% 0.30/0.62 ( ( P @ Y5 )
% 0.30/0.62 => ( ord_less_eq @ A @ X @ Y5 ) )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( P @ X3 )
% 0.30/0.62 => ( ! [Y4: A] :
% 0.30/0.62 ( ( P @ Y4 )
% 0.30/0.62 => ( ord_less_eq @ A @ X3 @ Y4 ) )
% 0.30/0.62 => ( Q @ X3 ) ) )
% 0.30/0.62 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI2_order
% 0.30/0.62 thf(fact_6114_Least__equality,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( order @ A )
% 0.30/0.62 => ! [P: A > $o,X: A] :
% 0.30/0.62 ( ( P @ X )
% 0.30/0.62 => ( ! [Y5: A] :
% 0.30/0.62 ( ( P @ Y5 )
% 0.30/0.62 => ( ord_less_eq @ A @ X @ Y5 ) )
% 0.30/0.62 => ( ( ord_Least @ A @ P )
% 0.30/0.62 = X ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_equality
% 0.30/0.62 thf(fact_6115_LeastI2__wellorder,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,A3: A,Q: A > $o] :
% 0.30/0.62 ( ( P @ A3 )
% 0.30/0.62 => ( ! [A8: A] :
% 0.30/0.62 ( ( P @ A8 )
% 0.30/0.62 => ( ! [B11: A] :
% 0.30/0.62 ( ( P @ B11 )
% 0.30/0.62 => ( ord_less_eq @ A @ A8 @ B11 ) )
% 0.30/0.62 => ( Q @ A8 ) ) )
% 0.30/0.62 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI2_wellorder
% 0.30/0.62 thf(fact_6116_LeastI2__wellorder__ex,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,Q: A > $o] :
% 0.30/0.62 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.62 => ( ! [A8: A] :
% 0.30/0.62 ( ( P @ A8 )
% 0.30/0.62 => ( ! [B11: A] :
% 0.30/0.62 ( ( P @ B11 )
% 0.30/0.62 => ( ord_less_eq @ A @ A8 @ B11 ) )
% 0.30/0.62 => ( Q @ A8 ) ) )
% 0.30/0.62 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI2_wellorder_ex
% 0.30/0.62 thf(fact_6117_not__less__Least,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [K: A,P: A > $o] :
% 0.30/0.62 ( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
% 0.30/0.62 => ~ ( P @ K ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % not_less_Least
% 0.30/0.62 thf(fact_6118_LeastI,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,K: A] :
% 0.30/0.62 ( ( P @ K )
% 0.30/0.62 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI
% 0.30/0.62 thf(fact_6119_LeastI2,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,A3: A,Q: A > $o] :
% 0.30/0.62 ( ( P @ A3 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( P @ X3 )
% 0.30/0.62 => ( Q @ X3 ) )
% 0.30/0.62 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI2
% 0.30/0.62 thf(fact_6120_LeastI__ex,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o] :
% 0.30/0.62 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.62 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI_ex
% 0.30/0.62 thf(fact_6121_LeastI2__ex,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ! [P: A > $o,Q: A > $o] :
% 0.30/0.62 ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( P @ X3 )
% 0.30/0.62 => ( Q @ X3 ) )
% 0.30/0.62 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % LeastI2_ex
% 0.30/0.62 thf(fact_6122_Least__Suc2,axiom,
% 0.30/0.62 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.30/0.62 ( ( P @ N )
% 0.30/0.62 => ( ( Q @ M )
% 0.30/0.62 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.62 => ( ! [K2: nat] :
% 0.30/0.62 ( ( P @ ( suc @ K2 ) )
% 0.30/0.62 = ( Q @ K2 ) )
% 0.30/0.62 => ( ( ord_Least @ nat @ P )
% 0.30/0.62 = ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_Suc2
% 0.30/0.62 thf(fact_6123_Least__Suc,axiom,
% 0.30/0.62 ! [P: nat > $o,N: nat] :
% 0.30/0.62 ( ( P @ N )
% 0.30/0.62 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.30/0.62 => ( ( ord_Least @ nat @ P )
% 0.30/0.62 = ( suc
% 0.30/0.62 @ ( ord_Least @ nat
% 0.30/0.62 @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_Suc
% 0.30/0.62 thf(fact_6124_Least__Min,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ! [P: A > $o] :
% 0.30/0.62 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.30/0.62 => ( ? [X_1: A] : ( P @ X_1 )
% 0.30/0.62 => ( ( ord_Least @ A @ P )
% 0.30/0.62 = ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Least_Min
% 0.30/0.62 thf(fact_6125_int_Opcr__cr__eq,axiom,
% 0.30/0.62 pcr_int = cr_int ).
% 0.30/0.62
% 0.30/0.62 % int.pcr_cr_eq
% 0.30/0.62 thf(fact_6126_Quotient__int,axiom,
% 0.30/0.62 quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% 0.30/0.62
% 0.30/0.62 % Quotient_int
% 0.30/0.62 thf(fact_6127_list__ex__length,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( list_ex @ A )
% 0.30/0.62 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.30/0.62 ? [N3: nat] :
% 0.30/0.62 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.62 & ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % list_ex_length
% 0.30/0.62 thf(fact_6128_map__of__zip__nth,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
% 0.30/0.62 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.30/0.62 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.62 => ( ( distinct @ A @ Xs )
% 0.30/0.62 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.62 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
% 0.30/0.62 = ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % map_of_zip_nth
% 0.30/0.62 thf(fact_6129_and_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.62 => ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % and.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6130_semilattice__neutr__order_Oaxioms_I1_J,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.62 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.62 => ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % semilattice_neutr_order.axioms(1)
% 0.30/0.62 thf(fact_6131_sup__bot_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.62 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sup_bot.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6132_inf__top_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.62 => ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inf_top.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6133_gcd__nat_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.62
% 0.30/0.62 % gcd_nat.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6134_or_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.62 => ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % or.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6135_max__nat_Osemilattice__neutr__axioms,axiom,
% 0.30/0.62 semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.62
% 0.30/0.62 % max_nat.semilattice_neutr_axioms
% 0.30/0.62 thf(fact_6136_map__upds__list__update2__drop,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys2: list @ B,Y3: B] :
% 0.30/0.62 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 0.30/0.62 => ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys2 @ I @ Y3 ) )
% 0.30/0.62 = ( map_upds @ A @ B @ M @ Xs @ Ys2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % map_upds_list_update2_drop
% 0.30/0.62 thf(fact_6137_natLess__def,axiom,
% 0.30/0.62 ( bNF_Ca8459412986667044542atLess
% 0.30/0.62 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % natLess_def
% 0.30/0.62 thf(fact_6138_relChain__def,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( ord @ B )
% 0.30/0.62 => ( ( bNF_Ca3754400796208372196lChain @ A @ B )
% 0.30/0.62 = ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.30/0.62 ! [I5: A,J3: A] :
% 0.30/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I5 @ J3 ) @ R5 )
% 0.30/0.62 => ( ord_less_eq @ B @ ( As @ I5 ) @ ( As @ J3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % relChain_def
% 0.30/0.62 thf(fact_6139_map__upds__append1,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M: A > ( option @ B ),X: A] :
% 0.30/0.62 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.30/0.62 => ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
% 0.30/0.62 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % map_upds_append1
% 0.30/0.62 thf(fact_6140_restrict__upd__same,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y3: B] :
% 0.30/0.62 ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y3 ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.62 = ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % restrict_upd_same
% 0.30/0.62 thf(fact_6141_restrict__complement__singleton__eq,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 0.30/0.62 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.62 = ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % restrict_complement_singleton_eq
% 0.30/0.62 thf(fact_6142_Restr__natLeq,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
% 0.30/0.62 @ ( product_Sigma @ nat @ nat
% 0.30/0.62 @ ( collect @ nat
% 0.30/0.62 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
% 0.30/0.62 @ ^ [Uu: nat] :
% 0.30/0.62 ( collect @ nat
% 0.30/0.62 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) )
% 0.30/0.62 = ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.62 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.62 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.62 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.62 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.62 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Restr_natLeq
% 0.30/0.62 thf(fact_6143_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( linorder @ A )
% 0.30/0.62 => ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
% 0.30/0.62 @ ^ [X2: A] : X2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % sorted_list_of_set.folding_insort_key_axioms
% 0.30/0.62 thf(fact_6144_natLeq__def,axiom,
% 0.30/0.62 ( bNF_Ca8665028551170535155natLeq
% 0.30/0.62 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % natLeq_def
% 0.30/0.62 thf(fact_6145_Restr__natLeq2,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
% 0.30/0.62 @ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.30/0.62 @ ^ [Uu: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 0.30/0.62 = ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.62 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.62 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.62 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.62 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.62 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Restr_natLeq2
% 0.30/0.62 thf(fact_6146_fun__cong__unused__0,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.62 ( ( zero @ B )
% 0.30/0.62 => ! [F: ( A > B ) > C,G: C] :
% 0.30/0.62 ( ( F
% 0.30/0.62 = ( ^ [X2: A > B] : G ) )
% 0.30/0.62 => ( ( F
% 0.30/0.62 @ ^ [X2: A] : ( zero_zero @ B ) )
% 0.30/0.62 = G ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fun_cong_unused_0
% 0.30/0.62 thf(fact_6147_natLeq__underS__less,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.30/0.62 = ( collect @ nat
% 0.30/0.62 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % natLeq_underS_less
% 0.30/0.62 thf(fact_6148_add_Ogroup__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( group_add @ A )
% 0.30/0.62 => ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % add.group_axioms
% 0.30/0.62 thf(fact_6149_inv__o__cancel,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.30/0.62 = ( id @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_o_cancel
% 0.30/0.62 thf(fact_6150_inv__into__f__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ A6 )
% 0.30/0.62 => ( ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ X ) )
% 0.30/0.62 = X ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_f_f
% 0.30/0.62 thf(fact_6151_inv__identity,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
% 0.30/0.62 @ ^ [A5: A] : A5 )
% 0.30/0.62 = ( ^ [A5: A] : A5 ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_identity
% 0.30/0.62 thf(fact_6152_inv__id,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
% 0.30/0.62 = ( id @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_id
% 0.30/0.62 thf(fact_6153_inv__into__image__cancel,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,S3: set @ A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ A6 )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ A6 )
% 0.30/0.62 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ ( image @ A @ B @ F @ S3 ) )
% 0.30/0.62 = S3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_image_cancel
% 0.30/0.62 thf(fact_6154_o__inv__o__cancel,axiom,
% 0.30/0.62 ! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
% 0.30/0.62 = G ) ) ).
% 0.30/0.62
% 0.30/0.62 % o_inv_o_cancel
% 0.30/0.62 thf(fact_6155_inv__into__def,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( hilbert_inv_into @ A @ B )
% 0.30/0.62 = ( ^ [A7: set @ A,F2: A > B,X2: B] :
% 0.30/0.62 ( fChoice @ A
% 0.30/0.62 @ ^ [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ A7 )
% 0.30/0.62 & ( ( F2 @ Y6 )
% 0.30/0.62 = X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_def
% 0.30/0.62 thf(fact_6156_inv__into__def2,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( hilbert_inv_into @ A @ B )
% 0.30/0.62 = ( ^ [A7: set @ A,F2: A > B,X2: B] :
% 0.30/0.62 ( fChoice @ A
% 0.30/0.62 @ ^ [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ A7 )
% 0.30/0.62 & ( ( F2 @ Y6 )
% 0.30/0.62 = X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_def2
% 0.30/0.62 thf(fact_6157_group_Oleft__cancel,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A,C2: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( ( F @ A3 @ B2 )
% 0.30/0.62 = ( F @ A3 @ C2 ) )
% 0.30/0.62 = ( B2 = C2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.left_cancel
% 0.30/0.62 thf(fact_6158_group_Oleft__inverse,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( F @ ( Inverse @ A3 ) @ A3 )
% 0.30/0.62 = Z3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.left_inverse
% 0.30/0.62 thf(fact_6159_group_Oright__cancel,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,B2: A,A3: A,C2: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( ( F @ B2 @ A3 )
% 0.30/0.62 = ( F @ C2 @ A3 ) )
% 0.30/0.62 = ( B2 = C2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.right_cancel
% 0.30/0.62 thf(fact_6160_group_Oright__inverse,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( F @ A3 @ ( Inverse @ A3 ) )
% 0.30/0.62 = Z3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.right_inverse
% 0.30/0.62 thf(fact_6161_group_Oinverse__unique,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( ( F @ A3 @ B2 )
% 0.30/0.62 = Z3 )
% 0.30/0.62 => ( ( Inverse @ A3 )
% 0.30/0.62 = B2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.inverse_unique
% 0.30/0.62 thf(fact_6162_group_Oinverse__inverse,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( Inverse @ ( Inverse @ A3 ) )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.inverse_inverse
% 0.30/0.62 thf(fact_6163_group_Oinverse__neutral,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( Inverse @ Z3 )
% 0.30/0.62 = Z3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.inverse_neutral
% 0.30/0.62 thf(fact_6164_group_Ogroup__left__neutral,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( F @ Z3 @ A3 )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.group_left_neutral
% 0.30/0.62 thf(fact_6165_group_Oinverse__distrib__swap,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.62 => ( ( Inverse @ ( F @ A3 @ B2 ) )
% 0.30/0.62 = ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % group.inverse_distrib_swap
% 0.30/0.62 thf(fact_6166_inv__into__injective,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,A6: set @ A,F: A > B,X: B,Y3: B] :
% 0.30/0.62 ( ( ( hilbert_inv_into @ A @ B @ A6 @ F @ X )
% 0.30/0.62 = ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 ) )
% 0.30/0.62 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A6 ) )
% 0.30/0.62 => ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A6 ) )
% 0.30/0.62 => ( X = Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_injective
% 0.30/0.62 thf(fact_6167_inv__into__into,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,X: A,F: B > A,A6: set @ B] :
% 0.30/0.62 ( ( member @ A @ X @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F @ X ) @ A6 ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_into
% 0.30/0.62 thf(fact_6168_f__inv__into__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,Y3: A,F: B > A,A6: set @ B] :
% 0.30/0.62 ( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( ( F @ ( hilbert_inv_into @ B @ A @ A6 @ F @ Y3 ) )
% 0.30/0.62 = Y3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % f_inv_into_f
% 0.30/0.62 thf(fact_6169_surj__f__inv__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A,Y3: A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ Y3 ) )
% 0.30/0.62 = Y3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % surj_f_inv_f
% 0.30/0.62 thf(fact_6170_surj__iff__all,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 = ( ! [X2: A] :
% 0.30/0.62 ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ X2 ) )
% 0.30/0.62 = X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % surj_iff_all
% 0.30/0.62 thf(fact_6171_image__f__inv__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ A6 ) )
% 0.30/0.62 = A6 ) ) ).
% 0.30/0.62
% 0.30/0.62 % image_f_inv_f
% 0.30/0.62 thf(fact_6172_surj__imp__inv__eq,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ! [X3: B] :
% 0.30/0.62 ( ( G @ ( F @ X3 ) )
% 0.30/0.62 = X3 )
% 0.30/0.62 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 0.30/0.62 = G ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % surj_imp_inv_eq
% 0.30/0.62 thf(fact_6173_inv__into__f__eq,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A,Y3: B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ A6 )
% 0.30/0.62 => ( ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ( ( F @ X )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 )
% 0.30/0.62 = X ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_f_eq
% 0.30/0.62 thf(fact_6174_inv__f__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 0.30/0.62 = X ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_f_f
% 0.30/0.62 thf(fact_6175_inv__f__eq,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,X: A,Y3: B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( ( F @ X )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y3 )
% 0.30/0.62 = X ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_f_eq
% 0.30/0.62 thf(fact_6176_inj__imp__inv__eq,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,F: A > B,G: B > A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ! [X3: B] :
% 0.30/0.62 ( ( F @ ( G @ X3 ) )
% 0.30/0.62 = X3 )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
% 0.30/0.62 = G ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_imp_inv_eq
% 0.30/0.62 thf(fact_6177_inv__equality,axiom,
% 0.30/0.62 ! [A: $tType,B: $tType,G: B > A,F: A > B] :
% 0.30/0.62 ( ! [X3: A] :
% 0.30/0.62 ( ( G @ ( F @ X3 ) )
% 0.30/0.62 = X3 )
% 0.30/0.62 => ( ! [Y5: B] :
% 0.30/0.62 ( ( F @ ( G @ Y5 ) )
% 0.30/0.62 = Y5 )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
% 0.30/0.62 = G ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_equality
% 0.30/0.62 thf(fact_6178_image__inv__into__cancel,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A,A6: set @ B,A11: set @ A,B13: set @ A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ A6 )
% 0.30/0.62 = A11 )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ A ) @ B13 @ A11 )
% 0.30/0.62 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B13 ) )
% 0.30/0.62 = B13 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % image_inv_into_cancel
% 0.30/0.62 thf(fact_6179_inv__def,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.62 ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 0.30/0.62 = ( ^ [Y6: A] :
% 0.30/0.62 ( fChoice @ B
% 0.30/0.62 @ ^ [X2: B] :
% 0.30/0.62 ( ( F @ X2 )
% 0.30/0.62 = Y6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_def
% 0.30/0.62 thf(fact_6180_inj__on__inv__into,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,B7: set @ A,F: B > A,A6: set @ B] :
% 0.30/0.62 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A6 ) )
% 0.30/0.62 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B7 ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_on_inv_into
% 0.30/0.62 thf(fact_6181_inj__transfer,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,P: A > $o,X: A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ! [Y5: B] :
% 0.30/0.62 ( ( member @ B @ Y5 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.30/0.62 => ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y5 ) ) )
% 0.30/0.62 => ( P @ X ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_transfer
% 0.30/0.62 thf(fact_6182_image__inv__f__f,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ A6 ) )
% 0.30/0.62 = A6 ) ) ).
% 0.30/0.62
% 0.30/0.62 % image_inv_f_f
% 0.30/0.62 thf(fact_6183_inj__imp__surj__inv,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_imp_surj_inv
% 0.30/0.62 thf(fact_6184_surj__imp__inj__inv,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % surj_imp_inj_inv
% 0.30/0.62 thf(fact_6185_inv__into__comp,axiom,
% 0.30/0.62 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A6: set @ C,X: B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) )
% 0.30/0.62 => ( ( inj_on @ C @ A @ G @ A6 )
% 0.30/0.62 => ( ( member @ B @ X @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) ) )
% 0.30/0.62 => ( ( hilbert_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 0.30/0.62 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A6 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image @ C @ A @ G @ A6 ) @ F ) @ X ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_into_comp
% 0.30/0.62 thf(fact_6186_inv__unique__comp,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 0.30/0.62 ( ( ( comp @ B @ A @ A @ F @ G )
% 0.30/0.62 = ( id @ A ) )
% 0.30/0.62 => ( ( ( comp @ A @ B @ B @ G @ F )
% 0.30/0.62 = ( id @ B ) )
% 0.30/0.62 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 0.30/0.62 = G ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inv_unique_comp
% 0.30/0.62 thf(fact_6187_surj__iff,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.62 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) )
% 0.30/0.62 = ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
% 0.30/0.62 = ( id @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % surj_iff
% 0.30/0.62 thf(fact_6188_inj__iff,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.62 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 = ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.30/0.62 = ( id @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inj_iff
% 0.30/0.62 thf(fact_6189_bijection_Oinv__comp__left,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.30/0.62 = ( id @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_comp_left
% 0.30/0.62 thf(fact_6190_bijection_Oinv__comp__right,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 0.30/0.62 = ( id @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_comp_right
% 0.30/0.62 thf(fact_6191_bijection_Oeq__iff,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( ( F @ A3 )
% 0.30/0.62 = ( F @ B2 ) )
% 0.30/0.62 = ( A3 = B2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.eq_iff
% 0.30/0.62 thf(fact_6192_bijection_OeqI,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( ( F @ A3 )
% 0.30/0.62 = ( F @ B2 ) )
% 0.30/0.62 => ( A3 = B2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.eqI
% 0.30/0.62 thf(fact_6193_bijection_Osurj,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.surj
% 0.30/0.62 thf(fact_6194_bijection_Oinj,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inj
% 0.30/0.62 thf(fact_6195_bijection_Oinv__right__eq__iff,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,B2: A,A3: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( B2
% 0.30/0.62 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
% 0.30/0.62 = ( ( F @ B2 )
% 0.30/0.62 = A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_right_eq_iff
% 0.30/0.62 thf(fact_6196_bijection_Oinv__left__eq__iff,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 0.30/0.62 = B2 )
% 0.30/0.62 = ( ( F @ B2 )
% 0.30/0.62 = A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_left_eq_iff
% 0.30/0.62 thf(fact_6197_bijection_Oeq__inv__iff,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 0.30/0.62 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
% 0.30/0.62 = ( A3 = B2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.eq_inv_iff
% 0.30/0.62 thf(fact_6198_bijection_Oinv__right,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_right
% 0.30/0.62 thf(fact_6199_bijection_Oinv__left,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ ( F @ A3 ) )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inv_left
% 0.30/0.62 thf(fact_6200_bijection_Oeq__invI,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 0.30/0.62 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
% 0.30/0.62 => ( A3 = B2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.eq_invI
% 0.30/0.62 thf(fact_6201_bijection_Osurj__inv,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.62 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.surj_inv
% 0.30/0.62 thf(fact_6202_bijection_Oinj__inv,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A] :
% 0.30/0.62 ( ( hilbert_bijection @ A @ F )
% 0.30/0.62 => ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bijection.inj_inv
% 0.30/0.62 thf(fact_6203_iteratesp_Omono,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [F: A > A] :
% 0.30/0.62 ( order_mono @ ( A > $o ) @ ( A > $o )
% 0.30/0.62 @ ^ [P6: A > $o,X2: A] :
% 0.30/0.62 ( ? [Y6: A] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( F @ Y6 ) )
% 0.30/0.62 & ( P6 @ Y6 ) )
% 0.30/0.62 | ? [M9: set @ A] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M9 ) )
% 0.30/0.62 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 0.30/0.62 & ! [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ M9 )
% 0.30/0.62 => ( P6 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iteratesp.mono
% 0.30/0.62 thf(fact_6204_finite__subset__wf,axiom,
% 0.30/0.62 ! [A: $tType,A6: set @ A] :
% 0.30/0.62 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( wf @ ( set @ A )
% 0.30/0.62 @ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.30/0.62 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.30/0.62 @ ^ [X7: set @ A,Y9: set @ A] :
% 0.30/0.62 ( ( ord_less @ ( set @ A ) @ X7 @ Y9 )
% 0.30/0.62 & ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % finite_subset_wf
% 0.30/0.62 thf(fact_6205_wf,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wellorder @ A )
% 0.30/0.62 => ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf
% 0.30/0.62 thf(fact_6206_ccpo__Sup__least,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [A6: set @ A,Z3: A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( member @ A @ X3 @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 0.30/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ccpo_Sup_least
% 0.30/0.62 thf(fact_6207_ccpo__Sup__upper,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [A6: set @ A,X: A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 0.30/0.62 => ( ( member @ A @ X @ A6 )
% 0.30/0.62 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ccpo_Sup_upper
% 0.30/0.62 thf(fact_6208_wf__iff__no__infinite__down__chain,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( wf @ A )
% 0.30/0.62 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ~ ? [F2: nat > A] :
% 0.30/0.62 ! [I5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I5 ) ) @ ( F2 @ I5 ) ) @ R5 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_iff_no_infinite_down_chain
% 0.30/0.62 thf(fact_6209_wf__no__infinite__down__chainE,axiom,
% 0.30/0.62 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 0.30/0.62 ( ( wf @ A @ R )
% 0.30/0.62 => ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K2 ) ) @ ( F @ K2 ) ) @ R ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_no_infinite_down_chainE
% 0.30/0.62 thf(fact_6210_chain__singleton,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % chain_singleton
% 0.30/0.62 thf(fact_6211_wf__if__measure,axiom,
% 0.30/0.62 ! [A: $tType,P: A > $o,F: A > nat,G: A > A] :
% 0.30/0.62 ( ! [X3: A] :
% 0.30/0.62 ( ( P @ X3 )
% 0.30/0.62 => ( ord_less @ nat @ ( F @ ( G @ X3 ) ) @ ( F @ X3 ) ) )
% 0.30/0.62 => ( wf @ A
% 0.30/0.62 @ ( collect @ ( product_prod @ A @ A )
% 0.30/0.62 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.62 @ ^ [Y6: A,X2: A] :
% 0.30/0.62 ( ( P @ X2 )
% 0.30/0.62 & ( Y6
% 0.30/0.62 = ( G @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_if_measure
% 0.30/0.62 thf(fact_6212_wf__bounded__measure,axiom,
% 0.30/0.62 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
% 0.30/0.62 ( ! [A8: A,B5: A] :
% 0.30/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
% 0.30/0.62 => ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
% 0.30/0.62 & ( ord_less_eq @ nat @ ( F @ B5 ) @ ( Ub @ A8 ) )
% 0.30/0.62 & ( ord_less @ nat @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
% 0.30/0.62 => ( wf @ A @ R ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_bounded_measure
% 0.30/0.62 thf(fact_6213_wf__linord__ex__has__least,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
% 0.30/0.62 ( ( wf @ A @ R )
% 0.30/0.62 => ( ! [X3: A,Y5: A] :
% 0.30/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y5 ) @ ( transitive_trancl @ A @ R ) )
% 0.30/0.62 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
% 0.30/0.62 => ( ( P @ K )
% 0.30/0.62 => ? [X3: B] :
% 0.30/0.62 ( ( P @ X3 )
% 0.30/0.62 & ! [Y4: B] :
% 0.30/0.62 ( ( P @ Y4 )
% 0.30/0.62 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y4 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_linord_ex_has_least
% 0.30/0.62 thf(fact_6214_reduction__pair__lemma,axiom,
% 0.30/0.62 ! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( fun_reduction_pair @ A @ P )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
% 0.30/0.62 => ( ( wf @ A @ S3 )
% 0.30/0.62 => ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % reduction_pair_lemma
% 0.30/0.62 thf(fact_6215_wf__bounded__set,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
% 0.30/0.62 ( ! [A8: A,B5: A] :
% 0.30/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
% 0.30/0.62 => ( ( finite_finite2 @ B @ ( Ub @ A8 ) )
% 0.30/0.62 & ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
% 0.30/0.62 & ( ord_less_eq @ ( set @ B ) @ ( F @ B5 ) @ ( Ub @ A8 ) )
% 0.30/0.62 & ( ord_less @ ( set @ B ) @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
% 0.30/0.62 => ( wf @ A @ R ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_bounded_set
% 0.30/0.62 thf(fact_6216_in__chain__finite,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [A6: set @ A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 0.30/0.62 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.62 => ( ( A6
% 0.30/0.62 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.62 => ( member @ A @ ( complete_Sup_Sup @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % in_chain_finite
% 0.30/0.62 thf(fact_6217_iteratesp__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ( ( comple7512665784863727008ratesp @ A )
% 0.30/0.62 = ( ^ [F2: A > A] :
% 0.30/0.62 ( complete_lattice_lfp @ ( A > $o )
% 0.30/0.62 @ ^ [P6: A > $o,X2: A] :
% 0.30/0.62 ( ? [Y6: A] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( F2 @ Y6 ) )
% 0.30/0.62 & ( P6 @ Y6 ) )
% 0.30/0.62 | ? [M9: set @ A] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M9 ) )
% 0.30/0.62 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 0.30/0.62 & ! [Y6: A] :
% 0.30/0.62 ( ( member @ A @ Y6 @ M9 )
% 0.30/0.62 => ( P6 @ Y6 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iteratesp_def
% 0.30/0.62 thf(fact_6218_wf__int__ge__less__than2,axiom,
% 0.30/0.62 ! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_int_ge_less_than2
% 0.30/0.62 thf(fact_6219_wf__int__ge__less__than,axiom,
% 0.30/0.62 ! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_int_ge_less_than
% 0.30/0.62 thf(fact_6220_wf__pair__less,axiom,
% 0.30/0.62 wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% 0.30/0.62
% 0.30/0.62 % wf_pair_less
% 0.30/0.62 thf(fact_6221_wf__less,axiom,
% 0.30/0.62 wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_less
% 0.30/0.62 thf(fact_6222_iteratesp_Osimps,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ( ( comple7512665784863727008ratesp @ A )
% 0.30/0.62 = ( ^ [F2: A > A,A5: A] :
% 0.30/0.62 ( ? [X2: A] :
% 0.30/0.62 ( ( A5
% 0.30/0.62 = ( F2 @ X2 ) )
% 0.30/0.62 & ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) )
% 0.30/0.62 | ? [M9: set @ A] :
% 0.30/0.62 ( ( A5
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M9 ) )
% 0.30/0.62 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 0.30/0.62 & ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ M9 )
% 0.30/0.62 => ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iteratesp.simps
% 0.30/0.62 thf(fact_6223_iteratesp_Ocases,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [F: A > A,A3: A] :
% 0.30/0.62 ( ( comple7512665784863727008ratesp @ A @ F @ A3 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( F @ X3 ) )
% 0.30/0.62 => ~ ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
% 0.30/0.62 => ~ ! [M8: set @ A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M8 ) )
% 0.30/0.62 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
% 0.30/0.62 => ~ ! [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ M8 )
% 0.30/0.62 => ( comple7512665784863727008ratesp @ A @ F @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iteratesp.cases
% 0.30/0.62 thf(fact_6224_iteratesp_OSup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [M7: set @ A,F: A > A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( member @ A @ X3 @ M7 )
% 0.30/0.62 => ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
% 0.30/0.62 => ( comple7512665784863727008ratesp @ A @ F @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iteratesp.Sup
% 0.30/0.62 thf(fact_6225_reduction__pair__def,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( fun_reduction_pair @ A )
% 0.30/0.62 = ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
% 0.30/0.62 ( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
% 0.30/0.62 & ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % reduction_pair_def
% 0.30/0.62 thf(fact_6226_admissible__chfin,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [P: A > $o] :
% 0.30/0.62 ( ! [S4: set @ A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S4 )
% 0.30/0.62 => ( finite_finite2 @ A @ S4 ) )
% 0.30/0.62 => ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % admissible_chfin
% 0.30/0.62 thf(fact_6227_reduction__pairI,axiom,
% 0.30/0.62 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( wf @ A @ R2 )
% 0.30/0.62 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 0.30/0.62 => ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % reduction_pairI
% 0.30/0.62 thf(fact_6228_wf__no__loop,axiom,
% 0.30/0.62 ! [B: $tType,R2: set @ ( product_prod @ B @ B )] :
% 0.30/0.62 ( ( ( relcomp @ B @ B @ B @ R2 @ R2 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.30/0.62 => ( wf @ B @ R2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % wf_no_loop
% 0.30/0.62 thf(fact_6229_union__comp__emptyL,axiom,
% 0.30/0.62 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.62 => ( ( ( relcomp @ A @ A @ A @ B7 @ C6 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.62 => ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) @ C6 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % union_comp_emptyL
% 0.30/0.62 thf(fact_6230_union__comp__emptyR,axiom,
% 0.30/0.62 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( ( relcomp @ A @ A @ A @ A6 @ B7 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.62 => ( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.30/0.62 => ( ( relcomp @ A @ A @ A @ A6 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B7 @ C6 ) )
% 0.30/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % union_comp_emptyR
% 0.30/0.62 thf(fact_6231_relpow_Osimps_I2_J,axiom,
% 0.30/0.62 ! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 )
% 0.30/0.62 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) @ R2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % relpow.simps(2)
% 0.30/0.62 thf(fact_6232_admissible__disj,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [P: A > $o,Q: A > $o] :
% 0.30/0.62 ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
% 0.30/0.62 => ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
% 0.30/0.62 => ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
% 0.30/0.62 @ ^ [X2: A] :
% 0.30/0.62 ( ( P @ X2 )
% 0.30/0.62 | ( Q @ X2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % admissible_disj
% 0.30/0.62 thf(fact_6233_max__ext__compat,axiom,
% 0.30/0.62 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 0.30/0.62 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % max_ext_compat
% 0.30/0.62 thf(fact_6234_min__ext__compat,axiom,
% 0.30/0.62 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 0.30/0.62 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % min_ext_compat
% 0.30/0.62 thf(fact_6235_ntrancl__Suc,axiom,
% 0.30/0.62 ! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( transitive_ntrancl @ A @ ( suc @ N ) @ R2 )
% 0.30/0.62 = ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ntrancl_Suc
% 0.30/0.62 thf(fact_6236_and_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.62 => ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % and.monoid_axioms
% 0.30/0.62 thf(fact_6237_monoid_Oright__neutral,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 0.30/0.62 ( ( monoid @ A @ F @ Z3 )
% 0.30/0.62 => ( ( F @ A3 @ Z3 )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % monoid.right_neutral
% 0.30/0.62 thf(fact_6238_monoid_Oleft__neutral,axiom,
% 0.30/0.62 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 0.30/0.62 ( ( monoid @ A @ F @ Z3 )
% 0.30/0.62 => ( ( F @ Z3 @ A3 )
% 0.30/0.62 = A3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % monoid.left_neutral
% 0.30/0.62 thf(fact_6239_relpow_Osimps_I1_J,axiom,
% 0.30/0.62 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.62 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 )
% 0.30/0.62 = ( id2 @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % relpow.simps(1)
% 0.30/0.62 thf(fact_6240_add_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( monoid_add @ A )
% 0.30/0.62 => ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % add.monoid_axioms
% 0.30/0.62 thf(fact_6241_mult_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( monoid_mult @ A )
% 0.30/0.62 => ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mult.monoid_axioms
% 0.30/0.62 thf(fact_6242_sup__bot_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.62 => ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sup_bot.monoid_axioms
% 0.30/0.62 thf(fact_6243_inf__top_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.62 => ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % inf_top.monoid_axioms
% 0.30/0.62 thf(fact_6244_gcd__nat_Omonoid__axioms,axiom,
% 0.30/0.62 monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.62
% 0.30/0.62 % gcd_nat.monoid_axioms
% 0.30/0.62 thf(fact_6245_or_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.62 => ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % or.monoid_axioms
% 0.30/0.62 thf(fact_6246_xor_Omonoid__axioms,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.62 => ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % xor.monoid_axioms
% 0.30/0.62 thf(fact_6247_max__nat_Omonoid__axioms,axiom,
% 0.30/0.62 monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.62
% 0.30/0.62 % max_nat.monoid_axioms
% 0.30/0.62 thf(fact_6248_pair__leq__def,axiom,
% 0.30/0.62 ( fun_pair_leq
% 0.30/0.62 = ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % pair_leq_def
% 0.30/0.62 thf(fact_6249_cr__integer__def,axiom,
% 0.30/0.62 ( code_cr_integer
% 0.30/0.62 = ( ^ [X2: int,Y6: code_integer] :
% 0.30/0.62 ( X2
% 0.30/0.62 = ( code_int_of_integer @ Y6 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % cr_integer_def
% 0.30/0.62 thf(fact_6250_bit__integer__def,axiom,
% 0.30/0.62 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 0.30/0.62 = ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bit_integer_def
% 0.30/0.62 thf(fact_6251_integer_Opcr__cr__eq,axiom,
% 0.30/0.62 code_pcr_integer = code_cr_integer ).
% 0.30/0.62
% 0.30/0.62 % integer.pcr_cr_eq
% 0.30/0.62 thf(fact_6252_arg__min__list_Osimps_I2_J,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( linorder @ B )
% 0.30/0.62 => ! [F: A > B,X: A,Y3: A,Zs: list @ A] :
% 0.30/0.62 ( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
% 0.30/0.62 = ( if @ A @ ( ord_less_eq @ B @ ( F @ X ) @ ( F @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % arg_min_list.simps(2)
% 0.30/0.62 thf(fact_6253_iterates_OSup,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [M7: set @ A,F: A > A] :
% 0.30/0.62 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( member @ A @ X3 @ M7 )
% 0.30/0.62 => ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 0.30/0.62 => ( member @ A @ ( complete_Sup_Sup @ A @ M7 ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iterates.Sup
% 0.30/0.62 thf(fact_6254_iterates_Osimps,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [A3: A,F: A > A] :
% 0.30/0.62 ( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
% 0.30/0.62 = ( ? [X2: A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( F @ X2 ) )
% 0.30/0.62 & ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 0.30/0.62 | ? [M9: set @ A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M9 ) )
% 0.30/0.62 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 0.30/0.62 & ! [X2: A] :
% 0.30/0.62 ( ( member @ A @ X2 @ M9 )
% 0.30/0.62 => ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iterates.simps
% 0.30/0.62 thf(fact_6255_iterates_Ocases,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [A3: A,F: A > A] :
% 0.30/0.62 ( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( F @ X3 ) )
% 0.30/0.62 => ~ ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 0.30/0.62 => ~ ! [M8: set @ A] :
% 0.30/0.62 ( ( A3
% 0.30/0.62 = ( complete_Sup_Sup @ A @ M8 ) )
% 0.30/0.62 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
% 0.30/0.62 => ~ ! [X4: A] :
% 0.30/0.62 ( ( member @ A @ X4 @ M8 )
% 0.30/0.62 => ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % iterates.cases
% 0.30/0.62 thf(fact_6256_arg__min__list_Oelims,axiom,
% 0.30/0.62 ! [B: $tType,A: $tType] :
% 0.30/0.62 ( ( linorder @ B )
% 0.30/0.62 => ! [X: A > B,Xa3: list @ A,Y3: A] :
% 0.30/0.62 ( ( ( arg_min_list @ A @ B @ X @ Xa3 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ! [X3: A] :
% 0.30/0.62 ( ( Xa3
% 0.30/0.62 = ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.30/0.62 => ( Y3 != X3 ) )
% 0.30/0.62 => ( ! [X3: A,Y5: A,Zs2: list @ A] :
% 0.30/0.62 ( ( Xa3
% 0.30/0.62 = ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
% 0.30/0.62 => ( Y3
% 0.30/0.62 != ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) )
% 0.30/0.62 => ~ ( ( Xa3
% 0.30/0.62 = ( nil @ A ) )
% 0.30/0.62 => ( Y3
% 0.30/0.62 != ( undefined @ A ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % arg_min_list.elims
% 0.30/0.62 thf(fact_6257_chain__iterates,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [F: A > A] :
% 0.30/0.62 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.62 => ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % chain_iterates
% 0.30/0.62 thf(fact_6258_iterates__le__f,axiom,
% 0.30/0.62 ! [A: $tType] :
% 0.30/0.62 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.62 => ! [X: A,F: A > A] :
% 0.30/0.62 ( ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F ) )
% 0.30/0.62 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.63 => ( ord_less_eq @ A @ X @ ( F @ X ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % iterates_le_f
% 0.30/0.63 thf(fact_6259_fixp__induct,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.63 => ! [P: A > $o,F: A > A] :
% 0.30/0.63 ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
% 0.30/0.63 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.63 => ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.63 => ( ! [X3: A] :
% 0.30/0.63 ( ( P @ X3 )
% 0.30/0.63 => ( P @ ( F @ X3 ) ) )
% 0.30/0.63 => ( P @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fixp_induct
% 0.30/0.63 thf(fact_6260_iterates__fixp,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.63 => ! [F: A > A] :
% 0.30/0.63 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.63 => ( member @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % iterates_fixp
% 0.30/0.63 thf(fact_6261_fixp__unfold,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.63 => ! [F: A > A] :
% 0.30/0.63 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.63 => ( ( comple115746919287870866o_fixp @ A @ F )
% 0.30/0.63 = ( F @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fixp_unfold
% 0.30/0.63 thf(fact_6262_fixp__lowerbound,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comple9053668089753744459l_ccpo @ A )
% 0.30/0.63 => ! [F: A > A,Z3: A] :
% 0.30/0.63 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.63 => ( ( ord_less_eq @ A @ ( F @ Z3 ) @ Z3 )
% 0.30/0.63 => ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ Z3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fixp_lowerbound
% 0.30/0.63 thf(fact_6263_arg__min__list_Opelims,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType] :
% 0.30/0.63 ( ( linorder @ B )
% 0.30/0.63 => ! [X: A > B,Xa3: list @ A,Y3: A] :
% 0.30/0.63 ( ( ( arg_min_list @ A @ B @ X @ Xa3 )
% 0.30/0.63 = Y3 )
% 0.30/0.63 => ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa3 ) )
% 0.30/0.63 => ( ! [X3: A] :
% 0.30/0.63 ( ( Xa3
% 0.30/0.63 = ( cons @ A @ X3 @ ( nil @ A ) ) )
% 0.30/0.63 => ( ( Y3 = X3 )
% 0.30/0.63 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
% 0.30/0.63 => ( ! [X3: A,Y5: A,Zs2: list @ A] :
% 0.30/0.63 ( ( Xa3
% 0.30/0.63 = ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
% 0.30/0.63 => ( ( Y3
% 0.30/0.63 = ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) )
% 0.30/0.63 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) ) )
% 0.30/0.63 => ~ ( ( Xa3
% 0.30/0.63 = ( nil @ A ) )
% 0.30/0.63 => ( ( Y3
% 0.30/0.63 = ( undefined @ A ) )
% 0.30/0.63 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_min_list.pelims
% 0.30/0.63 thf(fact_6264_sorted__sort__key,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType] :
% 0.30/0.63 ( ( linorder @ A )
% 0.30/0.63 => ! [F: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sorted_sort_key
% 0.30/0.63 thf(fact_6265_sorted__sort,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( linorder @ A )
% 0.30/0.63 => ! [Xs: list @ A] :
% 0.30/0.63 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 0.30/0.63 @ ( linorder_sort_key @ A @ A
% 0.30/0.63 @ ^ [X2: A] : X2
% 0.30/0.63 @ Xs ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sorted_sort
% 0.30/0.63 thf(fact_6266_sorted__sort__id,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( linorder @ A )
% 0.30/0.63 => ! [Xs: list @ A] :
% 0.30/0.63 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.63 => ( ( linorder_sort_key @ A @ A
% 0.30/0.63 @ ^ [X2: A] : X2
% 0.30/0.63 @ Xs )
% 0.30/0.63 = Xs ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sorted_sort_id
% 0.30/0.63 thf(fact_6267_length__removeAll__less,axiom,
% 0.30/0.63 ! [A: $tType,X: A,Xs: list @ A] :
% 0.30/0.63 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.30/0.63 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % length_removeAll_less
% 0.30/0.63 thf(fact_6268_card__Plus__conv__if,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
% 0.30/0.63 ( ( ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 & ( finite_finite2 @ B @ B7 ) )
% 0.30/0.63 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
% 0.30/0.63 = ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) )
% 0.30/0.63 & ( ~ ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 & ( finite_finite2 @ B @ B7 ) )
% 0.30/0.63 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
% 0.30/0.63 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % card_Plus_conv_if
% 0.30/0.63 thf(fact_6269_length__removeAll__less__eq,axiom,
% 0.30/0.63 ! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.63
% 0.30/0.63 % length_removeAll_less_eq
% 0.30/0.63 thf(fact_6270_and_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( bit_ri3973907225187159222ations @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % and.comm_monoid_axioms
% 0.30/0.63 thf(fact_6271_lcm__altdef__int,axiom,
% 0.30/0.63 ( ( gcd_lcm @ int )
% 0.30/0.63 = ( ^ [A5: int,B3: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A5 ) @ ( abs_abs @ int @ B3 ) ) @ ( gcd_gcd @ int @ A5 @ B3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_altdef_int
% 0.30/0.63 thf(fact_6272_lcm_Obottom__left__bottom,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.bottom_left_bottom
% 0.30/0.63 thf(fact_6273_lcm_Obottom__right__bottom,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.bottom_right_bottom
% 0.30/0.63 thf(fact_6274_lcm__neg1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.30/0.63 = ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg1
% 0.30/0.63 thf(fact_6275_lcm__neg2,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.30/0.63 = ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg2
% 0.30/0.63 thf(fact_6276_lcm__0__iff__int,axiom,
% 0.30/0.63 ! [M: int,N: int] :
% 0.30/0.63 ( ( ( gcd_lcm @ int @ M @ N )
% 0.30/0.63 = ( zero_zero @ int ) )
% 0.30/0.63 = ( ( M
% 0.30/0.63 = ( zero_zero @ int ) )
% 0.30/0.63 | ( N
% 0.30/0.63 = ( zero_zero @ int ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_0_iff_int
% 0.30/0.63 thf(fact_6277_lcm__neg__numeral__1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ring_gcd @ A )
% 0.30/0.63 => ! [N: num,A3: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
% 0.30/0.63 = ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg_numeral_1
% 0.30/0.63 thf(fact_6278_lcm__neg__numeral__2,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ring_gcd @ A )
% 0.30/0.63 => ! [A3: A,N: num] :
% 0.30/0.63 ( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.30/0.63 = ( gcd_lcm @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg_numeral_2
% 0.30/0.63 thf(fact_6279_lcm__eq__1__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 0.30/0.63 = ( one_one @ A ) )
% 0.30/0.63 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_eq_1_iff
% 0.30/0.63 thf(fact_6280_lcm__1__iff__int,axiom,
% 0.30/0.63 ! [M: int,N: int] :
% 0.30/0.63 ( ( ( gcd_lcm @ int @ M @ N )
% 0.30/0.63 = ( one_one @ int ) )
% 0.30/0.63 = ( ( ( M
% 0.30/0.63 = ( one_one @ int ) )
% 0.30/0.63 | ( M
% 0.30/0.63 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.30/0.63 & ( ( N
% 0.30/0.63 = ( one_one @ int ) )
% 0.30/0.63 | ( N
% 0.30/0.63 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_1_iff_int
% 0.30/0.63 thf(fact_6281_lcm__ge__0__int,axiom,
% 0.30/0.63 ! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_ge_0_int
% 0.30/0.63 thf(fact_6282_semilattice__neutr_Oaxioms_I2_J,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.63 ( ( semilattice_neutr @ A @ F @ Z3 )
% 0.30/0.63 => ( comm_monoid @ A @ F @ Z3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr.axioms(2)
% 0.30/0.63 thf(fact_6283_lcm__neg1__int,axiom,
% 0.30/0.63 ! [X: int,Y3: int] :
% 0.30/0.63 ( ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
% 0.30/0.63 = ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg1_int
% 0.30/0.63 thf(fact_6284_lcm__neg2__int,axiom,
% 0.30/0.63 ! [X: int,Y3: int] :
% 0.30/0.63 ( ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
% 0.30/0.63 = ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_neg2_int
% 0.30/0.63 thf(fact_6285_lcm__pos__int,axiom,
% 0.30/0.63 ! [M: int,N: int] :
% 0.30/0.63 ( ( M
% 0.30/0.63 != ( zero_zero @ int ) )
% 0.30/0.63 => ( ( N
% 0.30/0.63 != ( zero_zero @ int ) )
% 0.30/0.63 => ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ M @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_pos_int
% 0.30/0.63 thf(fact_6286_comm__monoid_Ocomm__neutral,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 0.30/0.63 ( ( comm_monoid @ A @ F @ Z3 )
% 0.30/0.63 => ( ( F @ A3 @ Z3 )
% 0.30/0.63 = A3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % comm_monoid.comm_neutral
% 0.30/0.63 thf(fact_6287_zero__eq__lcm__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( zero_zero @ A )
% 0.30/0.63 = ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 | ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_eq_lcm_iff
% 0.30/0.63 thf(fact_6288_lcm__eq__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 | ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_eq_0_iff
% 0.30/0.63 thf(fact_6289_add_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comm_monoid_add @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % add.comm_monoid_axioms
% 0.30/0.63 thf(fact_6290_mult_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comm_monoid_mult @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult.comm_monoid_axioms
% 0.30/0.63 thf(fact_6291_lcm__mult__unit1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 0.30/0.63 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_unit1
% 0.30/0.63 thf(fact_6292_lcm__mult__unit2,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 0.30/0.63 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_unit2
% 0.30/0.63 thf(fact_6293_lcm__div__unit1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 0.30/0.63 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_div_unit1
% 0.30/0.63 thf(fact_6294_lcm__div__unit2,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 0.30/0.63 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_div_unit2
% 0.30/0.63 thf(fact_6295_sup__bot_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( bounde4967611905675639751up_bot @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sup_bot.comm_monoid_axioms
% 0.30/0.63 thf(fact_6296_inf__top_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( bounde4346867609351753570nf_top @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inf_top.comm_monoid_axioms
% 0.30/0.63 thf(fact_6297_or_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % or.comm_monoid_axioms
% 0.30/0.63 thf(fact_6298_gcd__nat_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_nat.comm_monoid_axioms
% 0.30/0.63 thf(fact_6299_xor_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( bit_se359711467146920520ations @ A )
% 0.30/0.63 => ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % xor.comm_monoid_axioms
% 0.30/0.63 thf(fact_6300_lcm__cases__int,axiom,
% 0.30/0.63 ! [X: int,Y3: int,P: int > $o] :
% 0.30/0.63 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.63 => ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) )
% 0.30/0.63 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.30/0.63 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 0.30/0.63 => ( P @ ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 0.30/0.63 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 0.30/0.63 => ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
% 0.30/0.63 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 0.30/0.63 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 0.30/0.63 => ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 0.30/0.63 => ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_cases_int
% 0.30/0.63 thf(fact_6301_max__nat_Ocomm__monoid__axioms,axiom,
% 0.30/0.63 comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 0.30/0.63
% 0.30/0.63 % max_nat.comm_monoid_axioms
% 0.30/0.63 thf(fact_6302_lcm__unique__int,axiom,
% 0.30/0.63 ! [D2: int,A3: int,B2: int] :
% 0.30/0.63 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
% 0.30/0.63 & ( dvd_dvd @ int @ A3 @ D2 )
% 0.30/0.63 & ( dvd_dvd @ int @ B2 @ D2 )
% 0.30/0.63 & ! [E3: int] :
% 0.30/0.63 ( ( ( dvd_dvd @ int @ A3 @ E3 )
% 0.30/0.63 & ( dvd_dvd @ int @ B2 @ E3 ) )
% 0.30/0.63 => ( dvd_dvd @ int @ D2 @ E3 ) ) )
% 0.30/0.63 = ( D2
% 0.30/0.63 = ( gcd_lcm @ int @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_unique_int
% 0.30/0.63 thf(fact_6303_prod__gcd__lcm__int,axiom,
% 0.30/0.63 ! [M: int,N: int] :
% 0.30/0.63 ( ( times_times @ int @ ( abs_abs @ int @ M ) @ ( abs_abs @ int @ N ) )
% 0.30/0.63 = ( times_times @ int @ ( gcd_gcd @ int @ M @ N ) @ ( gcd_lcm @ int @ M @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % prod_gcd_lcm_int
% 0.30/0.63 thf(fact_6304_Lcm__int__set__eq__fold,axiom,
% 0.30/0.63 ! [Xs: list @ int] :
% 0.30/0.63 ( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
% 0.30/0.63 = ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_int_set_eq_fold
% 0.30/0.63 thf(fact_6305_Lcm__fin_Oeq__fold,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( semiring_gcd_Lcm_fin @ A )
% 0.30/0.63 = ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin.eq_fold
% 0.30/0.63 thf(fact_6306_lcm__0__iff__nat,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ( gcd_lcm @ nat @ M @ N )
% 0.30/0.63 = ( zero_zero @ nat ) )
% 0.30/0.63 = ( ( M
% 0.30/0.63 = ( zero_zero @ nat ) )
% 0.30/0.63 | ( N
% 0.30/0.63 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_0_iff_nat
% 0.30/0.63 thf(fact_6307_lcm__1__iff__nat,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ( gcd_lcm @ nat @ M @ N )
% 0.30/0.63 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.63 = ( ( M
% 0.30/0.63 = ( suc @ ( zero_zero @ nat ) ) )
% 0.30/0.63 & ( N
% 0.30/0.63 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_1_iff_nat
% 0.30/0.63 thf(fact_6308_lcm__int__int__eq,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.30/0.63 = ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_int_int_eq
% 0.30/0.63 thf(fact_6309_Lcm__UNIV,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_UNIV
% 0.30/0.63 thf(fact_6310_Lcm__empty,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_empty
% 0.30/0.63 thf(fact_6311_Lcm__1__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( one_one @ A ) )
% 0.30/0.63 = ( ! [X2: A] :
% 0.30/0.63 ( ( member @ A @ X2 @ A6 )
% 0.30/0.63 => ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_1_iff
% 0.30/0.63 thf(fact_6312_Lcm__fin_Oinfinite,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin.infinite
% 0.30/0.63 thf(fact_6313_Lcm__fin_Oempty,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin.empty
% 0.30/0.63 thf(fact_6314_is__unit__Lcm__fin__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) @ ( one_one @ A ) )
% 0.30/0.63 = ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_unit_Lcm_fin_iff
% 0.30/0.63 thf(fact_6315_lcm__nat__abs__right__eq,axiom,
% 0.30/0.63 ! [N: nat,K: int] :
% 0.30/0.63 ( ( gcd_lcm @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 0.30/0.63 = ( nat2 @ ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_nat_abs_right_eq
% 0.30/0.63 thf(fact_6316_lcm__nat__abs__left__eq,axiom,
% 0.30/0.63 ! [K: int,N: nat] :
% 0.30/0.63 ( ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 0.30/0.63 = ( nat2 @ ( gcd_lcm @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_nat_abs_left_eq
% 0.30/0.63 thf(fact_6317_prod__gcd__lcm__nat,axiom,
% 0.30/0.63 ( ( times_times @ nat )
% 0.30/0.63 = ( ^ [M2: nat,N3: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M2 @ N3 ) @ ( gcd_lcm @ nat @ M2 @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % prod_gcd_lcm_nat
% 0.30/0.63 thf(fact_6318_lcm__pos__nat,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.63 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.63 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_pos_nat
% 0.30/0.63 thf(fact_6319_Lcm__no__multiple,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ! [M6: A] :
% 0.30/0.63 ( ( M6
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ? [X4: A] :
% 0.30/0.63 ( ( member @ A @ X4 @ A6 )
% 0.30/0.63 & ~ ( dvd_dvd @ A @ X4 @ M6 ) ) )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_no_multiple
% 0.30/0.63 thf(fact_6320_Lcm__0__iff_H,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( ~ ? [L3: A] :
% 0.30/0.63 ( ( L3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 & ! [X2: A] :
% 0.30/0.63 ( ( member @ A @ X2 @ A6 )
% 0.30/0.63 => ( dvd_dvd @ A @ X2 @ L3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_0_iff'
% 0.30/0.63 thf(fact_6321_Lcm__eq__0__I,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( member @ A @ ( zero_zero @ A ) @ A6 )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_eq_0_I
% 0.30/0.63 thf(fact_6322_Lcm__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_0_iff
% 0.30/0.63 thf(fact_6323_Lcm__int__greater__eq__0,axiom,
% 0.30/0.63 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Lcm @ int @ K5 ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_int_greater_eq_0
% 0.30/0.63 thf(fact_6324_Lcm__fin__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin_0_iff
% 0.30/0.63 thf(fact_6325_lcm__nat__def,axiom,
% 0.30/0.63 ( ( gcd_lcm @ nat )
% 0.30/0.63 = ( ^ [X2: nat,Y6: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y6 ) @ ( gcd_gcd @ nat @ X2 @ Y6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_nat_def
% 0.30/0.63 thf(fact_6326_Lcm__fin__1__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 0.30/0.63 = ( one_one @ A ) )
% 0.30/0.63 = ( ! [X2: A] :
% 0.30/0.63 ( ( member @ A @ X2 @ A6 )
% 0.30/0.63 => ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
% 0.30/0.63 & ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin_1_iff
% 0.30/0.63 thf(fact_6327_Lcm__no__units,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ( ( gcd_Lcm @ A )
% 0.30/0.63 = ( ^ [A7: set @ A] :
% 0.30/0.63 ( gcd_Lcm @ A
% 0.30/0.63 @ ( minus_minus @ ( set @ A ) @ A7
% 0.30/0.63 @ ( collect @ A
% 0.30/0.63 @ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_no_units
% 0.30/0.63 thf(fact_6328_lcm__int__def,axiom,
% 0.30/0.63 ( ( gcd_lcm @ int )
% 0.30/0.63 = ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_int_def
% 0.30/0.63 thf(fact_6329_Lcm__set__eq__fold,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [Xs: list @ A] :
% 0.30/0.63 ( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
% 0.30/0.63 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_set_eq_fold
% 0.30/0.63 thf(fact_6330_Lcm__fin_Oset__eq__fold,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [Xs: list @ A] :
% 0.30/0.63 ( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
% 0.30/0.63 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin.set_eq_fold
% 0.30/0.63 thf(fact_6331_Lcm__fin__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( semiring_gcd_Lcm_fin @ A )
% 0.30/0.63 = ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin_def
% 0.30/0.63 thf(fact_6332_Lcm__int__def,axiom,
% 0.30/0.63 ( ( gcd_Lcm @ int )
% 0.30/0.63 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_int_def
% 0.30/0.63 thf(fact_6333_Lcm__int__eq,axiom,
% 0.30/0.63 ! [N6: set @ nat] :
% 0.30/0.63 ( ( gcd_Lcm @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
% 0.30/0.63 = ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N6 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_int_eq
% 0.30/0.63 thf(fact_6334_Lcm__eq__0__I__nat,axiom,
% 0.30/0.63 ! [A6: set @ nat] :
% 0.30/0.63 ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 0.30/0.63 => ( ( gcd_Lcm @ nat @ A6 )
% 0.30/0.63 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_eq_0_I_nat
% 0.30/0.63 thf(fact_6335_Lcm__0__iff__nat,axiom,
% 0.30/0.63 ! [A6: set @ nat] :
% 0.30/0.63 ( ( finite_finite2 @ nat @ A6 )
% 0.30/0.63 => ( ( ( gcd_Lcm @ nat @ A6 )
% 0.30/0.63 = ( zero_zero @ nat ) )
% 0.30/0.63 = ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_0_iff_nat
% 0.30/0.63 thf(fact_6336_Lcm__nat__infinite,axiom,
% 0.30/0.63 ! [M7: set @ nat] :
% 0.30/0.63 ( ~ ( finite_finite2 @ nat @ M7 )
% 0.30/0.63 => ( ( gcd_Lcm @ nat @ M7 )
% 0.30/0.63 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_nat_infinite
% 0.30/0.63 thf(fact_6337_Lcm__nat__empty,axiom,
% 0.30/0.63 ( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.63 = ( one_one @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_nat_empty
% 0.30/0.63 thf(fact_6338_Lcm__nat__set__eq__fold,axiom,
% 0.30/0.63 ! [Xs: list @ nat] :
% 0.30/0.63 ( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
% 0.30/0.63 = ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_nat_set_eq_fold
% 0.30/0.63 thf(fact_6339_Lcm__eq__Max__nat,axiom,
% 0.30/0.63 ! [M7: set @ nat] :
% 0.30/0.63 ( ( finite_finite2 @ nat @ M7 )
% 0.30/0.63 => ( ( M7
% 0.30/0.63 != ( bot_bot @ ( set @ nat ) ) )
% 0.30/0.63 => ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.63 => ( ! [M6: nat,N4: nat] :
% 0.30/0.63 ( ( member @ nat @ M6 @ M7 )
% 0.30/0.63 => ( ( member @ nat @ N4 @ M7 )
% 0.30/0.63 => ( member @ nat @ ( gcd_lcm @ nat @ M6 @ N4 ) @ M7 ) ) )
% 0.30/0.63 => ( ( gcd_Lcm @ nat @ M7 )
% 0.30/0.63 = ( lattic643756798349783984er_Max @ nat @ M7 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_eq_Max_nat
% 0.30/0.63 thf(fact_6340_Lcm__nat__def,axiom,
% 0.30/0.63 ( ( gcd_Lcm @ nat )
% 0.30/0.63 = ( ^ [M9: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M9 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M9 ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_nat_def
% 0.30/0.63 thf(fact_6341_Lcm__coprime_H,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( finite_card @ A @ A6 )
% 0.30/0.63 != ( zero_zero @ nat ) )
% 0.30/0.63 => ( ! [A8: A,B5: A] :
% 0.30/0.63 ( ( member @ A @ A8 @ A6 )
% 0.30/0.63 => ( ( member @ A @ B5 @ A6 )
% 0.30/0.63 => ( ( A8 != B5 )
% 0.30/0.63 => ( algebr8660921524188924756oprime @ A @ A8 @ B5 ) ) ) )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A
% 0.30/0.63 @ ( groups7121269368397514597t_prod @ A @ A
% 0.30/0.63 @ ^ [X2: A] : X2
% 0.30/0.63 @ A6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_coprime'
% 0.30/0.63 thf(fact_6342_normalize__idem,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_idem
% 0.30/0.63 thf(fact_6343_lcm_Onormalize__bottom,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.normalize_bottom
% 0.30/0.63 thf(fact_6344_normalize__eq__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( A3
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_eq_0_iff
% 0.30/0.63 thf(fact_6345_normalize__0,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_0
% 0.30/0.63 thf(fact_6346_normalize__mult__normalize__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult_normalize_right
% 0.30/0.63 thf(fact_6347_normalize__mult__normalize__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult_normalize_left
% 0.30/0.63 thf(fact_6348_gcd_Onormalize__bottom,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd.normalize_bottom
% 0.30/0.63 thf(fact_6349_normalize__1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_1
% 0.30/0.63 thf(fact_6350_dvd__normalize__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % dvd_normalize_iff
% 0.30/0.63 thf(fact_6351_normalize__dvd__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
% 0.30/0.63 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_dvd_iff
% 0.30/0.63 thf(fact_6352_coprime__normalize__left__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
% 0.30/0.63 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % coprime_normalize_left_iff
% 0.30/0.63 thf(fact_6353_coprime__normalize__right__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % coprime_normalize_right_iff
% 0.30/0.63 thf(fact_6354_gcd_Otop__left__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_gcd @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd.top_left_normalize
% 0.30/0.63 thf(fact_6355_gcd_Otop__right__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_gcd @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd.top_right_normalize
% 0.30/0.63 thf(fact_6356_lcm_Otop__right__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.top_right_normalize
% 0.30/0.63 thf(fact_6357_lcm_Otop__left__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.top_left_normalize
% 0.30/0.63 thf(fact_6358_normalize__mult__unit__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [B2: A,A3: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult_unit_right
% 0.30/0.63 thf(fact_6359_normalize__mult__unit__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult_unit_left
% 0.30/0.63 thf(fact_6360_lcm__mult__gcd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_gcd
% 0.30/0.63 thf(fact_6361_gcd__mult__lcm,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_mult_lcm
% 0.30/0.63 thf(fact_6362_lcm__mult__distrib_H,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_distrib'
% 0.30/0.63 thf(fact_6363_lcm__mult__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_right
% 0.30/0.63 thf(fact_6364_lcm__mult__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_left
% 0.30/0.63 thf(fact_6365_normalize__nat__def,axiom,
% 0.30/0.63 ( ( normal6383669964737779283malize @ nat )
% 0.30/0.63 = ( id @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_nat_def
% 0.30/0.63 thf(fact_6366_gcd__mult__distrib_H,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_mult_distrib'
% 0.30/0.63 thf(fact_6367_gcd__mult__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,C2: A,B2: A] :
% 0.30/0.63 ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_mult_right
% 0.30/0.63 thf(fact_6368_gcd__mult__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_mult_left
% 0.30/0.63 thf(fact_6369_dvd__normalize__div,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal6328177297339901930cative @ A )
% 0.30/0.63 => ! [B2: A,A3: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % dvd_normalize_div
% 0.30/0.63 thf(fact_6370_normalize__idem__imp__is__unit__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = A3 )
% 0.30/0.63 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 = ( A3
% 0.30/0.63 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_idem_imp_is_unit_iff
% 0.30/0.63 thf(fact_6371_is__unit__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_unit_normalize
% 0.30/0.63 thf(fact_6372_normalize__1__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( one_one @ A ) )
% 0.30/0.63 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_1_iff
% 0.30/0.63 thf(fact_6373_associated__unit,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associated_unit
% 0.30/0.63 thf(fact_6374_normalize__int__def,axiom,
% 0.30/0.63 ( ( normal6383669964737779283malize @ int )
% 0.30/0.63 = ( abs_abs @ int ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_int_def
% 0.30/0.63 thf(fact_6375_semilattice__neutr__set_OF_Ocong,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( lattic5214292709420241887eutr_F @ A )
% 0.30/0.63 = ( lattic5214292709420241887eutr_F @ A ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.F.cong
% 0.30/0.63 thf(fact_6376_associated__iff__dvd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 = ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.63 & ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associated_iff_dvd
% 0.30/0.63 thf(fact_6377_associated__eqI,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.63 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.63 => ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = A3 )
% 0.30/0.63 => ( ( ( normal6383669964737779283malize @ A @ B2 )
% 0.30/0.63 = B2 )
% 0.30/0.63 => ( A3 = B2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associated_eqI
% 0.30/0.63 thf(fact_6378_associatedD2,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associatedD2
% 0.30/0.63 thf(fact_6379_associatedD1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 => ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associatedD1
% 0.30/0.63 thf(fact_6380_associatedI,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.30/0.63 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % associatedI
% 0.30/0.63 thf(fact_6381_normalize__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal6328177297339901930cative @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult
% 0.30/0.63 thf(fact_6382_coprime__crossproduct,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,D2: A,B2: A,C2: A] :
% 0.30/0.63 ( ( algebr8660921524188924756oprime @ A @ A3 @ D2 )
% 0.30/0.63 => ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 0.30/0.63 => ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
% 0.30/0.63 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D2 ) ) )
% 0.30/0.63 = ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 & ( ( normal6383669964737779283malize @ A @ C2 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % coprime_crossproduct
% 0.30/0.63 thf(fact_6383_lcm__gcd__prod,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_gcd_prod
% 0.30/0.63 thf(fact_6384_Gcd__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [C2: A,A6: set @ A] :
% 0.30/0.63 ( ( gcd_Gcd @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Gcd_mult
% 0.30/0.63 thf(fact_6385_lcm__coprime,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.63 => ( ( gcd_lcm @ A @ A3 @ B2 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_coprime
% 0.30/0.63 thf(fact_6386_Lcm__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A,C2: A] :
% 0.30/0.63 ( ( A6
% 0.30/0.63 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 => ( ( gcd_Lcm @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_mult
% 0.30/0.63 thf(fact_6387_lcm__gcd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( ( gcd_lcm @ A )
% 0.30/0.63 = ( ^ [A5: A,B3: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B3 ) @ ( gcd_gcd @ A @ A5 @ B3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_gcd
% 0.30/0.63 thf(fact_6388_Lcm__fin__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A,B2: A] :
% 0.30/0.63 ( ( A6
% 0.30/0.63 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 => ( ( semiring_gcd_Lcm_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin_mult
% 0.30/0.63 thf(fact_6389_Gcd__fin__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A,B2: A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( semiring_gcd_Gcd_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Gcd_fin_mult
% 0.30/0.63 thf(fact_6390_gcd__lcm,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( B2
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_lcm
% 0.30/0.63 thf(fact_6391_semilattice__neutr__set_Oinsert__remove,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
% 0.30/0.63 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.insert_remove
% 0.30/0.63 thf(fact_6392_semilattice__neutr__set_Oremove,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( member @ A @ X @ A6 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 0.30/0.63 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.remove
% 0.30/0.63 thf(fact_6393_semilattice__neutr__set_Ointro,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.63 ( ( semilattice_neutr @ A @ F @ Z3 )
% 0.30/0.63 => ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.intro
% 0.30/0.63 thf(fact_6394_semilattice__neutr__set_Oaxioms,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.axioms
% 0.30/0.63 thf(fact_6395_semilattice__neutr__set__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A )
% 0.30/0.63 = ( semilattice_neutr @ A ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set_def
% 0.30/0.63 thf(fact_6396_semilattice__neutr__set_Oempty,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 = Z3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.empty
% 0.30/0.63 thf(fact_6397_semilattice__neutr__set_Oin__idem,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( member @ A @ X @ A6 )
% 0.30/0.63 => ( ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 0.30/0.63 = ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.in_idem
% 0.30/0.63 thf(fact_6398_semilattice__neutr__set_Oinfinite,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 0.30/0.63 = Z3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.infinite
% 0.30/0.63 thf(fact_6399_semilattice__neutr__set_Oeq__fold,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 0.30/0.63 = ( finite_fold @ A @ A @ F @ Z3 @ A6 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.eq_fold
% 0.30/0.63 thf(fact_6400_semilattice__neutr__set_Osubset,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.63 => ( ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 0.30/0.63 = ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.subset
% 0.30/0.63 thf(fact_6401_semilattice__neutr__set_Oinsert,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
% 0.30/0.63 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.insert
% 0.30/0.63 thf(fact_6402_semilattice__neutr__set_Ounion,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.63 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 0.30/0.63 = ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.union
% 0.30/0.63 thf(fact_6403_semilattice__neutr__set_Oclosed,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 0.30/0.63 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( A6
% 0.30/0.63 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.63 => ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.63 => ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_set.closed
% 0.30/0.63 thf(fact_6404_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Lcm_fin.bounded_quasi_semilattice_set_axioms
% 0.30/0.63 thf(fact_6405_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Gcd_fin.bounded_quasi_semilattice_set_axioms
% 0.30/0.63 thf(fact_6406_semilattice__order__neutr__set_Osubset__imp,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.63 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.63 => ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.subset_imp
% 0.30/0.63 thf(fact_6407_lcm_Obounded__quasi__semilattice__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm.bounded_quasi_semilattice_axioms
% 0.30/0.63 thf(fact_6408_semilattice__order__neutr__set_Oaxioms_I2_J,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.axioms(2)
% 0.30/0.63 thf(fact_6409_semilattice__order__neutr__set_Oaxioms_I1_J,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.axioms(1)
% 0.30/0.63 thf(fact_6410_semilattice__order__neutr__set_OboundedE,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 0.30/0.63 => ! [A10: A] :
% 0.30/0.63 ( ( member @ A @ A10 @ A6 )
% 0.30/0.63 => ( Less_eq @ X @ A10 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.boundedE
% 0.30/0.63 thf(fact_6411_semilattice__order__neutr__set_OboundedI,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ! [A8: A] :
% 0.30/0.63 ( ( member @ A @ A8 @ A6 )
% 0.30/0.63 => ( Less_eq @ X @ A8 ) )
% 0.30/0.63 => ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.boundedI
% 0.30/0.63 thf(fact_6412_semilattice__order__neutr__set_OcoboundedI,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.63 => ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.coboundedI
% 0.30/0.63 thf(fact_6413_semilattice__order__neutr__set_Obounded__iff,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 0.30/0.63 = ( ! [X2: A] :
% 0.30/0.63 ( ( member @ A @ X2 @ A6 )
% 0.30/0.63 => ( Less_eq @ X @ X2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.bounded_iff
% 0.30/0.63 thf(fact_6414_semilattice__order__neutr__set__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( lattic3600114342068043075tr_set @ A )
% 0.30/0.63 = ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.63 ( ( semila1105856199041335345_order @ A @ F2 @ Z7 @ Less_eq2 @ Less2 )
% 0.30/0.63 & ( lattic5652469242046573047tr_set @ A @ F2 @ Z7 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set_def
% 0.30/0.63 thf(fact_6415_semilattice__order__neutr__set_Ointro,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 0.30/0.63 => ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_neutr_set.intro
% 0.30/0.63 thf(fact_6416_gcd_Obounded__quasi__semilattice__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd.bounded_quasi_semilattice_axioms
% 0.30/0.63 thf(fact_6417_normalize__div,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
% 0.30/0.63 = ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_div
% 0.30/0.63 thf(fact_6418_normalize__unit__factor,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_unit_factor
% 0.30/0.63 thf(fact_6419_unit__factor__simps_I1_J,axiom,
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
% 0.30/0.63 = ( zero_zero @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_simps(1)
% 0.30/0.63 thf(fact_6420_unit__factor__idem,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.30/0.63 = ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_idem
% 0.30/0.63 thf(fact_6421_unit__factor__simps_I2_J,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
% 0.30/0.63 = ( one_one @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_simps(2)
% 0.30/0.63 thf(fact_6422_unit__factor__eq__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( unit_f5069060285200089521factor @ A @ A3 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( A3
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_eq_0_iff
% 0.30/0.63 thf(fact_6423_unit__factor__0,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semido2269285787275462019factor @ A )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
% 0.30/0.63 = ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_0
% 0.30/0.63 thf(fact_6424_unit__factor__1,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_1
% 0.30/0.63 thf(fact_6425_normalize__mult__unit__factor,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.30/0.63 = A3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_mult_unit_factor
% 0.30/0.63 thf(fact_6426_unit__factor__mult__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
% 0.30/0.63 = A3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_mult_normalize
% 0.30/0.63 thf(fact_6427_div__unit__factor,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % div_unit_factor
% 0.30/0.63 thf(fact_6428_div__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
% 0.30/0.63 = ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % div_normalize
% 0.30/0.63 thf(fact_6429_inv__unit__factor__eq__0__iff,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 = ( A3
% 0.30/0.63 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inv_unit_factor_eq_0_iff
% 0.30/0.63 thf(fact_6430_unit__factor__mult__unit__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semido2269285787275462019factor @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
% 0.30/0.63 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_mult_unit_right
% 0.30/0.63 thf(fact_6431_unit__factor__mult__unit__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semido2269285787275462019factor @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_mult_unit_left
% 0.30/0.63 thf(fact_6432_mult__one__div__unit__factor,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
% 0.30/0.63 = ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_one_div_unit_factor
% 0.30/0.63 thf(fact_6433_unit__factor__lcm,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 | ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 & ( ~ ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 | ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_lcm
% 0.30/0.63 thf(fact_6434_unit__factor__normalize,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_normalize
% 0.30/0.63 thf(fact_6435_normalize__unit__factor__eqI,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.30/0.63 => ( ( ( unit_f5069060285200089521factor @ A @ A3 )
% 0.30/0.63 = ( unit_f5069060285200089521factor @ A @ B2 ) )
% 0.30/0.63 => ( A3 = B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_unit_factor_eqI
% 0.30/0.63 thf(fact_6436_lcm__mult__distrib,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [K: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ K @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lcm_mult_distrib
% 0.30/0.63 thf(fact_6437_mult__lcm__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.63 = ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_lcm_right
% 0.30/0.63 thf(fact_6438_mult__lcm__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_lcm_left
% 0.30/0.63 thf(fact_6439_unit__factor__nat__def,axiom,
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ nat )
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if @ nat
% 0.30/0.63 @ ( N3
% 0.30/0.63 = ( zero_zero @ nat ) )
% 0.30/0.63 @ ( zero_zero @ nat )
% 0.30/0.63 @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_nat_def
% 0.30/0.63 thf(fact_6440_dvd__unit__factor__div,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal6328177297339901930cative @ A )
% 0.30/0.63 => ! [B2: A,A3: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % dvd_unit_factor_div
% 0.30/0.63 thf(fact_6441_unit__factor__self,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_self
% 0.30/0.63 thf(fact_6442_unit__factor__int__def,axiom,
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ int )
% 0.30/0.63 = ( sgn_sgn @ int ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_int_def
% 0.30/0.63 thf(fact_6443_unit__factor__mult,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal6328177297339901930cative @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_mult
% 0.30/0.63 thf(fact_6444_unit__factor__dvd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_dvd
% 0.30/0.63 thf(fact_6445_is__unit__unit__factor,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semido2269285787275462019factor @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 0.30/0.63 = A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_unit_unit_factor
% 0.30/0.63 thf(fact_6446_gcd__mult__distrib,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [K: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ K @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gcd_mult_distrib
% 0.30/0.63 thf(fact_6447_mult__gcd__right,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 )
% 0.30/0.63 = ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_gcd_right
% 0.30/0.63 thf(fact_6448_mult__gcd__left,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [C2: A,A3: A,B2: A] :
% 0.30/0.63 ( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_gcd_left
% 0.30/0.63 thf(fact_6449_unit__factor__is__unit,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semido2269285787275462019factor @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_is_unit
% 0.30/0.63 thf(fact_6450_unit__factor__gcd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A3: A,B2: A] :
% 0.30/0.63 ( ( ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 & ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 & ( ~ ( ( A3
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 & ( B2
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_gcd
% 0.30/0.63 thf(fact_6451_coprime__crossproduct_H,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiri6843258321239162965malize @ A )
% 0.30/0.63 => ! [B2: A,D2: A,A3: A,C2: A] :
% 0.30/0.63 ( ( B2
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( ( unit_f5069060285200089521factor @ A @ B2 )
% 0.30/0.63 = ( unit_f5069060285200089521factor @ A @ D2 ) )
% 0.30/0.63 => ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.30/0.63 => ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 0.30/0.63 => ( ( ( times_times @ A @ A3 @ D2 )
% 0.30/0.63 = ( times_times @ A @ B2 @ C2 ) )
% 0.30/0.63 = ( ( A3 = C2 )
% 0.30/0.63 & ( B2 = D2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % coprime_crossproduct'
% 0.30/0.63 thf(fact_6452_unit__factor__Lcm,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 & ( ( ( gcd_Lcm @ A @ A6 )
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_Lcm
% 0.30/0.63 thf(fact_6453_unit__factor__Gcd,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_Gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( ( ( gcd_Gcd @ A @ A6 )
% 0.30/0.63 = ( zero_zero @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 & ( ( ( gcd_Gcd @ A @ A6 )
% 0.30/0.63 != ( zero_zero @ A ) )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
% 0.30/0.63 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_Gcd
% 0.30/0.63 thf(fact_6454_normalize__idem__imp__unit__factor__eq,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( normal8620421768224518004emidom @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.30/0.63 = A3 )
% 0.30/0.63 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 0.30/0.63 = ( zero_neq_one_of_bool @ A
% 0.30/0.63 @ ( A3
% 0.30/0.63 != ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % normalize_idem_imp_unit_factor_eq
% 0.30/0.63 thf(fact_6455_unit__factor__Lcm__fin,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) )
% 0.30/0.63 = ( zero_neq_one_of_bool @ A
% 0.30/0.63 @ ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 0.30/0.63 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_Lcm_fin
% 0.30/0.63 thf(fact_6456_unit__factor__Gcd__fin,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_gcd @ A )
% 0.30/0.63 => ! [A6: set @ A] :
% 0.30/0.63 ( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) )
% 0.30/0.63 = ( zero_neq_one_of_bool @ A
% 0.30/0.63 @ ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 0.30/0.63 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unit_factor_Gcd_fin
% 0.30/0.63 thf(fact_6457_times__num__def,axiom,
% 0.30/0.63 ( ( times_times @ num )
% 0.30/0.63 = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % times_num_def
% 0.30/0.63 thf(fact_6458_nat__of__num__pos,axiom,
% 0.30/0.63 ! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_pos
% 0.30/0.63 thf(fact_6459_nat__of__num_Osimps_I2_J,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( nat_of_num @ ( bit0 @ X ) )
% 0.30/0.63 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num.simps(2)
% 0.30/0.63 thf(fact_6460_is__num__normalize_I5_J,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [X: A] :
% 0.30/0.63 ( ( neg_numeral_is_num @ A @ X )
% 0.30/0.63 => ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_normalize(5)
% 0.30/0.63 thf(fact_6461_is__num__numeral,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [K: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_numeral
% 0.30/0.63 thf(fact_6462_nat__of__num__neq__0,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( nat_of_num @ X )
% 0.30/0.63 != ( zero_zero @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_neq_0
% 0.30/0.63 thf(fact_6463_num__eq__iff,axiom,
% 0.30/0.63 ( ( ^ [Y: num,Z2: num] : ( Y = Z2 ) )
% 0.30/0.63 = ( ^ [X2: num,Y6: num] :
% 0.30/0.63 ( ( nat_of_num @ X2 )
% 0.30/0.63 = ( nat_of_num @ Y6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % num_eq_iff
% 0.30/0.63 thf(fact_6464_nat__of__num__numeral,axiom,
% 0.30/0.63 ( nat_of_num
% 0.30/0.63 = ( numeral_numeral @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_numeral
% 0.30/0.63 thf(fact_6465_nat__of__num__inverse,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( num_of_nat @ ( nat_of_num @ X ) )
% 0.30/0.63 = X ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_inverse
% 0.30/0.63 thf(fact_6466_is__num__normalize_I4_J,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_normalize(4)
% 0.30/0.63 thf(fact_6467_is__num__add__left__commute,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [X: A,Y3: A,Z3: A] :
% 0.30/0.63 ( ( neg_numeral_is_num @ A @ X )
% 0.30/0.63 => ( ( neg_numeral_is_num @ A @ Y3 )
% 0.30/0.63 => ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y3 @ Z3 ) )
% 0.30/0.63 = ( plus_plus @ A @ Y3 @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_add_left_commute
% 0.30/0.63 thf(fact_6468_is__num__add__commute,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [X: A,Y3: A] :
% 0.30/0.63 ( ( neg_numeral_is_num @ A @ X )
% 0.30/0.63 => ( ( neg_numeral_is_num @ A @ Y3 )
% 0.30/0.63 => ( ( plus_plus @ A @ X @ Y3 )
% 0.30/0.63 = ( plus_plus @ A @ Y3 @ X ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_add_commute
% 0.30/0.63 thf(fact_6469_is__num__normalize_I6_J,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [X: A,Y3: A] :
% 0.30/0.63 ( ( neg_numeral_is_num @ A @ X )
% 0.30/0.63 => ( ( neg_numeral_is_num @ A @ Y3 )
% 0.30/0.63 => ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num_normalize(6)
% 0.30/0.63 thf(fact_6470_nat__of__num__code_I1_J,axiom,
% 0.30/0.63 ( ( nat_of_num @ one2 )
% 0.30/0.63 = ( one_one @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_code(1)
% 0.30/0.63 thf(fact_6471_nat__of__num__add,axiom,
% 0.30/0.63 ! [X: num,Y3: num] :
% 0.30/0.63 ( ( nat_of_num @ ( plus_plus @ num @ X @ Y3 ) )
% 0.30/0.63 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_add
% 0.30/0.63 thf(fact_6472_nat__of__num__mult,axiom,
% 0.30/0.63 ! [X: num,Y3: num] :
% 0.30/0.63 ( ( nat_of_num @ ( times_times @ num @ X @ Y3 ) )
% 0.30/0.63 = ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_mult
% 0.30/0.63 thf(fact_6473_nat__of__num__inc,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( nat_of_num @ ( inc @ X ) )
% 0.30/0.63 = ( suc @ ( nat_of_num @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_inc
% 0.30/0.63 thf(fact_6474_less__num__def,axiom,
% 0.30/0.63 ( ( ord_less @ num )
% 0.30/0.63 = ( ^ [M2: num,N3: num] : ( ord_less @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_num_def
% 0.30/0.63 thf(fact_6475_less__eq__num__def,axiom,
% 0.30/0.63 ( ( ord_less_eq @ num )
% 0.30/0.63 = ( ^ [M2: num,N3: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_eq_num_def
% 0.30/0.63 thf(fact_6476_nat__of__num__code_I2_J,axiom,
% 0.30/0.63 ! [N: num] :
% 0.30/0.63 ( ( nat_of_num @ ( bit0 @ N ) )
% 0.30/0.63 = ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_code(2)
% 0.30/0.63 thf(fact_6477_nat__of__num__sqr,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( nat_of_num @ ( sqr @ X ) )
% 0.30/0.63 = ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_sqr
% 0.30/0.63 thf(fact_6478_nat__of__num_Osimps_I1_J,axiom,
% 0.30/0.63 ( ( nat_of_num @ one2 )
% 0.30/0.63 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num.simps(1)
% 0.30/0.63 thf(fact_6479_nat__of__num_Osimps_I3_J,axiom,
% 0.30/0.63 ! [X: num] :
% 0.30/0.63 ( ( nat_of_num @ ( bit1 @ X ) )
% 0.30/0.63 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num.simps(3)
% 0.30/0.63 thf(fact_6480_num__of__nat__inverse,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.30/0.63 => ( ( nat_of_num @ ( num_of_nat @ N ) )
% 0.30/0.63 = N ) ) ).
% 0.30/0.63
% 0.30/0.63 % num_of_nat_inverse
% 0.30/0.63 thf(fact_6481_plus__num__def,axiom,
% 0.30/0.63 ( ( plus_plus @ num )
% 0.30/0.63 = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % plus_num_def
% 0.30/0.63 thf(fact_6482_nat__of__num__code_I3_J,axiom,
% 0.30/0.63 ! [N: num] :
% 0.30/0.63 ( ( nat_of_num @ ( bit1 @ N ) )
% 0.30/0.63 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_num_code(3)
% 0.30/0.63 thf(fact_6483_is__num_Ocases,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ! [A3: A] :
% 0.30/0.63 ( ( neg_numeral_is_num @ A @ A3 )
% 0.30/0.63 => ( ( A3
% 0.30/0.63 != ( one_one @ A ) )
% 0.30/0.63 => ( ! [X3: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 = ( uminus_uminus @ A @ X3 ) )
% 0.30/0.63 => ~ ( neg_numeral_is_num @ A @ X3 ) )
% 0.30/0.63 => ~ ! [X3: A,Y5: A] :
% 0.30/0.63 ( ( A3
% 0.30/0.63 = ( plus_plus @ A @ X3 @ Y5 ) )
% 0.30/0.63 => ( ( neg_numeral_is_num @ A @ X3 )
% 0.30/0.63 => ~ ( neg_numeral_is_num @ A @ Y5 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num.cases
% 0.30/0.63 thf(fact_6484_is__num_Osimps,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( neg_numeral @ A )
% 0.30/0.63 => ( ( neg_numeral_is_num @ A )
% 0.30/0.63 = ( ^ [A5: A] :
% 0.30/0.63 ( ( A5
% 0.30/0.63 = ( one_one @ A ) )
% 0.30/0.63 | ? [X2: A] :
% 0.30/0.63 ( ( A5
% 0.30/0.63 = ( uminus_uminus @ A @ X2 ) )
% 0.30/0.63 & ( neg_numeral_is_num @ A @ X2 ) )
% 0.30/0.63 | ? [X2: A,Y6: A] :
% 0.30/0.63 ( ( A5
% 0.30/0.63 = ( plus_plus @ A @ X2 @ Y6 ) )
% 0.30/0.63 & ( neg_numeral_is_num @ A @ X2 )
% 0.30/0.63 & ( neg_numeral_is_num @ A @ Y6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_num.simps
% 0.30/0.63 thf(fact_6485_arg__max__nat__lemma,axiom,
% 0.30/0.63 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 0.30/0.63 ( ( P @ K )
% 0.30/0.63 => ( ! [Y5: A] :
% 0.30/0.63 ( ( P @ Y5 )
% 0.30/0.63 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 0.30/0.63 => ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) )
% 0.30/0.63 & ! [Y4: A] :
% 0.30/0.63 ( ( P @ Y4 )
% 0.30/0.63 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_nat_lemma
% 0.30/0.63 thf(fact_6486_arg__max__nat__le,axiom,
% 0.30/0.63 ! [A: $tType,P: A > $o,X: A,F: A > nat,B2: nat] :
% 0.30/0.63 ( ( P @ X )
% 0.30/0.63 => ( ! [Y5: A] :
% 0.30/0.63 ( ( P @ Y5 )
% 0.30/0.63 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 0.30/0.63 => ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_nat_le
% 0.30/0.63 thf(fact_6487_arg__max__natI,axiom,
% 0.30/0.63 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 0.30/0.63 ( ( P @ K )
% 0.30/0.63 => ( ! [Y5: A] :
% 0.30/0.63 ( ( P @ Y5 )
% 0.30/0.63 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 0.30/0.63 => ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_natI
% 0.30/0.63 thf(fact_6488_arg__max__equality,axiom,
% 0.30/0.63 ! [A: $tType,C: $tType] :
% 0.30/0.63 ( ( order @ A )
% 0.30/0.63 => ! [P: C > $o,K: C,F: C > A] :
% 0.30/0.63 ( ( P @ K )
% 0.30/0.63 => ( ! [X3: C] :
% 0.30/0.63 ( ( P @ X3 )
% 0.30/0.63 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ K ) ) )
% 0.30/0.63 => ( ( F @ ( lattices_ord_arg_max @ C @ A @ F @ P ) )
% 0.30/0.63 = ( F @ K ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_equality
% 0.30/0.63 thf(fact_6489_arg__maxI,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType] :
% 0.30/0.63 ( ( ord @ B )
% 0.30/0.63 => ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
% 0.30/0.63 ( ( P @ X )
% 0.30/0.63 => ( ! [Y5: A] :
% 0.30/0.63 ( ( P @ Y5 )
% 0.30/0.63 => ~ ( ord_less @ B @ ( F @ X ) @ ( F @ Y5 ) ) )
% 0.30/0.63 => ( ! [X3: A] :
% 0.30/0.63 ( ( P @ X3 )
% 0.30/0.63 => ( ! [Y4: A] :
% 0.30/0.63 ( ( P @ Y4 )
% 0.30/0.63 => ~ ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.63 => ( Q @ X3 ) ) )
% 0.30/0.63 => ( Q @ ( lattices_ord_arg_max @ A @ B @ F @ P ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_maxI
% 0.30/0.63 thf(fact_6490_arg__max__on__def,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType] :
% 0.30/0.63 ( ( ord @ A )
% 0.30/0.63 => ( ( lattic1883929316492267755max_on @ B @ A )
% 0.30/0.63 = ( ^ [F2: B > A,S7: set @ B] :
% 0.30/0.63 ( lattices_ord_arg_max @ B @ A @ F2
% 0.30/0.63 @ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_on_def
% 0.30/0.63 thf(fact_6491_arg__max__def,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType] :
% 0.30/0.63 ( ( ord @ A )
% 0.30/0.63 => ( ( lattices_ord_arg_max @ B @ A )
% 0.30/0.63 = ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751176901750rg_max @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arg_max_def
% 0.30/0.63 thf(fact_6492_is__arg__max__linorder,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType] :
% 0.30/0.63 ( ( linorder @ B )
% 0.30/0.63 => ( ( lattic501386751176901750rg_max @ A @ B )
% 0.30/0.63 = ( ^ [F2: A > B,P4: A > $o,X2: A] :
% 0.30/0.63 ( ( P4 @ X2 )
% 0.30/0.63 & ! [Y6: A] :
% 0.30/0.63 ( ( P4 @ Y6 )
% 0.30/0.63 => ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_arg_max_linorder
% 0.30/0.63 thf(fact_6493_is__arg__max__def,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType] :
% 0.30/0.63 ( ( ord @ A )
% 0.30/0.63 => ( ( lattic501386751176901750rg_max @ B @ A )
% 0.30/0.63 = ( ^ [F2: B > A,P4: B > $o,X2: B] :
% 0.30/0.63 ( ( P4 @ X2 )
% 0.30/0.63 & ~ ? [Y6: B] :
% 0.30/0.63 ( ( P4 @ Y6 )
% 0.30/0.63 & ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % is_arg_max_def
% 0.30/0.63 thf(fact_6494_int__of__integer__of__natural,axiom,
% 0.30/0.63 ! [N: code_natural] :
% 0.30/0.63 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 0.30/0.63 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_of_integer_of_natural
% 0.30/0.63 thf(fact_6495_integer__of__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural] :
% 0.30/0.63 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 0.30/0.63 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % integer_of_natural.rep_eq
% 0.30/0.63 thf(fact_6496_zero__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
% 0.30/0.63 = ( zero_zero @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_natural.rep_eq
% 0.30/0.63 thf(fact_6497_plus__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % plus_natural.rep_eq
% 0.30/0.63 thf(fact_6498_nat__of__natural__numeral,axiom,
% 0.30/0.63 ! [K: num] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
% 0.30/0.63 = ( numeral_numeral @ nat @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_numeral
% 0.30/0.63 thf(fact_6499_minus__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % minus_natural.rep_eq
% 0.30/0.63 thf(fact_6500_times__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % times_natural.rep_eq
% 0.30/0.63 thf(fact_6501_one__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( code_nat_of_natural @ ( one_one @ code_natural ) )
% 0.30/0.63 = ( one_one @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_natural.rep_eq
% 0.30/0.63 thf(fact_6502_modulo__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % modulo_natural.rep_eq
% 0.30/0.63 thf(fact_6503_divide__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % divide_natural.rep_eq
% 0.30/0.63 thf(fact_6504_nat__of__natural__min,axiom,
% 0.30/0.63 ! [K: code_natural,L: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
% 0.30/0.63 = ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_min
% 0.30/0.63 thf(fact_6505_nat__of__natural__max,axiom,
% 0.30/0.63 ! [K: code_natural,L: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
% 0.30/0.63 = ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_max
% 0.30/0.63 thf(fact_6506_division__segment__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
% 0.30/0.63 = ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % division_segment_natural.rep_eq
% 0.30/0.63 thf(fact_6507_euclidean__size__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( euclid6346220572633701492n_size @ code_natural )
% 0.30/0.63 = ( ^ [X2: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % euclidean_size_natural.rep_eq
% 0.30/0.63 thf(fact_6508_of__nat__of__natural,axiom,
% 0.30/0.63 ! [N: code_natural] :
% 0.30/0.63 ( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
% 0.30/0.63 = N ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_natural
% 0.30/0.63 thf(fact_6509_nat__of__natural__of__nat,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 0.30/0.63 = N ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_of_nat
% 0.30/0.63 thf(fact_6510_nat__of__natural__induct,axiom,
% 0.30/0.63 ! [Y3: nat,P: nat > $o] :
% 0.30/0.63 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ( ! [X3: code_natural] : ( P @ ( code_nat_of_natural @ X3 ) )
% 0.30/0.63 => ( P @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_induct
% 0.30/0.63 thf(fact_6511_nat__of__natural__cases,axiom,
% 0.30/0.63 ! [Y3: nat] :
% 0.30/0.63 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ~ ! [X3: code_natural] :
% 0.30/0.63 ( Y3
% 0.30/0.63 != ( code_nat_of_natural @ X3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_cases
% 0.30/0.63 thf(fact_6512_nat__of__natural,axiom,
% 0.30/0.63 ! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural
% 0.30/0.63 thf(fact_6513_set__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % set_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6514_unset__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unset_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6515_flip__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % flip_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6516_drop__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % drop_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6517_bit__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 0.30/0.63 = ( ^ [X2: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bit_natural.rep_eq
% 0.30/0.63 thf(fact_6518_take__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % take_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6519_or__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % or_natural.rep_eq
% 0.30/0.63 thf(fact_6520_xor__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % xor_natural.rep_eq
% 0.30/0.63 thf(fact_6521_and__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % and_natural.rep_eq
% 0.30/0.63 thf(fact_6522_mask__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
% 0.30/0.63 = ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % mask_natural.rep_eq
% 0.30/0.63 thf(fact_6523_push__bit__natural_Orep__eq,axiom,
% 0.30/0.63 ! [X: nat,Xa3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 = ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % push_bit_natural.rep_eq
% 0.30/0.63 thf(fact_6524_natural__eq__iff,axiom,
% 0.30/0.63 ( ( ^ [Y: code_natural,Z2: code_natural] : ( Y = Z2 ) )
% 0.30/0.63 = ( ^ [M2: code_natural,N3: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ M2 )
% 0.30/0.63 = ( code_nat_of_natural @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_eq_iff
% 0.30/0.63 thf(fact_6525_natural__eqI,axiom,
% 0.30/0.63 ! [M: code_natural,N: code_natural] :
% 0.30/0.63 ( ( ( code_nat_of_natural @ M )
% 0.30/0.63 = ( code_nat_of_natural @ N ) )
% 0.30/0.63 => ( M = N ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_eqI
% 0.30/0.63 thf(fact_6526_nat__of__natural__inject,axiom,
% 0.30/0.63 ! [X: code_natural,Y3: code_natural] :
% 0.30/0.63 ( ( ( code_nat_of_natural @ X )
% 0.30/0.63 = ( code_nat_of_natural @ Y3 ) )
% 0.30/0.63 = ( X = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_inject
% 0.30/0.63 thf(fact_6527_less__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( ord_less @ code_natural )
% 0.30/0.63 = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_natural.rep_eq
% 0.30/0.63 thf(fact_6528_less__eq__natural_Orep__eq,axiom,
% 0.30/0.63 ( ( ord_less_eq @ code_natural )
% 0.30/0.63 = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_eq_natural.rep_eq
% 0.30/0.63 thf(fact_6529_natural__of__integer_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_integer] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
% 0.30/0.63 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_integer.rep_eq
% 0.30/0.63 thf(fact_6530_natural__decr,axiom,
% 0.30/0.63 ! [N: code_natural] :
% 0.30/0.63 ( ( N
% 0.30/0.63 != ( zero_zero @ code_natural ) )
% 0.30/0.63 => ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_decr
% 0.30/0.63 thf(fact_6531_Nat_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_integer] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( code_Nat @ X ) )
% 0.30/0.63 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Nat.rep_eq
% 0.30/0.63 thf(fact_6532_Code__Numeral_Osize__nat__def,axiom,
% 0.30/0.63 code_size_nat = code_nat_of_natural ).
% 0.30/0.63
% 0.30/0.63 % Code_Numeral.size_nat_def
% 0.30/0.63 thf(fact_6533_natural__zero__minus__one,axiom,
% 0.30/0.63 ( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
% 0.30/0.63 = ( zero_zero @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_zero_minus_one
% 0.30/0.63 thf(fact_6534_log_Osimps,axiom,
% 0.30/0.63 ( log
% 0.30/0.63 = ( ^ [B3: code_natural,I5: code_natural] :
% 0.30/0.63 ( if @ code_natural
% 0.30/0.63 @ ( ( ord_less_eq @ code_natural @ B3 @ ( one_one @ code_natural ) )
% 0.30/0.63 | ( ord_less @ code_natural @ I5 @ B3 ) )
% 0.30/0.63 @ ( one_one @ code_natural )
% 0.30/0.63 @ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B3 @ ( divide_divide @ code_natural @ I5 @ B3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log.simps
% 0.30/0.63 thf(fact_6535_log_Oelims,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
% 0.30/0.63 ( ( ( log @ X @ Xa3 )
% 0.30/0.63 = Y3 )
% 0.30/0.63 => ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.30/0.63 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( one_one @ code_natural ) ) )
% 0.30/0.63 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.30/0.63 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log.elims
% 0.30/0.63 thf(fact_6536_minus__shift__def,axiom,
% 0.30/0.63 ( minus_shift
% 0.30/0.63 = ( ^ [R5: code_natural,K3: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K3 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R5 @ K3 ) @ L3 ) @ ( minus_minus @ code_natural @ K3 @ L3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % minus_shift_def
% 0.30/0.63 thf(fact_6537_natural_Osimps_I4_J,axiom,
% 0.30/0.63 ! [T: $tType,F1: T,F22: code_natural > T] :
% 0.30/0.63 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.30/0.63 = F1 ) ).
% 0.30/0.63
% 0.30/0.63 % natural.simps(4)
% 0.30/0.63 thf(fact_6538_range,axiom,
% 0.30/0.63 ! [K: code_natural,S: product_prod @ code_natural @ code_natural] :
% 0.30/0.63 ( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K )
% 0.30/0.63 => ( ord_less @ code_natural @ ( product_fst @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( range @ K @ S ) ) @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % range
% 0.30/0.63 thf(fact_6539_select__weight__member,axiom,
% 0.30/0.63 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A ),S: product_prod @ code_natural @ code_natural] :
% 0.30/0.63 ( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
% 0.30/0.63 => ( member @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select_weight @ A @ Xs @ S ) ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % select_weight_member
% 0.30/0.63 thf(fact_6540_select__weight__drop__zero,axiom,
% 0.30/0.63 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
% 0.30/0.63 ( ( select_weight @ A
% 0.30/0.63 @ ( filter @ ( product_prod @ code_natural @ A )
% 0.30/0.63 @ ( product_case_prod @ code_natural @ A @ $o
% 0.30/0.63 @ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
% 0.30/0.63 @ Xs ) )
% 0.30/0.63 = ( select_weight @ A @ Xs ) ) ).
% 0.30/0.63
% 0.30/0.63 % select_weight_drop_zero
% 0.30/0.63 thf(fact_6541_pick__member,axiom,
% 0.30/0.63 ! [A: $tType,I: code_natural,Xs: list @ ( product_prod @ code_natural @ A )] :
% 0.30/0.63 ( ( ord_less @ code_natural @ I @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
% 0.30/0.63 => ( member @ A @ ( pick @ A @ Xs @ I ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pick_member
% 0.30/0.63 thf(fact_6542_pick_Osimps,axiom,
% 0.30/0.63 ! [A: $tType,I: code_natural,X: product_prod @ code_natural @ A,Xs: list @ ( product_prod @ code_natural @ A )] :
% 0.30/0.63 ( ( ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
% 0.30/0.63 => ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
% 0.30/0.63 = ( product_snd @ code_natural @ A @ X ) ) )
% 0.30/0.63 & ( ~ ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
% 0.30/0.63 => ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
% 0.30/0.63 = ( pick @ A @ Xs @ ( minus_minus @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pick.simps
% 0.30/0.63 thf(fact_6543_pick__drop__zero,axiom,
% 0.30/0.63 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
% 0.30/0.63 ( ( pick @ A
% 0.30/0.63 @ ( filter @ ( product_prod @ code_natural @ A )
% 0.30/0.63 @ ( product_case_prod @ code_natural @ A @ $o
% 0.30/0.63 @ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
% 0.30/0.63 @ Xs ) )
% 0.30/0.63 = ( pick @ A @ Xs ) ) ).
% 0.30/0.63
% 0.30/0.63 % pick_drop_zero
% 0.30/0.63 thf(fact_6544_log_Opelims,axiom,
% 0.30/0.63 ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
% 0.30/0.63 ( ( ( log @ X @ Xa3 )
% 0.30/0.63 = Y3 )
% 0.30/0.63 => ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) )
% 0.30/0.63 => ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.30/0.63 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( one_one @ code_natural ) ) )
% 0.30/0.63 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.30/0.63 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) )
% 0.30/0.63 => ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log.pelims
% 0.30/0.63 thf(fact_6545_bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 0.30/0.63 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bit_natural_def
% 0.30/0.63 thf(fact_6546_pick__same,axiom,
% 0.30/0.63 ! [A: $tType,L: nat,Xs: list @ A] :
% 0.30/0.63 ( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.30/0.63 => ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
% 0.30/0.63 = ( nth @ A @ Xs @ L ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pick_same
% 0.30/0.63 thf(fact_6547_euclidean__size__natural__def,axiom,
% 0.30/0.63 ( ( euclid6346220572633701492n_size @ code_natural )
% 0.30/0.63 = ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % euclidean_size_natural_def
% 0.30/0.63 thf(fact_6548_natural__of__nat__of__natural__inverse,axiom,
% 0.30/0.63 ! [N: code_natural] :
% 0.30/0.63 ( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
% 0.30/0.63 = N ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_of_natural_inverse
% 0.30/0.63 thf(fact_6549_nat__of__natural__of__nat__inverse,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
% 0.30/0.63 = N ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_of_nat_inverse
% 0.30/0.63 thf(fact_6550_nat__of__natural__inverse,axiom,
% 0.30/0.63 ! [X: code_natural] :
% 0.30/0.63 ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
% 0.30/0.63 = X ) ).
% 0.30/0.63
% 0.30/0.63 % nat_of_natural_inverse
% 0.30/0.63 thf(fact_6551_natural__of__nat__of__nat,axiom,
% 0.30/0.63 ( code_natural_of_nat
% 0.30/0.63 = ( semiring_1_of_nat @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_of_nat
% 0.30/0.63 thf(fact_6552_natural__of__nat__inverse,axiom,
% 0.30/0.63 ! [Y3: nat] :
% 0.30/0.63 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y3 ) )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_inverse
% 0.30/0.63 thf(fact_6553_natural__of__nat__cases,axiom,
% 0.30/0.63 ! [X: code_natural] :
% 0.30/0.63 ~ ! [Y5: nat] :
% 0.30/0.63 ( ( X
% 0.30/0.63 = ( code_natural_of_nat @ Y5 ) )
% 0.30/0.63 => ~ ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_cases
% 0.30/0.63 thf(fact_6554_natural__of__nat__induct,axiom,
% 0.30/0.63 ! [P: code_natural > $o,X: code_natural] :
% 0.30/0.63 ( ! [Y5: nat] :
% 0.30/0.63 ( ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ( P @ ( code_natural_of_nat @ Y5 ) ) )
% 0.30/0.63 => ( P @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_induct
% 0.30/0.63 thf(fact_6555_natural__of__nat__inject,axiom,
% 0.30/0.63 ! [X: nat,Y3: nat] :
% 0.30/0.63 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.63 => ( ( ( code_natural_of_nat @ X )
% 0.30/0.63 = ( code_natural_of_nat @ Y3 ) )
% 0.30/0.63 = ( X = Y3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_nat_inject
% 0.30/0.63 thf(fact_6556_minus__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( minus_minus @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % minus_natural.abs_eq
% 0.30/0.63 thf(fact_6557_one__natural__def,axiom,
% 0.30/0.63 ( ( one_one @ code_natural )
% 0.30/0.63 = ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_natural_def
% 0.30/0.63 thf(fact_6558_divide__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( divide_divide @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % divide_natural.abs_eq
% 0.30/0.63 thf(fact_6559_plus__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( plus_plus @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % plus_natural.abs_eq
% 0.30/0.63 thf(fact_6560_zero__natural__def,axiom,
% 0.30/0.63 ( ( zero_zero @ code_natural )
% 0.30/0.63 = ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_natural_def
% 0.30/0.63 thf(fact_6561_less__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( ord_less @ nat @ Xa3 @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_natural.abs_eq
% 0.30/0.63 thf(fact_6562_less__eq__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( ord_less_eq @ nat @ Xa3 @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_eq_natural.abs_eq
% 0.30/0.63 thf(fact_6563_times__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( times_times @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % times_natural.abs_eq
% 0.30/0.63 thf(fact_6564_modulo__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % modulo_natural.abs_eq
% 0.30/0.63 thf(fact_6565_take__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % take_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6566_xor__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % xor_natural.abs_eq
% 0.30/0.63 thf(fact_6567_or__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % or_natural.abs_eq
% 0.30/0.63 thf(fact_6568_and__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % and_natural.abs_eq
% 0.30/0.63 thf(fact_6569_bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % bit_natural.abs_eq
% 0.30/0.63 thf(fact_6570_push__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % push_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6571_mask__natural_Oabs__eq,axiom,
% 0.30/0.63 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 0.30/0.63 = ( ^ [X2: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mask_natural.abs_eq
% 0.30/0.63 thf(fact_6572_drop__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se4197421643247451524op_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % drop_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6573_euclidean__size__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% 0.30/0.63
% 0.30/0.63 % euclidean_size_natural.abs_eq
% 0.30/0.63 thf(fact_6574_flip__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % flip_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6575_unset__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se2638667681897837118et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unset_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6576_division__segment__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % division_segment_natural.abs_eq
% 0.30/0.63 thf(fact_6577_set__bit__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [Xa3: nat,X: nat] :
% 0.30/0.63 ( ( bit_se5668285175392031749et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa3 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % set_bit_natural.abs_eq
% 0.30/0.63 thf(fact_6578_integer__of__natural_Oabs__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % integer_of_natural.abs_eq
% 0.30/0.63 thf(fact_6579_natural__of__integer_Oabs__eq,axiom,
% 0.30/0.63 ! [X: int] :
% 0.30/0.63 ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_integer.abs_eq
% 0.30/0.63 thf(fact_6580_Nat_Oabs__eq,axiom,
% 0.30/0.63 ! [X: int] :
% 0.30/0.63 ( ( code_Nat @ ( code_integer_of_int @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Nat.abs_eq
% 0.30/0.63 thf(fact_6581_mask__natural__def,axiom,
% 0.30/0.63 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 0.30/0.63 = ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mask_natural_def
% 0.30/0.63 thf(fact_6582_case__natural__def,axiom,
% 0.30/0.63 ! [T: $tType] :
% 0.30/0.63 ( ( code_case_natural @ T )
% 0.30/0.63 = ( ^ [F12: T,F23: code_natural > T] :
% 0.30/0.63 ( code_rec_natural @ T @ F12
% 0.30/0.63 @ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % case_natural_def
% 0.30/0.63 thf(fact_6583_natural_Osimps_I6_J,axiom,
% 0.30/0.63 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 0.30/0.63 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.30/0.63 = F1 ) ).
% 0.30/0.63
% 0.30/0.63 % natural.simps(6)
% 0.30/0.63 thf(fact_6584_rec__natural__def,axiom,
% 0.30/0.63 ! [T: $tType] :
% 0.30/0.63 ( ( code_rec_natural @ T )
% 0.30/0.63 = ( ^ [F12: T,F23: code_natural > T > T,X2: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % rec_natural_def
% 0.30/0.63 thf(fact_6585_Nat__def,axiom,
% 0.30/0.63 ( code_Nat
% 0.30/0.63 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % Nat_def
% 0.30/0.63 thf(fact_6586_natural__of__integer__def,axiom,
% 0.30/0.63 ( code_n4118661773612635043nteger
% 0.30/0.63 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural_of_integer_def
% 0.30/0.63 thf(fact_6587_division__segment__natural__def,axiom,
% 0.30/0.63 ( ( euclid7384307370059645450egment @ code_natural )
% 0.30/0.63 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % division_segment_natural_def
% 0.30/0.63 thf(fact_6588_integer__of__natural__def,axiom,
% 0.30/0.63 ( code_i5400310926305786745atural
% 0.30/0.63 = ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % integer_of_natural_def
% 0.30/0.63 thf(fact_6589_set__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se5668285175392031749et_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % set_bit_natural_def
% 0.30/0.63 thf(fact_6590_unset__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se2638667681897837118et_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unset_bit_natural_def
% 0.30/0.63 thf(fact_6591_take__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se2584673776208193580ke_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % take_bit_natural_def
% 0.30/0.63 thf(fact_6592_push__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se4730199178511100633sh_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % push_bit_natural_def
% 0.30/0.63 thf(fact_6593_drop__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se4197421643247451524op_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % drop_bit_natural_def
% 0.30/0.63 thf(fact_6594_flip__bit__natural__def,axiom,
% 0.30/0.63 ( ( bit_se8732182000553998342ip_bit @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % flip_bit_natural_def
% 0.30/0.63 thf(fact_6595_and__natural__def,axiom,
% 0.30/0.63 ( ( bit_se5824344872417868541ns_and @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % and_natural_def
% 0.30/0.63 thf(fact_6596_xor__natural__def,axiom,
% 0.30/0.63 ( ( bit_se5824344971392196577ns_xor @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % xor_natural_def
% 0.30/0.63 thf(fact_6597_plus__natural__def,axiom,
% 0.30/0.63 ( ( plus_plus @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % plus_natural_def
% 0.30/0.63 thf(fact_6598_minus__natural__def,axiom,
% 0.30/0.63 ( ( minus_minus @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % minus_natural_def
% 0.30/0.63 thf(fact_6599_divide__natural__def,axiom,
% 0.30/0.63 ( ( divide_divide @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % divide_natural_def
% 0.30/0.63 thf(fact_6600_times__natural__def,axiom,
% 0.30/0.63 ( ( times_times @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % times_natural_def
% 0.30/0.63 thf(fact_6601_modulo__natural__def,axiom,
% 0.30/0.63 ( ( modulo_modulo @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % modulo_natural_def
% 0.30/0.63 thf(fact_6602_or__natural__def,axiom,
% 0.30/0.63 ( ( bit_se1065995026697491101ons_or @ code_natural )
% 0.30/0.63 = ( 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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % or_natural_def
% 0.30/0.63 thf(fact_6603_Code__Numeral_OSuc__def,axiom,
% 0.30/0.63 ( code_Suc
% 0.30/0.63 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 0.30/0.63
% 0.30/0.63 % Code_Numeral.Suc_def
% 0.30/0.63 thf(fact_6604_cr__natural__def,axiom,
% 0.30/0.63 ( code_cr_natural
% 0.30/0.63 = ( ^ [X2: nat,Y6: code_natural] :
% 0.30/0.63 ( X2
% 0.30/0.63 = ( code_nat_of_natural @ Y6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % cr_natural_def
% 0.30/0.63 thf(fact_6605_natural_Oinject,axiom,
% 0.30/0.63 ! [Natural: code_natural,Natural2: code_natural] :
% 0.30/0.63 ( ( ( code_Suc @ Natural )
% 0.30/0.63 = ( code_Suc @ Natural2 ) )
% 0.30/0.63 = ( Natural = Natural2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.inject
% 0.30/0.63 thf(fact_6606_natural_Osimps_I7_J,axiom,
% 0.30/0.63 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 0.30/0.63 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.30/0.63 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.simps(7)
% 0.30/0.63 thf(fact_6607_natural_Osimps_I5_J,axiom,
% 0.30/0.63 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 0.30/0.63 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.30/0.63 = ( F22 @ Natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.simps(5)
% 0.30/0.63 thf(fact_6608_Suc_Orep__eq,axiom,
% 0.30/0.63 ! [X: code_natural] :
% 0.30/0.63 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 0.30/0.63 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Suc.rep_eq
% 0.30/0.63 thf(fact_6609_natural_Oexhaust,axiom,
% 0.30/0.63 ! [Y3: code_natural] :
% 0.30/0.63 ( ( Y3
% 0.30/0.63 != ( zero_zero @ code_natural ) )
% 0.30/0.63 => ~ ! [Natural3: code_natural] :
% 0.30/0.63 ( Y3
% 0.30/0.63 != ( code_Suc @ Natural3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.exhaust
% 0.30/0.63 thf(fact_6610_natural_Odistinct_I1_J,axiom,
% 0.30/0.63 ! [Natural2: code_natural] :
% 0.30/0.63 ( ( zero_zero @ code_natural )
% 0.30/0.63 != ( code_Suc @ Natural2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.distinct(1)
% 0.30/0.63 thf(fact_6611_natural_Odistinct_I2_J,axiom,
% 0.30/0.63 ! [Natural4: code_natural] :
% 0.30/0.63 ( ( code_Suc @ Natural4 )
% 0.30/0.63 != ( zero_zero @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % natural.distinct(2)
% 0.30/0.63 thf(fact_6612_Suc_Oabs__eq,axiom,
% 0.30/0.63 ! [X: nat] :
% 0.30/0.63 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 0.30/0.63 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Suc.abs_eq
% 0.30/0.63 thf(fact_6613_Suc__natural__minus__one,axiom,
% 0.30/0.63 ! [N: code_natural] :
% 0.30/0.63 ( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
% 0.30/0.63 = N ) ).
% 0.30/0.63
% 0.30/0.63 % Suc_natural_minus_one
% 0.30/0.63 thf(fact_6614_Nat_Otransfer,axiom,
% 0.30/0.63 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% 0.30/0.63
% 0.30/0.63 % Nat.transfer
% 0.30/0.63 thf(fact_6615_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.30/0.63 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 0.30/0.63 ( ( 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 ) ) ) )
% 0.30/0.63 => ( ( ( ord_less @ nat @ B2 @ A3 )
% 0.30/0.63 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 0.30/0.63 = Acc2 ) )
% 0.30/0.63 & ( ~ ( ord_less @ nat @ B2 @ A3 )
% 0.30/0.63 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 0.30/0.63 = ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fold_atLeastAtMost_nat.psimps
% 0.30/0.63 thf(fact_6616_Suc_Otransfer,axiom,
% 0.30/0.63 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 0.30/0.63
% 0.30/0.63 % Suc.transfer
% 0.30/0.63 thf(fact_6617_natural_Opcr__cr__eq,axiom,
% 0.30/0.63 code_pcr_natural = code_cr_natural ).
% 0.30/0.63
% 0.30/0.63 % natural.pcr_cr_eq
% 0.30/0.63 thf(fact_6618_natural_Oid__abs__transfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ code_pcr_natural
% 0.30/0.63 @ ^ [X2: nat] : X2
% 0.30/0.63 @ code_natural_of_nat ) ).
% 0.30/0.63
% 0.30/0.63 % natural.id_abs_transfer
% 0.30/0.63 thf(fact_6619_less__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 0.30/0.63 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.63 @ ( ord_less @ nat )
% 0.30/0.63 @ ( ord_less @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_natural.transfer
% 0.30/0.63 thf(fact_6620_plus__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % plus_natural.transfer
% 0.30/0.63 thf(fact_6621_minus__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % minus_natural.transfer
% 0.30/0.63 thf(fact_6622_zero__natural_Otransfer,axiom,
% 0.30/0.63 code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% 0.30/0.63
% 0.30/0.63 % zero_natural.transfer
% 0.30/0.63 thf(fact_6623_one__natural_Otransfer,axiom,
% 0.30/0.63 code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% 0.30/0.63
% 0.30/0.63 % one_natural.transfer
% 0.30/0.63 thf(fact_6624_natural_Obi__total,axiom,
% 0.30/0.63 bi_total @ nat @ code_natural @ code_pcr_natural ).
% 0.30/0.63
% 0.30/0.63 % natural.bi_total
% 0.30/0.63 thf(fact_6625_natural_Orep__transfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ^ [X2: nat] : X2
% 0.30/0.63 @ code_nat_of_natural ) ).
% 0.30/0.63
% 0.30/0.63 % natural.rep_transfer
% 0.30/0.63 thf(fact_6626_less__eq__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 0.30/0.63 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.63 @ ( ord_less_eq @ nat )
% 0.30/0.63 @ ( ord_less_eq @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_eq_natural.transfer
% 0.30/0.63 thf(fact_6627_divide__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % divide_natural.transfer
% 0.30/0.63 thf(fact_6628_times__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % times_natural.transfer
% 0.30/0.63 thf(fact_6629_modulo__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % modulo_natural.transfer
% 0.30/0.63 thf(fact_6630_fold__atLeastAtMost__nat_Opinduct,axiom,
% 0.30/0.63 ! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
% 0.30/0.63 ( ( 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 ) ) ) )
% 0.30/0.63 => ( ! [F3: nat > A > A,A8: nat,B5: nat,Acc3: A] :
% 0.30/0.63 ( ( 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 ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A8 @ ( product_Pair @ nat @ A @ B5 @ Acc3 ) ) ) )
% 0.30/0.63 => ( ( ~ ( ord_less @ nat @ B5 @ A8 )
% 0.30/0.63 => ( P @ F3 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B5 @ ( F3 @ A8 @ Acc3 ) ) )
% 0.30/0.63 => ( P @ F3 @ A8 @ B5 @ Acc3 ) ) )
% 0.30/0.63 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fold_atLeastAtMost_nat.pinduct
% 0.30/0.63 thf(fact_6631_take__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se2584673776208193580ke_bit @ nat )
% 0.30/0.63 @ ( bit_se2584673776208193580ke_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % take_bit_natural.transfer
% 0.30/0.63 thf(fact_6632_or__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % or_natural.transfer
% 0.30/0.63 thf(fact_6633_xor__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % xor_natural.transfer
% 0.30/0.63 thf(fact_6634_and__natural_Otransfer,axiom,
% 0.30/0.63 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 ) ).
% 0.30/0.63
% 0.30/0.63 % and_natural.transfer
% 0.30/0.63 thf(fact_6635_bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
% 0.30/0.63 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 0.30/0.63 @ ( bit_se5641148757651400278ts_bit @ nat )
% 0.30/0.63 @ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % bit_natural.transfer
% 0.30/0.63 thf(fact_6636_mask__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ code_pcr_natural
% 0.30/0.63 @ ( bit_se2239418461657761734s_mask @ nat )
% 0.30/0.63 @ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % mask_natural.transfer
% 0.30/0.63 thf(fact_6637_push__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se4730199178511100633sh_bit @ nat )
% 0.30/0.63 @ ( bit_se4730199178511100633sh_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % push_bit_natural.transfer
% 0.30/0.63 thf(fact_6638_drop__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se4197421643247451524op_bit @ nat )
% 0.30/0.63 @ ( bit_se4197421643247451524op_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % drop_bit_natural.transfer
% 0.30/0.63 thf(fact_6639_set__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se5668285175392031749et_bit @ nat )
% 0.30/0.63 @ ( bit_se5668285175392031749et_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % set_bit_natural.transfer
% 0.30/0.63 thf(fact_6640_division__segment__natural_Otransfer,axiom,
% 0.30/0.63 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% 0.30/0.63
% 0.30/0.63 % division_segment_natural.transfer
% 0.30/0.63 thf(fact_6641_unset__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se2638667681897837118et_bit @ nat )
% 0.30/0.63 @ ( bit_se2638667681897837118et_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % unset_bit_natural.transfer
% 0.30/0.63 thf(fact_6642_flip__bit__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.30/0.63 @ ( bit_se8732182000553998342ip_bit @ nat )
% 0.30/0.63 @ ( bit_se8732182000553998342ip_bit @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % flip_bit_natural.transfer
% 0.30/0.63 thf(fact_6643_euclidean__size__natural_Otransfer,axiom,
% 0.30/0.63 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 0.30/0.63 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.63 @ ( euclid6346220572633701492n_size @ nat )
% 0.30/0.63 @ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% 0.30/0.63
% 0.30/0.63 % euclidean_size_natural.transfer
% 0.30/0.63 thf(fact_6644_integer__of__natural_Otransfer,axiom,
% 0.30/0.63 bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% 0.30/0.63
% 0.30/0.63 % integer_of_natural.transfer
% 0.30/0.63 thf(fact_6645_natural__of__integer_Otransfer,axiom,
% 0.30/0.63 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% 0.30/0.63
% 0.30/0.63 % natural_of_integer.transfer
% 0.30/0.63 thf(fact_6646_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.30/0.63 ! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
% 0.30/0.63 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
% 0.30/0.63 = Y3 )
% 0.30/0.63 => ( ( 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) )
% 0.30/0.63 => ~ ( ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
% 0.30/0.63 => ( Y3 = Xc ) )
% 0.30/0.63 & ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) )
% 0.30/0.63 => ~ ( 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fold_atLeastAtMost_nat.pelims
% 0.30/0.63 thf(fact_6647_prod_Ocomm__monoid__list__set__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comm_monoid_mult @ A )
% 0.30/0.63 => ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % prod.comm_monoid_list_set_axioms
% 0.30/0.63 thf(fact_6648_prod_Oreindex__bij__betw__not__neutral,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.63 ( ( comm_monoid_mult @ A )
% 0.30/0.63 => ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
% 0.30/0.63 ( ( finite_finite2 @ B @ S5 )
% 0.30/0.63 => ( ( finite_finite2 @ C @ T5 )
% 0.30/0.63 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.63 => ( ! [A8: B] :
% 0.30/0.63 ( ( member @ B @ A8 @ S5 )
% 0.30/0.63 => ( ( G @ ( H2 @ A8 ) )
% 0.30/0.63 = ( one_one @ A ) ) )
% 0.30/0.63 => ( ! [B5: C] :
% 0.30/0.63 ( ( member @ C @ B5 @ T5 )
% 0.30/0.63 => ( ( G @ B5 )
% 0.30/0.63 = ( one_one @ A ) ) )
% 0.30/0.63 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.30/0.63 @ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
% 0.30/0.63 @ S3 )
% 0.30/0.63 = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % prod.reindex_bij_betw_not_neutral
% 0.30/0.63 thf(fact_6649_bij__betw__of__nat,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semiring_char_0 @ A )
% 0.30/0.63 => ! [N6: set @ nat,A6: set @ A] :
% 0.30/0.63 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A6 )
% 0.30/0.63 = ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
% 0.30/0.63 = A6 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_of_nat
% 0.30/0.63 thf(fact_6650_bij__betw__Suc,axiom,
% 0.30/0.63 ! [M7: set @ nat,N6: set @ nat] :
% 0.30/0.63 ( ( bij_betw @ nat @ nat @ suc @ M7 @ N6 )
% 0.30/0.63 = ( ( image @ nat @ nat @ suc @ M7 )
% 0.30/0.63 = N6 ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_Suc
% 0.30/0.63 thf(fact_6651_bijection__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( hilbert_bijection @ A )
% 0.30/0.63 = ( ^ [F2: A > A] : ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bijection_def
% 0.30/0.63 thf(fact_6652_bijection_Ointro,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( hilbert_bijection @ A @ F ) ) ).
% 0.30/0.63
% 0.30/0.63 % bijection.intro
% 0.30/0.63 thf(fact_6653_bijection_Obij,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A] :
% 0.30/0.63 ( ( hilbert_bijection @ A @ F )
% 0.30/0.63 => ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bijection.bij
% 0.30/0.63 thf(fact_6654_bij__betw__funpow,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A,S3: set @ A,N: nat] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ S3 @ S3 )
% 0.30/0.63 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ S3 @ S3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_funpow
% 0.30/0.63 thf(fact_6655_bij__fn,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_fn
% 0.30/0.63 thf(fact_6656_bij__uminus,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ab_group_add @ A )
% 0.30/0.63 => ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_uminus
% 0.30/0.63 thf(fact_6657_bij__imp__bij__inv,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_imp_bij_inv
% 0.30/0.63 thf(fact_6658_bij__inv__eq__iff,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType,P5: A > B,X: A,Y3: B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ P5 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( X
% 0.30/0.63 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P5 @ Y3 ) )
% 0.30/0.63 = ( ( P5 @ X )
% 0.30/0.63 = Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_inv_eq_iff
% 0.30/0.63 thf(fact_6659_inv__inv__eq,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) )
% 0.30/0.63 = F ) ) ).
% 0.30/0.63
% 0.30/0.63 % inv_inv_eq
% 0.30/0.63 thf(fact_6660_bij__betw__inv__into__right,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType,F: A > B,A6: set @ A,A11: set @ B,A4: B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 0.30/0.63 => ( ( member @ B @ A4 @ A11 )
% 0.30/0.63 => ( ( F @ ( hilbert_inv_into @ A @ B @ A6 @ F @ A4 ) )
% 0.30/0.63 = A4 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_inv_into_right
% 0.30/0.63 thf(fact_6661_bij__betw__inv__into__left,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 0.30/0.63 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.63 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ A3 ) )
% 0.30/0.63 = A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_inv_into_left
% 0.30/0.63 thf(fact_6662_inv__into__inv__into__eq,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 0.30/0.63 => ( ( member @ A @ A3 @ A6 )
% 0.30/0.63 => ( ( hilbert_inv_into @ B @ A @ A11 @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ A3 )
% 0.30/0.63 = ( F @ A3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inv_into_inv_into_eq
% 0.30/0.63 thf(fact_6663_bij__betw__inv__into,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,B7: set @ B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ A6 @ B7 )
% 0.30/0.63 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B7 @ A6 ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_inv_into
% 0.30/0.63 thf(fact_6664_bij__betw__inv__into__subset,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,B7: set @ A,B13: set @ B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 0.30/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 0.30/0.63 => ( ( ( image @ A @ B @ F @ B7 )
% 0.30/0.63 = B13 )
% 0.30/0.63 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B13 @ B7 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_betw_inv_into_subset
% 0.30/0.63 thf(fact_6665_bij__image__Compl__eq,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.30/0.63 = ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_image_Compl_eq
% 0.30/0.63 thf(fact_6666_o__inv__distrib,axiom,
% 0.30/0.63 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > A] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F @ G ) )
% 0.30/0.63 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % o_inv_distrib
% 0.30/0.63 thf(fact_6667_inv__fn,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 0.30/0.63 = ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inv_fn
% 0.30/0.63 thf(fact_6668_mono__inv,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType] :
% 0.30/0.63 ( ( ( linorder @ A )
% 0.30/0.63 & ( linorder @ B ) )
% 0.30/0.63 => ! [F: A > B] :
% 0.30/0.63 ( ( order_mono @ A @ B @ F )
% 0.30/0.63 => ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mono_inv
% 0.30/0.63 thf(fact_6669_bijection_Obij__inv,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A] :
% 0.30/0.63 ( ( hilbert_bijection @ A @ F )
% 0.30/0.63 => ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bijection.bij_inv
% 0.30/0.63 thf(fact_6670_bij__image__Collect__eq,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType,F: A > B,P: A > $o] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( image @ A @ B @ F @ ( collect @ A @ P ) )
% 0.30/0.63 = ( collect @ B
% 0.30/0.63 @ ^ [Y6: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_image_Collect_eq
% 0.30/0.63 thf(fact_6671_bij__int__decode,axiom,
% 0.30/0.63 bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_int_decode
% 0.30/0.63 thf(fact_6672_bij__int__encode,axiom,
% 0.30/0.63 bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_int_encode
% 0.30/0.63 thf(fact_6673_infinite__imp__bij__betw2,axiom,
% 0.30/0.63 ! [A: $tType,A6: set @ A,A3: A] :
% 0.30/0.63 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % infinite_imp_bij_betw2
% 0.30/0.63 thf(fact_6674_infinite__imp__bij__betw,axiom,
% 0.30/0.63 ! [A: $tType,A6: set @ A,A3: A] :
% 0.30/0.63 ( ~ ( finite_finite2 @ A @ A6 )
% 0.30/0.63 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % infinite_imp_bij_betw
% 0.30/0.63 thf(fact_6675_ex__bij__betw__finite__nat,axiom,
% 0.30/0.63 ! [A: $tType,M7: set @ A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ M7 )
% 0.30/0.63 => ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ex_bij_betw_finite_nat
% 0.30/0.63 thf(fact_6676_ex__bij__betw__nat__finite,axiom,
% 0.30/0.63 ! [A: $tType,M7: set @ A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ M7 )
% 0.30/0.63 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ex_bij_betw_nat_finite
% 0.30/0.63 thf(fact_6677_mono__bij__Inf,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType] :
% 0.30/0.63 ( ( ( comple5582772986160207858norder @ A )
% 0.30/0.63 & ( comple5582772986160207858norder @ B ) )
% 0.30/0.63 => ! [F: A > B,A6: set @ A] :
% 0.30/0.63 ( ( order_mono @ A @ B @ F )
% 0.30/0.63 => ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( F @ ( complete_Inf_Inf @ A @ A6 ) )
% 0.30/0.63 = ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mono_bij_Inf
% 0.30/0.63 thf(fact_6678_bij__prod__decode,axiom,
% 0.30/0.63 bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_prod_decode
% 0.30/0.63 thf(fact_6679_inj__imp__bij__betw__inv,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B,M7: set @ A] :
% 0.30/0.63 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ M7 ) @ M7 ) ) ).
% 0.30/0.63
% 0.30/0.63 % inj_imp_bij_betw_inv
% 0.30/0.63 thf(fact_6680_ex__bij__betw__nat__finite__1,axiom,
% 0.30/0.63 ! [A: $tType,M7: set @ A] :
% 0.30/0.63 ( ( finite_finite2 @ A @ M7 )
% 0.30/0.63 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ex_bij_betw_nat_finite_1
% 0.30/0.63 thf(fact_6681_bij__list__decode,axiom,
% 0.30/0.63 bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_list_decode
% 0.30/0.63 thf(fact_6682_bij__list__encode,axiom,
% 0.30/0.63 bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_list_encode
% 0.30/0.63 thf(fact_6683_bij__prod__encode,axiom,
% 0.30/0.63 bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_prod_encode
% 0.30/0.63 thf(fact_6684_bij__vimage__eq__inv__image,axiom,
% 0.30/0.63 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 0.30/0.63 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.63 => ( ( vimage @ A @ B @ F @ A6 )
% 0.30/0.63 = ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ A6 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % bij_vimage_eq_inv_image
% 0.30/0.63 thf(fact_6685_fn__o__inv__fn__is__id,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
% 0.30/0.63 = ( ^ [X2: A] : X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fn_o_inv_fn_is_id
% 0.30/0.63 thf(fact_6686_inv__fn__o__fn__is__id,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A,N: nat] :
% 0.30/0.63 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 0.30/0.63 = ( ^ [X2: A] : X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inv_fn_o_fn_is_id
% 0.30/0.63 thf(fact_6687_sum_Oreindex__bij__betw__not__neutral,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.63 ( ( comm_monoid_add @ A )
% 0.30/0.63 => ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
% 0.30/0.63 ( ( finite_finite2 @ B @ S5 )
% 0.30/0.63 => ( ( finite_finite2 @ C @ T5 )
% 0.30/0.63 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.30/0.63 => ( ! [A8: B] :
% 0.30/0.63 ( ( member @ B @ A8 @ S5 )
% 0.30/0.63 => ( ( G @ ( H2 @ A8 ) )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ! [B5: C] :
% 0.30/0.63 ( ( member @ C @ B5 @ T5 )
% 0.30/0.63 => ( ( G @ B5 )
% 0.30/0.63 = ( zero_zero @ A ) ) )
% 0.30/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.30/0.63 @ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
% 0.30/0.63 @ S3 )
% 0.30/0.63 = ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.reindex_bij_betw_not_neutral
% 0.30/0.63 thf(fact_6688_sum_Ocomm__monoid__list__set__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( comm_monoid_add @ A )
% 0.30/0.63 => ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.comm_monoid_list_set_axioms
% 0.30/0.63 thf(fact_6689_prod_H__def,axiom,
% 0.30/0.63 ! [C: $tType,A: $tType] :
% 0.30/0.63 ( ( comm_monoid_mult @ A )
% 0.30/0.63 => ( ( groups1962203154675924110t_prod @ C @ A )
% 0.30/0.63 = ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % prod'_def
% 0.30/0.63 thf(fact_6690_natLeq__on__wo__rel,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( bNF_Wellorder_wo_rel @ nat
% 0.30/0.63 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.63 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.63 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.63 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.63 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.63 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % natLeq_on_wo_rel
% 0.30/0.63 thf(fact_6691_sum_H__def,axiom,
% 0.30/0.63 ! [C: $tType,A: $tType] :
% 0.30/0.63 ( ( comm_monoid_add @ A )
% 0.30/0.63 => ( ( groups1027152243600224163dd_sum @ C @ A )
% 0.30/0.63 = ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum'_def
% 0.30/0.63 thf(fact_6692_Heap_Osize__gen__o__map,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 0.30/0.63 ( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
% 0.30/0.63 = ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.size_gen_o_map
% 0.30/0.63 thf(fact_6693_max_Osemilattice__order__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( linorder @ A )
% 0.30/0.63 => ( semilattice_order @ A @ ( ord_max @ A )
% 0.30/0.63 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.63 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % max.semilattice_order_axioms
% 0.30/0.63 thf(fact_6694_Heap_Oinj__map,axiom,
% 0.30/0.63 ! [B: $tType,A: $tType,F: A > B] :
% 0.30/0.63 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.63 => ( inj_on @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Time_Heap @ A ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.inj_map
% 0.30/0.63 thf(fact_6695_semilattice__order__set_Oaxioms_I1_J,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_set.axioms(1)
% 0.30/0.63 thf(fact_6696_Heap_Omap__comp,axiom,
% 0.30/0.63 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
% 0.30/0.63 ( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 0.30/0.63 = ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.map_comp
% 0.30/0.63 thf(fact_6697_semilattice__neutr__order_Oaxioms_I2_J,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 0.30/0.63 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_order.axioms(2)
% 0.30/0.63 thf(fact_6698_Heap_Omap__ident,axiom,
% 0.30/0.63 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 0.30/0.63 ( ( heap_Time_map_Heap @ A @ A
% 0.30/0.63 @ ^ [X2: A] : X2
% 0.30/0.63 @ T2 )
% 0.30/0.63 = T2 ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.map_ident
% 0.30/0.63 thf(fact_6699_semilattice__order_Omono,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A,D2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ C2 )
% 0.30/0.63 => ( ( Less_eq @ B2 @ D2 )
% 0.30/0.63 => ( Less_eq @ ( F @ A3 @ B2 ) @ ( F @ C2 @ D2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.mono
% 0.30/0.63 thf(fact_6700_semilattice__order_OorderE,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 => ( A3
% 0.30/0.63 = ( F @ A3 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.orderE
% 0.30/0.63 thf(fact_6701_semilattice__order_OorderI,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( A3
% 0.30/0.63 = ( F @ A3 @ B2 ) )
% 0.30/0.63 => ( Less_eq @ A3 @ B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.orderI
% 0.30/0.63 thf(fact_6702_semilattice__order_Oabsorb1,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 => ( ( F @ A3 @ B2 )
% 0.30/0.63 = A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb1
% 0.30/0.63 thf(fact_6703_semilattice__order_Oabsorb2,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ B2 @ A3 )
% 0.30/0.63 => ( ( F @ A3 @ B2 )
% 0.30/0.63 = B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb2
% 0.30/0.63 thf(fact_6704_semilattice__order_Oabsorb3,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ A3 @ B2 )
% 0.30/0.63 => ( ( F @ A3 @ B2 )
% 0.30/0.63 = A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb3
% 0.30/0.63 thf(fact_6705_semilattice__order_Oabsorb4,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ B2 @ A3 )
% 0.30/0.63 => ( ( F @ A3 @ B2 )
% 0.30/0.63 = B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb4
% 0.30/0.63 thf(fact_6706_semilattice__order_OboundedE,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
% 0.30/0.63 => ~ ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 => ~ ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.boundedE
% 0.30/0.63 thf(fact_6707_semilattice__order_OboundedI,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 => ( ( Less_eq @ A3 @ C2 )
% 0.30/0.63 => ( Less_eq @ A3 @ ( F @ B2 @ C2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.boundedI
% 0.30/0.63 thf(fact_6708_semilattice__order_Oorder__iff,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 = ( A3
% 0.30/0.63 = ( F @ A3 @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.order_iff
% 0.30/0.63 thf(fact_6709_semilattice__order_Ocobounded1,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( Less_eq @ ( F @ A3 @ B2 ) @ A3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.cobounded1
% 0.30/0.63 thf(fact_6710_semilattice__order_Ocobounded2,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( Less_eq @ ( F @ A3 @ B2 ) @ B2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.cobounded2
% 0.30/0.63 thf(fact_6711_semilattice__order_Oabsorb__iff1,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 = ( ( F @ A3 @ B2 )
% 0.30/0.63 = A3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb_iff1
% 0.30/0.63 thf(fact_6712_semilattice__order_Oabsorb__iff2,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ B2 @ A3 )
% 0.30/0.63 = ( ( F @ A3 @ B2 )
% 0.30/0.63 = B2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.absorb_iff2
% 0.30/0.63 thf(fact_6713_semilattice__order_Obounded__iff,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
% 0.30/0.63 = ( ( Less_eq @ A3 @ B2 )
% 0.30/0.63 & ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.bounded_iff
% 0.30/0.63 thf(fact_6714_semilattice__order_OcoboundedI1,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ A3 @ C2 )
% 0.30/0.63 => ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.coboundedI1
% 0.30/0.63 thf(fact_6715_semilattice__order_OcoboundedI2,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less_eq @ B2 @ C2 )
% 0.30/0.63 => ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.coboundedI2
% 0.30/0.63 thf(fact_6716_semilattice__order_Ostrict__boundedE,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ A3 @ ( F @ B2 @ C2 ) )
% 0.30/0.63 => ~ ( ( Less @ A3 @ B2 )
% 0.30/0.63 => ~ ( Less @ A3 @ C2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.strict_boundedE
% 0.30/0.63 thf(fact_6717_semilattice__order_Ostrict__order__iff,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ A3 @ B2 )
% 0.30/0.63 = ( ( A3
% 0.30/0.63 = ( F @ A3 @ B2 ) )
% 0.30/0.63 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.strict_order_iff
% 0.30/0.63 thf(fact_6718_semilattice__order_Ostrict__coboundedI1,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ A3 @ C2 )
% 0.30/0.63 => ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.strict_coboundedI1
% 0.30/0.63 thf(fact_6719_semilattice__order_Ostrict__coboundedI2,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( Less @ B2 @ C2 )
% 0.30/0.63 => ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order.strict_coboundedI2
% 0.30/0.63 thf(fact_6720_Heap_Omap__id,axiom,
% 0.30/0.63 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 0.30/0.63 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
% 0.30/0.63 = T2 ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.map_id
% 0.30/0.63 thf(fact_6721_Heap_Omap__id0,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
% 0.30/0.63 = ( id @ ( heap_Time_Heap @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Heap.map_id0
% 0.30/0.63 thf(fact_6722_semilattice__order__set__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( lattic4895041142388067077er_set @ A )
% 0.30/0.63 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.63 ( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
% 0.30/0.63 & ( lattic149705377957585745ce_set @ A @ F2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_set_def
% 0.30/0.63 thf(fact_6723_semilattice__order__set_Ointro,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.63 => ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_order_set.intro
% 0.30/0.63 thf(fact_6724_semilattice__neutr__order__def,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semila1105856199041335345_order @ A )
% 0.30/0.63 = ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.63 ( ( semilattice_neutr @ A @ F2 @ Z7 )
% 0.30/0.63 & ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_order_def
% 0.30/0.63 thf(fact_6725_semilattice__neutr__order_Ointro,axiom,
% 0.30/0.63 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.63 ( ( semilattice_neutr @ A @ F @ Z3 )
% 0.30/0.63 => ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.63 => ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % semilattice_neutr_order.intro
% 0.30/0.63 thf(fact_6726_inf_Osemilattice__order__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semilattice_inf @ A )
% 0.30/0.63 => ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inf.semilattice_order_axioms
% 0.30/0.63 thf(fact_6727_min_Osemilattice__order__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( linorder @ A )
% 0.30/0.63 => ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % min.semilattice_order_axioms
% 0.30/0.63 thf(fact_6728_sup_Osemilattice__order__axioms,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( semilattice_sup @ A )
% 0.30/0.63 => ( semilattice_order @ A @ ( sup_sup @ A )
% 0.30/0.63 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.63 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sup.semilattice_order_axioms
% 0.30/0.63 thf(fact_6729_lexordp__eq_Osimps,axiom,
% 0.30/0.63 ! [A: $tType] :
% 0.30/0.63 ( ( ord @ A )
% 0.30/0.63 => ( ( ord_lexordp_eq @ A )
% 0.30/0.63 = ( ^ [A12: list @ A,A23: list @ A] :
% 0.30/0.63 ( ? [Ys3: list @ A] :
% 0.30/0.63 ( ( A12
% 0.30/0.63 = ( nil @ A ) )
% 0.30/0.63 & ( A23 = Ys3 ) )
% 0.30/0.63 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.63 ( ( A12
% 0.30/0.63 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.64 & ( A23
% 0.30/0.64 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.64 & ( ord_less @ A @ X2 @ Y6 ) )
% 0.30/0.64 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 0.30/0.64 ( ( A12
% 0.30/0.64 = ( cons @ A @ X2 @ Xs3 ) )
% 0.30/0.64 & ( A23
% 0.30/0.64 = ( cons @ A @ Y6 @ Ys3 ) )
% 0.30/0.64 & ~ ( ord_less @ A @ X2 @ Y6 )
% 0.30/0.64 & ~ ( ord_less @ A @ Y6 @ X2 )
% 0.30/0.64 & ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lexordp_eq.simps
% 0.30/0.64 thf(fact_6730_lexordp__eq__simps_I4_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ord @ A )
% 0.30/0.64 => ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
% 0.30/0.64 ( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
% 0.30/0.64 = ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.64 | ( ~ ( ord_less @ A @ Y3 @ X )
% 0.30/0.64 & ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lexordp_eq_simps(4)
% 0.30/0.64 thf(fact_6731_lexordp__eq_OCons,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ord @ A )
% 0.30/0.64 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.64 ( ( ord_less @ A @ X @ Y3 )
% 0.30/0.64 => ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lexordp_eq.Cons
% 0.30/0.64 thf(fact_6732_lexordp__eq_OCons__eq,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ord @ A )
% 0.30/0.64 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 0.30/0.64 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.30/0.64 => ( ~ ( ord_less @ A @ Y3 @ X )
% 0.30/0.64 => ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 0.30/0.64 => ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lexordp_eq.Cons_eq
% 0.30/0.64 thf(fact_6733_lexordp__eq_Ocases,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ord @ A )
% 0.30/0.64 => ! [A1: list @ A,A22: list @ A] :
% 0.30/0.64 ( ( ord_lexordp_eq @ A @ A1 @ A22 )
% 0.30/0.64 => ( ( A1
% 0.30/0.64 != ( nil @ A ) )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ? [Xs2: list @ A] :
% 0.30/0.64 ( A1
% 0.30/0.64 = ( cons @ A @ X3 @ Xs2 ) )
% 0.30/0.64 => ! [Y5: A] :
% 0.30/0.64 ( ? [Ys4: list @ A] :
% 0.30/0.64 ( A22
% 0.30/0.64 = ( cons @ A @ Y5 @ Ys4 ) )
% 0.30/0.64 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 0.30/0.64 => ~ ! [X3: A,Y5: A,Xs2: list @ A] :
% 0.30/0.64 ( ( A1
% 0.30/0.64 = ( cons @ A @ X3 @ Xs2 ) )
% 0.30/0.64 => ! [Ys4: list @ A] :
% 0.30/0.64 ( ( A22
% 0.30/0.64 = ( cons @ A @ Y5 @ Ys4 ) )
% 0.30/0.64 => ( ~ ( ord_less @ A @ X3 @ Y5 )
% 0.30/0.64 => ( ~ ( ord_less @ A @ Y5 @ X3 )
% 0.30/0.64 => ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lexordp_eq.cases
% 0.30/0.64 thf(fact_6734_bdd__below__primitive__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( ( condit1013018076250108175_below @ A )
% 0.30/0.64 = ( condit16957441358409770ng_bdd @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bdd_below_primitive_def
% 0.30/0.64 thf(fact_6735_Quotient__natural,axiom,
% 0.30/0.64 ( quotient @ nat @ code_natural
% 0.30/0.64 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 0.30/0.64 @ code_natural_of_nat
% 0.30/0.64 @ code_nat_of_natural
% 0.30/0.64 @ code_cr_natural ) ).
% 0.30/0.64
% 0.30/0.64 % Quotient_natural
% 0.30/0.64 thf(fact_6736_bdd__above__primitive__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( ( condit941137186595557371_above @ A )
% 0.30/0.64 = ( condit16957441358409770ng_bdd @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bdd_above_primitive_def
% 0.30/0.64 thf(fact_6737_card_Ofolding__on__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
% 0.30/0.64 @ ^ [Uu: A] : suc ) ).
% 0.30/0.64
% 0.30/0.64 % card.folding_on_axioms
% 0.30/0.64 thf(fact_6738_bdd__below_Opreordering__bdd__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( condit622319405099724424ng_bdd @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bdd_below.preordering_bdd_axioms
% 0.30/0.64 thf(fact_6739_bdd__above_Opreordering__bdd__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bdd_above.preordering_bdd_axioms
% 0.30/0.64 thf(fact_6740_Quotient__integer,axiom,
% 0.30/0.64 ( quotient @ int @ code_integer
% 0.30/0.64 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 0.30/0.64 @ code_integer_of_int
% 0.30/0.64 @ code_int_of_integer
% 0.30/0.64 @ code_cr_integer ) ).
% 0.30/0.64
% 0.30/0.64 % Quotient_integer
% 0.30/0.64 thf(fact_6741_type__definition__integer,axiom,
% 0.30/0.64 type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % type_definition_integer
% 0.30/0.64 thf(fact_6742_finite__subset__Union__chain,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ A,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 0.30/0.64 ( ( finite_finite2 @ A @ A6 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
% 0.30/0.64 => ( ( B12
% 0.30/0.64 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 0.30/0.64 => ~ ! [B9: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ B9 @ B12 )
% 0.30/0.64 => ~ ( ord_less_eq @ ( set @ A ) @ A6 @ B9 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % finite_subset_Union_chain
% 0.30/0.64 thf(fact_6743_wait__def,axiom,
% 0.30/0.64 ( heap_Time_wait
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( heap_Time_Heap2 @ product_unit
% 0.30/0.64 @ ^ [H4: 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 @ H4 @ N3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % wait_def
% 0.30/0.64 thf(fact_6744_subset_Ochain__empty,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_empty
% 0.30/0.64 thf(fact_6745_subset__chain__def,axiom,
% 0.30/0.64 ! [A: $tType,A16: set @ ( set @ A ),C7: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C7 )
% 0.30/0.64 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C7 @ A16 )
% 0.30/0.64 & ! [X2: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X2 @ C7 )
% 0.30/0.64 => ! [Y6: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Y6 @ C7 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y6 )
% 0.30/0.64 | ( ord_less_eq @ ( set @ A ) @ Y6 @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_chain_def
% 0.30/0.64 thf(fact_6746_subset__Zorn,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.64 ( ! [C8: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
% 0.30/0.64 => ? [X4: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X4 @ A6 )
% 0.30/0.64 & ! [Xa2: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Xa2 @ C8 )
% 0.30/0.64 => ( ord_less_eq @ ( set @ A ) @ Xa2 @ X4 ) ) ) )
% 0.30/0.64 => ? [X3: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 0.30/0.64 & ! [Xa: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Xa @ A6 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 0.30/0.64 => ( Xa = X3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_Zorn
% 0.30/0.64 thf(fact_6747_subset__chain__insert,axiom,
% 0.30/0.64 ! [A: $tType,A16: set @ ( set @ A ),B7: set @ A,B12: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ B7 @ B12 ) )
% 0.30/0.64 = ( ( member @ ( set @ A ) @ B7 @ A16 )
% 0.30/0.64 & ! [X2: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X2 @ B12 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X2 @ B7 )
% 0.30/0.64 | ( ord_less_eq @ ( set @ A ) @ B7 @ X2 ) ) )
% 0.30/0.64 & ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_chain_insert
% 0.30/0.64 thf(fact_6748_subset__Zorn_H,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.64 ( ! [C8: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
% 0.30/0.64 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) @ A6 ) )
% 0.30/0.64 => ? [X3: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 0.30/0.64 & ! [Xa: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Xa @ A6 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 0.30/0.64 => ( Xa = X3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_Zorn'
% 0.30/0.64 thf(fact_6749_subset_Ochain__extend,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),Z3: set @ A] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ( member @ ( set @ A ) @ Z3 @ A6 )
% 0.30/0.64 => ( ! [X3: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X3 @ C6 )
% 0.30/0.64 => ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ X3
% 0.30/0.64 @ Z3 ) )
% 0.30/0.64 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C6 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_extend
% 0.30/0.64 thf(fact_6750_subset_Ochain__total,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X: set @ A,Y3: set @ A] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ( member @ ( set @ A ) @ X @ C6 )
% 0.30/0.64 => ( ( member @ ( set @ A ) @ Y3 @ C6 )
% 0.30/0.64 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ X
% 0.30/0.64 @ Y3 )
% 0.30/0.64 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ Y3
% 0.30/0.64 @ X ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_total
% 0.30/0.64 thf(fact_6751_subset_OchainI,axiom,
% 0.30/0.64 ! [A: $tType,C6: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.64 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
% 0.30/0.64 => ( ! [X3: set @ A,Y5: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X3 @ C6 )
% 0.30/0.64 => ( ( member @ ( set @ A ) @ Y5 @ C6 )
% 0.30/0.64 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ X3
% 0.30/0.64 @ Y5 )
% 0.30/0.64 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ Y5
% 0.30/0.64 @ X3 ) ) ) )
% 0.30/0.64 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chainI
% 0.30/0.64 thf(fact_6752_subset_Ochain__def,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
% 0.30/0.64 & ! [X2: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X2 @ C6 )
% 0.30/0.64 => ! [Y6: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Y6 @ C6 )
% 0.30/0.64 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ X2
% 0.30/0.64 @ Y6 )
% 0.30/0.64 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 0.30/0.64 @ Y6
% 0.30/0.64 @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_def
% 0.30/0.64 thf(fact_6753_uminus__unit__def,axiom,
% 0.30/0.64 ( ( uminus_uminus @ product_unit )
% 0.30/0.64 = ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% 0.30/0.64
% 0.30/0.64 % uminus_unit_def
% 0.30/0.64 thf(fact_6754_subset__Zorn__nonempty,axiom,
% 0.30/0.64 ! [A: $tType,A16: set @ ( set @ A )] :
% 0.30/0.64 ( ( A16
% 0.30/0.64 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ! [C9: set @ ( set @ A )] :
% 0.30/0.64 ( ( C9
% 0.30/0.64 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C9 )
% 0.30/0.64 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) @ A16 ) ) )
% 0.30/0.64 => ? [X3: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ X3 @ A16 )
% 0.30/0.64 & ! [Xa: set @ A] :
% 0.30/0.64 ( ( member @ ( set @ A ) @ Xa @ A16 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 0.30/0.64 => ( Xa = X3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_Zorn_nonempty
% 0.30/0.64 thf(fact_6755_dependent__wellorder__choice,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( wellorder @ A )
% 0.30/0.64 => ! [P: ( A > B ) > A > B > $o] :
% 0.30/0.64 ( ! [R3: B,F3: A > B,G5: A > B,X3: A] :
% 0.30/0.64 ( ! [Y4: A] :
% 0.30/0.64 ( ( ord_less @ A @ Y4 @ X3 )
% 0.30/0.64 => ( ( F3 @ Y4 )
% 0.30/0.64 = ( G5 @ Y4 ) ) )
% 0.30/0.64 => ( ( P @ F3 @ X3 @ R3 )
% 0.30/0.64 = ( P @ G5 @ X3 @ R3 ) ) )
% 0.30/0.64 => ( ! [X3: A,F3: A > B] :
% 0.30/0.64 ( ! [Y4: A] :
% 0.30/0.64 ( ( ord_less @ A @ Y4 @ X3 )
% 0.30/0.64 => ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
% 0.30/0.64 => ? [X_1: B] : ( P @ F3 @ X3 @ X_1 ) )
% 0.30/0.64 => ? [F3: A > B] :
% 0.30/0.64 ! [X4: A] : ( P @ F3 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % dependent_wellorder_choice
% 0.30/0.64 thf(fact_6756_Union__in__chain,axiom,
% 0.30/0.64 ! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 0.30/0.64 ( ( finite_finite2 @ ( set @ A ) @ B12 )
% 0.30/0.64 => ( ( B12
% 0.30/0.64 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 0.30/0.64 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Union_in_chain
% 0.30/0.64 thf(fact_6757_Inter__in__chain,axiom,
% 0.30/0.64 ! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 0.30/0.64 ( ( finite_finite2 @ ( set @ A ) @ B12 )
% 0.30/0.64 => ( ( B12
% 0.30/0.64 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 0.30/0.64 => ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Inter_in_chain
% 0.30/0.64 thf(fact_6758_chain__subset__alt__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( chain_subset @ A )
% 0.30/0.64 = ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % chain_subset_alt_def
% 0.30/0.64 thf(fact_6759_chains__alt__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( chains @ A )
% 0.30/0.64 = ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % chains_alt_def
% 0.30/0.64 thf(fact_6760_sup2CI,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
% 0.30/0.64 ( ( ~ ( B7 @ X @ Y3 )
% 0.30/0.64 => ( A6 @ X @ Y3 ) )
% 0.30/0.64 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup2CI
% 0.30/0.64 thf(fact_6761_sup2I2,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
% 0.30/0.64 ( ( B7 @ X @ Y3 )
% 0.30/0.64 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup2I2
% 0.30/0.64 thf(fact_6762_sup2I1,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,A6: A > B > $o,X: A,Y3: B,B7: A > B > $o] :
% 0.30/0.64 ( ( A6 @ X @ Y3 )
% 0.30/0.64 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup2I1
% 0.30/0.64 thf(fact_6763_sup2E,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,A6: A > B > $o,B7: A > B > $o,X: A,Y3: B] :
% 0.30/0.64 ( ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 )
% 0.30/0.64 => ( ~ ( A6 @ X @ Y3 )
% 0.30/0.64 => ( B7 @ X @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup2E
% 0.30/0.64 thf(fact_6764_pred__on_Onot__maxchain__Some,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ A,P: A > A > $o,C6: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A6 @ P @ C6 )
% 0.30/0.64 => ( ~ ( pred_maxchain @ A @ A6 @ P @ C6 )
% 0.30/0.64 => ( ( pred_chain @ A @ A6 @ P
% 0.30/0.64 @ ( fChoice @ ( set @ A )
% 0.30/0.64 @ ^ [D7: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A6 @ P @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ C6
% 0.30/0.64 @ ( fChoice @ ( set @ A )
% 0.30/0.64 @ ^ [D7: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A6 @ P @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pred_on.not_maxchain_Some
% 0.30/0.64 thf(fact_6765_subset__maxchain__max,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X8: set @ A] :
% 0.30/0.64 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ( member @ ( set @ A ) @ X8 @ A6 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ X8 )
% 0.30/0.64 => ( ( complete_Sup_Sup @ ( set @ A ) @ C6 )
% 0.30/0.64 = X8 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset_maxchain_max
% 0.30/0.64 thf(fact_6766_subset_OHausdorff,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.64 ? [X_12: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X_12 ) ).
% 0.30/0.64
% 0.30/0.64 % subset.Hausdorff
% 0.30/0.64 thf(fact_6767_subset_Omaxchain__def,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 = ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 & ~ ? [S7: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ S7 )
% 0.30/0.64 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ S7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.maxchain_def
% 0.30/0.64 thf(fact_6768_subset_Omaxchain__imp__chain,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.maxchain_imp_chain
% 0.30/0.64 thf(fact_6769_subset_Onot__maxchain__Some,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) )
% 0.30/0.64 @ ( fChoice @ ( set @ ( set @ A ) )
% 0.30/0.64 @ ^ [D7: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) )
% 0.30/0.64 & ( ord_less @ ( set @ ( set @ A ) ) @ C6
% 0.30/0.64 @ ( fChoice @ ( set @ ( set @ A ) )
% 0.30/0.64 @ ^ [D7: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.not_maxchain_Some
% 0.30/0.64 thf(fact_6770_pred__on_Omaxchain__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( pred_maxchain @ A )
% 0.30/0.64 = ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A7 @ P4 @ C10 )
% 0.30/0.64 & ~ ? [S7: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A7 @ P4 @ S7 )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ C10 @ S7 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pred_on.maxchain_def
% 0.30/0.64 thf(fact_6771_pred__on_Osuc__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( pred_suc @ A )
% 0.30/0.64 = ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
% 0.30/0.64 ( if @ ( set @ A )
% 0.30/0.64 @ ( ~ ( pred_chain @ A @ A7 @ P4 @ C10 )
% 0.30/0.64 | ( pred_maxchain @ A @ A7 @ P4 @ C10 ) )
% 0.30/0.64 @ C10
% 0.30/0.64 @ ( fChoice @ ( set @ A )
% 0.30/0.64 @ ^ [D7: set @ A] :
% 0.30/0.64 ( ( pred_chain @ A @ A7 @ P4 @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ C10 @ D7 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pred_on.suc_def
% 0.30/0.64 thf(fact_6772_subset_Osuc__def,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 | ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
% 0.30/0.64 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 = C6 ) )
% 0.30/0.64 & ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 | ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
% 0.30/0.64 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 = ( fChoice @ ( set @ ( set @ A ) )
% 0.30/0.64 @ ^ [D7: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 0.30/0.64 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.suc_def
% 0.30/0.64 thf(fact_6773_subset_Onot__chain__suc,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 0.30/0.64 ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 = X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.not_chain_suc
% 0.30/0.64 thf(fact_6774_subset_Omaxchain__suc,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 = X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.maxchain_suc
% 0.30/0.64 thf(fact_6775_subset_Ochain__suc,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_suc
% 0.30/0.64 thf(fact_6776_subset_Osubset__suc,axiom,
% 0.30/0.64 ! [A: $tType,X8: set @ ( set @ A ),Y7: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.64 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 0.30/0.64 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.subset_suc
% 0.30/0.64 thf(fact_6777_subset_Osuc__subset,axiom,
% 0.30/0.64 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.suc_subset
% 0.30/0.64 thf(fact_6778_subset_Osuc__in__carrier,axiom,
% 0.30/0.64 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.64 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 )
% 0.30/0.64 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.suc_in_carrier
% 0.30/0.64 thf(fact_6779_subset_Ochain__sucD,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.64 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 )
% 0.30/0.64 & ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.chain_sucD
% 0.30/0.64 thf(fact_6780_subset_Osuc__not__equals,axiom,
% 0.30/0.64 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 0.30/0.64 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 0.30/0.64 != C6 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.suc_not_equals
% 0.30/0.64 thf(fact_6781_alloc__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap @ A )
% 0.30/0.64 => ( ( array_alloc @ A )
% 0.30/0.64 = ( ^ [Xs3: list @ A,H4: heap_ext @ product_unit] :
% 0.30/0.64 ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H4 ) )
% 0.30/0.64 @ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H4 ) ) @ Xs3
% 0.30/0.64 @ ( lim_update @ product_unit
% 0.30/0.64 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H4 ) @ ( one_one @ nat ) )
% 0.30/0.64 @ H4 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % alloc_def
% 0.30/0.64 thf(fact_6782_Code__Numeral_Osub__def,axiom,
% 0.30/0.64 ( code_sub
% 0.30/0.64 = ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
% 0.30/0.64 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Code_Numeral.sub_def
% 0.30/0.64 thf(fact_6783_present__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap @ A )
% 0.30/0.64 => ( ( array_present @ A )
% 0.30/0.64 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A5 ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % present_def
% 0.30/0.64 thf(fact_6784_array_Osize_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,X: nat] :
% 0.30/0.64 ( ( size_size @ ( array @ A ) @ ( array2 @ A @ X ) )
% 0.30/0.64 = ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % array.size(2)
% 0.30/0.64 thf(fact_6785_array_Osize__gen,axiom,
% 0.30/0.64 ! [A: $tType,Xa3: A > nat,X: nat] :
% 0.30/0.64 ( ( size_array @ A @ Xa3 @ ( array2 @ A @ X ) )
% 0.30/0.64 = ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % array.size_gen
% 0.30/0.64 thf(fact_6786_noteq__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( ( heap @ A )
% 0.30/0.64 & ( heap @ B ) )
% 0.30/0.64 => ( ( array_noteq @ A @ B )
% 0.30/0.64 = ( ^ [R5: array @ A,S6: array @ B] :
% 0.30/0.64 ( ( ( typerep_typerep @ A @ ( type2 @ A ) )
% 0.30/0.64 != ( typerep_typerep @ B @ ( type2 @ B ) ) )
% 0.30/0.64 | ( ( addr_of_array @ A @ R5 )
% 0.30/0.64 != ( addr_of_array @ B @ S6 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % noteq_def
% 0.30/0.64 thf(fact_6787_get__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap @ A )
% 0.30/0.64 => ( ( array_get @ A )
% 0.30/0.64 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H4 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % get_def
% 0.30/0.64 thf(fact_6788_Heap_Oempty__def,axiom,
% 0.30/0.64 ( empty
% 0.30/0.64 = ( heap_ext2 @ product_unit
% 0.30/0.64 @ ^ [Uu: typerep,Uv: nat] : ( nil @ nat )
% 0.30/0.64 @ ^ [Uu: typerep,Uv: nat] : ( zero_zero @ nat )
% 0.30/0.64 @ ( zero_zero @ nat )
% 0.30/0.64 @ product_Unity ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.empty_def
% 0.30/0.64 thf(fact_6789_coset__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( coset @ A )
% 0.30/0.64 = ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % coset_def
% 0.30/0.64 thf(fact_6790_compl__coset,axiom,
% 0.30/0.64 ! [A: $tType,Xs: list @ A] :
% 0.30/0.64 ( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
% 0.30/0.64 = ( set2 @ A @ Xs ) ) ).
% 0.30/0.64
% 0.30/0.64 % compl_coset
% 0.30/0.64 thf(fact_6791_Nitpick_Otranclp__unfold,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( transitive_tranclp @ A )
% 0.30/0.64 = ( ^ [R5: A > A > $o,A5: A,B3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Nitpick.tranclp_unfold
% 0.30/0.64 thf(fact_6792_Field__natLeq__on,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( field2 @ nat
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.64 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.64 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
% 0.30/0.64 = ( collect @ nat
% 0.30/0.64 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Field_natLeq_on
% 0.30/0.64 thf(fact_6793_less__nat__rel,axiom,
% 0.30/0.64 ( ( ord_less @ nat )
% 0.30/0.64 = ( transitive_tranclp @ nat
% 0.30/0.64 @ ^ [M2: nat,N3: nat] :
% 0.30/0.64 ( N3
% 0.30/0.64 = ( suc @ M2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_nat_rel
% 0.30/0.64 thf(fact_6794_tranclp__power,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( transitive_tranclp @ A )
% 0.30/0.64 = ( ^ [P4: A > A > $o,X2: A,Y6: A] :
% 0.30/0.64 ? [N3: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.30/0.64 & ( compow @ ( A > A > $o ) @ N3 @ P4 @ X2 @ Y6 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tranclp_power
% 0.30/0.64 thf(fact_6795_underS__Field3,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.30/0.64 ( ( ( field2 @ A @ R )
% 0.30/0.64 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.64 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % underS_Field3
% 0.30/0.64 thf(fact_6796_underS__Field2,axiom,
% 0.30/0.64 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.64 ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.30/0.64 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % underS_Field2
% 0.30/0.64 thf(fact_6797_natLeq__on__Well__order,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( order_well_order_on @ nat
% 0.30/0.64 @ ( field2 @ nat
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.64 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.64 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.64 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.64 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % natLeq_on_Well_order
% 0.30/0.64 thf(fact_6798_Quotient3__int,axiom,
% 0.30/0.64 quotient3 @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ ).
% 0.30/0.64
% 0.30/0.64 % Quotient3_int
% 0.30/0.64 thf(fact_6799_natLeq__on__well__order__on,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( order_well_order_on @ nat
% 0.30/0.64 @ ( collect @ nat
% 0.30/0.64 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [X2: nat,Y6: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ X2 @ N )
% 0.30/0.64 & ( ord_less @ nat @ Y6 @ N )
% 0.30/0.64 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % natLeq_on_well_order_on
% 0.30/0.64 thf(fact_6800_embedS__Field,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 0.30/0.64 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % embedS_Field
% 0.30/0.64 thf(fact_6801_list__all__length,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( list_all @ A )
% 0.30/0.64 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.30/0.64 ! [N3: nat] :
% 0.30/0.64 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.30/0.64 => ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % list_all_length
% 0.30/0.64 thf(fact_6802_ofilter__subset__embedS,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ A6 )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ B7 )
% 0.30/0.64 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.64 = ( bNF_Wellorder_embedS @ A @ A
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A6
% 0.30/0.64 @ ^ [Uu: A] : A6 ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ B7
% 0.30/0.64 @ ^ [Uu: A] : B7 ) )
% 0.30/0.64 @ ( id @ A ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ofilter_subset_embedS
% 0.30/0.64 thf(fact_6803_embedS__iff,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 0.30/0.64 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 0.30/0.64 = ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % embedS_iff
% 0.30/0.64 thf(fact_6804_ofilterIncl__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( bNF_We413866401316099525erIncl @ A )
% 0.30/0.64 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 0.30/0.64 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.30/0.64 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.30/0.64 @ ^ [A7: set @ A,B6: set @ A] :
% 0.30/0.64 ( ( order_ofilter @ A @ R5 @ A7 )
% 0.30/0.64 & ( A7
% 0.30/0.64 != ( field2 @ A @ R5 ) )
% 0.30/0.64 & ( order_ofilter @ A @ R5 @ B6 )
% 0.30/0.64 & ( B6
% 0.30/0.64 != ( field2 @ A @ R5 ) )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ofilterIncl_def
% 0.30/0.64 thf(fact_6805_ofilter__subset__embedS__iso,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ A6 )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ B7 )
% 0.30/0.64 => ( ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.64 = ( bNF_Wellorder_embedS @ A @ A
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A6
% 0.30/0.64 @ ^ [Uu: A] : A6 ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ B7
% 0.30/0.64 @ ^ [Uu: A] : B7 ) )
% 0.30/0.64 @ ( id @ A ) ) )
% 0.30/0.64 & ( ( A6 = B7 )
% 0.30/0.64 = ( bNF_Wellorder_iso @ A @ A
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A6
% 0.30/0.64 @ ^ [Uu: A] : A6 ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ B7
% 0.30/0.64 @ ^ [Uu: A] : B7 ) )
% 0.30/0.64 @ ( id @ A ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ofilter_subset_embedS_iso
% 0.30/0.64 thf(fact_6806_ofilter__ordLess,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ A6 )
% 0.30/0.64 => ( ( ord_less @ ( set @ A ) @ A6 @ ( field2 @ A @ R ) )
% 0.30/0.64 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.30/0.64 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A6
% 0.30/0.64 @ ^ [Uu: A] : A6 ) )
% 0.30/0.64 @ R )
% 0.30/0.64 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ofilter_ordLess
% 0.30/0.64 thf(fact_6807_ofilter__subset__ordLess,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ A6 )
% 0.30/0.64 => ( ( order_ofilter @ A @ R @ B7 )
% 0.30/0.64 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.64 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.30/0.64 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A6
% 0.30/0.64 @ ^ [Uu: A] : A6 ) )
% 0.30/0.64 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.30/0.64 @ ( product_Sigma @ A @ A @ B7
% 0.30/0.64 @ ^ [Uu: A] : B7 ) ) )
% 0.30/0.64 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ofilter_subset_ordLess
% 0.30/0.64 thf(fact_6808_bsqr__ofilter,axiom,
% 0.30/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),D4: set @ ( product_prod @ A @ A )] :
% 0.30/0.64 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.30/0.64 => ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R ) @ D4 )
% 0.30/0.64 => ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D4
% 0.30/0.64 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
% 0.30/0.64 @ ^ [Uu: A] : ( field2 @ A @ R ) ) )
% 0.30/0.64 => ( ~ ? [A8: A] :
% 0.30/0.64 ( ( field2 @ A @ R )
% 0.30/0.64 = ( order_under @ A @ R @ A8 ) )
% 0.30/0.64 => ? [A9: set @ A] :
% 0.30/0.64 ( ( order_ofilter @ A @ R @ A9 )
% 0.30/0.64 & ( ord_less @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 0.30/0.64 & ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D4
% 0.30/0.64 @ ( product_Sigma @ A @ A @ A9
% 0.30/0.64 @ ^ [Uu: A] : A9 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bsqr_ofilter
% 0.30/0.64 thf(fact_6809_internalize__ordLess,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.30/0.64 ( ( member @ ( 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 ) )
% 0.30/0.64 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 0.30/0.64 ( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 0.30/0.64 & ( member @ ( 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 ) )
% 0.30/0.64 & ( 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 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % internalize_ordLess
% 0.30/0.64 thf(fact_6810_of__int__code_I1_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ring_1 @ A )
% 0.30/0.64 => ! [K: num] :
% 0.30/0.64 ( ( ring_1_of_int @ A @ ( neg @ K ) )
% 0.30/0.64 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_int_code(1)
% 0.30/0.64 thf(fact_6811_type__definition__natural,axiom,
% 0.30/0.64 type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % type_definition_natural
% 0.30/0.64 thf(fact_6812_less__int__code_I3_J,axiom,
% 0.30/0.64 ! [L: num] :
% 0.30/0.64 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(3)
% 0.30/0.64 thf(fact_6813_less__int__code_I7_J,axiom,
% 0.30/0.64 ! [K: num] : ( ord_less @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(7)
% 0.30/0.64 thf(fact_6814_less__eq__int__code_I7_J,axiom,
% 0.30/0.64 ! [K: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(7)
% 0.30/0.64 thf(fact_6815_less__eq__int__code_I3_J,axiom,
% 0.30/0.64 ! [L: num] :
% 0.30/0.64 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(3)
% 0.30/0.64 thf(fact_6816_nat__code_I1_J,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( nat2 @ ( neg @ K ) )
% 0.30/0.64 = ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % nat_code(1)
% 0.30/0.64 thf(fact_6817_plus__int__code_I6_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( plus_plus @ int @ ( neg @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % plus_int_code(6)
% 0.30/0.64 thf(fact_6818_less__eq__int__code_I9_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ( ( ord_less_eq @ int @ ( neg @ K ) @ ( neg @ L ) )
% 0.30/0.64 = ( ord_less_eq @ num @ L @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(9)
% 0.30/0.64 thf(fact_6819_less__int__code_I9_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ( ( ord_less @ int @ ( neg @ K ) @ ( neg @ L ) )
% 0.30/0.64 = ( ord_less @ num @ L @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(9)
% 0.30/0.64 thf(fact_6820_Int_Osub__code_I4_J,axiom,
% 0.30/0.64 ! [N: num] :
% 0.30/0.64 ( ( sub @ one2 @ ( bit0 @ N ) )
% 0.30/0.64 = ( neg @ ( bitM @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(4)
% 0.30/0.64 thf(fact_6821_Int_Osub__code_I5_J,axiom,
% 0.30/0.64 ! [N: num] :
% 0.30/0.64 ( ( sub @ one2 @ ( bit1 @ N ) )
% 0.30/0.64 = ( neg @ ( bit0 @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(5)
% 0.30/0.64 thf(fact_6822_Int_Osub__code_I1_J,axiom,
% 0.30/0.64 ( ( sub @ one2 @ one2 )
% 0.30/0.64 = ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(1)
% 0.30/0.64 thf(fact_6823_Int_Osub__def,axiom,
% 0.30/0.64 ( sub
% 0.30/0.64 = ( ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_def
% 0.30/0.64 thf(fact_6824_minus__int__code_I6_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( minus_minus @ int @ ( neg @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( sub @ N @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_int_code(6)
% 0.30/0.64 thf(fact_6825_Int_Osub__code_I9_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.64 = ( minus_minus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(9)
% 0.30/0.64 thf(fact_6826_Int_Osub__code_I8_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.64 = ( plus_plus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(8)
% 0.30/0.64 thf(fact_6827_Int_Odup__code_I1_J,axiom,
% 0.30/0.64 ( ( dup @ ( zero_zero @ int ) )
% 0.30/0.64 = ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.dup_code(1)
% 0.30/0.64 thf(fact_6828_Int_Odup__def,axiom,
% 0.30/0.64 ( dup
% 0.30/0.64 = ( ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.dup_def
% 0.30/0.64 thf(fact_6829_Int_Odup__code_I3_J,axiom,
% 0.30/0.64 ! [N: num] :
% 0.30/0.64 ( ( dup @ ( neg @ N ) )
% 0.30/0.64 = ( neg @ ( bit0 @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.dup_code(3)
% 0.30/0.64 thf(fact_6830_Int_Osub__code_I6_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.64 = ( dup @ ( sub @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(6)
% 0.30/0.64 thf(fact_6831_Int_Osub__code_I7_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.64 = ( dup @ ( sub @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(7)
% 0.30/0.64 thf(fact_6832_Int_Osub__code_I2_J,axiom,
% 0.30/0.64 ! [M: num] :
% 0.30/0.64 ( ( sub @ ( bit0 @ M ) @ one2 )
% 0.30/0.64 = ( pos @ ( bitM @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(2)
% 0.30/0.64 thf(fact_6833_Int_Osub__code_I3_J,axiom,
% 0.30/0.64 ! [M: num] :
% 0.30/0.64 ( ( sub @ ( bit1 @ M ) @ one2 )
% 0.30/0.64 = ( pos @ ( bit0 @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.sub_code(3)
% 0.30/0.64 thf(fact_6834_less__int__code_I8_J,axiom,
% 0.30/0.64 ! [K: num,L: num] : ( ord_less @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(8)
% 0.30/0.64 thf(fact_6835_less__int__code_I6_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ~ ( ord_less @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(6)
% 0.30/0.64 thf(fact_6836_less__eq__int__code_I6_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ~ ( ord_less_eq @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(6)
% 0.30/0.64 thf(fact_6837_less__eq__int__code_I8_J,axiom,
% 0.30/0.64 ! [K: num,L: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(8)
% 0.30/0.64 thf(fact_6838_Int_ONeg__def,axiom,
% 0.30/0.64 ( neg
% 0.30/0.64 = ( ^ [N3: num] : ( uminus_uminus @ int @ ( pos @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.Neg_def
% 0.30/0.64 thf(fact_6839_uminus__int__code_I2_J,axiom,
% 0.30/0.64 ! [M: num] :
% 0.30/0.64 ( ( uminus_uminus @ int @ ( pos @ M ) )
% 0.30/0.64 = ( neg @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % uminus_int_code(2)
% 0.30/0.64 thf(fact_6840_uminus__int__code_I3_J,axiom,
% 0.30/0.64 ! [M: num] :
% 0.30/0.64 ( ( uminus_uminus @ int @ ( neg @ M ) )
% 0.30/0.64 = ( pos @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % uminus_int_code(3)
% 0.30/0.64 thf(fact_6841_nat__code_I3_J,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( nat2 @ ( pos @ K ) )
% 0.30/0.64 = ( nat_of_num @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % nat_code(3)
% 0.30/0.64 thf(fact_6842_times__int__code_I3_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( times_times @ int @ ( pos @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % times_int_code(3)
% 0.30/0.64 thf(fact_6843_Int_OPos__def,axiom,
% 0.30/0.64 ( pos
% 0.30/0.64 = ( numeral_numeral @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.Pos_def
% 0.30/0.64 thf(fact_6844_less__eq__int__code_I2_J,axiom,
% 0.30/0.64 ! [L: num] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(2)
% 0.30/0.64 thf(fact_6845_less__eq__int__code_I4_J,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ~ ( ord_less_eq @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(4)
% 0.30/0.64 thf(fact_6846_of__int__code_I3_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ring_1 @ A )
% 0.30/0.64 => ! [K: num] :
% 0.30/0.64 ( ( ring_1_of_int @ A @ ( pos @ K ) )
% 0.30/0.64 = ( numeral_numeral @ A @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_int_code(3)
% 0.30/0.64 thf(fact_6847_one__int__code,axiom,
% 0.30/0.64 ( ( one_one @ int )
% 0.30/0.64 = ( pos @ one2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % one_int_code
% 0.30/0.64 thf(fact_6848_plus__int__code_I3_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( plus_plus @ int @ ( pos @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % plus_int_code(3)
% 0.30/0.64 thf(fact_6849_less__int__code_I4_J,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ~ ( ord_less @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(4)
% 0.30/0.64 thf(fact_6850_less__int__code_I2_J,axiom,
% 0.30/0.64 ! [L: num] : ( ord_less @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(2)
% 0.30/0.64 thf(fact_6851_less__eq__int__code_I5_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ( ( ord_less_eq @ int @ ( pos @ K ) @ ( pos @ L ) )
% 0.30/0.64 = ( ord_less_eq @ num @ K @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_eq_int_code(5)
% 0.30/0.64 thf(fact_6852_less__int__code_I5_J,axiom,
% 0.30/0.64 ! [K: num,L: num] :
% 0.30/0.64 ( ( ord_less @ int @ ( pos @ K ) @ ( pos @ L ) )
% 0.30/0.64 = ( ord_less @ num @ K @ L ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_int_code(5)
% 0.30/0.64 thf(fact_6853_minus__int__code_I3_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( minus_minus @ int @ ( pos @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( sub @ M @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_int_code(3)
% 0.30/0.64 thf(fact_6854_Int_Odup__code_I2_J,axiom,
% 0.30/0.64 ! [N: num] :
% 0.30/0.64 ( ( dup @ ( pos @ N ) )
% 0.30/0.64 = ( pos @ ( bit0 @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Int.dup_code(2)
% 0.30/0.64 thf(fact_6855_minus__int__code_I4_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( minus_minus @ int @ ( pos @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_int_code(4)
% 0.30/0.64 thf(fact_6856_minus__int__code_I5_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( minus_minus @ int @ ( neg @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_int_code(5)
% 0.30/0.64 thf(fact_6857_times__int__code_I4_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( times_times @ int @ ( pos @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % times_int_code(4)
% 0.30/0.64 thf(fact_6858_times__int__code_I5_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( times_times @ int @ ( neg @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % times_int_code(5)
% 0.30/0.64 thf(fact_6859_times__int__code_I6_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( times_times @ int @ ( neg @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % times_int_code(6)
% 0.30/0.64 thf(fact_6860_plus__int__code_I5_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( plus_plus @ int @ ( neg @ M ) @ ( pos @ N ) )
% 0.30/0.64 = ( sub @ N @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % plus_int_code(5)
% 0.30/0.64 thf(fact_6861_plus__int__code_I4_J,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( plus_plus @ int @ ( pos @ M ) @ ( neg @ N ) )
% 0.30/0.64 = ( sub @ M @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % plus_int_code(4)
% 0.30/0.64 thf(fact_6862_prod__list_Ocomm__monoid__list__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid_mult @ A )
% 0.30/0.64 => ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % prod_list.comm_monoid_list_axioms
% 0.30/0.64 thf(fact_6863_Least__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ord @ A )
% 0.30/0.64 => ( ( ord_Least @ A )
% 0.30/0.64 = ( ^ [P4: A > $o] :
% 0.30/0.64 ( the @ A
% 0.30/0.64 @ ^ [X2: A] :
% 0.30/0.64 ( ( P4 @ X2 )
% 0.30/0.64 & ! [Y6: A] :
% 0.30/0.64 ( ( P4 @ Y6 )
% 0.30/0.64 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Least_def
% 0.30/0.64 thf(fact_6864_Ball__def__raw,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ball @ A )
% 0.30/0.64 = ( ^ [A7: set @ A,P4: A > $o] :
% 0.30/0.64 ! [X2: A] :
% 0.30/0.64 ( ( member @ A @ X2 @ A7 )
% 0.30/0.64 => ( P4 @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Ball_def_raw
% 0.30/0.64 thf(fact_6865_sum__list_Ocomm__monoid__list__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid_add @ A )
% 0.30/0.64 => ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_list.comm_monoid_list_axioms
% 0.30/0.64 thf(fact_6866_Greatest__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( order @ A )
% 0.30/0.64 => ( ( order_Greatest @ A )
% 0.30/0.64 = ( ^ [P4: A > $o] :
% 0.30/0.64 ( the @ A
% 0.30/0.64 @ ^ [X2: A] :
% 0.30/0.64 ( ( P4 @ X2 )
% 0.30/0.64 & ! [Y6: A] :
% 0.30/0.64 ( ( P4 @ Y6 )
% 0.30/0.64 => ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Greatest_def
% 0.30/0.64 thf(fact_6867_ord_OLeast__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( least @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
% 0.30/0.64 ( the @ A
% 0.30/0.64 @ ^ [X2: A] :
% 0.30/0.64 ( ( P4 @ X2 )
% 0.30/0.64 & ! [Y6: A] :
% 0.30/0.64 ( ( P4 @ Y6 )
% 0.30/0.64 => ( Less_eq2 @ X2 @ Y6 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ord.Least_def
% 0.30/0.64 thf(fact_6868_ord_OLeast_Ocong,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( least @ A )
% 0.30/0.64 = ( least @ A ) ) ).
% 0.30/0.64
% 0.30/0.64 % ord.Least.cong
% 0.30/0.64 thf(fact_6869_GreatestI__nat,axiom,
% 0.30/0.64 ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.64 ( ( P @ K )
% 0.30/0.64 => ( ! [Y5: nat] :
% 0.30/0.64 ( ( P @ Y5 )
% 0.30/0.64 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 0.30/0.64 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % GreatestI_nat
% 0.30/0.64 thf(fact_6870_Greatest__le__nat,axiom,
% 0.30/0.64 ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.64 ( ( P @ K )
% 0.30/0.64 => ( ! [Y5: nat] :
% 0.30/0.64 ( ( P @ Y5 )
% 0.30/0.64 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 0.30/0.64 => ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Greatest_le_nat
% 0.30/0.64 thf(fact_6871_GreatestI__ex__nat,axiom,
% 0.30/0.64 ! [P: nat > $o,B2: nat] :
% 0.30/0.64 ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.64 => ( ! [Y5: nat] :
% 0.30/0.64 ( ( P @ Y5 )
% 0.30/0.64 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 0.30/0.64 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % GreatestI_ex_nat
% 0.30/0.64 thf(fact_6872_GreatestI2__order,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( order @ A )
% 0.30/0.64 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.30/0.64 ( ( P @ X )
% 0.30/0.64 => ( ! [Y5: A] :
% 0.30/0.64 ( ( P @ Y5 )
% 0.30/0.64 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( P @ X3 )
% 0.30/0.64 => ( ! [Y4: A] :
% 0.30/0.64 ( ( P @ Y4 )
% 0.30/0.64 => ( ord_less_eq @ A @ Y4 @ X3 ) )
% 0.30/0.64 => ( Q @ X3 ) ) )
% 0.30/0.64 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % GreatestI2_order
% 0.30/0.64 thf(fact_6873_Greatest__equality,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( order @ A )
% 0.30/0.64 => ! [P: A > $o,X: A] :
% 0.30/0.64 ( ( P @ X )
% 0.30/0.64 => ( ! [Y5: A] :
% 0.30/0.64 ( ( P @ Y5 )
% 0.30/0.64 => ( ord_less_eq @ A @ Y5 @ X ) )
% 0.30/0.64 => ( ( order_Greatest @ A @ P )
% 0.30/0.64 = X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Greatest_equality
% 0.30/0.64 thf(fact_6874_sum__encode__def,axiom,
% 0.30/0.64 ( nat_sum_encode
% 0.30/0.64 = ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.30/0.64 @ ^ [B3: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_encode_def
% 0.30/0.64 thf(fact_6875_disjnt__ge__max,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ! [Y7: set @ A,X8: set @ A] :
% 0.30/0.64 ( ( finite_finite2 @ A @ Y7 )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( member @ A @ X3 @ X8 )
% 0.30/0.64 => ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y7 ) @ X3 ) )
% 0.30/0.64 => ( disjnt @ A @ X8 @ Y7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % disjnt_ge_max
% 0.30/0.64 thf(fact_6876_inj__sum__encode,axiom,
% 0.30/0.64 ! [A6: set @ ( sum_sum @ nat @ nat )] : ( inj_on @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ A6 ) ).
% 0.30/0.64
% 0.30/0.64 % inj_sum_encode
% 0.30/0.64 thf(fact_6877_sum__encode__eq,axiom,
% 0.30/0.64 ! [X: sum_sum @ nat @ nat,Y3: sum_sum @ nat @ nat] :
% 0.30/0.64 ( ( ( nat_sum_encode @ X )
% 0.30/0.64 = ( nat_sum_encode @ Y3 ) )
% 0.30/0.64 = ( X = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_encode_eq
% 0.30/0.64 thf(fact_6878_bij__sum__encode,axiom,
% 0.30/0.64 bij_betw @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % bij_sum_encode
% 0.30/0.64 thf(fact_6879_surj__sum__encode,axiom,
% 0.30/0.64 ( ( image @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
% 0.30/0.64 = ( top_top @ ( set @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % surj_sum_encode
% 0.30/0.64 thf(fact_6880_int__encode__def,axiom,
% 0.30/0.64 ( nat_int_encode
% 0.30/0.64 = ( ^ [I5: int] : ( nat_sum_encode @ ( if @ ( sum_sum @ nat @ nat ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I5 ) @ ( sum_Inl @ nat @ nat @ ( nat2 @ I5 ) ) @ ( sum_Inr @ nat @ nat @ ( nat2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ I5 ) @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % int_encode_def
% 0.30/0.64 thf(fact_6881_ordering__top_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 0.30/0.64 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.64 => ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top.axioms(2)
% 0.30/0.64 thf(fact_6882_prod_OPlus,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.64 ( ( comm_monoid_mult @ A )
% 0.30/0.64 => ! [A6: set @ B,B7: set @ C,G: ( sum_sum @ B @ C ) > A] :
% 0.30/0.64 ( ( finite_finite2 @ B @ A6 )
% 0.30/0.64 => ( ( finite_finite2 @ C @ B7 )
% 0.30/0.64 => ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A6 @ B7 ) )
% 0.30/0.64 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B7 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % prod.Plus
% 0.30/0.64 thf(fact_6883_ordering__top__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ordering_top_axioms @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Top2: A] :
% 0.30/0.64 ! [A5: A] : ( Less_eq2 @ A5 @ Top2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top_axioms_def
% 0.30/0.64 thf(fact_6884_ordering__top__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Top: A] :
% 0.30/0.64 ( ! [A8: A] : ( Less_eq @ A8 @ Top )
% 0.30/0.64 => ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top_axioms.intro
% 0.30/0.64 thf(fact_6885_ndepth__Push__Node__aux,axiom,
% 0.30/0.64 ! [A: $tType,I: nat,F: nat > ( sum_sum @ A @ nat ),K: nat] :
% 0.30/0.64 ( ( ( case_nat @ ( sum_sum @ A @ nat ) @ ( sum_Inr @ nat @ A @ ( suc @ I ) ) @ F @ K )
% 0.30/0.64 = ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) )
% 0.30/0.64 => ( ord_less_eq @ nat
% 0.30/0.64 @ ( suc
% 0.30/0.64 @ ( ord_Least @ nat
% 0.30/0.64 @ ^ [X2: nat] :
% 0.30/0.64 ( ( F @ X2 )
% 0.30/0.64 = ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) )
% 0.30/0.64 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % ndepth_Push_Node_aux
% 0.30/0.64 thf(fact_6886_le__sum__encode__Inl,axiom,
% 0.30/0.64 ! [X: nat,Y3: nat] :
% 0.30/0.64 ( ( ord_less_eq @ nat @ X @ Y3 )
% 0.30/0.64 => ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inl @ nat @ nat @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_sum_encode_Inl
% 0.30/0.64 thf(fact_6887_le__sum__encode__Inr,axiom,
% 0.30/0.64 ! [X: nat,Y3: nat] :
% 0.30/0.64 ( ( ord_less_eq @ nat @ X @ Y3 )
% 0.30/0.64 => ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inr @ nat @ nat @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_sum_encode_Inr
% 0.30/0.64 thf(fact_6888_sum_Osize_I3_J,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,X13: A] :
% 0.30/0.64 ( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X13 ) )
% 0.30/0.64 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.size(3)
% 0.30/0.64 thf(fact_6889_sum_Osize_I4_J,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,X22: B] :
% 0.30/0.64 ( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X22 ) )
% 0.30/0.64 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.size(4)
% 0.30/0.64 thf(fact_6890_sum_Osize__neq,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
% 0.30/0.64 ( ( size_size @ ( sum_sum @ A @ B ) @ X )
% 0.30/0.64 != ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.size_neq
% 0.30/0.64 thf(fact_6891_prod_Osize__neq,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,X: product_prod @ A @ B] :
% 0.30/0.64 ( ( size_size @ ( product_prod @ A @ B ) @ X )
% 0.30/0.64 != ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % prod.size_neq
% 0.30/0.64 thf(fact_6892_sum_Osize__gen_I1_J,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,Xa3: A > nat,X: B > nat,X13: A] :
% 0.30/0.64 ( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inl @ A @ B @ X13 ) )
% 0.30/0.64 = ( plus_plus @ nat @ ( Xa3 @ X13 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.size_gen(1)
% 0.30/0.64 thf(fact_6893_sum_Osize__gen_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,Xa3: A > nat,X: B > nat,X22: B] :
% 0.30/0.64 ( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inr @ B @ A @ X22 ) )
% 0.30/0.64 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.size_gen(2)
% 0.30/0.64 thf(fact_6894_sum__decode__def,axiom,
% 0.30/0.64 ( nat_sum_decode
% 0.30/0.64 = ( ^ [N3: nat] : ( if @ ( sum_sum @ nat @ nat ) @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( sum_Inl @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( sum_Inr @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_decode_def
% 0.30/0.64 thf(fact_6895_sum__decode__inverse,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
% 0.30/0.64 = N ) ).
% 0.30/0.64
% 0.30/0.64 % sum_decode_inverse
% 0.30/0.64 thf(fact_6896_sum__encode__inverse,axiom,
% 0.30/0.64 ! [X: sum_sum @ nat @ nat] :
% 0.30/0.64 ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
% 0.30/0.64 = X ) ).
% 0.30/0.64
% 0.30/0.64 % sum_encode_inverse
% 0.30/0.64 thf(fact_6897_sum__decode__eq,axiom,
% 0.30/0.64 ! [X: nat,Y3: nat] :
% 0.30/0.64 ( ( ( nat_sum_decode @ X )
% 0.30/0.64 = ( nat_sum_decode @ Y3 ) )
% 0.30/0.64 = ( X = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_decode_eq
% 0.30/0.64 thf(fact_6898_inj__sum__decode,axiom,
% 0.30/0.64 ! [A6: set @ nat] : ( inj_on @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ A6 ) ).
% 0.30/0.64
% 0.30/0.64 % inj_sum_decode
% 0.30/0.64 thf(fact_6899_surj__sum__decode,axiom,
% 0.30/0.64 ( ( image @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
% 0.30/0.64 = ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % surj_sum_decode
% 0.30/0.64 thf(fact_6900_bij__sum__decode,axiom,
% 0.30/0.64 bij_betw @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % bij_sum_decode
% 0.30/0.64 thf(fact_6901_int__decode__def,axiom,
% 0.30/0.64 ( nat_int_decode
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
% 0.30/0.64 @ ^ [B3: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B3 ) ) @ ( one_one @ int ) )
% 0.30/0.64 @ ( nat_sum_decode @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % int_decode_def
% 0.30/0.64 thf(fact_6902_Node__def,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType] :
% 0.30/0.64 ( ( old_Node @ B @ A )
% 0.30/0.64 = ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
% 0.30/0.64 @ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
% 0.30/0.64 ? [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat,K3: nat] :
% 0.30/0.64 ( ( P6
% 0.30/0.64 = ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X2 ) )
% 0.30/0.64 & ( ( F2 @ K3 )
% 0.30/0.64 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Node_def
% 0.30/0.64 thf(fact_6903_Node__K0__I,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
% 0.30/0.64 ( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
% 0.30/0.64 @ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
% 0.30/0.64 @ ^ [K3: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
% 0.30/0.64 @ A3 )
% 0.30/0.64 @ ( old_Node @ A @ B ) ) ).
% 0.30/0.64
% 0.30/0.64 % Node_K0_I
% 0.30/0.64 thf(fact_6904_Push__neq__K0,axiom,
% 0.30/0.64 ! [A: $tType,K: nat,F: nat > ( sum_sum @ A @ nat )] :
% 0.30/0.64 ( ( old_Push @ A @ ( sum_Inr @ nat @ A @ ( suc @ K ) ) @ F )
% 0.30/0.64 != ( ^ [Z7: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Push_neq_K0
% 0.30/0.64 thf(fact_6905_ndepth__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( old_ndepth @ A @ B )
% 0.30/0.64 = ( ^ [N3: old_node @ A @ B] :
% 0.30/0.64 ( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
% 0.30/0.64 @ ^ [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat] :
% 0.30/0.64 ( ord_Least @ nat
% 0.30/0.64 @ ^ [K3: nat] :
% 0.30/0.64 ( ( F2 @ K3 )
% 0.30/0.64 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
% 0.30/0.64 @ ( old_Rep_Node @ A @ B @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ndepth_def
% 0.30/0.64 thf(fact_6906_ndepth__K0,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
% 0.30/0.64 ( ( old_ndepth @ A @ B
% 0.30/0.64 @ ( old_Abs_Node @ B @ A
% 0.30/0.64 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 0.30/0.64 @ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 0.30/0.64 @ X ) ) )
% 0.30/0.64 = ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % ndepth_K0
% 0.30/0.64 thf(fact_6907_ndepth__Push__Node,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,I: nat,N: old_node @ A @ B] :
% 0.30/0.64 ( ( old_ndepth @ A @ B @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ I ) ) @ N ) )
% 0.30/0.64 = ( suc @ ( old_ndepth @ A @ B @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ndepth_Push_Node
% 0.30/0.64 thf(fact_6908_Atom__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( old_Atom @ A @ B )
% 0.30/0.64 = ( ^ [X2: sum_sum @ A @ nat] :
% 0.30/0.64 ( insert @ ( old_node @ A @ B )
% 0.30/0.64 @ ( old_Abs_Node @ B @ A
% 0.30/0.64 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 0.30/0.64 @ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 0.30/0.64 @ X2 ) )
% 0.30/0.64 @ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Atom_def
% 0.30/0.64 thf(fact_6909_Scons__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( old_Scons @ A @ B )
% 0.30/0.64 = ( ^ [M9: set @ ( old_node @ A @ B ),N7: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Scons_def
% 0.30/0.64 thf(fact_6910_ntrunc__0,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M7 )
% 0.30/0.64 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_0
% 0.30/0.64 thf(fact_6911_ntrunc__Atom,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,K: nat,A3: sum_sum @ A @ nat] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Atom @ A @ B @ A3 ) )
% 0.30/0.64 = ( old_Atom @ A @ B @ A3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_Atom
% 0.30/0.64 thf(fact_6912_ntrunc__Scons,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B ),N6: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Scons @ A @ B @ M7 @ N6 ) )
% 0.30/0.64 = ( old_Scons @ A @ B @ ( old_ntrunc @ A @ B @ K @ M7 ) @ ( old_ntrunc @ A @ B @ K @ N6 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_Scons
% 0.30/0.64 thf(fact_6913_ntrunc__def,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B )
% 0.30/0.64 = ( ^ [K3: nat,N7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( collect @ ( old_node @ A @ B )
% 0.30/0.64 @ ^ [N3: old_node @ A @ B] :
% 0.30/0.64 ( ( member @ ( old_node @ A @ B ) @ N3 @ N7 )
% 0.30/0.64 & ( ord_less @ nat @ ( old_ndepth @ A @ B @ N3 ) @ K3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_def
% 0.30/0.64 thf(fact_6914_ntrunc__one__In0,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M7 ) )
% 0.30/0.64 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_one_In0
% 0.30/0.64 thf(fact_6915_ntrunc__one__In1,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M7 ) )
% 0.30/0.64 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_one_In1
% 0.30/0.64 thf(fact_6916_ntrunc__In0,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In0 @ A @ B @ M7 ) )
% 0.30/0.64 = ( old_In0 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_In0
% 0.30/0.64 thf(fact_6917_ntrunc__In1,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In1 @ A @ B @ M7 ) )
% 0.30/0.64 = ( old_In1 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_In1
% 0.30/0.64 thf(fact_6918_ntrunc__Leaf,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,K: nat,A3: A] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Leaf @ A @ B @ A3 ) )
% 0.30/0.64 = ( old_Leaf @ A @ B @ A3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_Leaf
% 0.30/0.64 thf(fact_6919_In1__def,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType] :
% 0.30/0.64 ( ( old_In1 @ A @ B )
% 0.30/0.64 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % In1_def
% 0.30/0.64 thf(fact_6920_ntrunc__Numb,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,K: nat,I: nat] :
% 0.30/0.64 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Numb @ A @ B @ I ) )
% 0.30/0.64 = ( old_Numb @ A @ B @ I ) ) ).
% 0.30/0.64
% 0.30/0.64 % ntrunc_Numb
% 0.30/0.64 thf(fact_6921_In0__def,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType] :
% 0.30/0.64 ( ( old_In0 @ A @ B )
% 0.30/0.64 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % In0_def
% 0.30/0.64 thf(fact_6922_ordering__top_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( ordering_top_axioms @ A @ Less_eq @ Top )
% 0.30/0.64 => ( ordering_top @ A @ Less_eq @ Less @ Top ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top.intro
% 0.30/0.64 thf(fact_6923_ordering__top__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ordering_top @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,Top2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.64 & ( ordering_top_axioms @ A @ Less_eq2 @ Top2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top_def
% 0.30/0.64 thf(fact_6924_ordering__top_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 0.30/0.64 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.30/0.64 => ( ordering @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_top.axioms(1)
% 0.30/0.64 thf(fact_6925_ordering__strictI,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 = ( ( Less @ A8 @ B5 )
% 0.30/0.64 | ( A8 = B5 ) ) )
% 0.30/0.64 => ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 => ~ ( Less @ B5 @ A8 ) )
% 0.30/0.64 => ( ! [A8: A] :
% 0.30/0.64 ~ ( Less @ A8 @ A8 )
% 0.30/0.64 => ( ! [A8: A,B5: A,C3: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 => ( ( Less @ B5 @ C3 )
% 0.30/0.64 => ( Less @ A8 @ C3 ) ) )
% 0.30/0.64 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_strictI
% 0.30/0.64 thf(fact_6926_ordering_Onot__eq__order__implies__strict,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( A3 != B2 )
% 0.30/0.64 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 => ( Less @ A3 @ B2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.not_eq_order_implies_strict
% 0.30/0.64 thf(fact_6927_ordering_Ostrict__implies__not__eq,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 => ( A3 != B2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.strict_implies_not_eq
% 0.30/0.64 thf(fact_6928_ordering_Ostrict__iff__order,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 = ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.strict_iff_order
% 0.30/0.64 thf(fact_6929_ordering_Oorder__iff__strict,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 = ( ( Less @ A3 @ B2 )
% 0.30/0.64 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.order_iff_strict
% 0.30/0.64 thf(fact_6930_ordering_Oantisym,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 => ( ( Less_eq @ B2 @ A3 )
% 0.30/0.64 => ( A3 = B2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.antisym
% 0.30/0.64 thf(fact_6931_ordering_Oeq__iff,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( A3 = B2 )
% 0.30/0.64 = ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 & ( Less_eq @ B2 @ A3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.eq_iff
% 0.30/0.64 thf(fact_6932_ordering__dualI,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( ordering @ A
% 0.30/0.64 @ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
% 0.30/0.64 @ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
% 0.30/0.64 => ( ordering @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_dualI
% 0.30/0.64 thf(fact_6933_order_Oordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( order @ A )
% 0.30/0.64 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % order.ordering_axioms
% 0.30/0.64 thf(fact_6934_dual__order_Oordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( order @ A )
% 0.30/0.64 => ( ordering @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % dual_order.ordering_axioms
% 0.30/0.64 thf(fact_6935_nth__item_Oelims,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
% 0.30/0.64 ( ( ( nth_item @ A @ X )
% 0.30/0.64 = Y3 )
% 0.30/0.64 => ( ( ( X
% 0.30/0.64 = ( zero_zero @ nat ) )
% 0.30/0.64 => ( Y3
% 0.30/0.64 != ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
% 0.30/0.64 => ~ ! [N4: nat] :
% 0.30/0.64 ( ( X
% 0.30/0.64 = ( suc @ N4 ) )
% 0.30/0.64 => ( Y3
% 0.30/0.64 != ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.30/0.64 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 0.30/0.64 @ ( nat_prod_decode @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ( nat_sum_decode @ N4 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.elims
% 0.30/0.64 thf(fact_6936_group_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 0.30/0.64 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.64 => ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ).
% 0.30/0.64
% 0.30/0.64 % group.axioms(2)
% 0.30/0.64 thf(fact_6937_group__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 0.30/0.64 ( ! [A8: A] :
% 0.30/0.64 ( ( F @ Z3 @ A8 )
% 0.30/0.64 = A8 )
% 0.30/0.64 => ( ! [A8: A] :
% 0.30/0.64 ( ( F @ ( Inverse @ A8 ) @ A8 )
% 0.30/0.64 = Z3 )
% 0.30/0.64 => ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % group_axioms.intro
% 0.30/0.64 thf(fact_6938_group__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( group_axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
% 0.30/0.64 ( ! [A5: A] :
% 0.30/0.64 ( ( F2 @ Z7 @ A5 )
% 0.30/0.64 = A5 )
% 0.30/0.64 & ! [A5: A] :
% 0.30/0.64 ( ( F2 @ ( Inverse2 @ A5 ) @ A5 )
% 0.30/0.64 = Z7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % group_axioms_def
% 0.30/0.64 thf(fact_6939_nth__item_Osimps_I1_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 0.30/0.64 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.simps(1)
% 0.30/0.64 thf(fact_6940_nth__item_Osimps_I2_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ! [N: nat] :
% 0.30/0.64 ( ( nth_item @ A @ ( suc @ N ) )
% 0.30/0.64 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.30/0.64 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 0.30/0.64 @ ( nat_prod_decode @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ( nat_sum_decode @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.simps(2)
% 0.30/0.64 thf(fact_6941_nth__item_Opelims,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
% 0.30/0.64 ( ( ( nth_item @ A @ X )
% 0.30/0.64 = Y3 )
% 0.30/0.64 => ( ( accp @ nat @ nth_item_rel @ X )
% 0.30/0.64 => ( ( ( X
% 0.30/0.64 = ( zero_zero @ nat ) )
% 0.30/0.64 => ( ( Y3
% 0.30/0.64 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
% 0.30/0.64 => ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
% 0.30/0.64 => ~ ! [N4: nat] :
% 0.30/0.64 ( ( X
% 0.30/0.64 = ( suc @ N4 ) )
% 0.30/0.64 => ( ( Y3
% 0.30/0.64 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.30/0.64 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 0.30/0.64 @ ( nat_prod_decode @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ( nat_sum_decode @ N4 ) ) )
% 0.30/0.64 => ~ ( accp @ nat @ nth_item_rel @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.pelims
% 0.30/0.64 thf(fact_6942_nth__item_Opsimps_I2_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ! [N: nat] :
% 0.30/0.64 ( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
% 0.30/0.64 => ( ( nth_item @ A @ ( suc @ N ) )
% 0.30/0.64 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.30/0.64 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.30/0.64 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 0.30/0.64 @ ( nat_prod_decode @ J3 ) )
% 0.30/0.64 @ ( nat_sum_decode @ I5 ) )
% 0.30/0.64 @ ( nat_sum_decode @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.psimps(2)
% 0.30/0.64 thf(fact_6943_nth__item_Opsimps_I1_J,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( countable @ A )
% 0.30/0.64 => ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 0.30/0.64 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 0.30/0.64 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nth_item.psimps(1)
% 0.30/0.64 thf(fact_6944_group_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 0.30/0.64 ( ( semigroup @ A @ F )
% 0.30/0.64 => ( ( group_axioms @ A @ F @ Z3 @ Inverse )
% 0.30/0.64 => ( group @ A @ F @ Z3 @ Inverse ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % group.intro
% 0.30/0.64 thf(fact_6945_group__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( group @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
% 0.30/0.64 ( ( semigroup @ A @ F2 )
% 0.30/0.64 & ( group_axioms @ A @ F2 @ Z7 @ Inverse2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % group_def
% 0.30/0.64 thf(fact_6946_sup_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_sup @ A )
% 0.30/0.64 => ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup.semigroup_axioms
% 0.30/0.64 thf(fact_6947_semigroup_Oassoc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
% 0.30/0.64 ( ( semigroup @ A @ F )
% 0.30/0.64 => ( ( F @ ( F @ A3 @ B2 ) @ C2 )
% 0.30/0.64 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semigroup.assoc
% 0.30/0.64 thf(fact_6948_semigroup_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ! [A8: A,B5: A,C3: A] :
% 0.30/0.64 ( ( F @ ( F @ A8 @ B5 ) @ C3 )
% 0.30/0.64 = ( F @ A8 @ ( F @ B5 @ C3 ) ) )
% 0.30/0.64 => ( semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semigroup.intro
% 0.30/0.64 thf(fact_6949_semigroup__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semigroup @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A] :
% 0.30/0.64 ! [A5: A,B3: A,C4: A] :
% 0.30/0.64 ( ( F2 @ ( F2 @ A5 @ B3 ) @ C4 )
% 0.30/0.64 = ( F2 @ A5 @ ( F2 @ B3 @ C4 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semigroup_def
% 0.30/0.64 thf(fact_6950_group_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 0.30/0.64 ( ( group @ A @ F @ Z3 @ Inverse )
% 0.30/0.64 => ( semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % group.axioms(1)
% 0.30/0.64 thf(fact_6951_monoid_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( monoid @ A @ F @ Z3 )
% 0.30/0.64 => ( semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid.axioms(1)
% 0.30/0.64 thf(fact_6952_min_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % min.semigroup_axioms
% 0.30/0.64 thf(fact_6953_max_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % max.semigroup_axioms
% 0.30/0.64 thf(fact_6954_add_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semigroup_add @ A )
% 0.30/0.64 => ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % add.semigroup_axioms
% 0.30/0.64 thf(fact_6955_mult_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semigroup_mult @ A )
% 0.30/0.64 => ( semigroup @ A @ ( times_times @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % mult.semigroup_axioms
% 0.30/0.64 thf(fact_6956_inf_Osemigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_inf @ A )
% 0.30/0.64 => ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % inf.semigroup_axioms
% 0.30/0.64 thf(fact_6957_monoid_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( semigroup @ A @ F )
% 0.30/0.64 => ( ( monoid_axioms @ A @ F @ Z3 )
% 0.30/0.64 => ( monoid @ A @ F @ Z3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid.intro
% 0.30/0.64 thf(fact_6958_monoid__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( monoid @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A] :
% 0.30/0.64 ( ( semigroup @ A @ F2 )
% 0.30/0.64 & ( monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid_def
% 0.30/0.64 thf(fact_6959_monoid__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( monoid_axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A] :
% 0.30/0.64 ( ! [A5: A] :
% 0.30/0.64 ( ( F2 @ Z7 @ A5 )
% 0.30/0.64 = A5 )
% 0.30/0.64 & ! [A5: A] :
% 0.30/0.64 ( ( F2 @ A5 @ Z7 )
% 0.30/0.64 = A5 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid_axioms_def
% 0.30/0.64 thf(fact_6960_monoid__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ! [A8: A] :
% 0.30/0.64 ( ( F @ Z3 @ A8 )
% 0.30/0.64 = A8 )
% 0.30/0.64 => ( ! [A8: A] :
% 0.30/0.64 ( ( F @ A8 @ Z3 )
% 0.30/0.64 = A8 )
% 0.30/0.64 => ( monoid_axioms @ A @ F @ Z3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid_axioms.intro
% 0.30/0.64 thf(fact_6961_monoid_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( monoid @ A @ F @ Z3 )
% 0.30/0.64 => ( monoid_axioms @ A @ F @ Z3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoid.axioms(2)
% 0.30/0.64 thf(fact_6962_nths__nths,axiom,
% 0.30/0.64 ! [A: $tType,Xs: list @ A,A6: set @ nat,B7: set @ nat] :
% 0.30/0.64 ( ( nths @ A @ ( nths @ A @ Xs @ A6 ) @ B7 )
% 0.30/0.64 = ( nths @ A @ Xs
% 0.30/0.64 @ ( collect @ nat
% 0.30/0.64 @ ^ [I5: nat] :
% 0.30/0.64 ( ( member @ nat @ I5 @ A6 )
% 0.30/0.64 & ( member @ nat
% 0.30/0.64 @ ( finite_card @ nat
% 0.30/0.64 @ ( collect @ nat
% 0.30/0.64 @ ^ [I9: nat] :
% 0.30/0.64 ( ( member @ nat @ I9 @ A6 )
% 0.30/0.64 & ( ord_less @ nat @ I9 @ I5 ) ) ) )
% 0.30/0.64 @ B7 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nths_nths
% 0.30/0.64 thf(fact_6963_ordering_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ordering_axioms @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.axioms(2)
% 0.30/0.64 thf(fact_6964_ordering__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 = ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 & ( A8 != B5 ) ) )
% 0.30/0.64 => ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 => ( ( Less_eq @ B5 @ A8 )
% 0.30/0.64 => ( A8 = B5 ) ) )
% 0.30/0.64 => ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_axioms.intro
% 0.30/0.64 thf(fact_6965_ordering__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ordering_axioms @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ( ! [A5: A,B3: A] :
% 0.30/0.64 ( ( Less2 @ A5 @ B3 )
% 0.30/0.64 = ( ( Less_eq2 @ A5 @ B3 )
% 0.30/0.64 & ( A5 != B3 ) ) )
% 0.30/0.64 & ! [A5: A,B3: A] :
% 0.30/0.64 ( ( Less_eq2 @ A5 @ B3 )
% 0.30/0.64 => ( ( Less_eq2 @ B3 @ A5 )
% 0.30/0.64 => ( A5 = B3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_axioms_def
% 0.30/0.64 thf(fact_6966_Bex__def__raw,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( bex @ A )
% 0.30/0.64 = ( ^ [A7: set @ A,P4: A > $o] :
% 0.30/0.64 ? [X2: A] :
% 0.30/0.64 ( ( member @ A @ X2 @ A7 )
% 0.30/0.64 & ( P4 @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Bex_def_raw
% 0.30/0.64 thf(fact_6967_ordering__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ordering @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq2 )
% 0.30/0.64 & ( ordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering_def
% 0.30/0.64 thf(fact_6968_ordering_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq )
% 0.30/0.64 => ( ( ordering_axioms @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ordering @ A @ Less_eq @ Less ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.intro
% 0.30/0.64 thf(fact_6969_dual__order_Opartial__preordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( partial_preordering @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % dual_order.partial_preordering_axioms
% 0.30/0.64 thf(fact_6970_order_Opartial__preordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % order.partial_preordering_axioms
% 0.30/0.64 thf(fact_6971_partial__preordering__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( partial_preordering @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o] :
% 0.30/0.64 ( ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
% 0.30/0.64 & ! [A5: A,B3: A,C4: A] :
% 0.30/0.64 ( ( Less_eq2 @ A5 @ B3 )
% 0.30/0.64 => ( ( Less_eq2 @ B3 @ C4 )
% 0.30/0.64 => ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % partial_preordering_def
% 0.30/0.64 thf(fact_6972_partial__preordering_Otrans,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq )
% 0.30/0.64 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 => ( ( Less_eq @ B2 @ C2 )
% 0.30/0.64 => ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % partial_preordering.trans
% 0.30/0.64 thf(fact_6973_partial__preordering_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o] :
% 0.30/0.64 ( ! [A8: A] : ( Less_eq @ A8 @ A8 )
% 0.30/0.64 => ( ! [A8: A,B5: A,C3: A] :
% 0.30/0.64 ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 => ( ( Less_eq @ B5 @ C3 )
% 0.30/0.64 => ( Less_eq @ A8 @ C3 ) ) )
% 0.30/0.64 => ( partial_preordering @ A @ Less_eq ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % partial_preordering.intro
% 0.30/0.64 thf(fact_6974_partial__preordering_Orefl,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,A3: A] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq )
% 0.30/0.64 => ( Less_eq @ A3 @ A3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % partial_preordering.refl
% 0.30/0.64 thf(fact_6975_ordering_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( ordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( partial_preordering @ A @ Less_eq ) ) ).
% 0.30/0.64
% 0.30/0.64 % ordering.axioms(1)
% 0.30/0.64 thf(fact_6976_Abs__int__cases,axiom,
% 0.30/0.64 ! [X: int] :
% 0.30/0.64 ~ ! [Y5: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ( ( X
% 0.30/0.64 = ( abs_int @ Y5 ) )
% 0.30/0.64 => ~ ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Abs_int_cases
% 0.30/0.64 thf(fact_6977_Abs__int__induct,axiom,
% 0.30/0.64 ! [P: int > $o,X: int] :
% 0.30/0.64 ( ! [Y5: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ( P @ ( abs_int @ Y5 ) ) )
% 0.30/0.64 => ( P @ X ) ) ).
% 0.30/0.64
% 0.30/0.64 % Abs_int_induct
% 0.30/0.64 thf(fact_6978_Abs__int__inject,axiom,
% 0.30/0.64 ! [X: set @ ( product_prod @ nat @ nat ),Y3: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ X
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ( ( ( abs_int @ X )
% 0.30/0.64 = ( abs_int @ Y3 ) )
% 0.30/0.64 = ( X = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Abs_int_inject
% 0.30/0.64 thf(fact_6979_Abs__Integ__def,axiom,
% 0.30/0.64 ( abs_Integ
% 0.30/0.64 = ( quot_abs @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_int ) ) ).
% 0.30/0.64
% 0.30/0.64 % Abs_Integ_def
% 0.30/0.64 thf(fact_6980_Abs__int__inverse,axiom,
% 0.30/0.64 ! [Y3: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ( ( rep_int @ ( abs_int @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % Abs_int_inverse
% 0.30/0.64 thf(fact_6981_Rep__int__inject,axiom,
% 0.30/0.64 ! [X: int,Y3: int] :
% 0.30/0.64 ( ( ( rep_int @ X )
% 0.30/0.64 = ( rep_int @ Y3 ) )
% 0.30/0.64 = ( X = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_int_inject
% 0.30/0.64 thf(fact_6982_Rep__int__inverse,axiom,
% 0.30/0.64 ! [X: int] :
% 0.30/0.64 ( ( abs_int @ ( rep_int @ X ) )
% 0.30/0.64 = X ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_int_inverse
% 0.30/0.64 thf(fact_6983_Rep__int,axiom,
% 0.30/0.64 ! [X: int] :
% 0.30/0.64 ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ ( rep_int @ X )
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_int
% 0.30/0.64 thf(fact_6984_Rep__int__cases,axiom,
% 0.30/0.64 ! [Y3: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ~ ! [X3: int] :
% 0.30/0.64 ( Y3
% 0.30/0.64 != ( rep_int @ X3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_int_cases
% 0.30/0.64 thf(fact_6985_Rep__int__induct,axiom,
% 0.30/0.64 ! [Y3: set @ ( product_prod @ nat @ nat ),P: ( set @ ( product_prod @ nat @ nat ) ) > $o] :
% 0.30/0.64 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 0.30/0.64 => ( ! [X3: int] : ( P @ ( rep_int @ X3 ) )
% 0.30/0.64 => ( P @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_int_induct
% 0.30/0.64 thf(fact_6986_Rep__Integ__def,axiom,
% 0.30/0.64 ( rep_Integ
% 0.30/0.64 = ( quot_rep @ int @ ( product_prod @ nat @ nat ) @ rep_int ) ) ).
% 0.30/0.64
% 0.30/0.64 % Rep_Integ_def
% 0.30/0.64 thf(fact_6987_type__definition__int,axiom,
% 0.30/0.64 ( type_definition @ int @ ( set @ ( product_prod @ nat @ nat ) ) @ rep_int @ abs_int
% 0.30/0.64 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 0.30/0.64 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 0.30/0.64 ? [X2: product_prod @ nat @ nat] :
% 0.30/0.64 ( ( intrel @ X2 @ X2 )
% 0.30/0.64 & ( C4
% 0.30/0.64 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % type_definition_int
% 0.30/0.64 thf(fact_6988_Domainp__cr__rat,axiom,
% 0.30/0.64 ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
% 0.30/0.64 = ( ^ [X2: product_prod @ int @ int] :
% 0.30/0.64 ( ( product_snd @ int @ int @ X2 )
% 0.30/0.64 != ( zero_zero @ int ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Domainp_cr_rat
% 0.30/0.64 thf(fact_6989_semilattice__neutr_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( ( comm_monoid @ A @ F @ Z3 )
% 0.30/0.64 => ( semilattice_neutr @ A @ F @ Z3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_neutr.intro
% 0.30/0.64 thf(fact_6990_inf_Osemilattice__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_inf @ A )
% 0.30/0.64 => ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % inf.semilattice_axioms
% 0.30/0.64 thf(fact_6991_semilattice__order_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.64 => ( semilattice @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order.axioms(1)
% 0.30/0.64 thf(fact_6992_max_Osemilattice__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % max.semilattice_axioms
% 0.30/0.64 thf(fact_6993_min_Osemilattice__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % min.semilattice_axioms
% 0.30/0.64 thf(fact_6994_semilattice_Oright__idem,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( ( F @ ( F @ A3 @ B2 ) @ B2 )
% 0.30/0.64 = ( F @ A3 @ B2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.right_idem
% 0.30/0.64 thf(fact_6995_semilattice_Oleft__idem,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( ( F @ A3 @ ( F @ A3 @ B2 ) )
% 0.30/0.64 = ( F @ A3 @ B2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.left_idem
% 0.30/0.64 thf(fact_6996_semilattice_Oidem,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,A3: A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( ( F @ A3 @ A3 )
% 0.30/0.64 = A3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.idem
% 0.30/0.64 thf(fact_6997_semilattice__set_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( lattic149705377957585745ce_set @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_set.intro
% 0.30/0.64 thf(fact_6998_semilattice__set_Oaxioms,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.30/0.64 => ( semilattice @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_set.axioms
% 0.30/0.64 thf(fact_6999_semilattice__set__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( lattic149705377957585745ce_set @ A )
% 0.30/0.64 = ( semilattice @ A ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_set_def
% 0.30/0.64 thf(fact_7000_semilattice__neutr_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( semilattice_neutr @ A @ F @ Z3 )
% 0.30/0.64 => ( semilattice @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_neutr.axioms(1)
% 0.30/0.64 thf(fact_7001_sup_Osemilattice__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_sup @ A )
% 0.30/0.64 => ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup.semilattice_axioms
% 0.30/0.64 thf(fact_7002_semilattice__neutr__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_neutr @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A] :
% 0.30/0.64 ( ( semilattice @ A @ F2 )
% 0.30/0.64 & ( comm_monoid @ A @ F2 @ Z7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_neutr_def
% 0.30/0.64 thf(fact_7003_semilattice__order_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less )
% 0.30/0.64 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order.intro
% 0.30/0.64 thf(fact_7004_semilattice__order__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_order @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ( ( semilattice @ A @ F2 )
% 0.30/0.64 & ( semila6385135966242565138axioms @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order_def
% 0.30/0.64 thf(fact_7005_semilattice__order__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,F: A > A > A,Less: A > A > $o] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 = ( A8
% 0.30/0.64 = ( F @ A8 @ B5 ) ) )
% 0.30/0.64 => ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 = ( ( A8
% 0.30/0.64 = ( F @ A8 @ B5 ) )
% 0.30/0.64 & ( A8 != B5 ) ) )
% 0.30/0.64 => ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order_axioms.intro
% 0.30/0.64 thf(fact_7006_semilattice__order__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semila6385135966242565138axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ( ! [A5: A,B3: A] :
% 0.30/0.64 ( ( Less_eq2 @ A5 @ B3 )
% 0.30/0.64 = ( A5
% 0.30/0.64 = ( F2 @ A5 @ B3 ) ) )
% 0.30/0.64 & ! [A5: A,B3: A] :
% 0.30/0.64 ( ( Less2 @ A5 @ B3 )
% 0.30/0.64 = ( ( A5
% 0.30/0.64 = ( F2 @ A5 @ B3 ) )
% 0.30/0.64 & ( A5 != B3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order_axioms_def
% 0.30/0.64 thf(fact_7007_semilattice__order_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.30/0.64 => ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_order.axioms(2)
% 0.30/0.64 thf(fact_7008_prod__list_Omonoid__list__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( monoid_mult @ A )
% 0.30/0.64 => ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % prod_list.monoid_list_axioms
% 0.30/0.64 thf(fact_7009_comm__monoid_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( comm_monoid @ A @ F @ Z3 )
% 0.30/0.64 => ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid.axioms(2)
% 0.30/0.64 thf(fact_7010_comm__monoid__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid_axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A] :
% 0.30/0.64 ! [A5: A] :
% 0.30/0.64 ( ( F2 @ A5 @ Z7 )
% 0.30/0.64 = A5 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_axioms_def
% 0.30/0.64 thf(fact_7011_comm__monoid__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ! [A8: A] :
% 0.30/0.64 ( ( F @ A8 @ Z3 )
% 0.30/0.64 = A8 )
% 0.30/0.64 => ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_axioms.intro
% 0.30/0.64 thf(fact_7012_sum__list_Omonoid__list__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( monoid_add @ A )
% 0.30/0.64 => ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum_list.monoid_list_axioms
% 0.30/0.64 thf(fact_7013_comm__monoid__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A,Z7: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F2 )
% 0.30/0.64 & ( comm_monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_def
% 0.30/0.64 thf(fact_7014_comm__monoid_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( ( comm_monoid_axioms @ A @ F @ Z3 )
% 0.30/0.64 => ( comm_monoid @ A @ F @ Z3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid.intro
% 0.30/0.64 thf(fact_7015_semilattice_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( abel_semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.axioms(1)
% 0.30/0.64 thf(fact_7016_sup_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_sup @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sup.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7017_comm__monoid_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A] :
% 0.30/0.64 ( ( comm_monoid @ A @ F @ Z3 )
% 0.30/0.64 => ( abel_semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid.axioms(1)
% 0.30/0.64 thf(fact_7018_abel__semigroup_Ocommute,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( ( F @ A3 @ B2 )
% 0.30/0.64 = ( F @ B2 @ A3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup.commute
% 0.30/0.64 thf(fact_7019_abel__semigroup_Oleft__commute,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( ( F @ B2 @ ( F @ A3 @ C2 ) )
% 0.30/0.64 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup.left_commute
% 0.30/0.64 thf(fact_7020_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.64 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.64 => ( abel_semigroup @ A @ Conj ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra.axioms(1)
% 0.30/0.64 thf(fact_7021_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.64 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.64 => ( abel_semigroup @ A @ Disj ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra.axioms(2)
% 0.30/0.64 thf(fact_7022_min_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % min.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7023_add_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ab_semigroup_add @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % add.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7024_max_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( linorder @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % max.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7025_mult_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( ab_semigroup_mult @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % mult.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7026_inf_Oabel__semigroup__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_inf @ A )
% 0.30/0.64 => ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % inf.abel_semigroup_axioms
% 0.30/0.64 thf(fact_7027_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( semigroup @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup.axioms(1)
% 0.30/0.64 thf(fact_7028_abel__semigroup__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( abel_semigroup @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A] :
% 0.30/0.64 ( ( semigroup @ A @ F2 )
% 0.30/0.64 & ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup_def
% 0.30/0.64 thf(fact_7029_abel__semigroup_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( semigroup @ A @ F )
% 0.30/0.64 => ( ( abel_s757365448890700780axioms @ A @ F )
% 0.30/0.64 => ( abel_semigroup @ A @ F ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup.intro
% 0.30/0.64 thf(fact_7030_abel__semigroup__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( abel_s757365448890700780axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A] :
% 0.30/0.64 ! [A5: A,B3: A] :
% 0.30/0.64 ( ( F2 @ A5 @ B3 )
% 0.30/0.64 = ( F2 @ B3 @ A5 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup_axioms_def
% 0.30/0.64 thf(fact_7031_abel__semigroup__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( F @ A8 @ B5 )
% 0.30/0.64 = ( F @ B5 @ A8 ) )
% 0.30/0.64 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup_axioms.intro
% 0.30/0.64 thf(fact_7032_abel__semigroup_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % abel_semigroup.axioms(2)
% 0.30/0.64 thf(fact_7033_semilattice_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F )
% 0.30/0.64 => ( ( semilattice_axioms @ A @ F )
% 0.30/0.64 => ( semilattice @ A @ F ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.intro
% 0.30/0.64 thf(fact_7034_semilattice__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ F2 )
% 0.30/0.64 & ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_def
% 0.30/0.64 thf(fact_7035_semilattice__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ! [A8: A] :
% 0.30/0.64 ( ( F @ A8 @ A8 )
% 0.30/0.64 = A8 )
% 0.30/0.64 => ( semilattice_axioms @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_axioms.intro
% 0.30/0.64 thf(fact_7036_semilattice__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( semilattice_axioms @ A )
% 0.30/0.64 = ( ^ [F2: A > A > A] :
% 0.30/0.64 ! [A5: A] :
% 0.30/0.64 ( ( F2 @ A5 @ A5 )
% 0.30/0.64 = A5 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice_axioms_def
% 0.30/0.64 thf(fact_7037_semilattice_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A] :
% 0.30/0.64 ( ( semilattice @ A @ F )
% 0.30/0.64 => ( semilattice_axioms @ A @ F ) ) ).
% 0.30/0.64
% 0.30/0.64 % semilattice.axioms(2)
% 0.30/0.64 thf(fact_7038_abstract__boolean__algebra__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( boolea2506097494486148201lgebra @ A )
% 0.30/0.64 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ Conj2 )
% 0.30/0.64 & ( abel_semigroup @ A @ Disj2 )
% 0.30/0.64 & ( boolea6902313364301356556axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra_def
% 0.30/0.64 thf(fact_7039_abstract__boolean__algebra_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.64 ( ( abel_semigroup @ A @ Conj )
% 0.30/0.64 => ( ( abel_semigroup @ A @ Disj )
% 0.30/0.64 => ( ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.64 => ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra.intro
% 0.30/0.64 thf(fact_7040_abstract__boolean__algebra__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( boolea6902313364301356556axioms @ A )
% 0.30/0.64 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
% 0.30/0.64 ( ! [X2: A,Y6: A,Z7: A] :
% 0.30/0.64 ( ( Conj2 @ X2 @ ( Disj2 @ Y6 @ Z7 ) )
% 0.30/0.64 = ( Disj2 @ ( Conj2 @ X2 @ Y6 ) @ ( Conj2 @ X2 @ Z7 ) ) )
% 0.30/0.64 & ! [X2: A,Y6: A,Z7: A] :
% 0.30/0.64 ( ( Disj2 @ X2 @ ( Conj2 @ Y6 @ Z7 ) )
% 0.30/0.64 = ( Conj2 @ ( Disj2 @ X2 @ Y6 ) @ ( Disj2 @ X2 @ Z7 ) ) )
% 0.30/0.64 & ! [X2: A] :
% 0.30/0.64 ( ( Conj2 @ X2 @ One )
% 0.30/0.64 = X2 )
% 0.30/0.64 & ! [X2: A] :
% 0.30/0.64 ( ( Disj2 @ X2 @ Zero2 )
% 0.30/0.64 = X2 )
% 0.30/0.64 & ! [X2: A] :
% 0.30/0.64 ( ( Conj2 @ X2 @ ( Compl2 @ X2 ) )
% 0.30/0.64 = Zero2 )
% 0.30/0.64 & ! [X2: A] :
% 0.30/0.64 ( ( Disj2 @ X2 @ ( Compl2 @ X2 ) )
% 0.30/0.64 = One ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra_axioms_def
% 0.30/0.64 thf(fact_7041_abstract__boolean__algebra__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,One2: A,Zero: A,Compl: A > A] :
% 0.30/0.64 ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.64 ( ( Conj @ X3 @ ( Disj @ Y5 @ Z4 ) )
% 0.30/0.64 = ( Disj @ ( Conj @ X3 @ Y5 ) @ ( Conj @ X3 @ Z4 ) ) )
% 0.30/0.64 => ( ! [X3: A,Y5: A,Z4: A] :
% 0.30/0.64 ( ( Disj @ X3 @ ( Conj @ Y5 @ Z4 ) )
% 0.30/0.64 = ( Conj @ ( Disj @ X3 @ Y5 ) @ ( Disj @ X3 @ Z4 ) ) )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( Conj @ X3 @ One2 )
% 0.30/0.64 = X3 )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( Disj @ X3 @ Zero )
% 0.30/0.64 = X3 )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( Conj @ X3 @ ( Compl @ X3 ) )
% 0.30/0.64 = Zero )
% 0.30/0.64 => ( ! [X3: A] :
% 0.30/0.64 ( ( Disj @ X3 @ ( Compl @ X3 ) )
% 0.30/0.64 = One2 )
% 0.30/0.64 => ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra_axioms.intro
% 0.30/0.64 thf(fact_7042_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
% 0.30/0.64 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 0.30/0.64 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 0.30/0.64 => ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % abstract_boolean_algebra.axioms(3)
% 0.30/0.64 thf(fact_7043_typerep_Osize__neq,axiom,
% 0.30/0.64 ! [X: typerep] :
% 0.30/0.64 ( ( size_size @ typerep @ X )
% 0.30/0.64 != ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % typerep.size_neq
% 0.30/0.64 thf(fact_7044_tuple__isomorphism_Osize__neq,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism @ A @ B @ C] :
% 0.30/0.64 ( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X )
% 0.30/0.64 != ( zero_zero @ nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % tuple_isomorphism.size_neq
% 0.30/0.64 thf(fact_7045_tuple__isomorphism_Osize_I2_J,axiom,
% 0.30/0.64 ! [B: $tType,C: $tType,A: $tType,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
% 0.30/0.64 ( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
% 0.30/0.64 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tuple_isomorphism.size(2)
% 0.30/0.64 thf(fact_7046_typerep_Osize_I2_J,axiom,
% 0.30/0.64 ! [X13: literal,X22: list @ typerep] :
% 0.30/0.64 ( ( size_size @ typerep @ ( typerep3 @ X13 @ X22 ) )
% 0.30/0.64 = ( plus_plus @ nat @ ( size_list @ typerep @ ( size_size @ typerep ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % typerep.size(2)
% 0.30/0.64 thf(fact_7047_typerep_Osize__gen,axiom,
% 0.30/0.64 ! [X13: literal,X22: list @ typerep] :
% 0.30/0.64 ( ( size_typerep @ ( typerep3 @ X13 @ X22 ) )
% 0.30/0.64 = ( plus_plus @ nat @ ( size_list @ typerep @ size_typerep @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % typerep.size_gen
% 0.30/0.64 thf(fact_7048_tuple__isomorphism_Osize__gen,axiom,
% 0.30/0.64 ! [B: $tType,C: $tType,A: $tType,Xb3: A > nat,Xa3: B > nat,X: C > nat,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
% 0.30/0.64 ( ( tuple_9181185373184732606rphism @ A @ B @ C @ Xb3 @ Xa3 @ X @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
% 0.30/0.64 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tuple_isomorphism.size_gen
% 0.30/0.64 thf(fact_7049_typerep__fun__box__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( ( typerep2 @ A )
% 0.30/0.64 & ( typerep2 @ B ) )
% 0.30/0.64 => ( ( typerep_typerep @ ( fun_box @ A @ B ) )
% 0.30/0.64 = ( ^ [T7: itself @ ( fun_box @ A @ B )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( cons @ typerep @ ( typerep_typerep @ B @ ( type2 @ B ) ) @ ( nil @ typerep ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % typerep_fun_box_def
% 0.30/0.64 thf(fact_7050_typerep__word__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( typerep2 @ A )
% 0.30/0.64 => ( ( typerep_typerep @ ( word @ A ) )
% 0.30/0.64 = ( ^ [T7: itself @ ( word @ A )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( nil @ typerep ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % typerep_word_def
% 0.30/0.64 thf(fact_7051_dual__order_Opreordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( preordering @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % dual_order.preordering_axioms
% 0.30/0.64 thf(fact_7052_irreflp__greater,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( irreflp @ A
% 0.30/0.64 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % irreflp_greater
% 0.30/0.64 thf(fact_7053_preordering_Oaxioms_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( partial_preordering @ A @ Less_eq ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.axioms(1)
% 0.30/0.64 thf(fact_7054_irreflp__less,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % irreflp_less
% 0.30/0.64 thf(fact_7055_preordering__dualI,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( preordering @ A
% 0.30/0.64 @ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
% 0.30/0.64 @ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
% 0.30/0.64 => ( preordering @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering_dualI
% 0.30/0.64 thf(fact_7056_preordering_Oasym,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 => ~ ( Less @ B2 @ A3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.asym
% 0.30/0.64 thf(fact_7057_preordering_Oirrefl,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ~ ( Less @ A3 @ A3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.irrefl
% 0.30/0.64 thf(fact_7058_preordering_Ostrict__trans,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 => ( ( Less @ B2 @ C2 )
% 0.30/0.64 => ( Less @ A3 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.strict_trans
% 0.30/0.64 thf(fact_7059_preordering_Ostrict__trans1,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 => ( ( Less @ B2 @ C2 )
% 0.30/0.64 => ( Less @ A3 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.strict_trans1
% 0.30/0.64 thf(fact_7060_preordering_Ostrict__trans2,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 => ( ( Less_eq @ B2 @ C2 )
% 0.30/0.64 => ( Less @ A3 @ C2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.strict_trans2
% 0.30/0.64 thf(fact_7061_preordering_Ostrict__iff__not,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 = ( ( Less_eq @ A3 @ B2 )
% 0.30/0.64 & ~ ( Less_eq @ B2 @ A3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.strict_iff_not
% 0.30/0.64 thf(fact_7062_preordering_Ostrict__implies__order,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( ( Less @ A3 @ B2 )
% 0.30/0.64 => ( Less_eq @ A3 @ B2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.strict_implies_order
% 0.30/0.64 thf(fact_7063_preordering__strictI,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 = ( ( Less @ A8 @ B5 )
% 0.30/0.64 | ( A8 = B5 ) ) )
% 0.30/0.64 => ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 => ~ ( Less @ B5 @ A8 ) )
% 0.30/0.64 => ( ! [A8: A] :
% 0.30/0.64 ~ ( Less @ A8 @ A8 )
% 0.30/0.64 => ( ! [A8: A,B5: A,C3: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 => ( ( Less @ B5 @ C3 )
% 0.30/0.64 => ( Less @ A8 @ C3 ) ) )
% 0.30/0.64 => ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering_strictI
% 0.30/0.64 thf(fact_7064_order_Opreordering__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preorder @ A )
% 0.30/0.64 => ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % order.preordering_axioms
% 0.30/0.64 thf(fact_7065_preordering__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preordering @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq2 )
% 0.30/0.64 & ( preordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering_def
% 0.30/0.64 thf(fact_7066_preordering_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( partial_preordering @ A @ Less_eq )
% 0.30/0.64 => ( ( preordering_axioms @ A @ Less_eq @ Less )
% 0.30/0.64 => ( preordering @ A @ Less_eq @ Less ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.intro
% 0.30/0.64 thf(fact_7067_preordering__axioms__def,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( preordering_axioms @ A )
% 0.30/0.64 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.30/0.64 ! [A5: A,B3: A] :
% 0.30/0.64 ( ( Less2 @ A5 @ B3 )
% 0.30/0.64 = ( ( Less_eq2 @ A5 @ B3 )
% 0.30/0.64 & ~ ( Less_eq2 @ B3 @ A5 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering_axioms_def
% 0.30/0.64 thf(fact_7068_preordering__axioms_Ointro,axiom,
% 0.30/0.64 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 0.30/0.64 ( ! [A8: A,B5: A] :
% 0.30/0.64 ( ( Less @ A8 @ B5 )
% 0.30/0.64 = ( ( Less_eq @ A8 @ B5 )
% 0.30/0.64 & ~ ( Less_eq @ B5 @ A8 ) ) )
% 0.30/0.64 => ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering_axioms.intro
% 0.30/0.64 thf(fact_7069_preordering_Oaxioms_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.64 ( ( preordering @ A @ Less_eq @ Less )
% 0.30/0.64 => ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% 0.30/0.64
% 0.30/0.64 % preordering.axioms(2)
% 0.30/0.64 thf(fact_7070_comm__monoid__set_Ozero__middle,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ Z3 @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.zero_middle
% 0.30/0.64 thf(fact_7071_Heap_Oset__map,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
% 0.30/0.64 ( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 0.30/0.64 = ( image @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.set_map
% 0.30/0.64 thf(fact_7072_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastAtMost_shift_0
% 0.30/0.64 thf(fact_7073_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.64 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_atMost_pred_shift
% 0.30/0.64 thf(fact_7074_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ( comp @ nat @ A @ nat @ G
% 0.30/0.64 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.30/0.64 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_lessThan_pred_shift
% 0.30/0.64 thf(fact_7075_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast0_lessThan_Suc_shift
% 0.30/0.64 thf(fact_7076_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast0_atMost_Suc_shift
% 0.30/0.64 thf(fact_7077_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastLessThan_shift_0
% 0.30/0.64 thf(fact_7078_comm__monoid__add__class_Osum__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( comm_monoid_add @ A )
% 0.30/0.64 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_add_class.sum_def
% 0.30/0.64 thf(fact_7079_comm__monoid__set_OatMost__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.64 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.64 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atMost_shift
% 0.30/0.64 thf(fact_7080_comm__monoid__set_OlessThan__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.64 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.lessThan_Suc_shift
% 0.30/0.64 thf(fact_7081_comm__monoid__set_OatMost__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.30/0.64 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atMost_Suc_shift
% 0.30/0.64 thf(fact_7082_sum_Ocomm__monoid__set__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid_add @ A )
% 0.30/0.64 => ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sum.comm_monoid_set_axioms
% 0.30/0.64 thf(fact_7083_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast0_atMost_Suc
% 0.30/0.64 thf(fact_7084_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast0_lessThan_Suc
% 0.30/0.64 thf(fact_7085_comm__monoid__set_Onested__swap,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.64 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.nested_swap
% 0.30/0.64 thf(fact_7086_comm__monoid__set_OatLeast1__atMost__eq,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast1_atMost_eq
% 0.30/0.64 thf(fact_7087_comm__monoid__set_Oin__pairs__0,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.in_pairs_0
% 0.30/0.64 thf(fact_7088_comm__monoid__set_Ocl__ivl__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.64 = Z3 ) )
% 0.30/0.64 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.cl_ivl_Suc
% 0.30/0.64 thf(fact_7089_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less @ nat @ M @ N )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 0.30/0.64 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_Suc_lessThan
% 0.30/0.64 thf(fact_7090_comm__monoid__set_Oop__ivl__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.64 = Z3 ) )
% 0.30/0.64 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.op_ivl_Suc
% 0.30/0.64 thf(fact_7091_comm__monoid__set_Ohead__if,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ( ord_less @ nat @ N @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = Z3 ) )
% 0.30/0.64 & ( ~ ( ord_less @ nat @ N @ M )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.head_if
% 0.30/0.64 thf(fact_7092_comm__monoid__set_Oivl__cong,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType] :
% 0.30/0.64 ( ( ord @ B )
% 0.30/0.64 => ! [F: A > A > A,Z3: A,A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( A3 = C2 )
% 0.30/0.64 => ( ( B2 = D2 )
% 0.30/0.64 => ( ! [X3: B] :
% 0.30/0.64 ( ( ord_less_eq @ B @ C2 @ X3 )
% 0.30/0.64 => ( ( ord_less @ B @ X3 @ D2 )
% 0.30/0.64 => ( ( G @ X3 )
% 0.30/0.64 = ( H2 @ X3 ) ) ) )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.ivl_cong
% 0.30/0.64 thf(fact_7093_comm__monoid__set_Onested__swap_H,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [J3: nat] :
% 0.30/0.64 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.nested_swap'
% 0.30/0.64 thf(fact_7094_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.shift_bounds_cl_Suc_ivl
% 0.30/0.64 thf(fact_7095_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.shift_bounds_Suc_ivl
% 0.30/0.64 thf(fact_7096_comm__monoid__set_OlessThan__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.lessThan_Suc
% 0.30/0.64 thf(fact_7097_comm__monoid__set_OatMost__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atMost_Suc
% 0.30/0.64 thf(fact_7098_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
% 0.30/0.64 thf(fact_7099_comm__monoid__set_OatLeastLessThan__rev,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 0.30/0.64 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastLessThan_rev
% 0.30/0.64 thf(fact_7100_prod_Ocomm__monoid__set__axioms,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( comm_monoid_mult @ A )
% 0.30/0.64 => ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % prod.comm_monoid_set_axioms
% 0.30/0.64 thf(fact_7101_comm__monoid__set_Onat__diff__reindex,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.nat_diff_reindex
% 0.30/0.64 thf(fact_7102_comm__monoid__set_Onat__group,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,K: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [M2: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.nat_group
% 0.30/0.64 thf(fact_7103_comm__monoid__set_Oub__add__nat,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A,P5: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.ub_add_nat
% 0.30/0.64 thf(fact_7104_comm__monoid__set_Ohead,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.head
% 0.30/0.64 thf(fact_7105_comm__monoid__set_OatLeast__Suc__atMost,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_Suc_atMost
% 0.30/0.64 thf(fact_7106_comm__monoid__set_Onat__ivl__Suc_H,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.nat_ivl_Suc'
% 0.30/0.64 thf(fact_7107_comm__monoid__set_OatLeastLessThan__Suc,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,A3: nat,B2: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastLessThan_Suc
% 0.30/0.64 thf(fact_7108_comm__monoid__set_OatLeastLessThan__concat,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,P5: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.30/0.64 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeastLessThan_concat
% 0.30/0.64 thf(fact_7109_comm__monoid__set_Olast__plus,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 0.30/0.64 = ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.last_plus
% 0.30/0.64 thf(fact_7110_comm__monoid__set_OSuc__reindex__ivl,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.64 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.64 = ( F @ ( G @ M )
% 0.30/0.64 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.Suc_reindex_ivl
% 0.30/0.64 thf(fact_7111_Heap_Oinj__map__strong,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa3: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
% 0.30/0.64 ( ! [Z4: A,Za: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.30/0.64 => ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa3 ) )
% 0.30/0.64 => ( ( ( F @ Z4 )
% 0.30/0.64 = ( Fa @ Za ) )
% 0.30/0.64 => ( Z4 = Za ) ) ) )
% 0.30/0.64 => ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.30/0.64 = ( heap_Time_map_Heap @ A @ B @ Fa @ Xa3 ) )
% 0.30/0.64 => ( X = Xa3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.inj_map_strong
% 0.30/0.64 thf(fact_7112_Heap_Omap__cong0,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.30/0.64 ( ! [Z4: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.30/0.64 => ( ( F @ Z4 )
% 0.30/0.64 = ( G @ Z4 ) ) )
% 0.30/0.64 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.30/0.64 = ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.map_cong0
% 0.30/0.64 thf(fact_7113_Heap_Omap__cong,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.30/0.64 ( ( X = Ya )
% 0.30/0.64 => ( ! [Z4: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.30/0.64 => ( ( F @ Z4 )
% 0.30/0.64 = ( G @ Z4 ) ) )
% 0.30/0.64 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.30/0.64 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.map_cong
% 0.30/0.64 thf(fact_7114_comm__monoid__set_OIf__cases,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,F: A > A > A,Z3: A,A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( finite_finite2 @ B @ A6 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z3
% 0.30/0.64 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 0.30/0.64 @ A6 )
% 0.30/0.64 = ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.If_cases
% 0.30/0.64 thf(fact_7115_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
% 0.30/0.64 thf(fact_7116_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_Suc_atMost_Suc_shift
% 0.30/0.64 thf(fact_7117_comm__monoid__mult__class_Oprod__def,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( comm_monoid_mult @ A )
% 0.30/0.64 => ( ( groups7121269368397514597t_prod @ B @ A )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_mult_class.prod_def
% 0.30/0.64 thf(fact_7118_comm__monoid__set_Oin__pairs,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 0.30/0.64 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.in_pairs
% 0.30/0.64 thf(fact_7119_comm__monoid__set_Otriangle__reindex,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [K3: nat] :
% 0.30/0.64 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.64 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.triangle_reindex
% 0.30/0.64 thf(fact_7120_comm__monoid__set_Otriangle__reindex__eq,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.30/0.64 @ ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.64 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.64 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [K3: nat] :
% 0.30/0.64 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 0.30/0.64 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ K3 ) )
% 0.30/0.64 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.triangle_reindex_eq
% 0.30/0.64 thf(fact_7121_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_int_atMost_int_shift
% 0.30/0.64 thf(fact_7122_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
% 0.30/0.64 ! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
% 0.30/0.64 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 0.30/0.64 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.30/0.64 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % comm_monoid_set.atLeast_int_lessThan_int_shift
% 0.30/0.64 thf(fact_7123_Heap_Oin__rel,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType] :
% 0.30/0.64 ( ( heap_Time_rel_Heap @ A @ B )
% 0.30/0.64 = ( ^ [R6: A > B > $o,A5: heap_Time_Heap @ A,B3: heap_Time_Heap @ B] :
% 0.30/0.64 ? [Z7: heap_Time_Heap @ ( product_prod @ A @ B )] :
% 0.30/0.64 ( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z7
% 0.30/0.64 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 0.30/0.64 @ ^ [X2: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 0.30/0.64 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z7 )
% 0.30/0.64 = A5 )
% 0.30/0.64 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z7 )
% 0.30/0.64 = B3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.in_rel
% 0.30/0.64 thf(fact_7124_Heap_Opred__set,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap_Time_pred_Heap @ A )
% 0.30/0.64 = ( ^ [P4: A > $o,X2: heap_Time_Heap @ A] :
% 0.30/0.64 ! [Y6: A] :
% 0.30/0.64 ( ( member @ A @ Y6 @ ( heap_Time_set_Heap @ A @ X2 ) )
% 0.30/0.64 => ( P4 @ Y6 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_set
% 0.30/0.64 thf(fact_7125_Heap_ODomainp__rel,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,R2: A > B > $o] :
% 0.30/0.64 ( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) )
% 0.30/0.64 = ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.Domainp_rel
% 0.30/0.64 thf(fact_7126_Heap_Opred__True,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap_Time_pred_Heap @ A
% 0.30/0.64 @ ^ [Uu: A] : $true )
% 0.30/0.64 = ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_True
% 0.30/0.64 thf(fact_7127_Heap_Orel__eq,axiom,
% 0.30/0.64 ! [A: $tType] :
% 0.30/0.64 ( ( heap_Time_rel_Heap @ A @ A
% 0.30/0.64 @ ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 0.30/0.64 = ( ^ [Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] : ( Y = Z2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_eq
% 0.30/0.64 thf(fact_7128_Heap_Orel__refl,axiom,
% 0.30/0.64 ! [B: $tType,Ra: B > B > $o,X: heap_Time_Heap @ B] :
% 0.30/0.64 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 0.30/0.64 => ( heap_Time_rel_Heap @ B @ B @ Ra @ X @ X ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_refl
% 0.30/0.64 thf(fact_7129_Heap_Orel__transp,axiom,
% 0.30/0.64 ! [A: $tType,R2: A > A > $o] :
% 0.30/0.64 ( ( transp @ A @ R2 )
% 0.30/0.64 => ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_transp
% 0.30/0.64 thf(fact_7130_Heap_Opred__mono,axiom,
% 0.30/0.64 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 0.30/0.64 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 0.30/0.64 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_mono
% 0.30/0.64 thf(fact_7131_Heap_Orel__mono,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,R2: A > B > $o,Ra: A > B > $o] :
% 0.30/0.64 ( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra )
% 0.30/0.64 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_mono
% 0.30/0.64 thf(fact_7132_Heap_Obi__total__rel,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,R2: A > B > $o] :
% 0.30/0.64 ( ( bi_total @ A @ B @ R2 )
% 0.30/0.64 => ( bi_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.bi_total_rel
% 0.30/0.64 thf(fact_7133_Heap_Orel__transfer,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.30/0.64 ( 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 )
% 0.30/0.64 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.30/0.64 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.30/0.64 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) ) )
% 0.30/0.64 @ ( 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 )
% 0.30/0.64 @ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
% 0.30/0.64 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) ) )
% 0.30/0.64 @ ( heap_Time_rel_Heap @ A @ B )
% 0.30/0.64 @ ( heap_Time_rel_Heap @ C @ D ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_transfer
% 0.30/0.64 thf(fact_7134_Heap_Opred__transfer,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,R2: A > B > $o] :
% 0.30/0.64 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
% 0.30/0.64 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
% 0.30/0.64 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.64 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R2 )
% 0.30/0.64 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 0.30/0.64 @ ( heap_Time_pred_Heap @ A )
% 0.30/0.64 @ ( heap_Time_pred_Heap @ B ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_transfer
% 0.30/0.64 thf(fact_7135_Heap_Omap__transfer,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,F7: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F7 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F7 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E4 ) > ( heap_Time_Heap @ F7 ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F7 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E4 ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F7 ) @ ( heap_Time_rel_Heap @ A @ E4 @ Rb ) @ ( heap_Time_rel_Heap @ B @ F7 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E4 @ F7 ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.map_transfer
% 0.30/0.64 thf(fact_7136_Heap_Omap__cong__pred,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.30/0.64 ( ( X = Ya )
% 0.30/0.64 => ( ( heap_Time_pred_Heap @ A
% 0.30/0.64 @ ^ [Z7: A] :
% 0.30/0.64 ( ( F @ Z7 )
% 0.30/0.64 = ( G @ Z7 ) )
% 0.30/0.64 @ Ya )
% 0.30/0.64 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.30/0.64 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.map_cong_pred
% 0.30/0.64 thf(fact_7137_Heap_Orel__map_I1_J,axiom,
% 0.30/0.64 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B] :
% 0.30/0.64 ( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y3 )
% 0.30/0.64 = ( heap_Time_rel_Heap @ A @ B
% 0.30/0.64 @ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
% 0.30/0.64 @ X
% 0.30/0.64 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_map(1)
% 0.30/0.64 thf(fact_7138_Heap_Orel__map_I2_J,axiom,
% 0.30/0.64 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y3: heap_Time_Heap @ B] :
% 0.30/0.64 ( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y3 ) )
% 0.30/0.64 = ( heap_Time_rel_Heap @ A @ B
% 0.30/0.64 @ ^ [X2: A,Y6: B] : ( Sa @ X2 @ ( G @ Y6 ) )
% 0.30/0.64 @ X
% 0.30/0.64 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_map(2)
% 0.30/0.64 thf(fact_7139_Heap_OQuotient,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,R2: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 0.30/0.64 ( ( quotient @ A @ B @ R2 @ Abs @ Rep @ T6 )
% 0.30/0.64 => ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.Quotient
% 0.30/0.64 thf(fact_7140_Heap_Opred__map,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
% 0.30/0.64 ( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
% 0.30/0.64 = ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_map
% 0.30/0.64 thf(fact_7141_Heap_Opred__mono__strong,axiom,
% 0.30/0.64 ! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
% 0.30/0.64 ( ( heap_Time_pred_Heap @ A @ P @ X )
% 0.30/0.64 => ( ! [Z4: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.30/0.64 => ( ( P @ Z4 )
% 0.30/0.64 => ( Pa @ Z4 ) ) )
% 0.30/0.64 => ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_mono_strong
% 0.30/0.64 thf(fact_7142_Heap_Orel__refl__strong,axiom,
% 0.30/0.64 ! [A: $tType,X: heap_Time_Heap @ A,Ra: A > A > $o] :
% 0.30/0.64 ( ! [Z4: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.30/0.64 => ( Ra @ Z4 @ Z4 ) )
% 0.30/0.64 => ( heap_Time_rel_Heap @ A @ A @ Ra @ X @ X ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_refl_strong
% 0.30/0.64 thf(fact_7143_Heap_Orel__mono__strong,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,R2: A > B > $o,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Ra: A > B > $o] :
% 0.30/0.64 ( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
% 0.30/0.64 => ( ! [Z4: A,Yb: B] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.30/0.64 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y3 ) )
% 0.30/0.64 => ( ( R2 @ Z4 @ Yb )
% 0.30/0.64 => ( Ra @ Z4 @ Yb ) ) ) )
% 0.30/0.64 => ( heap_Time_rel_Heap @ A @ B @ Ra @ X @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_mono_strong
% 0.30/0.64 thf(fact_7144_Heap_Opred__cong,axiom,
% 0.30/0.64 ! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
% 0.30/0.64 ( ( X = Ya )
% 0.30/0.64 => ( ! [Z4: A] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.30/0.64 => ( ( P @ Z4 )
% 0.30/0.64 = ( Pa @ Z4 ) ) )
% 0.30/0.64 => ( ( heap_Time_pred_Heap @ A @ P @ X )
% 0.30/0.64 = ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.pred_cong
% 0.30/0.64 thf(fact_7145_Heap_Orel__cong,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Xa3: heap_Time_Heap @ B,R2: A > B > $o,Ra: A > B > $o] :
% 0.30/0.64 ( ( X = Ya )
% 0.30/0.64 => ( ( Y3 = Xa3 )
% 0.30/0.64 => ( ! [Z4: A,Yb: B] :
% 0.30/0.64 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.30/0.64 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa3 ) )
% 0.30/0.64 => ( ( R2 @ Z4 @ Yb )
% 0.30/0.64 = ( Ra @ Z4 @ Yb ) ) ) )
% 0.30/0.64 => ( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
% 0.30/0.64 = ( heap_Time_rel_Heap @ A @ B @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Heap.rel_cong
% 0.30/0.64 thf(fact_7146_Fract__def,axiom,
% 0.30/0.64 ( fract
% 0.30/0.64 = ( 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 )
% 0.30/0.64 @ ^ [A5: int,B3: int] :
% 0.30/0.64 ( if @ ( product_prod @ int @ int )
% 0.30/0.64 @ ( B3
% 0.30/0.64 = ( zero_zero @ int ) )
% 0.30/0.64 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.30/0.64 @ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Fract_def
% 0.30/0.64 thf(fact_7147_subset_Osuc__Union__closed__suc,axiom,
% 0.30/0.64 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 0.30/0.64 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.64 => ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 )
% 0.30/0.64 = Y7 )
% 0.30/0.64 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % subset.suc_Union_closed_suc
% 0.30/0.64 thf(fact_7148_predicate2I,axiom,
% 0.30/0.64 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.30/0.64 ( ! [X3: A,Y5: B] :
% 0.30/0.64 ( ( P @ X3 @ Y5 )
% 0.30/0.64 => ( Q @ X3 @ Y5 ) )
% 0.30/0.64 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.30/0.64
% 0.30/0.64 % predicate2I
% 0.30/0.64 thf(fact_7149_predicate1I,axiom,
% 0.30/0.64 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.64 ( ! [X3: A] :
% 0.30/0.64 ( ( P @ X3 )
% 0.30/0.64 => ( Q @ X3 ) )
% 0.30/0.64 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.30/0.64
% 0.30/0.64 % predicate1I
% 0.30/0.64 thf(fact_7150_of__int__eq__id,axiom,
% 0.30/0.64 ( ( ring_1_of_int @ int )
% 0.30/0.64 = ( id @ int ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_int_eq_id
% 0.30/0.64 thf(fact_7151_predicate1D,axiom,
% 0.30/0.64 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.30/0.64 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.64 => ( ( P @ X )
% 0.30/0.64 => ( Q @ X ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % predicate1D
% 0.30/0.64 thf(fact_7152_predicate2D,axiom,
% 0.30/0.64 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
% 0.30/0.64 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.65 => ( ( P @ X @ Y3 )
% 0.30/0.65 => ( Q @ X @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % predicate2D
% 0.30/0.65 thf(fact_7153_rev__predicate1D,axiom,
% 0.30/0.65 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.30/0.65 ( ( P @ X )
% 0.30/0.65 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.30/0.65 => ( Q @ X ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % rev_predicate1D
% 0.30/0.65 thf(fact_7154_rev__predicate2D,axiom,
% 0.30/0.65 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
% 0.30/0.65 ( ( P @ X @ Y3 )
% 0.30/0.65 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.65 => ( Q @ X @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % rev_predicate2D
% 0.30/0.65 thf(fact_7155_subset_Osuc__Union__closed_Ocases,axiom,
% 0.30/0.65 ! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ! [X10: set @ ( set @ A )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 0.30/0.65 => ~ ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 0.30/0.65 => ~ ! [X10: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
% 0.30/0.65 => ~ ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed.cases
% 0.30/0.65 thf(fact_7156_subset_Osuc__Union__closed_Osimps,axiom,
% 0.30/0.65 ! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 = ( ? [X7: set @ ( set @ A )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 0.30/0.65 & ( member @ ( set @ ( set @ A ) ) @ X7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 0.30/0.65 | ? [X7: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
% 0.30/0.65 & ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X7 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed.simps
% 0.30/0.65 thf(fact_7157_subset_Osuc__Union__closed_Osuc,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( member @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed.suc
% 0.30/0.65 thf(fact_7158_subset_Oeq__suc__Union,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.65 = X8 )
% 0.30/0.65 = ( X8
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.eq_suc_Union
% 0.30/0.65 thf(fact_7159_subset_Osuc__Union__closed__Union,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_Union
% 0.30/0.65 thf(fact_7160_subset_Osuc__Union__closed__in__carrier,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_in_carrier
% 0.30/0.65 thf(fact_7161_subset_Osuc__Union__closed__total,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 0.30/0.65 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_total
% 0.30/0.65 thf(fact_7162_subset_Osuc__Union__closed__induct,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Q: ( set @ ( set @ A ) ) > $o] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ! [X10: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( Q @ X10 )
% 0.30/0.65 => ( Q @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) ) ) )
% 0.30/0.65 => ( ! [X10: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ! [X4: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X4 @ X10 )
% 0.30/0.65 => ( Q @ X4 ) )
% 0.30/0.65 => ( Q @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) ) ) )
% 0.30/0.65 => ( Q @ X8 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_induct
% 0.30/0.65 thf(fact_7163_subset_Osuc__Union__closed__cases,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ! [Y10: set @ ( set @ A )] :
% 0.30/0.65 ( ( X8
% 0.30/0.65 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y10 ) )
% 0.30/0.65 => ~ ( member @ ( set @ ( set @ A ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 0.30/0.65 => ~ ! [Y10: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( X8
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Y10 ) )
% 0.30/0.65 => ~ ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_cases
% 0.30/0.65 thf(fact_7164_subset_Osuc__Union__closed_OUnion,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed.Union
% 0.30/0.65 thf(fact_7165_subset_Osuc__Union__closed_Ointros_I2_J,axiom,
% 0.30/0.65 ! [A: $tType,Xa5: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Xa5 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed.intros(2)
% 0.30/0.65 thf(fact_7166_subset_Osuc__Union__closed__chain,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_chain
% 0.30/0.65 thf(fact_7167_subset_Osuc__Union__closed__empty,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_empty
% 0.30/0.65 thf(fact_7168_natural_Odomain,axiom,
% 0.30/0.65 ! [P: nat > $o] :
% 0.30/0.65 ( ( ( domainp @ nat @ nat
% 0.30/0.65 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.65 = P )
% 0.30/0.65 => ( ( domainp @ nat @ code_natural @ code_pcr_natural )
% 0.30/0.65 = P ) ) ).
% 0.30/0.65
% 0.30/0.65 % natural.domain
% 0.30/0.65 thf(fact_7169_integer_Odomain,axiom,
% 0.30/0.65 ! [P: int > $o] :
% 0.30/0.65 ( ( ( domainp @ int @ int
% 0.30/0.65 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 0.30/0.65 = P )
% 0.30/0.65 => ( ( domainp @ int @ code_integer @ code_pcr_integer )
% 0.30/0.65 = P ) ) ).
% 0.30/0.65
% 0.30/0.65 % integer.domain
% 0.30/0.65 thf(fact_7170_subset_Osuc__Union__closed__subsetD,axiom,
% 0.30/0.65 ! [A: $tType,Y7: set @ ( set @ A ),X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 )
% 0.30/0.65 => ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( X8 = Y7 )
% 0.30/0.65 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_subsetD
% 0.30/0.65 thf(fact_7171_subset_Osuc__Union__closed__total_H,axiom,
% 0.30/0.65 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ! [Z9: set @ ( set @ A )] :
% 0.30/0.65 ( ( member @ ( set @ ( set @ A ) ) @ Z9 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 0.30/0.65 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Z9 @ Y7 )
% 0.30/0.65 => ( ( Z9 = Y7 )
% 0.30/0.65 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Z9 ) @ Y7 ) ) ) )
% 0.30/0.65 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 0.30/0.65 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_total'
% 0.30/0.65 thf(fact_7172_length__remdups__leq,axiom,
% 0.30/0.65 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.30/0.65
% 0.30/0.65 % length_remdups_leq
% 0.30/0.65 thf(fact_7173_subset_Osuc__Union__closed__def,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
% 0.30/0.65 = ( collect @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closed_def
% 0.30/0.65 thf(fact_7174_sorted__remdups,axiom,
% 0.30/0.65 ! [A: $tType] :
% 0.30/0.65 ( ( linorder @ A )
% 0.30/0.65 => ! [Xs: list @ A] :
% 0.30/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.30/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sorted_remdups
% 0.30/0.65 thf(fact_7175_subset_Osuc__Union__closedp__suc__Union__closed__eq,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A )] :
% 0.30/0.65 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
% 0.30/0.65 = ( ^ [X2: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ X2 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp_suc_Union_closed_eq
% 0.30/0.65 thf(fact_7176_subset_Osuc__Union__closedp_Osuc,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 0.30/0.65 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 0.30/0.65 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp.suc
% 0.30/0.65 thf(fact_7177_subset_Osuc__Union__closedp_Osimps,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
% 0.30/0.65 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
% 0.30/0.65 = ( ? [X7: set @ ( set @ A )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 0.30/0.65 & ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 0.30/0.65 | ? [X7: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
% 0.30/0.65 & ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X7 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp.simps
% 0.30/0.65 thf(fact_7178_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A ),Xa5: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ Xa5 )
% 0.30/0.65 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp.intros(2)
% 0.30/0.65 thf(fact_7179_subset_Osuc__Union__closedp_OUnion,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X8 )
% 0.30/0.65 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp.Union
% 0.30/0.65 thf(fact_7180_subset_Osuc__Union__closedp_Ocases,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
% 0.30/0.65 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
% 0.30/0.65 => ( ! [X10: set @ ( set @ A )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 0.30/0.65 => ~ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 0.30/0.65 => ~ ! [X10: set @ ( set @ ( set @ A ) )] :
% 0.30/0.65 ( ( A3
% 0.30/0.65 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
% 0.30/0.65 => ~ ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X10 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset.suc_Union_closedp.cases
% 0.30/0.65 thf(fact_7181_int_Odomain,axiom,
% 0.30/0.65 ! [DR1: nat > $o,DR2: nat > $o] :
% 0.30/0.65 ( ( ( domainp @ nat @ nat
% 0.30/0.65 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.65 = DR1 )
% 0.30/0.65 => ( ( ( domainp @ nat @ nat
% 0.30/0.65 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 0.30/0.65 = DR2 )
% 0.30/0.65 => ( ( domainp @ ( product_prod @ nat @ nat ) @ int @ pcr_int )
% 0.30/0.65 = ( basic_pred_prod @ nat @ nat @ DR1 @ DR2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int.domain
% 0.30/0.65 thf(fact_7182_Heap_Ocase,axiom,
% 0.30/0.65 ! [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 ) ) )] :
% 0.30/0.65 ( ( heap_Time_case_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 0.30/0.65 = ( F @ X ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap.case
% 0.30/0.65 thf(fact_7183_Heap_Ocase__distrib,axiom,
% 0.30/0.65 ! [C: $tType,B: $tType,A: $tType,H2: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( H2 @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
% 0.30/0.65 = ( heap_Time_case_Heap @ A @ C
% 0.30/0.65 @ ^ [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H2 @ ( F @ X2 ) )
% 0.30/0.65 @ Heap ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap.case_distrib
% 0.30/0.65 thf(fact_7184_Heap_Orec,axiom,
% 0.30/0.65 ! [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 ) ) )] :
% 0.30/0.65 ( ( heap_Time_rec_Heap @ A @ C @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 0.30/0.65 = ( F @ X ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap.rec
% 0.30/0.65 thf(fact_7185_enumerate__Suc__eq,axiom,
% 0.30/0.65 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.30/0.65 ( ( enumerate @ A @ ( suc @ N ) @ Xs )
% 0.30/0.65 = ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % enumerate_Suc_eq
% 0.30/0.65 thf(fact_7186_Heap__ordE,axiom,
% 0.30/0.65 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 0.30/0.65 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 0.30/0.65 => ( ( ( heap_Time_execute @ A @ X @ H2 )
% 0.30/0.65 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.65 => ( ( heap_Time_execute @ A @ X @ H2 )
% 0.30/0.65 = ( heap_Time_execute @ A @ Y3 @ H2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap_ordE
% 0.30/0.65 thf(fact_7187_Heap__ordI,axiom,
% 0.30/0.65 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
% 0.30/0.65 ( ! [H6: heap_ext @ product_unit] :
% 0.30/0.65 ( ( ( heap_Time_execute @ A @ X @ H6 )
% 0.30/0.65 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.30/0.65 | ( ( heap_Time_execute @ A @ X @ H6 )
% 0.30/0.65 = ( heap_Time_execute @ A @ Y3 @ H6 ) ) )
% 0.30/0.65 => ( heap_Time_Heap_ord @ A @ X @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap_ordI
% 0.30/0.65 thf(fact_7188_heap_Oleq__refl,axiom,
% 0.30/0.65 ! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% 0.30/0.65
% 0.30/0.65 % heap.leq_refl
% 0.30/0.65 thf(fact_7189_heap_Oleq__trans,axiom,
% 0.30/0.65 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,Z3: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 0.30/0.65 => ( ( heap_Time_Heap_ord @ A @ Y3 @ Z3 )
% 0.30/0.65 => ( heap_Time_Heap_ord @ A @ X @ Z3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.leq_trans
% 0.30/0.65 thf(fact_7190_heap_Oleq__antisym,axiom,
% 0.30/0.65 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 0.30/0.65 => ( ( heap_Time_Heap_ord @ A @ Y3 @ X )
% 0.30/0.65 => ( X = Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.leq_antisym
% 0.30/0.65 thf(fact_7191_heap_Olub__least,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),Z3: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
% 0.30/0.65 => ( ! [X3: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( member @ ( heap_Time_Heap @ A ) @ X3 @ A6 )
% 0.30/0.65 => ( heap_Time_Heap_ord @ A @ X3 @ Z3 ) )
% 0.30/0.65 => ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A6 ) @ Z3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.lub_least
% 0.30/0.65 thf(fact_7192_heap_Olub__upper,axiom,
% 0.30/0.65 ! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
% 0.30/0.65 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
% 0.30/0.65 => ( ( member @ ( heap_Time_Heap @ A ) @ X @ A6 )
% 0.30/0.65 => ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A6 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.lub_upper
% 0.30/0.65 thf(fact_7193_heap_Oconst__mono,axiom,
% 0.30/0.65 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
% 0.30/0.65 ( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
% 0.30/0.65 @ ^ [F2: B] : C2 ) ).
% 0.30/0.65
% 0.30/0.65 % heap.const_mono
% 0.30/0.65 thf(fact_7194_heap__interpretation,axiom,
% 0.30/0.65 ! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap_interpretation
% 0.30/0.65 thf(fact_7195_Heap__Time__Monad_Obind__mono,axiom,
% 0.30/0.65 ! [C: $tType,D: $tType,B: $tType,A: $tType,B7: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C6: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
% 0.30/0.65 ( ( 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 ) @ B7 )
% 0.30/0.65 => ( ! [Y5: 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 ) @ ( C6 @ Y5 ) )
% 0.30/0.65 => ( 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 )
% 0.30/0.65 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 0.30/0.65 ( heap_Time_bind @ C @ D @ ( B7 @ F2 )
% 0.30/0.65 @ ^ [Y6: C] : ( C6 @ Y6 @ F2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Heap_Time_Monad.bind_mono
% 0.30/0.65 thf(fact_7196_admissible__heap,axiom,
% 0.30/0.65 ! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
% 0.30/0.65 ( 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 ) )
% 0.30/0.65 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 0.30/0.65 ! [X2: A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: B,N3: nat] :
% 0.30/0.65 ( ( heap_Time_effect @ B @ ( F2 @ X2 ) @ H4 @ H5 @ R5 @ N3 )
% 0.30/0.65 => ( P @ X2 @ H4 @ H5 @ R5 @ N3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % admissible_heap
% 0.30/0.65 thf(fact_7197_heap_Ofixp__induct__uc,axiom,
% 0.30/0.65 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
% 0.30/0.65 ( ! [X3: B] :
% 0.30/0.65 ( 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 )
% 0.30/0.65 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
% 0.30/0.65 => ( ( F
% 0.30/0.65 = ( C6
% 0.30/0.65 @ ( 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 ) )
% 0.30/0.65 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
% 0.30/0.65 => ( ! [F3: B > ( heap_Time_Heap @ A )] :
% 0.30/0.65 ( ( U4 @ ( C6 @ F3 ) )
% 0.30/0.65 = F3 )
% 0.30/0.65 => ( ( 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 )
% 0.30/0.65 => ( ( P
% 0.30/0.65 @ ^ [Uu: B] :
% 0.30/0.65 ( heap_Time_Heap2 @ A
% 0.30/0.65 @ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
% 0.30/0.65 => ( ! [F3: C] :
% 0.30/0.65 ( ( P @ ( U4 @ F3 ) )
% 0.30/0.65 => ( P @ ( U4 @ ( F4 @ F3 ) ) ) )
% 0.30/0.65 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.fixp_induct_uc
% 0.30/0.65 thf(fact_7198_heap_Omono__body__fixp,axiom,
% 0.30/0.65 ! [A: $tType,B: $tType,F4: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
% 0.30/0.65 ( ! [X3: B] :
% 0.30/0.65 ( 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 )
% 0.30/0.65 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F4 @ F2 @ X3 ) )
% 0.30/0.65 => ( ( 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 ) ) @ F4 )
% 0.30/0.65 = ( F4 @ ( 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 ) ) @ F4 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.mono_body_fixp
% 0.30/0.65 thf(fact_7199_heap_Ofixp__rule__uc,axiom,
% 0.30/0.65 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
% 0.30/0.65 ( ! [X3: B] :
% 0.30/0.65 ( 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 )
% 0.30/0.65 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
% 0.30/0.65 => ( ( F
% 0.30/0.65 = ( C6
% 0.30/0.65 @ ( 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 ) )
% 0.30/0.65 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
% 0.30/0.65 => ( ! [F3: C] :
% 0.30/0.65 ( ( C6 @ ( U4 @ F3 ) )
% 0.30/0.65 = F3 )
% 0.30/0.65 => ( F
% 0.30/0.65 = ( F4 @ F ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % heap.fixp_rule_uc
% 0.30/0.65
% 0.30/0.65 % Subclasses (3)
% 0.30/0.65 thf(subcl_Heap_Oheap___HOL_Otype,axiom,
% 0.30/0.65 ! [A: $tType] :
% 0.30/0.65 ( ( heap @ A )
% 0.30/0.65 => ( type @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
% 0.30/0.65 ! [A: $tType] :
% 0.30/0.65 ( ( heap @ A )
% 0.30/0.65 => ( typerep2 @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
% 0.30/0.65 ! [A: $tType] :
% 0.30/0.65 ( ( heap @ A )
% 0.30/0.65 => ( countable @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % Type constructors (657)
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
% 0.30/0.65 bounded_lattice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
% 0.30/0.65 finite_finite @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( finite_finite @ A17 )
% 0.30/0.65 & ( finite_finite @ A18 ) )
% 0.30/0.65 => ( finite_finite @ ( product_prod @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( finite_finite @ A17 )
% 0.30/0.65 => ( finite_finite @ ( option @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( finite_finite @ A17 )
% 0.30/0.65 & ( finite_finite @ A18 ) )
% 0.30/0.65 => ( finite_finite @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
% 0.30/0.65 finite_finite @ char ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_5,axiom,
% 0.30/0.65 bounded_lattice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 0.30/0.65 finite_finite @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Obounded__lattice_7,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( finite_finite @ A17 )
% 0.30/0.65 => ( finite_finite @ ( set @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Obounded__lattice_9,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bounded_lattice @ A18 )
% 0.30/0.65 => ( bounded_lattice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Finite__Set_Ofinite_10,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( finite_finite @ A17 )
% 0.30/0.65 & ( finite_finite @ A18 ) )
% 0.30/0.65 => ( finite_finite @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( comple6319245703460814977attice @ A18 )
% 0.30/0.65 => ( condit1219197933456340205attice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( comple592849572758109894attice @ A18 )
% 0.30/0.65 => ( comple592849572758109894attice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( comple489889107523837845lgebra @ A18 )
% 0.30/0.65 => ( comple489889107523837845lgebra @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bounded_lattice @ A18 )
% 0.30/0.65 => ( bounde4967611905675639751up_bot @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bounded_lattice @ A18 )
% 0.30/0.65 => ( bounde4346867609351753570nf_top @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( comple6319245703460814977attice @ A18 )
% 0.30/0.65 => ( comple6319245703460814977attice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( boolea8198339166811842893lgebra @ A18 )
% 0.30/0.65 => ( boolea8198339166811842893lgebra @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bounded_lattice @ A18 )
% 0.30/0.65 => ( bounded_lattice_top @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bounded_lattice @ A18 )
% 0.30/0.65 => ( bounded_lattice_bot @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( comple6319245703460814977attice @ A18 )
% 0.30/0.65 => ( comple9053668089753744459l_ccpo @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( semilattice_sup @ A18 )
% 0.30/0.65 => ( semilattice_sup @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( semilattice_inf @ A18 )
% 0.30/0.65 => ( semilattice_inf @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( distrib_lattice @ A18 )
% 0.30/0.65 => ( distrib_lattice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( order_top @ A18 )
% 0.30/0.65 => ( order_top @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( order_bot @ A18 )
% 0.30/0.65 => ( order_bot @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Countable_Ocountable,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( finite_finite @ A17 )
% 0.30/0.65 & ( countable @ A18 ) )
% 0.30/0.65 => ( countable @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( preorder @ A18 )
% 0.30/0.65 => ( preorder @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( lattice @ A18 )
% 0.30/0.65 => ( lattice @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Typerep_Otyperep,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 ) )
% 0.30/0.65 => ( typerep2 @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( order @ A18 )
% 0.30/0.65 => ( order @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( top @ A18 )
% 0.30/0.65 => ( top @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ord @ A18 )
% 0.30/0.65 => ( ord @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( bot @ A18 )
% 0.30/0.65 => ( bot @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( uminus @ A18 )
% 0.30/0.65 => ( uminus @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_fun___Groups_Ominus,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( minus @ A18 )
% 0.30/0.65 => ( minus @ ( A17 > A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/0.65 condit6923001295902523014norder @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 0.30/0.65 condit1219197933456340205attice @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
% 0.30/0.65 bit_un5681908812861735899ations @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 0.30/0.65 semiri1453513574482234551roduct @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 0.30/0.65 euclid5411537665997757685th_nat @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
% 0.30/0.65 euclid8789492081693882211th_nat @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.30/0.65 ordere1937475149494474687imp_le @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
% 0.30/0.65 euclid3128863361964157862miring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
% 0.30/0.65 euclid4440199948858584721cancel @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
% 0.30/0.65 normal6328177297339901930cative @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 0.30/0.65 unique1627219031080169319umeral @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
% 0.30/0.65 euclid8851590272496341667cancel @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.30/0.65 semiri6575147826004484403cancel @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.30/0.65 strict9044650504122735259up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.30/0.65 ordere580206878836729694up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.30/0.65 ordere2412721322843649153imp_le @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
% 0.30/0.65 bit_se359711467146920520ations @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
% 0.30/0.65 linord2810124833399127020strict @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.30/0.65 strict7427464778891057005id_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.30/0.65 ordere8940638589300402666id_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
% 0.30/0.65 euclid3725896446679973847miring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
% 0.30/0.65 linord715952674999750819strict @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
% 0.30/0.65 linord4140545234300271783up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
% 0.30/0.65 bit_ri3973907225187159222ations @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.30/0.65 semiri2026040879449505780visors @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.30/0.65 linord181362715937106298miring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
% 0.30/0.65 euclid5891614535332579305n_ring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
% 0.30/0.65 semido2269285787275462019factor @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
% 0.30/0.65 linord8928482502909563296strict @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 0.30/0.65 semiri3467727345109120633visors @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.30/0.65 ordere6658533253407199908up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 0.30/0.65 ordere166539214618696060dd_abs @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
% 0.30/0.65 semiri6843258321239162965malize @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
% 0.30/0.65 ordere6911136660526730532id_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.30/0.65 linord5086331880401160121up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.30/0.65 cancel2418104881723323429up_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
% 0.30/0.65 ring_15535105094025558882visors @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.30/0.65 cancel1802427076303600483id_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
% 0.30/0.65 linord4710134922213307826strict @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
% 0.30/0.65 comm_s4317794764714335236cancel @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
% 0.30/0.65 bit_semiring_bits @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
% 0.30/0.65 ordere2520102378445227354miring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
% 0.30/0.65 normal8620421768224518004emidom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
% 0.30/0.65 linord6961819062388156250ring_1 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.30/0.65 ordered_ab_group_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.30/0.65 cancel_semigroup_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
% 0.30/0.65 linordered_semiring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
% 0.30/0.65 ordered_semiring_0 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.30/0.65 linordered_semidom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_12,axiom,
% 0.30/0.65 semilattice_sup @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_13,axiom,
% 0.30/0.65 semilattice_inf @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Lattices_Odistrib__lattice_14,axiom,
% 0.30/0.65 distrib_lattice @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
% 0.30/0.65 ab_semigroup_mult @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
% 0.30/0.65 semiring_1_cancel @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
% 0.30/0.65 algebraic_semidom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
% 0.30/0.65 comm_monoid_mult @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.30/0.65 ab_semigroup_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
% 0.30/0.65 ordered_semiring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
% 0.30/0.65 ordered_ring_abs @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
% 0.30/0.65 semiring_parity @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.30/0.65 comm_monoid_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
% 0.30/0.65 semiring_modulo @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
% 0.30/0.65 linordered_ring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.30/0.65 linordered_idom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 0.30/0.65 comm_semiring_1 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 0.30/0.65 comm_semiring_0 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
% 0.30/0.65 semigroup_mult @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
% 0.30/0.65 semidom_modulo @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 0.30/0.65 semidom_divide @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
% 0.30/0.65 semiring_numeral @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.30/0.65 semigroup_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
% 0.30/0.65 zero_less_one @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
% 0.30/0.65 comm_semiring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.30/0.65 semiring_char_0 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 0.30/0.65 ab_group_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Countable_Ocountable_15,axiom,
% 0.30/0.65 countable @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 0.30/0.65 zero_neq_one @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
% 0.30/0.65 ordered_ring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
% 0.30/0.65 idom_abs_sgn @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
% 0.30/0.65 ring_parity @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Opreorder_16,axiom,
% 0.30/0.65 preorder @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.30/0.65 linorder @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 0.30/0.65 monoid_mult @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
% 0.30/0.65 idom_modulo @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
% 0.30/0.65 idom_divide @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 0.30/0.65 comm_ring_1 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 0.30/0.65 monoid_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.30/0.65 semiring_1 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
% 0.30/0.65 semiring_0 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.30/0.65 no_top @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.30/0.65 no_bot @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Lattices_Olattice_17,axiom,
% 0.30/0.65 lattice @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.30/0.65 group_add @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
% 0.30/0.65 semiring_gcd @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
% 0.30/0.65 semiring_Gcd @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Typerep_Otyperep_18,axiom,
% 0.30/0.65 typerep2 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
% 0.30/0.65 mult_zero @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
% 0.30/0.65 comm_ring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Oorder_19,axiom,
% 0.30/0.65 order @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.30/0.65 neg_numeral @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.30/0.65 ring_char_0 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemiring,axiom,
% 0.30/0.65 semiring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Osemidom,axiom,
% 0.30/0.65 semidom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Orderings_Oord_20,axiom,
% 0.30/0.65 ord @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ouminus_21,axiom,
% 0.30/0.65 uminus @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.30/0.65 ring_1 @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
% 0.30/0.65 abs_if @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ominus_22,axiom,
% 0.30/0.65 minus @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
% 0.30/0.65 ring_gcd @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Power_Opower,axiom,
% 0.30/0.65 power @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 0.30/0.65 numeral @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.30/0.65 zero @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oplus,axiom,
% 0.30/0.65 plus @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oring,axiom,
% 0.30/0.65 ring @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Oidom,axiom,
% 0.30/0.65 idom @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.30/0.65 one @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Rings_Odvd,axiom,
% 0.30/0.65 dvd @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Int_Oint___Heap_Oheap,axiom,
% 0.30/0.65 heap @ int ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_23,axiom,
% 0.30/0.65 condit6923001295902523014norder @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
% 0.30/0.65 condit1219197933456340205attice @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_25,axiom,
% 0.30/0.65 bit_un5681908812861735899ations @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_26,axiom,
% 0.30/0.65 semiri1453513574482234551roduct @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_27,axiom,
% 0.30/0.65 euclid5411537665997757685th_nat @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28,axiom,
% 0.30/0.65 ordere1937475149494474687imp_le @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_29,axiom,
% 0.30/0.65 euclid3128863361964157862miring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_30,axiom,
% 0.30/0.65 euclid4440199948858584721cancel @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_31,axiom,
% 0.30/0.65 normal6328177297339901930cative @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_32,axiom,
% 0.30/0.65 unique1627219031080169319umeral @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_33,axiom,
% 0.30/0.65 semiri6575147826004484403cancel @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_34,axiom,
% 0.30/0.65 strict9044650504122735259up_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 0.30/0.65 ordere1170586879665033532d_diff @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_35,axiom,
% 0.30/0.65 ordere580206878836729694up_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_36,axiom,
% 0.30/0.65 ordere2412721322843649153imp_le @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_37,axiom,
% 0.30/0.65 bit_se359711467146920520ations @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_38,axiom,
% 0.30/0.65 linord2810124833399127020strict @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_39,axiom,
% 0.30/0.65 strict7427464778891057005id_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_40,axiom,
% 0.30/0.65 ordere8940638589300402666id_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.30/0.65 canoni5634975068530333245id_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_41,axiom,
% 0.30/0.65 euclid3725896446679973847miring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_42,axiom,
% 0.30/0.65 linord4140545234300271783up_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43,axiom,
% 0.30/0.65 semiri2026040879449505780visors @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44,axiom,
% 0.30/0.65 linord181362715937106298miring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_45,axiom,
% 0.30/0.65 semido2269285787275462019factor @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46,axiom,
% 0.30/0.65 linord8928482502909563296strict @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47,axiom,
% 0.30/0.65 semiri3467727345109120633visors @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48,axiom,
% 0.30/0.65 ordere6658533253407199908up_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49,axiom,
% 0.30/0.65 semiri6843258321239162965malize @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50,axiom,
% 0.30/0.65 ordere6911136660526730532id_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51,axiom,
% 0.30/0.65 cancel2418104881723323429up_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
% 0.30/0.65 cancel1802427076303600483id_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
% 0.30/0.65 comm_s4317794764714335236cancel @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
% 0.30/0.65 bit_semiring_bits @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55,axiom,
% 0.30/0.65 ordere2520102378445227354miring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Onormalization__semidom_56,axiom,
% 0.30/0.65 normal8620421768224518004emidom @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_57,axiom,
% 0.30/0.65 cancel_semigroup_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Olinordered__semiring_58,axiom,
% 0.30/0.65 linordered_semiring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_59,axiom,
% 0.30/0.65 ordered_semiring_0 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_60,axiom,
% 0.30/0.65 linordered_semidom @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_61,axiom,
% 0.30/0.65 semilattice_sup @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_62,axiom,
% 0.30/0.65 semilattice_inf @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_63,axiom,
% 0.30/0.65 distrib_lattice @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
% 0.30/0.65 ab_semigroup_mult @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
% 0.30/0.65 semiring_1_cancel @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
% 0.30/0.65 algebraic_semidom @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,axiom,
% 0.30/0.65 comm_monoid_mult @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.30/0.65 comm_monoid_diff @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_68,axiom,
% 0.30/0.65 ab_semigroup_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
% 0.30/0.65 ordered_semiring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
% 0.30/0.65 semiring_parity @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
% 0.30/0.65 comm_monoid_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
% 0.30/0.65 semiring_modulo @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
% 0.30/0.65 comm_semiring_1 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
% 0.30/0.65 comm_semiring_0 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
% 0.30/0.65 semigroup_mult @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
% 0.30/0.65 semidom_modulo @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
% 0.30/0.65 semidom_divide @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
% 0.30/0.65 semiring_numeral @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
% 0.30/0.65 semigroup_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
% 0.30/0.65 zero_less_one @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
% 0.30/0.65 comm_semiring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.30/0.65 wellorder @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
% 0.30/0.65 order_bot @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
% 0.30/0.65 semiring_char_0 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
% 0.30/0.65 countable @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
% 0.30/0.65 zero_neq_one @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
% 0.30/0.65 preorder @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
% 0.30/0.65 linorder @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
% 0.30/0.65 monoid_mult @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
% 0.30/0.65 monoid_add @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
% 0.30/0.65 semiring_1 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
% 0.30/0.65 semiring_0 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
% 0.30/0.65 no_top @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
% 0.30/0.65 lattice @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
% 0.30/0.65 semiring_gcd @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
% 0.30/0.65 semiring_Gcd @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Typerep_Otyperep_96,axiom,
% 0.30/0.65 typerep2 @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
% 0.30/0.65 mult_zero @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
% 0.30/0.65 order @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
% 0.30/0.65 semiring @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
% 0.30/0.65 semidom @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Oord_101,axiom,
% 0.30/0.65 ord @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Orderings_Obot_102,axiom,
% 0.30/0.65 bot @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ominus_103,axiom,
% 0.30/0.65 minus @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Power_Opower_104,axiom,
% 0.30/0.65 power @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Num_Onumeral_105,axiom,
% 0.30/0.65 numeral @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Ozero_106,axiom,
% 0.30/0.65 zero @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oplus_107,axiom,
% 0.30/0.65 plus @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Groups_Oone_108,axiom,
% 0.30/0.65 one @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Rings_Odvd_109,axiom,
% 0.30/0.65 dvd @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Heap_Oheap_110,axiom,
% 0.30/0.65 heap @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.30/0.65 size @ nat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Orderings_Opreorder_111,axiom,
% 0.30/0.65 preorder @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Orderings_Olinorder_112,axiom,
% 0.30/0.65 linorder @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Typerep_Otyperep_113,axiom,
% 0.30/0.65 typerep2 @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Orderings_Oorder_114,axiom,
% 0.30/0.65 order @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Orderings_Oord_115,axiom,
% 0.30/0.65 ord @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Groups_Oplus_116,axiom,
% 0.30/0.65 plus @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Num_Onum___Nat_Osize_117,axiom,
% 0.30/0.65 size @ num ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_118,axiom,
% 0.30/0.65 semiri1453513574482234551roduct @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_119,axiom,
% 0.30/0.65 ordere1937475149494474687imp_le @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_120,axiom,
% 0.30/0.65 semiri6575147826004484403cancel @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_121,axiom,
% 0.30/0.65 strict9044650504122735259up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_122,axiom,
% 0.30/0.65 ordere580206878836729694up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_123,axiom,
% 0.30/0.65 ordere2412721322843649153imp_le @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_124,axiom,
% 0.30/0.65 linord2810124833399127020strict @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_125,axiom,
% 0.30/0.65 strict7427464778891057005id_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_126,axiom,
% 0.30/0.65 ordere8940638589300402666id_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
% 0.30/0.65 archim462609752435547400_field @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_127,axiom,
% 0.30/0.65 linord715952674999750819strict @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_128,axiom,
% 0.30/0.65 linord4140545234300271783up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_129,axiom,
% 0.30/0.65 semiri2026040879449505780visors @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_130,axiom,
% 0.30/0.65 linord181362715937106298miring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_131,axiom,
% 0.30/0.65 linord8928482502909563296strict @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_132,axiom,
% 0.30/0.65 semiri3467727345109120633visors @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_133,axiom,
% 0.30/0.65 ordere6658533253407199908up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_134,axiom,
% 0.30/0.65 ordere166539214618696060dd_abs @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
% 0.30/0.65 archim2362893244070406136eiling @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_135,axiom,
% 0.30/0.65 ordere6911136660526730532id_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_136,axiom,
% 0.30/0.65 linord5086331880401160121up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_137,axiom,
% 0.30/0.65 cancel2418104881723323429up_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_138,axiom,
% 0.30/0.65 ring_15535105094025558882visors @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_139,axiom,
% 0.30/0.65 cancel1802427076303600483id_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_140,axiom,
% 0.30/0.65 linord4710134922213307826strict @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_141,axiom,
% 0.30/0.65 comm_s4317794764714335236cancel @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_142,axiom,
% 0.30/0.65 ordere2520102378445227354miring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_143,axiom,
% 0.30/0.65 linord6961819062388156250ring_1 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_144,axiom,
% 0.30/0.65 ordered_ab_group_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_145,axiom,
% 0.30/0.65 cancel_semigroup_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__semiring_146,axiom,
% 0.30/0.65 linordered_semiring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_147,axiom,
% 0.30/0.65 ordered_semiring_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_148,axiom,
% 0.30/0.65 linordered_semidom @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
% 0.30/0.65 dense_linorder @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_149,axiom,
% 0.30/0.65 semilattice_sup @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_150,axiom,
% 0.30/0.65 semilattice_inf @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_151,axiom,
% 0.30/0.65 distrib_lattice @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_152,axiom,
% 0.30/0.65 ab_semigroup_mult @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_153,axiom,
% 0.30/0.65 semiring_1_cancel @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_154,axiom,
% 0.30/0.65 comm_monoid_mult @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_155,axiom,
% 0.30/0.65 ab_semigroup_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
% 0.30/0.65 linordered_field @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oordered__semiring_156,axiom,
% 0.30/0.65 ordered_semiring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_157,axiom,
% 0.30/0.65 ordered_ring_abs @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_158,axiom,
% 0.30/0.65 comm_monoid_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__ring_159,axiom,
% 0.30/0.65 linordered_ring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Olinordered__idom_160,axiom,
% 0.30/0.65 linordered_idom @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_161,axiom,
% 0.30/0.65 comm_semiring_1 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_162,axiom,
% 0.30/0.65 comm_semiring_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
% 0.30/0.65 dense_order @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_163,axiom,
% 0.30/0.65 semigroup_mult @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemidom__divide_164,axiom,
% 0.30/0.65 semidom_divide @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Num_Osemiring__numeral_165,axiom,
% 0.30/0.65 semiring_numeral @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Osemigroup__add_166,axiom,
% 0.30/0.65 semigroup_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
% 0.30/0.65 field_abs_sgn @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
% 0.30/0.65 division_ring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ozero__less__one_167,axiom,
% 0.30/0.65 zero_less_one @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_168,axiom,
% 0.30/0.65 comm_semiring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_169,axiom,
% 0.30/0.65 semiring_char_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oab__group__add_170,axiom,
% 0.30/0.65 ab_group_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
% 0.30/0.65 field_char_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Countable_Ocountable_171,axiom,
% 0.30/0.65 countable @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_172,axiom,
% 0.30/0.65 zero_neq_one @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oordered__ring_173,axiom,
% 0.30/0.65 ordered_ring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_174,axiom,
% 0.30/0.65 idom_abs_sgn @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Opreorder_175,axiom,
% 0.30/0.65 preorder @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Olinorder_176,axiom,
% 0.30/0.65 linorder @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Omonoid__mult_177,axiom,
% 0.30/0.65 monoid_mult @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oidom__divide_178,axiom,
% 0.30/0.65 idom_divide @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_179,axiom,
% 0.30/0.65 comm_ring_1 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Omonoid__add_180,axiom,
% 0.30/0.65 monoid_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__1_181,axiom,
% 0.30/0.65 semiring_1 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring__0_182,axiom,
% 0.30/0.65 semiring_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Ono__top_183,axiom,
% 0.30/0.65 no_top @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Ono__bot_184,axiom,
% 0.30/0.65 no_bot @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Lattices_Olattice_185,axiom,
% 0.30/0.65 lattice @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ogroup__add_186,axiom,
% 0.30/0.65 group_add @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Typerep_Otyperep_187,axiom,
% 0.30/0.65 typerep2 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Omult__zero_188,axiom,
% 0.30/0.65 mult_zero @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Ocomm__ring_189,axiom,
% 0.30/0.65 comm_ring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Oorder_190,axiom,
% 0.30/0.65 order @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Num_Oneg__numeral_191,axiom,
% 0.30/0.65 neg_numeral @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Nat_Oring__char__0_192,axiom,
% 0.30/0.65 ring_char_0 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemiring_193,axiom,
% 0.30/0.65 semiring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
% 0.30/0.65 inverse @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Osemidom_194,axiom,
% 0.30/0.65 semidom @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Orderings_Oord_195,axiom,
% 0.30/0.65 ord @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ouminus_196,axiom,
% 0.30/0.65 uminus @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oring__1_197,axiom,
% 0.30/0.65 ring_1 @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oabs__if_198,axiom,
% 0.30/0.65 abs_if @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ominus_199,axiom,
% 0.30/0.65 minus @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Fields_Ofield,axiom,
% 0.30/0.65 field @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Power_Opower_200,axiom,
% 0.30/0.65 power @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Num_Onumeral_201,axiom,
% 0.30/0.65 numeral @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Ozero_202,axiom,
% 0.30/0.65 zero @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oplus_203,axiom,
% 0.30/0.65 plus @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oring_204,axiom,
% 0.30/0.65 ring @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Oidom_205,axiom,
% 0.30/0.65 idom @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Groups_Oone_206,axiom,
% 0.30/0.65 one @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Rat_Orat___Rings_Odvd_207,axiom,
% 0.30/0.65 dvd @ rat ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_208,axiom,
% 0.30/0.65 ! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_209,axiom,
% 0.30/0.65 ! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_210,axiom,
% 0.30/0.65 ! [A17: $tType] : ( comple489889107523837845lgebra @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_211,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_212,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_213,axiom,
% 0.30/0.65 ! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_214,axiom,
% 0.30/0.65 ! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_215,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bounded_lattice_top @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_216,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bounded_lattice_bot @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_217,axiom,
% 0.30/0.65 ! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_218,axiom,
% 0.30/0.65 ! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_219,axiom,
% 0.30/0.65 ! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_220,axiom,
% 0.30/0.65 ! [A17: $tType] : ( distrib_lattice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Oorder__top_221,axiom,
% 0.30/0.65 ! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Oorder__bot_222,axiom,
% 0.30/0.65 ! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Countable_Ocountable_223,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( finite_finite @ A17 )
% 0.30/0.65 => ( countable @ ( set @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Opreorder_224,axiom,
% 0.30/0.65 ! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Lattices_Olattice_225,axiom,
% 0.30/0.65 ! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Typerep_Otyperep_226,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( set @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Oorder_227,axiom,
% 0.30/0.65 ! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Otop_228,axiom,
% 0.30/0.65 ! [A17: $tType] : ( top @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Oord_229,axiom,
% 0.30/0.65 ! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Orderings_Obot_230,axiom,
% 0.30/0.65 ! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Groups_Ouminus_231,axiom,
% 0.30/0.65 ! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Set_Oset___Groups_Ominus_232,axiom,
% 0.30/0.65 ! [A17: $tType] : ( minus @ ( set @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233,axiom,
% 0.30/0.65 condit1219197933456340205attice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_234,axiom,
% 0.30/0.65 comple592849572758109894attice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_235,axiom,
% 0.30/0.65 comple489889107523837845lgebra @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_236,axiom,
% 0.30/0.65 bounde4967611905675639751up_bot @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_237,axiom,
% 0.30/0.65 bounde4346867609351753570nf_top @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_238,axiom,
% 0.30/0.65 comple6319245703460814977attice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_239,axiom,
% 0.30/0.65 boolea8198339166811842893lgebra @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_240,axiom,
% 0.30/0.65 bounded_lattice_top @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_241,axiom,
% 0.30/0.65 bounded_lattice_bot @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_242,axiom,
% 0.30/0.65 comple9053668089753744459l_ccpo @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_243,axiom,
% 0.30/0.65 semilattice_sup @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_244,axiom,
% 0.30/0.65 semilattice_inf @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_245,axiom,
% 0.30/0.65 distrib_lattice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
% 0.30/0.65 order_top @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
% 0.30/0.65 order_bot @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
% 0.30/0.65 countable @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
% 0.30/0.65 preorder @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
% 0.30/0.65 linorder @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Lattices_Olattice_251,axiom,
% 0.30/0.65 lattice @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Typerep_Otyperep_252,axiom,
% 0.30/0.65 typerep2 @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
% 0.30/0.65 order @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Otop_254,axiom,
% 0.30/0.65 top @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Oord_255,axiom,
% 0.30/0.65 ord @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Orderings_Obot_256,axiom,
% 0.30/0.65 bot @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
% 0.30/0.65 uminus @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Groups_Ominus_258,axiom,
% 0.30/0.65 minus @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_HOL_Obool___Heap_Oheap_259,axiom,
% 0.30/0.65 heap @ $o ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_List_Olist___Countable_Ocountable_260,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( countable @ A17 )
% 0.30/0.65 => ( countable @ ( list @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_List_Olist___Typerep_Otyperep_261,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( list @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_List_Olist___Heap_Oheap_262,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( heap @ A17 )
% 0.30/0.65 => ( heap @ ( list @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_List_Olist___Nat_Osize_263,axiom,
% 0.30/0.65 ! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap_Oarray___Countable_Ocountable_264,axiom,
% 0.30/0.65 ! [A17: $tType] : ( countable @ ( array @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap_Oarray___Typerep_Otyperep_265,axiom,
% 0.30/0.65 ! [A17: $tType] : ( typerep2 @ ( array @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap_Oarray___Heap_Oheap_266,axiom,
% 0.30/0.65 ! [A17: $tType] : ( heap @ ( array @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap_Oarray___Nat_Osize_267,axiom,
% 0.30/0.65 ! [A17: $tType] : ( size @ ( array @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Ochar___Countable_Ocountable_268,axiom,
% 0.30/0.65 countable @ char ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Ochar___Typerep_Otyperep_269,axiom,
% 0.30/0.65 typerep2 @ char ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Ochar___Heap_Oheap_270,axiom,
% 0.30/0.65 heap @ char ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Ochar___Nat_Osize_271,axiom,
% 0.30/0.65 size @ char ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nitpick_Oword___Typerep_Otyperep_272,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( word @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Sum__Type_Osum___Countable_Ocountable_273,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( countable @ A17 )
% 0.30/0.65 & ( countable @ A18 ) )
% 0.30/0.65 => ( countable @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_274,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 ) )
% 0.30/0.65 => ( typerep2 @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Sum__Type_Osum___Heap_Oheap_275,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( heap @ A17 )
% 0.30/0.65 & ( heap @ A18 ) )
% 0.30/0.65 => ( heap @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Sum__Type_Osum___Nat_Osize_276,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] : ( size @ ( sum_sum @ A17 @ A18 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Option_Ooption___Countable_Ocountable_277,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( countable @ A17 )
% 0.30/0.65 => ( countable @ ( option @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Option_Ooption___Typerep_Otyperep_278,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( option @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Option_Ooption___Heap_Oheap_279,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( heap @ A17 )
% 0.30/0.65 => ( heap @ ( option @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Option_Ooption___Nat_Osize_280,axiom,
% 0.30/0.65 ! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Groups_Osemigroup__add_281,axiom,
% 0.30/0.65 semigroup_add @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Countable_Ocountable_282,axiom,
% 0.30/0.65 countable @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Orderings_Opreorder_283,axiom,
% 0.30/0.65 preorder @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Orderings_Olinorder_284,axiom,
% 0.30/0.65 linorder @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Groups_Omonoid__add_285,axiom,
% 0.30/0.65 monoid_add @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Typerep_Otyperep_286,axiom,
% 0.30/0.65 typerep2 @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Orderings_Oorder_287,axiom,
% 0.30/0.65 order @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Orderings_Oord_288,axiom,
% 0.30/0.65 ord @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Groups_Ozero_289,axiom,
% 0.30/0.65 zero @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Groups_Oplus_290,axiom,
% 0.30/0.65 plus @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Heap_Oheap_291,axiom,
% 0.30/0.65 heap @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_String_Oliteral___Nat_Osize_292,axiom,
% 0.30/0.65 size @ literal ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Nitpick_Ofun__box___Typerep_Otyperep_293,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 ) )
% 0.30/0.65 => ( typerep2 @ ( fun_box @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Typerep_Otyperep___Countable_Ocountable_294,axiom,
% 0.30/0.65 countable @ typerep ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_295,axiom,
% 0.30/0.65 typerep2 @ typerep ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Typerep_Otyperep___Heap_Oheap_296,axiom,
% 0.30/0.65 heap @ typerep ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Typerep_Otyperep___Nat_Osize_297,axiom,
% 0.30/0.65 size @ typerep ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_298,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 ) )
% 0.30/0.65 => ( typerep2 @ ( old_node @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Oprod___Countable_Ocountable_299,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( countable @ A17 )
% 0.30/0.65 & ( countable @ A18 ) )
% 0.30/0.65 => ( countable @ ( product_prod @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_300,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 ) )
% 0.30/0.65 => ( typerep2 @ ( product_prod @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Oprod___Heap_Oheap_301,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] :
% 0.30/0.65 ( ( ( heap @ A17 )
% 0.30/0.65 & ( heap @ A18 ) )
% 0.30/0.65 => ( heap @ ( product_prod @ A17 @ A18 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Oprod___Nat_Osize_302,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType] : ( size @ ( product_prod @ A17 @ A18 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_303,axiom,
% 0.30/0.65 condit6923001295902523014norder @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_304,axiom,
% 0.30/0.65 condit1219197933456340205attice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_305,axiom,
% 0.30/0.65 comple592849572758109894attice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_306,axiom,
% 0.30/0.65 comple489889107523837845lgebra @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_307,axiom,
% 0.30/0.65 bounde4967611905675639751up_bot @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_308,axiom,
% 0.30/0.65 bounde4346867609351753570nf_top @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
% 0.30/0.65 comple5582772986160207858norder @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_309,axiom,
% 0.30/0.65 comple6319245703460814977attice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_310,axiom,
% 0.30/0.65 boolea8198339166811842893lgebra @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_311,axiom,
% 0.30/0.65 bounded_lattice_top @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_312,axiom,
% 0.30/0.65 bounded_lattice_bot @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_313,axiom,
% 0.30/0.65 comple9053668089753744459l_ccpo @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_314,axiom,
% 0.30/0.65 semilattice_sup @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_315,axiom,
% 0.30/0.65 semilattice_inf @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_316,axiom,
% 0.30/0.65 distrib_lattice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_317,axiom,
% 0.30/0.65 wellorder @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_318,axiom,
% 0.30/0.65 order_top @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_319,axiom,
% 0.30/0.65 order_bot @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Countable_Ocountable_320,axiom,
% 0.30/0.65 countable @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_321,axiom,
% 0.30/0.65 preorder @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_322,axiom,
% 0.30/0.65 linorder @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Lattices_Olattice_323,axiom,
% 0.30/0.65 lattice @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_324,axiom,
% 0.30/0.65 typerep2 @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Oorder_325,axiom,
% 0.30/0.65 order @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Otop_326,axiom,
% 0.30/0.65 top @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Oord_327,axiom,
% 0.30/0.65 ord @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Orderings_Obot_328,axiom,
% 0.30/0.65 bot @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Groups_Ouminus_329,axiom,
% 0.30/0.65 uminus @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Groups_Ominus_330,axiom,
% 0.30/0.65 minus @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Product__Type_Ounit___Heap_Oheap_331,axiom,
% 0.30/0.65 heap @ product_unit ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_332,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( heap_ext @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_333,axiom,
% 0.30/0.65 bit_un5681908812861735899ations @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_334,axiom,
% 0.30/0.65 semiri1453513574482234551roduct @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_335,axiom,
% 0.30/0.65 euclid5411537665997757685th_nat @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_336,axiom,
% 0.30/0.65 euclid8789492081693882211th_nat @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_337,axiom,
% 0.30/0.65 ordere1937475149494474687imp_le @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_338,axiom,
% 0.30/0.65 euclid3128863361964157862miring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_339,axiom,
% 0.30/0.65 euclid4440199948858584721cancel @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_340,axiom,
% 0.30/0.65 unique1627219031080169319umeral @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_341,axiom,
% 0.30/0.65 euclid8851590272496341667cancel @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_342,axiom,
% 0.30/0.65 semiri6575147826004484403cancel @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_343,axiom,
% 0.30/0.65 strict9044650504122735259up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_344,axiom,
% 0.30/0.65 ordere580206878836729694up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_345,axiom,
% 0.30/0.65 ordere2412721322843649153imp_le @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_346,axiom,
% 0.30/0.65 bit_se359711467146920520ations @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_347,axiom,
% 0.30/0.65 linord2810124833399127020strict @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_348,axiom,
% 0.30/0.65 strict7427464778891057005id_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_349,axiom,
% 0.30/0.65 ordere8940638589300402666id_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_350,axiom,
% 0.30/0.65 euclid3725896446679973847miring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_351,axiom,
% 0.30/0.65 linord715952674999750819strict @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_352,axiom,
% 0.30/0.65 linord4140545234300271783up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_353,axiom,
% 0.30/0.65 bit_ri3973907225187159222ations @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_354,axiom,
% 0.30/0.65 semiri2026040879449505780visors @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_355,axiom,
% 0.30/0.65 linord181362715937106298miring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_356,axiom,
% 0.30/0.65 euclid5891614535332579305n_ring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_357,axiom,
% 0.30/0.65 linord8928482502909563296strict @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_358,axiom,
% 0.30/0.65 semiri3467727345109120633visors @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_359,axiom,
% 0.30/0.65 ordere6658533253407199908up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_360,axiom,
% 0.30/0.65 ordere166539214618696060dd_abs @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_361,axiom,
% 0.30/0.65 ordere6911136660526730532id_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_362,axiom,
% 0.30/0.65 linord5086331880401160121up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_363,axiom,
% 0.30/0.65 cancel2418104881723323429up_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_364,axiom,
% 0.30/0.65 ring_15535105094025558882visors @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_365,axiom,
% 0.30/0.65 cancel1802427076303600483id_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_366,axiom,
% 0.30/0.65 linord4710134922213307826strict @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_367,axiom,
% 0.30/0.65 comm_s4317794764714335236cancel @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_368,axiom,
% 0.30/0.65 bit_semiring_bits @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_369,axiom,
% 0.30/0.65 ordere2520102378445227354miring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_370,axiom,
% 0.30/0.65 linord6961819062388156250ring_1 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_371,axiom,
% 0.30/0.65 ordered_ab_group_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_372,axiom,
% 0.30/0.65 cancel_semigroup_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_373,axiom,
% 0.30/0.65 linordered_semiring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_374,axiom,
% 0.30/0.65 ordered_semiring_0 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_375,axiom,
% 0.30/0.65 linordered_semidom @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_376,axiom,
% 0.30/0.65 ab_semigroup_mult @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_377,axiom,
% 0.30/0.65 semiring_1_cancel @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_378,axiom,
% 0.30/0.65 algebraic_semidom @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_379,axiom,
% 0.30/0.65 comm_monoid_mult @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_380,axiom,
% 0.30/0.65 ab_semigroup_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_381,axiom,
% 0.30/0.65 ordered_semiring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_382,axiom,
% 0.30/0.65 ordered_ring_abs @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_383,axiom,
% 0.30/0.65 semiring_parity @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_384,axiom,
% 0.30/0.65 comm_monoid_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_385,axiom,
% 0.30/0.65 semiring_modulo @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_386,axiom,
% 0.30/0.65 linordered_ring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_387,axiom,
% 0.30/0.65 linordered_idom @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_388,axiom,
% 0.30/0.65 comm_semiring_1 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_389,axiom,
% 0.30/0.65 comm_semiring_0 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_390,axiom,
% 0.30/0.65 semigroup_mult @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_391,axiom,
% 0.30/0.65 semidom_modulo @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_392,axiom,
% 0.30/0.65 semidom_divide @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_393,axiom,
% 0.30/0.65 semiring_numeral @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_394,axiom,
% 0.30/0.65 semigroup_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_395,axiom,
% 0.30/0.65 zero_less_one @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_396,axiom,
% 0.30/0.65 comm_semiring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_397,axiom,
% 0.30/0.65 semiring_char_0 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_398,axiom,
% 0.30/0.65 ab_group_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_399,axiom,
% 0.30/0.65 zero_neq_one @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_400,axiom,
% 0.30/0.65 ordered_ring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_401,axiom,
% 0.30/0.65 idom_abs_sgn @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_402,axiom,
% 0.30/0.65 ring_parity @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_403,axiom,
% 0.30/0.65 preorder @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_404,axiom,
% 0.30/0.65 linorder @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_405,axiom,
% 0.30/0.65 monoid_mult @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_406,axiom,
% 0.30/0.65 idom_modulo @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_407,axiom,
% 0.30/0.65 idom_divide @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_408,axiom,
% 0.30/0.65 comm_ring_1 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_409,axiom,
% 0.30/0.65 monoid_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_410,axiom,
% 0.30/0.65 semiring_1 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_411,axiom,
% 0.30/0.65 semiring_0 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_412,axiom,
% 0.30/0.65 group_add @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_413,axiom,
% 0.30/0.65 typerep2 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
% 0.30/0.65 mult_zero @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
% 0.30/0.65 comm_ring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
% 0.30/0.65 order @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
% 0.30/0.65 neg_numeral @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
% 0.30/0.65 ring_char_0 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
% 0.30/0.65 semiring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_420,axiom,
% 0.30/0.65 semidom @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_421,axiom,
% 0.30/0.65 ord @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_422,axiom,
% 0.30/0.65 uminus @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_423,axiom,
% 0.30/0.65 ring_1 @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_424,axiom,
% 0.30/0.65 abs_if @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_425,axiom,
% 0.30/0.65 minus @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Power_Opower_426,axiom,
% 0.30/0.65 power @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_427,axiom,
% 0.30/0.65 numeral @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_428,axiom,
% 0.30/0.65 zero @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_429,axiom,
% 0.30/0.65 plus @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_430,axiom,
% 0.30/0.65 ring @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_431,axiom,
% 0.30/0.65 idom @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_432,axiom,
% 0.30/0.65 one @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_433,axiom,
% 0.30/0.65 dvd @ code_integer ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_434,axiom,
% 0.30/0.65 bit_un5681908812861735899ations @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_435,axiom,
% 0.30/0.65 euclid5411537665997757685th_nat @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_436,axiom,
% 0.30/0.65 ordere1937475149494474687imp_le @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_437,axiom,
% 0.30/0.65 euclid3128863361964157862miring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_438,axiom,
% 0.30/0.65 euclid4440199948858584721cancel @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_439,axiom,
% 0.30/0.65 semiri6575147826004484403cancel @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_440,axiom,
% 0.30/0.65 strict9044650504122735259up_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_441,axiom,
% 0.30/0.65 ordere580206878836729694up_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_442,axiom,
% 0.30/0.65 ordere2412721322843649153imp_le @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_443,axiom,
% 0.30/0.65 bit_se359711467146920520ations @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_444,axiom,
% 0.30/0.65 linord2810124833399127020strict @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_445,axiom,
% 0.30/0.65 strict7427464778891057005id_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_446,axiom,
% 0.30/0.65 ordere8940638589300402666id_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_447,axiom,
% 0.30/0.65 euclid3725896446679973847miring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_448,axiom,
% 0.30/0.65 linord4140545234300271783up_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_449,axiom,
% 0.30/0.65 semiri2026040879449505780visors @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_450,axiom,
% 0.30/0.65 linord181362715937106298miring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_451,axiom,
% 0.30/0.65 linord8928482502909563296strict @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_452,axiom,
% 0.30/0.65 semiri3467727345109120633visors @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_453,axiom,
% 0.30/0.65 ordere6658533253407199908up_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_454,axiom,
% 0.30/0.65 ordere6911136660526730532id_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_455,axiom,
% 0.30/0.65 cancel2418104881723323429up_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_456,axiom,
% 0.30/0.65 cancel1802427076303600483id_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_457,axiom,
% 0.30/0.65 comm_s4317794764714335236cancel @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_458,axiom,
% 0.30/0.65 bit_semiring_bits @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_459,axiom,
% 0.30/0.65 ordere2520102378445227354miring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_460,axiom,
% 0.30/0.65 cancel_semigroup_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_461,axiom,
% 0.30/0.65 linordered_semiring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_462,axiom,
% 0.30/0.65 ordered_semiring_0 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_463,axiom,
% 0.30/0.65 linordered_semidom @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_464,axiom,
% 0.30/0.65 ab_semigroup_mult @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_465,axiom,
% 0.30/0.65 semiring_1_cancel @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_466,axiom,
% 0.30/0.65 algebraic_semidom @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_467,axiom,
% 0.30/0.65 comm_monoid_mult @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_468,axiom,
% 0.30/0.65 comm_monoid_diff @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_469,axiom,
% 0.30/0.65 ab_semigroup_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_470,axiom,
% 0.30/0.65 ordered_semiring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_471,axiom,
% 0.30/0.65 semiring_parity @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_472,axiom,
% 0.30/0.65 comm_monoid_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_473,axiom,
% 0.30/0.65 semiring_modulo @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_474,axiom,
% 0.30/0.65 comm_semiring_1 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_475,axiom,
% 0.30/0.65 comm_semiring_0 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_476,axiom,
% 0.30/0.65 semigroup_mult @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_477,axiom,
% 0.30/0.65 semidom_modulo @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_478,axiom,
% 0.30/0.65 semidom_divide @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_479,axiom,
% 0.30/0.65 semiring_numeral @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_480,axiom,
% 0.30/0.65 semigroup_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_481,axiom,
% 0.30/0.65 zero_less_one @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_482,axiom,
% 0.30/0.65 comm_semiring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_483,axiom,
% 0.30/0.65 semiring_char_0 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_484,axiom,
% 0.30/0.65 zero_neq_one @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_485,axiom,
% 0.30/0.65 preorder @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_486,axiom,
% 0.30/0.65 linorder @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_487,axiom,
% 0.30/0.65 monoid_mult @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_488,axiom,
% 0.30/0.65 monoid_add @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_489,axiom,
% 0.30/0.65 semiring_1 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_490,axiom,
% 0.30/0.65 semiring_0 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_491,axiom,
% 0.30/0.65 typerep2 @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_492,axiom,
% 0.30/0.65 mult_zero @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_493,axiom,
% 0.30/0.65 order @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_494,axiom,
% 0.30/0.65 semiring @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_495,axiom,
% 0.30/0.65 semidom @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_496,axiom,
% 0.30/0.65 ord @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_497,axiom,
% 0.30/0.65 minus @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Power_Opower_498,axiom,
% 0.30/0.65 power @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_499,axiom,
% 0.30/0.65 numeral @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_500,axiom,
% 0.30/0.65 zero @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_501,axiom,
% 0.30/0.65 plus @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Groups_Oone_502,axiom,
% 0.30/0.65 one @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_503,axiom,
% 0.30/0.65 dvd @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Code__Numeral_Onatural___Nat_Osize_504,axiom,
% 0.30/0.65 size @ code_natural ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_505,axiom,
% 0.30/0.65 ! [A17: $tType] :
% 0.30/0.65 ( ( typerep2 @ A17 )
% 0.30/0.65 => ( typerep2 @ ( heap_Time_Heap @ A17 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_506,axiom,
% 0.30/0.65 ! [A17: $tType] : ( size @ ( heap_Time_Heap @ A17 ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_507,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType,A19: $tType] :
% 0.30/0.65 ( ( ( typerep2 @ A17 )
% 0.30/0.65 & ( typerep2 @ A18 )
% 0.30/0.65 & ( typerep2 @ A19 ) )
% 0.30/0.65 => ( typerep2 @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(tcon_Record_Otuple__isomorphism___Nat_Osize_508,axiom,
% 0.30/0.65 ! [A17: $tType,A18: $tType,A19: $tType] : ( size @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ).
% 0.30/0.65
% 0.30/0.65 % Helper facts (4)
% 0.30/0.65 thf(help_If_3_1_T,axiom,
% 0.30/0.65 ! [P: $o] :
% 0.30/0.65 ( ( P = $true )
% 0.30/0.65 | ( P = $false ) ) ).
% 0.30/0.65
% 0.30/0.65 thf(help_If_2_1_T,axiom,
% 0.30/0.65 ! [A: $tType,X: A,Y3: A] :
% 0.30/0.65 ( ( if @ A @ $false @ X @ Y3 )
% 0.30/0.65 = Y3 ) ).
% 0.30/0.65
% 0.30/0.65 thf(help_If_1_1_T,axiom,
% 0.30/0.65 ! [A: $tType,X: A,Y3: A] :
% 0.30/0.65 ( ( if @ A @ $true @ X @ Y3 )
% 0.30/0.65 = X ) ).
% 0.30/0.65
% 0.30/0.65 thf(help_fChoice_1_1_T,axiom,
% 0.30/0.65 ! [A: $tType,P: A > $o] :
% 0.30/0.65 ( ( P @ ( fChoice @ A @ P ) )
% 0.30/0.65 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Free types (1)
% 0.30/0.65 thf(tfree_0,hypothesis,
% 0.30/0.65 heap @ a ).
% 0.30/0.65
% 0.30/0.65 % Conjectures (2)
% 0.30/0.65 thf(conj_0,hypothesis,
% 0.30/0.65 $true ).
% 0.30/0.65
% 0.30/0.65 thf(conj_1,conjecture,
% 0.30/0.65 thesis ).
% 0.30/0.65
% 0.30/0.65 %------------------------------------------------------------------------------
% 0.30/0.65 ------------------------
% 0.33/5.20 ---- Embedded in THF ---
% 0.33/5.20 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.33/5.20 %%% Generated on Thu Feb 26 01:51:57 EST 2026
% 0.33/5.20 %%% using '$$dhol' embedding, version 1.3.0.
% 0.33/5.20 %%% Logic specification used:
% 0.33/5.20 %%% thf(spec, logic, $$dhol).
% 0.33/5.20
% 0.33/5.20 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2DTF31297.p
% 0.33/5.20 thf(ty_t_Record_Otuple__isomorphism, type, tuple_isomorphism: ($tType > ($tType > ($tType > $tType)))).
% 0.33/5.20 thf(ty_t_Heap__Time__Monad_OHeap, type, heap_Time_Heap: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_Code__Numeral_Onatural, type, code_natural: $tType).
% 0.33/5.20 thf(ty_t_Code__Numeral_Ointeger, type, code_integer: $tType).
% 0.33/5.20 thf(ty_t_Heap_Oheap_Oheap__ext, type, heap_ext: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 0.33/5.20 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 0.33/5.20 thf(ty_t_Old__Datatype_Onode, type, old_node: ($tType > ($tType > $tType))).
% 0.33/5.20 thf(ty_t_Nitpick_Ofun__box, type, fun_box: ($tType > ($tType > $tType))).
% 0.33/5.20 thf(ty_t_Typerep_Otyperep, type, typerep: $tType).
% 0.33/5.20 thf(ty_t_String_Oliteral, type, literal: $tType).
% 0.33/5.20 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 0.33/5.20 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_Nitpick_Oword, type, word: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_String_Ochar, type, char: $tType).
% 0.33/5.20 thf(ty_t_Heap_Oarray, type, array: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.33/5.20 thf(ty_t_Rat_Orat, type, rat: $tType).
% 0.33/5.20 thf(ty_t_Num_Onum, type, num: $tType).
% 0.33/5.20 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.33/5.20 thf(ty_t_Int_Oint, type, int: $tType).
% 0.33/5.20 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 0.33/5.20 thf(ty_tf_a, type, a: $tType).
% 0.33/5.20 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Heap_Oheap, type, heap: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Odvd, type, dvd: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Power_Opower, type, power: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_GCD_Oring__gcd, type, ring_gcd: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oabs__if, type, abs_if: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemidom, type, semidom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Oinverse, type, inverse: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring, type, semiring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Typerep_Otyperep, type, typerep2: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_GCD_Osemiring__Gcd, type, semiring_Gcd: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_GCD_Osemiring__gcd, type, semiring_gcd: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__0, type, semiring_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oidom__divide, type, idom_divide: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oidom__modulo, type, idom_modulo: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Parity_Oring__parity, type, ring_parity: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oidom__abs__sgn, type, idom_abs_sgn: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oordered__ring, type, ordered_ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__semiring, type, comm_semiring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Ofield__abs__sgn, type, field_abs_sgn: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Num_Osemiring__numeral, type, semiring_numeral: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemidom__modulo, type, semidom_modulo: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__modulo, type, semiring_modulo: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Parity_Osemiring__parity, type, semiring_parity: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oordered__ring__abs, type, ordered_ring_abs: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oordered__semiring, type, ordered_semiring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oalgebraic__semidom, type, algebraic_semidom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__1__cancel, type, semiring_1_cancel: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oordered__semiring__0, type, ordered_semiring_0: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__semiring, type, linordered_semiring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Complete__Partial__Order_Occpo, type, comple9053668089753744459l_ccpo: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Enum_Ofinite__distrib__lattice, type, finite8700451911770168679attice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__semiring__1, type, linord6961819062388156250ring_1: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Onormalization__semidom, type, normal8620421768224518004emidom: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oordered__comm__semiring, type, ordere2520102378445227354miring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Bit__Operations_Osemiring__bits, type, bit_semiring_bits: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Ocomm__semiring__1__cancel, type, comm_s4317794764714335236cancel: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord4710134922213307826strict: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1802427076303600483id_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Oring__1__no__zero__divisors, type, ring_15535105094025558882visors: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel2418104881723323429up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord5086331880401160121up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere6911136660526730532id_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim2362893244070406136eiling: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize, type, semiri6843258321239162965malize: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere166539214618696060dd_abs: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere6658533253407199908up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri3467727345109120633visors: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Boolean__Algebras_Oboolean__algebra, type, boolea8198339166811842893lgebra: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__semiring__strict, type, linord8928482502909563296strict: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemidom__divide__unit__factor, type, semido2269285787275462019factor: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Oeuclidean__ring, type, euclid5891614535332579305n_ring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord181362715937106298miring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri2026040879449505780visors: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Bit__Operations_Oring__bit__operations, type, bit_ri3973907225187159222ations: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple6319245703460814977attice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Olinordered__ab__semigroup__add, type, linord4140545234300271783up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__semiring__1__strict, type, linord715952674999750819strict: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim462609752435547400_field: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple5582772986160207858norder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Obounded__semilattice__inf__top, type, bounde4346867609351753570nf_top: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde4967611905675639751up_bot: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring, type, euclid3725896446679973847miring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni5634975068530333245id_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere8940638589300402666id_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict7427464778891057005id_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Olinordered__comm__semiring__strict, type, linord2810124833399127020strict: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Bit__Operations_Osemiring__bit__operations, type, bit_se359711467146920520ations: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere2412721322843649153imp_le: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere580206878836729694up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff, type, ordere1170586879665033532d_diff: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict9044650504122735259up_add: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri6575147826004484403cancel: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel, type, euclid8851590272496341667cancel: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1627219031080169319umeral: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra, type, comple489889107523837845lgebra: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice, type, comple592849572758109894attice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Rings_Onormalization__semidom__multiplicative, type, normal6328177297339901930cative: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel, type, euclid4440199948858584721cancel: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring, type, euclid3128863361964157862miring: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere1937475149494474687imp_le: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit5016429287641298734tinuum: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat, type, euclid8789492081693882211th_nat: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat, type, euclid5411537665997757685th_nat: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri1453513574482234551roduct: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, type, bit_un5681908812861735899ations: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit1219197933456340205attice: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit6923001295902523014norder: (!> [A:$tType]: ($o))).
% 0.33/5.20 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 0.33/5.20 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor, type, archim6421214686448440834_floor: (!> [A:$tType]: ((A > int)))).
% 0.33/5.20 thf(sy_c_Archimedean__Field_Ofrac, type, archimedean_frac: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))).
% 0.33/5.20 thf(sy_c_Array__Time_Oalloc, type, array_alloc: (!> [A:$tType]: (((list @ A) > ((heap_ext @ product_unit) > ((product_prod @ (array @ A)) @ (heap_ext @ product_unit))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Ofreeze, type, array_freeze: (!> [A:$tType]: (((array @ A) > (heap_Time_Heap @ (list @ A)))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oget, type, array_get: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_Array__Time_Olen, type, array_len: (!> [A:$tType]: (((array @ A) > (heap_Time_Heap @ nat))))).
% 0.33/5.20 thf(sy_c_Array__Time_Olength, type, array_length: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > nat))))).
% 0.33/5.20 thf(sy_c_Array__Time_Omake, type, array_make: (!> [A:$tType]: ((nat > ((nat > A) > (heap_Time_Heap @ (array @ A))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Omap__entry, type, array_map_entry: (!> [A:$tType]: ((nat > ((A > A) > ((array @ A) > (heap_Time_Heap @ (array @ A)))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Onew, type, array_new: (!> [A:$tType]: ((nat > (A > (heap_Time_Heap @ (array @ A))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Onoteq, type, array_noteq: (!> [A:$tType,B:$tType]: (((array @ A) > ((array @ B) > $o))))).
% 0.33/5.20 thf(sy_c_Array__Time_Onth, type, array_nth: (!> [A:$tType]: (((array @ A) > (nat > (heap_Time_Heap @ A)))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oof__list, type, array_of_list: (!> [A:$tType]: (((list @ A) > (heap_Time_Heap @ (array @ A)))))).
% 0.33/5.20 thf(sy_c_Array__Time_Opresent, type, array_present: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oset, type, array_set: (!> [A:$tType]: (((array @ A) > ((list @ A) > ((heap_ext @ product_unit) > (heap_ext @ product_unit))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oswap, type, array_swap: (!> [A:$tType]: ((nat > (A > ((array @ A) > (heap_Time_Heap @ A))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oupd, type, array_upd: (!> [A:$tType]: ((nat > (A > ((array @ A) > (heap_Time_Heap @ (array @ A)))))))).
% 0.33/5.20 thf(sy_c_Array__Time_Oupdate, type, array_update: (!> [A:$tType]: (((array @ A) > (nat > (A > ((heap_ext @ product_unit) > (heap_ext @ product_unit)))))))).
% 0.33/5.20 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq, type, bNF_Ca8665028551170535155natLeq: (set @ ((product_prod @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess, type, bNF_Ca8459412986667044542atLess: (set @ ((product_prod @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca3754400796208372196lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 0.33/5.20 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Constructions_Obsqr, type, bNF_Wellorder_bsqr: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl, type, bNF_We413866401316099525erIncl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Constructions_OordIso, type, bNF_Wellorder_ordIso: (!> [A:$tType,A2:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A2) @ A2))))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Constructions_OordLess, type, bNF_We4044943003108391690rdLess: (!> [A:$tType,A2:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A2) @ A2))))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Embedding_Oembed, type, bNF_Wellorder_embed: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Embedding_OembedS, type, bNF_Wellorder_embedS: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Embedding_Oiso, type, bNF_Wellorder_iso: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 0.33/5.20 thf(sy_c_BNF__Wellorder__Relation_Owo__rel, type, bNF_Wellorder_wo_rel: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 0.33/5.20 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod, type, basic_BNF_size_prod: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((product_prod @ A) @ B) > nat)))))).
% 0.33/5.20 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum, type, basic_BNF_size_sum: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((sum_sum @ A) @ B) > nat)))))).
% 0.33/5.20 thf(sy_c_Basic__BNFs_Opred__prod, type, basic_pred_prod: (!> [A:$tType,B:$tType]: (((A > $o) > ((B > $o) > (((product_prod @ A) @ B) > $o)))))).
% 0.33/5.20 thf(sy_c_Binomial_Obinomial, type, binomial: (nat > (nat > nat))).
% 0.33/5.20 thf(sy_c_Binomial_Ogbinomial, type, gbinomial: (!> [A:$tType]: ((A > (nat > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oand__int__rel, type, bit_and_int_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oand__not__num, type, bit_and_not_num: (num > (num > (option @ num)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oconcat__bit, type, bit_concat_bit: (nat > (int > (int > int)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oor__not__num__neg, type, bit_or_not_num_neg: (num > (num > num))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot, type, bit_ri4277139882892585799ns_not: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit, type, bit_ri4674362597316999326ke_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand, type, bit_se5824344872417868541ns_and: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit, type, bit_se4197421643247451524op_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit, type, bit_se8732182000553998342ip_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask, type, bit_se2239418461657761734s_mask: (!> [A:$tType]: ((nat > A)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor, type, bit_se1065995026697491101ons_or: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit, type, bit_se4730199178511100633sh_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit, type, bit_se5668285175392031749et_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit, type, bit_se2584673776208193580ke_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit, type, bit_se2638667681897837118et_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor, type, bit_se5824344971392196577ns_xor: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit, type, bit_se5641148757651400278ts_bit: (!> [A:$tType]: ((A > (nat > $o))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit, type, bit_se6407376104438227557le_bit: (!> [A:$tType]: (((itself @ A) > (nat > $o))))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Otake__bit__num, type, bit_take_bit_num: (nat > (num > (option @ num)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num, type, bit_un7362597486090784418nd_num: (num > (num > (option @ num)))).
% 0.33/5.20 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num, type, bit_un2480387367778600638or_num: (num > (num > (option @ num)))).
% 0.33/5.20 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra, type, boolea2506097494486148201lgebra: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > $o)))))))).
% 0.33/5.20 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__axioms, type, boolea6902313364301356556axioms: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > $o)))))))).
% 0.33/5.20 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff, type, boolea3799213064322606851m_diff: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > ((A > (A > A)) > $o))))))))).
% 0.33/5.20 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms, type, boolea5476839437570043046axioms: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > ((A > (A > A)) > $o))))))).
% 0.33/5.20 thf(sy_c_Code__Numeral_ONat, type, code_Nat: (code_integer > code_natural)).
% 0.33/5.20 thf(sy_c_Code__Numeral_ONeg, type, code_Neg: (num > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_OPos, type, code_Pos: (num > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_OSuc, type, code_Suc: (code_natural > code_natural)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Obit__cut__integer, type, code_bit_cut_integer: (code_integer > ((product_prod @ code_integer) @ $o))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ocr__integer, type, code_cr_integer: (int > (code_integer > $o))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ocr__natural, type, code_cr_natural: (nat > (code_natural > $o))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Odivmod__abs, type, code_divmod_abs: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Odivmod__integer, type, code_divmod_integer: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Odup, type, code_dup: (code_integer > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer, type, code_int_of_integer: (code_integer > int)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int, type, code_integer_of_int: (int > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ointeger__of__nat, type, code_integer_of_nat: (nat > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ointeger__of__natural, type, code_i5400310926305786745atural: (code_natural > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Ointeger__of__num, type, code_integer_of_num: (num > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onat__of__integer, type, code_nat_of_integer: (code_integer > nat)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural_Ocase__natural, type, code_case_natural: (!> [T:$tType]: ((T > ((code_natural > T) > (code_natural > T)))))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural, type, code_nat_of_natural: (code_natural > nat)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat, type, code_natural_of_nat: (nat > code_natural)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural_Orec__natural, type, code_rec_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > T)))))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural, type, code_rec_set_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > (T > $o))))))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onatural__of__integer, type, code_n4118661773612635043nteger: (code_integer > code_natural)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onegative, type, code_negative: (num > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Onum__of__integer, type, code_num_of_integer: (code_integer > num)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Opcr__integer, type, code_pcr_integer: (int > (code_integer > $o))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Opcr__natural, type, code_pcr_natural: (nat > (code_natural > $o))).
% 0.33/5.20 thf(sy_c_Code__Numeral_Opositive, type, code_positive: (num > code_integer)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Osize__nat, type, code_size_nat: (code_natural > nat)).
% 0.33/5.20 thf(sy_c_Code__Numeral_Osub, type, code_sub: (num > (num > code_integer))).
% 0.33/5.20 thf(sy_c_Complete__Lattices_OInf__class_OInf, type, complete_Inf_Inf: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible, type, comple1908693960933563346ssible: (!> [A:$tType]: ((((set @ A) > A) > ((A > (A > $o)) > ((A > $o) > $o)))))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp, type, comple187402453842119260l_fixp: (!> [A:$tType]: ((((set @ A) > A) > ((A > (A > $o)) > ((A > A) > A)))))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp, type, comple115746919287870866o_fixp: (!> [A:$tType]: (((A > A) > A)))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates, type, comple6359979572994053840erates: (!> [A:$tType]: (((A > A) > (set @ A))))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp, type, comple7512665784863727008ratesp: (!> [A:$tType]: (((A > A) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Ochain, type, comple1602240252501008431_chain: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Complete__Partial__Order_Omonotone, type, comple7038119648293358887notone: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > (B > $o)) > ((A > B) > $o)))))).
% 0.33/5.20 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above, type, condit941137186595557371_above: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.33/5.20 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below, type, condit1013018076250108175_below: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.33/5.20 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd, type, condit622319405099724424ng_bdd: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 0.33/5.20 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd, type, condit16957441358409770ng_bdd: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Countable_Ofrom__nat, type, from_nat: (!> [A:$tType]: ((nat > A)))).
% 0.33/5.20 thf(sy_c_Countable_Onth__item, type, nth_item: (!> [A:$tType]: ((nat > (set @ ((old_node @ A) @ product_unit)))))).
% 0.33/5.20 thf(sy_c_Countable_Onth__item__rel, type, nth_item_rel: (nat > (nat > $o))).
% 0.33/5.20 thf(sy_c_Divides_Oadjust__div, type, adjust_div: (((product_prod @ int) @ int) > int)).
% 0.33/5.20 thf(sy_c_Divides_Oadjust__mod, type, adjust_mod: (int > (int > int))).
% 0.33/5.20 thf(sy_c_Divides_Odivmod__nat, type, divmod_nat: (nat > (nat > ((product_prod @ nat) @ nat)))).
% 0.33/5.20 thf(sy_c_Divides_Oeucl__rel__int, type, eucl_rel_int: (int > (int > (((product_prod @ int) @ int) > $o)))).
% 0.33/5.20 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux, type, unique5940410009612947441es_aux: (!> [A:$tType]: ((((product_prod @ A) @ A) > $o)))).
% 0.33/5.20 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod, type, unique8689654367752047608divmod: (!> [A:$tType]: ((num > (num > ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step, type, unique1321980374590559556d_step: (!> [A:$tType]: ((num > (((product_prod @ A) @ A) > ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size, type, euclid6346220572633701492n_size: (!> [A:$tType]: ((A > nat)))).
% 0.33/5.20 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment, type, euclid7384307370059645450egment: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer, type, comm_s3205402744901411588hammer: (!> [A:$tType]: ((A > (nat > A))))).
% 0.33/5.20 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact, type, semiring_char_0_fact: (!> [A:$tType]: ((nat > A)))).
% 0.33/5.20 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 0.33/5.20 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.33/5.20 thf(sy_c_Finite__Set_Ofold, type, finite_fold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > B)))))).
% 0.33/5.20 thf(sy_c_Finite__Set_Ofolding__on, type, finite_folding_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))).
% 0.33/5.20 thf(sy_c_Finite__Set_Ofolding__on_OF, type, finite_folding_F: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > B)))))).
% 0.33/5.20 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 0.33/5.20 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 0.33/5.20 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 0.33/5.20 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Fun_Omap__fun, type, map_fun: (!> [C:$tType,A:$tType,B:$tType,D:$tType]: (((C > A) > ((B > D) > ((A > B) > (C > D))))))).
% 0.33/5.20 thf(sy_c_Fun__Def_Ois__measure, type, fun_is_measure: (!> [A:$tType]: (((A > nat) > $o)))).
% 0.33/5.20 thf(sy_c_Fun__Def_Omax__strict, type, fun_max_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.20 thf(sy_c_Fun__Def_Omax__weak, type, fun_max_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.20 thf(sy_c_Fun__Def_Omin__strict, type, fun_min_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.20 thf(sy_c_Fun__Def_Omin__weak, type, fun_min_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.20 thf(sy_c_Fun__Def_Opair__leq, type, fun_pair_leq: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))).
% 0.33/5.20 thf(sy_c_Fun__Def_Opair__less, type, fun_pair_less: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))).
% 0.33/5.20 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 0.33/5.20 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 0.33/5.20 thf(sy_c_GCD_OGcd__class_OGcd, type, gcd_Gcd: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_GCD_OGcd__class_OLcm, type, gcd_Lcm: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_GCD_Obezw, type, bezw: (nat > (nat > ((product_prod @ int) @ int)))).
% 0.33/5.20 thf(sy_c_GCD_Obezw__rel, type, bezw_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 0.33/5.20 thf(sy_c_GCD_Obounded__quasi__semilattice, type, bounde8507323023520639062attice: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((A > A) > $o))))))).
% 0.33/5.20 thf(sy_c_GCD_Obounded__quasi__semilattice__set, type, bounde6485984586167503788ce_set: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((A > A) > $o))))))).
% 0.33/5.20 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF, type, bounde2362111253966948842tice_F: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((set @ A) > A))))))).
% 0.33/5.20 thf(sy_c_GCD_Ogcd__class_Ogcd, type, gcd_gcd: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_GCD_Ogcd__class_Olcm, type, gcd_lcm: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_GCD_Ogcd__nat__rel, type, gcd_nat_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 0.33/5.20 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char, type, semiri4206861660011772517g_char: (!> [A:$tType]: (((itself @ A) > nat)))).
% 0.33/5.20 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin, type, semiring_gcd_Gcd_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin, type, semiring_gcd_Lcm_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Groups_Oabel__semigroup, type, abel_semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Groups_Oabel__semigroup__axioms, type, abel_s757365448890700780axioms: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Groups_Ocomm__monoid, type, comm_monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups_Ocomm__monoid__axioms, type, comm_monoid_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups_Ogroup, type, group: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > A) > $o)))))).
% 0.33/5.20 thf(sy_c_Groups_Ogroup__axioms, type, group_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > A) > $o)))))).
% 0.33/5.20 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Groups_Omonoid, type, monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups_Omonoid__axioms, type, monoid_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 0.33/5.20 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum, type, groups7311177749621191930dd_sum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H, type, groups1027152243600224163dd_sum: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod, type, groups7121269368397514597t_prod: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H, type, groups1962203154675924110t_prod: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__set, type, groups778175481326437816id_set: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF, type, groups_comm_monoid_F: (!> [A:$tType,B:$tType]: (((A > (A > A)) > (A > ((B > A) > ((set @ B) > A))))))).
% 0.33/5.20 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG, type, groups_comm_monoid_G: (!> [A:$tType,B:$tType]: (((A > (A > A)) > (A > ((B > A) > ((set @ B) > A))))))).
% 0.33/5.20 thf(sy_c_Groups__List_Ocomm__monoid__list, type, groups1828464146339083142d_list: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups__List_Ocomm__monoid__list__set, type, groups4802862169904069756st_set: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum, type, groups4207007520872428315er_sum: (!> [B:$tType,A:$tType]: (((B > A) > (A > ((list @ B) > A)))))).
% 0.33/5.20 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list, type, groups8242544230860333062m_list: (!> [A:$tType]: (((list @ A) > A)))).
% 0.33/5.20 thf(sy_c_Groups__List_Omonoid__list, type, groups_monoid_list: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Groups__List_Omonoid__list_OF, type, groups_monoid_F: (!> [A:$tType]: (((A > (A > A)) > (A > ((list @ A) > A)))))).
% 0.33/5.20 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list, type, groups5270119922927024881d_list: (!> [A:$tType]: (((list @ A) > A)))).
% 0.33/5.20 thf(sy_c_HOL_ONO__MATCH, type, nO_MATCH: (!> [A:$tType,B:$tType]: ((A > (B > $o))))).
% 0.33/5.20 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 0.33/5.20 thf(sy_c_HOL_OUniq, type, uniq: (!> [A:$tType]: (((A > $o) > $o)))).
% 0.33/5.20 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 0.33/5.20 thf(sy_c_Heap_Oaddr__of__array, type, addr_of_array: (!> [A:$tType]: (((array @ A) > nat)))).
% 0.33/5.20 thf(sy_c_Heap_Oarray_OArray, type, array2: (!> [A:$tType]: ((nat > (array @ A))))).
% 0.33/5.20 thf(sy_c_Heap_Oarray_Osize__array, type, size_array: (!> [A:$tType]: (((A > nat) > ((array @ A) > nat))))).
% 0.33/5.20 thf(sy_c_Heap_Oempty, type, empty: (heap_ext @ product_unit)).
% 0.33/5.20 thf(sy_c_Heap_Oheap_Oarrays, type, arrays: (!> [Z:$tType]: (((heap_ext @ Z) > (typerep > (nat > (list @ nat))))))).
% 0.33/5.20 thf(sy_c_Heap_Oheap_Oheap__ext, type, heap_ext2: (!> [Z:$tType]: (((typerep > (nat > (list @ nat))) > ((typerep > (nat > nat)) > (nat > (Z > (heap_ext @ Z)))))))).
% 0.33/5.20 thf(sy_c_Heap_Oheap_Olim, type, lim: (!> [Z:$tType]: (((heap_ext @ Z) > nat)))).
% 0.33/5.20 thf(sy_c_Heap_Oheap_Olim__update, type, lim_update: (!> [Z:$tType]: (((nat > nat) > ((heap_ext @ Z) > (heap_ext @ Z)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_OHeap, type, heap_Time_Heap2: (!> [A:$tType]: ((((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > (heap_Time_Heap @ A))))).
% 0.33/5.20 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))))).
% 0.33/5.20 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)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap, type, heap_Time_pred_Heap: (!> [A:$tType]: (((A > $o) > ((heap_Time_Heap @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap, type, heap_Time_rec_Heap: (!> [A:$tType,C:$tType]: (((((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C) > ((heap_Time_Heap @ A) > C))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap, type, heap_Time_rel_Heap: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap, type, heap_Time_set_Heap: (!> [A:$tType]: (((heap_Time_Heap @ A) > (set @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap, type, heap_Time_size_Heap: (!> [A:$tType]: (((A > nat) > ((heap_Time_Heap @ A) > nat))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap__lub, type, heap_Time_Heap_lub: (!> [A:$tType]: (((set @ (heap_Time_Heap @ A)) > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OHeap__ord, type, heap_Time_Heap_ord: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_Time_Heap @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oassert, type, heap_Time_assert: (!> [A:$tType]: (((A > $o) > (A > (heap_Time_Heap @ A)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Obind, type, heap_Time_bind: (!> [A:$tType,B:$tType]: (((heap_Time_Heap @ A) > ((A > (heap_Time_Heap @ B)) > (heap_Time_Heap @ B)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oeffect, type, heap_Time_effect: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > (A > (nat > $o)))))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oexecute, type, heap_Time_execute: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oguard, type, heap_Time_guard: (!> [A:$tType]: ((((heap_ext @ product_unit) > $o) > (((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (heap_Time_Heap @ A)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oheap, type, heap_Time_heap: (!> [A:$tType]: ((((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Olift, type, heap_Time_lift: (!> [A:$tType,B:$tType]: (((A > B) > (A > (heap_Time_Heap @ B)))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oraise, type, heap_Time_raise: (!> [A:$tType]: (((list @ char) > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oreturn, type, heap_Time_return: (!> [A:$tType]: ((A > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Osuccess, type, heap_Time_success: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > $o))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Otap, type, heap_Time_tap: (!> [A:$tType]: ((((heap_ext @ product_unit) > A) > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OtimeFrame, type, heap_Time_timeFrame: (!> [A:$tType]: ((nat > ((option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel, type, heap_T5500966940807335491me_rel: (!> [A:$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)))) > $o))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Oureturn, type, heap_Time_ureturn: (!> [A:$tType]: ((A > (heap_Time_Heap @ A))))).
% 0.33/5.20 thf(sy_c_Heap__Time__Monad_Owait, type, heap_Time_wait: (nat > (heap_Time_Heap @ product_unit))).
% 0.33/5.20 thf(sy_c_Hilbert__Choice_Obijection, type, hilbert_bijection: (!> [A:$tType]: (((A > A) > $o)))).
% 0.33/5.20 thf(sy_c_Hilbert__Choice_Oinv__into, type, hilbert_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 0.33/5.20 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 0.33/5.20 thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp, type, complete_lattice_gfp: (!> [A:$tType]: (((A > A) > A)))).
% 0.33/5.20 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp, type, complete_lattice_lfp: (!> [A:$tType]: (((A > A) > A)))).
% 0.33/5.20 thf(sy_c_Int_OAbs__Integ, type, abs_Integ: (((product_prod @ nat) @ nat) > int)).
% 0.33/5.20 thf(sy_c_Int_ONeg, type, neg: (num > int)).
% 0.33/5.20 thf(sy_c_Int_OPos, type, pos: (num > int)).
% 0.33/5.20 thf(sy_c_Int_ORep__Integ, type, rep_Integ: (int > ((product_prod @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_Int_Ocr__int, type, cr_int: (((product_prod @ nat) @ nat) > (int > $o))).
% 0.33/5.20 thf(sy_c_Int_Odup, type, dup: (int > int)).
% 0.33/5.20 thf(sy_c_Int_Oint_OAbs__int, type, abs_int: ((set @ ((product_prod @ nat) @ nat)) > int)).
% 0.33/5.20 thf(sy_c_Int_Oint_ORep__int, type, rep_int: (int > (set @ ((product_prod @ nat) @ nat)))).
% 0.33/5.20 thf(sy_c_Int_Oint__ge__less__than, type, int_ge_less_than: (int > (set @ ((product_prod @ int) @ int)))).
% 0.33/5.20 thf(sy_c_Int_Oint__ge__less__than2, type, int_ge_less_than2: (int > (set @ ((product_prod @ int) @ int)))).
% 0.33/5.20 thf(sy_c_Int_Ointrel, type, intrel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 0.33/5.20 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 0.33/5.20 thf(sy_c_Int_Opcr__int, type, pcr_int: (((product_prod @ nat) @ nat) > (int > $o))).
% 0.33/5.20 thf(sy_c_Int_Opower__int, type, power_int: (!> [A:$tType]: ((A > (int > A))))).
% 0.33/5.20 thf(sy_c_Int_Oring__1__class_OInts, type, ring_1_Ints: (!> [A:$tType]: ((set @ A)))).
% 0.33/5.20 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 0.33/5.20 thf(sy_c_Int_Osub, type, sub: (num > (num > int))).
% 0.33/5.20 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice, type, semilattice: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice__axioms, type, semilattice_axioms: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice__neutr, type, semilattice_neutr: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice__neutr__order, type, semila1105856199041335345_order: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > (A > $o)) > ((A > (A > $o)) > $o))))))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice__order, type, semilattice_order: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 0.33/5.20 thf(sy_c_Lattices_Osemilattice__order__axioms, type, semila6385135966242565138axioms: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 0.33/5.20 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Olinorder_OMax, type, lattices_Max: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > A))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Olinorder_OMin, type, lattices_Min: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > A))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Olinorder__class_OMax, type, lattic643756798349783984er_Max: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Olinorder__class_OMin, type, lattic643756798350308766er_Min: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Oarg__max, type, lattices_ord_arg_max: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > B))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on, type, lattic1883929316492267755max_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Oarg__min, type, lattices_ord_arg_min: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > B))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on, type, lattic7623131987881927897min_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max, type, lattic501386751176901750rg_max: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > (B > $o)))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min, type, lattic501386751177426532rg_min: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > (B > $o)))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin, type, lattic7752659483105999362nf_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__neutr__set, type, lattic5652469242046573047tr_set: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF, type, lattic5214292709420241887eutr_F: (!> [A:$tType]: (((A > (A > A)) > (A > ((set @ A) > A)))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__order__neutr__set, type, lattic3600114342068043075tr_set: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > (A > $o)) > ((A > (A > $o)) > $o))))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__order__set, type, lattic4895041142388067077er_set: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__set, type, lattic149705377957585745ce_set: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__set_OF, type, lattic1715443433743089157tice_F: (!> [A:$tType]: (((A > (A > A)) > ((set @ A) > A))))).
% 0.33/5.20 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin, type, lattic5882676163264333800up_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 0.33/5.20 thf(sy_c_Lifting_OQuotient, type, quotient: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > ((A > (B > $o)) > $o))))))).
% 0.33/5.20 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oarg__min__list, type, arg_min_list: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > A))))).
% 0.33/5.20 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)) > $o))))).
% 0.33/5.20 thf(sy_c_List_Obutlast, type, butlast: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Ocoset, type, coset: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 0.33/5.20 thf(sy_c_List_Ocount__list, type, count_list: (!> [A:$tType]: (((list @ A) > (A > nat))))).
% 0.33/5.20 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 0.33/5.20 thf(sy_c_List_Odrop, type, drop: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_OdropWhile, type, dropWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oenumerate, type, enumerate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ ((product_prod @ nat) @ A))))))).
% 0.33/5.20 thf(sy_c_List_Ofilter, type, filter: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Ofind, type, find: (!> [A:$tType]: (((A > $o) > ((list @ A) > (option @ A)))))).
% 0.33/5.20 thf(sy_c_List_Ofold, type, fold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))).
% 0.33/5.20 thf(sy_c_List_Ofolding__insort__key, type, folding_insort_key: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > ((set @ B) > ((B > A) > $o))))))).
% 0.33/5.20 thf(sy_c_List_Ofoldr, type, foldr: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))).
% 0.33/5.20 thf(sy_c_List_Ogen__length, type, gen_length: (!> [A:$tType]: ((nat > ((list @ A) > nat))))).
% 0.33/5.20 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 0.33/5.20 thf(sy_c_List_Olenlex, type, lenlex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Olex, type, lex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Olexord, type, lexord: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Olinorder__class_Oinsort__insert__key, type, linord329482645794927042rt_key: (!> [B:$tType,A:$tType]: (((B > A) > (B > ((list @ B) > (list @ B))))))).
% 0.33/5.20 thf(sy_c_List_Olinorder__class_Oinsort__key, type, linorder_insort_key: (!> [B:$tType,A:$tType]: (((B > A) > (B > ((list @ B) > (list @ B))))))).
% 0.33/5.20 thf(sy_c_List_Olinorder__class_Osort__key, type, linorder_sort_key: (!> [B:$tType,A:$tType]: (((B > A) > ((list @ B) > (list @ B)))))).
% 0.33/5.20 thf(sy_c_List_Olinorder__class_Osorted__list__of__set, type, linord4507533701916653071of_set: (!> [A:$tType]: (((set @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 0.33/5.20 thf(sy_c_List_Olist_Ohd, type, hd: (!> [A:$tType]: (((list @ A) > A)))).
% 0.33/5.20 thf(sy_c_List_Olist_Olist__all, type, list_all: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 0.33/5.20 thf(sy_c_List_Olist_Olist__all2, type, list_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))).
% 0.33/5.20 thf(sy_c_List_Olist_Omap, type, map: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((list @ A) > (list @ Aa)))))).
% 0.33/5.20 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 0.33/5.20 thf(sy_c_List_Olist_Osize__list, type, size_list: (!> [A:$tType]: (((A > nat) > ((list @ A) > nat))))).
% 0.33/5.20 thf(sy_c_List_Olist_Otl, type, tl: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 0.33/5.20 thf(sy_c_List_Olist__update, type, list_update: (!> [A:$tType]: (((list @ A) > (nat > (A > (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Olistrel, type, listrel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (list @ A)) @ (list @ B))))))).
% 0.33/5.20 thf(sy_c_List_Olistrel1, type, listrel1: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Omeasures, type, measures: (!> [A:$tType]: (((list @ (A > nat)) > (set @ ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_List_On__lists, type, n_lists: (!> [A:$tType]: ((nat > ((list @ A) > (list @ (list @ A))))))).
% 0.33/5.20 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 0.33/5.20 thf(sy_c_List_Onths, type, nths: (!> [A:$tType]: (((list @ A) > ((set @ nat) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oord__class_Olexordp, type, ord_lexordp: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 0.33/5.20 thf(sy_c_List_Oord__class_Olexordp__eq, type, ord_lexordp_eq: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 0.33/5.20 thf(sy_c_List_Oproduct, type, product: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))).
% 0.33/5.20 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oremdups, type, remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Oremdups__adj, type, remdups_adj: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Oremove1, type, remove1: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_OremoveAll, type, removeAll: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oreplicate, type, replicate: (!> [A:$tType]: ((nat > (A > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Orotate, type, rotate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Orotate1, type, rotate1: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 0.33/5.20 thf(sy_c_List_Osorted__wrt, type, sorted_wrt: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 0.33/5.20 thf(sy_c_List_Otake, type, take: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_OtakeWhile, type, takeWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Otranspose, type, transpose: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 0.33/5.20 thf(sy_c_List_Oupt, type, upt: (nat > (nat > (list @ nat)))).
% 0.33/5.20 thf(sy_c_List_Oupto, type, upto: (int > (int > (list @ int)))).
% 0.33/5.20 thf(sy_c_List_Oupto__aux, type, upto_aux: (int > (int > ((list @ int) > (list @ int))))).
% 0.33/5.20 thf(sy_c_List_Oupto__rel, type, upto_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 0.33/5.20 thf(sy_c_List_Ozip, type, zip: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))).
% 0.33/5.20 thf(sy_c_Map_Omap__of, type, map_of: (!> [A:$tType,B:$tType]: (((list @ ((product_prod @ A) @ B)) > (A > (option @ B)))))).
% 0.33/5.20 thf(sy_c_Map_Omap__upds, type, map_upds: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > ((list @ B) > (A > (option @ B)))))))).
% 0.33/5.20 thf(sy_c_Map_Orestrict__map, type, restrict_map: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((set @ A) > (A > (option @ B))))))).
% 0.33/5.20 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 0.33/5.20 thf(sy_c_Nat_Ocompow, type, compow: (!> [A:$tType]: ((nat > (A > A))))).
% 0.33/5.20 thf(sy_c_Nat_Ofunpow, type, funpow: (!> [A:$tType]: ((nat > ((A > A) > (A > A)))))).
% 0.33/5.20 thf(sy_c_Nat_Onat_Ocase__nat, type, case_nat: (!> [A:$tType]: ((A > ((nat > A) > (nat > A)))))).
% 0.33/5.20 thf(sy_c_Nat_Onat_Opred, type, pred: (nat > nat)).
% 0.33/5.20 thf(sy_c_Nat_Oold_Onat_Orec__nat, type, rec_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > T)))))).
% 0.33/5.20 thf(sy_c_Nat_Oold_Onat_Orec__set__nat, type, rec_set_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > (T > $o))))))).
% 0.33/5.20 thf(sy_c_Nat_Osemiring__1__class_ONats, type, semiring_1_Nats: (!> [A:$tType]: ((set @ A)))).
% 0.33/5.20 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 0.33/5.20 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri8178284476397505188at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 0.33/5.20 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oint__decode, type, nat_int_decode: (nat > int)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oint__encode, type, nat_int_encode: (int > nat)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Olist__decode, type, nat_list_decode: (nat > (list @ nat))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Olist__decode__rel, type, nat_list_decode_rel: (nat > (nat > $o))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Olist__encode, type, nat_list_encode: ((list @ nat) > nat)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Olist__encode__rel, type, nat_list_encode_rel: ((list @ nat) > ((list @ nat) > $o))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oprod__decode, type, nat_prod_decode: (nat > ((product_prod @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oprod__decode__aux, type, nat_prod_decode_aux: (nat > (nat > ((product_prod @ nat) @ nat)))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel, type, nat_pr5047031295181774490ux_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oprod__encode, type, nat_prod_encode: (((product_prod @ nat) @ nat) > nat)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oset__decode, type, nat_set_decode: (nat > (set @ nat))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Oset__encode, type, nat_set_encode: ((set @ nat) > nat)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Osum__decode, type, nat_sum_decode: (nat > ((sum_sum @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Osum__encode, type, nat_sum_encode: (((sum_sum @ nat) @ nat) > nat)).
% 0.33/5.20 thf(sy_c_Nat__Bijection_Otriangle, type, nat_triangle: (nat > nat)).
% 0.33/5.20 thf(sy_c_Num_OBitM, type, bitM: (num > num)).
% 0.33/5.20 thf(sy_c_Num_Oinc, type, inc: (num > num)).
% 0.33/5.20 thf(sy_c_Num_Onat__of__num, type, nat_of_num: (num > nat)).
% 0.33/5.20 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec, type, neg_numeral_dbl_dec: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc, type, neg_numeral_dbl_inc: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Num_Oneg__numeral__class_Ois__num, type, neg_numeral_is_num: (!> [A:$tType]: ((A > $o)))).
% 0.33/5.20 thf(sy_c_Num_Oneg__numeral__class_Osub, type, neg_numeral_sub: (!> [A:$tType]: ((num > (num > A))))).
% 0.33/5.20 thf(sy_c_Num_Onum_OBit0, type, bit0: (num > num)).
% 0.33/5.20 thf(sy_c_Num_Onum_OBit1, type, bit1: (num > num)).
% 0.33/5.20 thf(sy_c_Num_Onum_OOne, type, one2: num).
% 0.33/5.20 thf(sy_c_Num_Onum_Ocase__num, type, case_num: (!> [A:$tType]: ((A > ((num > A) > ((num > A) > (num > A))))))).
% 0.33/5.20 thf(sy_c_Num_Onum_Orec__num, type, rec_num: (!> [A:$tType]: ((A > ((num > (A > A)) > ((num > (A > A)) > (num > A))))))).
% 0.33/5.20 thf(sy_c_Num_Onum_Osize__num, type, size_num: (num > nat)).
% 0.33/5.20 thf(sy_c_Num_Onum__of__nat, type, num_of_nat: (nat > num)).
% 0.33/5.20 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 0.33/5.20 thf(sy_c_Num_Opow, type, pow: (num > (num > num))).
% 0.33/5.20 thf(sy_c_Num_Opred__numeral, type, pred_numeral: (num > nat)).
% 0.33/5.20 thf(sy_c_Num_Oring__1__class_Oiszero, type, ring_1_iszero: (!> [A:$tType]: ((A > $o)))).
% 0.33/5.20 thf(sy_c_Num_Osqr, type, sqr: (num > num)).
% 0.33/5.20 thf(sy_c_Old__Datatype_OAtom, type, old_Atom: (!> [A:$tType,B:$tType]: ((((sum_sum @ A) @ nat) > (set @ ((old_node @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_OIn0, type, old_In0: (!> [A:$tType,B:$tType]: (((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_OIn1, type, old_In1: (!> [A:$tType,B:$tType]: (((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_OLeaf, type, old_Leaf: (!> [A:$tType,B:$tType]: ((A > (set @ ((old_node @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_ONode, type, old_Node: (!> [B:$tType,A:$tType]: ((set @ ((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_ONumb, type, old_Numb: (!> [A:$tType,B:$tType]: ((nat > (set @ ((old_node @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_OPush, type, old_Push: (!> [B:$tType]: ((((sum_sum @ B) @ nat) > ((nat > ((sum_sum @ B) @ nat)) > (nat > ((sum_sum @ B) @ nat))))))).
% 0.33/5.20 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)))))).
% 0.33/5.20 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))))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_Ondepth, type, old_ndepth: (!> [A:$tType,B:$tType]: ((((old_node @ A) @ B) > nat)))).
% 0.33/5.20 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))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_Onode_ORep__Node, type, old_Rep_Node: (!> [A:$tType,B:$tType]: ((((old_node @ A) @ B) > ((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)))))).
% 0.33/5.20 thf(sy_c_Old__Datatype_Ontrunc, type, old_ntrunc: (!> [A:$tType,B:$tType]: ((nat > ((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B))))))).
% 0.33/5.20 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))).
% 0.33/5.20 thf(sy_c_Option_Ooption_OSome, type, some: (!> [A:$tType]: ((A > (option @ A))))).
% 0.33/5.20 thf(sy_c_Option_Ooption_Ocase__option, type, case_option: (!> [B:$tType,A:$tType]: ((B > ((A > B) > ((option @ A) > B)))))).
% 0.33/5.20 thf(sy_c_Option_Ooption_Osize__option, type, size_option: (!> [A:$tType]: (((A > nat) > ((option @ A) > nat))))).
% 0.33/5.20 thf(sy_c_Option_Ooption_Othe, type, the2: (!> [A:$tType]: (((option @ A) > A)))).
% 0.33/5.20 thf(sy_c_Order__Relation_Oofilter, type, order_ofilter: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Order__Relation_Ounder, type, order_under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Order__Relation_OunderS, type, order_underS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Order__Relation_Owell__order__on, type, order_well_order_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 0.33/5.20 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord__class_OLeast, type, ord_Least: (!> [A:$tType]: (((A > $o) > A)))).
% 0.33/5.20 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord__class_Omax, type, ord_max: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Orderings_Oord__class_Omin, type, ord_min: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 0.33/5.20 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 0.33/5.20 thf(sy_c_Orderings_Oorder__class_Omono, type, order_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 0.33/5.20 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Oordering__axioms, type, ordering_axioms: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Oordering__top, type, ordering_top: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > $o)))))).
% 0.33/5.20 thf(sy_c_Orderings_Oordering__top__axioms, type, ordering_top_axioms: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Opartial__preordering, type, partial_preordering: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 0.33/5.20 thf(sy_c_Orderings_Opreordering, type, preordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Opreordering__axioms, type, preordering_axioms: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 0.33/5.20 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 0.33/5.20 thf(sy_c_Partial__Function_Ofun__lub, type, partial_fun_lub: (!> [C:$tType,B:$tType,A:$tType]: ((((set @ C) > B) > ((set @ (A > C)) > (A > B)))))).
% 0.33/5.20 thf(sy_c_Partial__Function_Ofun__ord, type, partial_fun_ord: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((C > A) > ((C > B) > $o)))))).
% 0.33/5.20 thf(sy_c_Partial__Function_Opartial__function__definitions, type, partia7178651479351089652itions: (!> [A:$tType]: (((A > (A > $o)) > (((set @ A) > A) > $o))))).
% 0.33/5.20 thf(sy_c_Power_Opower_Opower, type, power2: (!> [A:$tType]: ((A > ((A > (A > A)) > (A > (nat > A))))))).
% 0.33/5.20 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 0.33/5.20 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 0.33/5.20 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 0.33/5.20 thf(sy_c_Product__Type_OUnity, type, product_Unity: product_unit).
% 0.33/5.20 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 0.33/5.20 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 0.33/5.20 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 0.33/5.20 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 0.33/5.20 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 0.33/5.20 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 0.33/5.20 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 0.33/5.20 thf(sy_c_Quotient_Oquot__type_Oabs, type, quot_abs: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > (((set @ A) > B) > (A > B)))))).
% 0.33/5.20 thf(sy_c_Quotient_Oquot__type_Orep, type, quot_rep: (!> [B:$tType,A:$tType]: (((B > (set @ A)) > (B > A))))).
% 0.33/5.20 thf(sy_c_Random_Olog, type, log: (code_natural > (code_natural > code_natural))).
% 0.33/5.20 thf(sy_c_Random_Olog__rel, type, log_rel: (((product_prod @ code_natural) @ code_natural) > (((product_prod @ code_natural) @ code_natural) > $o))).
% 0.33/5.20 thf(sy_c_Random_Ominus__shift, type, minus_shift: (code_natural > (code_natural > (code_natural > code_natural)))).
% 0.33/5.20 thf(sy_c_Random_Opick, type, pick: (!> [A:$tType]: (((list @ ((product_prod @ code_natural) @ A)) > (code_natural > A))))).
% 0.33/5.20 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))))).
% 0.33/5.20 thf(sy_c_Random_Oselect__weight, type, select_weight: (!> [A:$tType]: (((list @ ((product_prod @ code_natural) @ A)) > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ A) @ ((product_prod @ code_natural) @ code_natural))))))).
% 0.33/5.20 thf(sy_c_Rat_OAbs__Rat, type, abs_Rat: (((product_prod @ int) @ int) > rat)).
% 0.33/5.20 thf(sy_c_Rat_OFract, type, fract: (int > (int > rat))).
% 0.33/5.20 thf(sy_c_Rat_OFrct, type, frct: (((product_prod @ int) @ int) > rat)).
% 0.33/5.20 thf(sy_c_Rat_ORep__Rat, type, rep_Rat: (rat > ((product_prod @ int) @ int))).
% 0.33/5.20 thf(sy_c_Rat_Ofield__char__0__class_ORats, type, field_char_0_Rats: (!> [A:$tType]: ((set @ A)))).
% 0.33/5.20 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat, type, field_char_0_of_rat: (!> [A:$tType]: ((rat > A)))).
% 0.33/5.20 thf(sy_c_Rat_Onormalize, type, normalize: (((product_prod @ int) @ int) > ((product_prod @ int) @ int))).
% 0.33/5.20 thf(sy_c_Rat_Oof__int, type, of_int: (int > rat)).
% 0.33/5.20 thf(sy_c_Rat_Opcr__rat, type, pcr_rat: (((product_prod @ int) @ int) > (rat > $o))).
% 0.33/5.20 thf(sy_c_Rat_Opositive, type, positive: (rat > $o)).
% 0.33/5.20 thf(sy_c_Rat_Oquotient__of, type, quotient_of: (rat > ((product_prod @ int) @ int))).
% 0.33/5.20 thf(sy_c_Rat_Oratrel, type, ratrel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 0.33/5.20 thf(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism, type, tuple_1188178415141063261rphism: (!> [A:$tType,B:$tType,C:$tType]: (((A > ((product_prod @ B) @ C)) > ((((product_prod @ B) @ C) > A) > (((tuple_isomorphism @ A) @ B) @ C)))))).
% 0.33/5.20 thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism, type, tuple_9181185373184732606rphism: (!> [A:$tType,B:$tType,C:$tType]: (((A > nat) > ((B > nat) > ((C > nat) > ((((tuple_isomorphism @ A) @ B) @ C) > nat))))))).
% 0.33/5.20 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Relation_OField, type, field2: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ A))))).
% 0.33/5.20 thf(sy_c_Relation_OId, type, id2: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 0.33/5.20 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 0.33/5.20 thf(sy_c_Relation_Oirreflp, type, irreflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 0.33/5.20 thf(sy_c_Relation_Orelcomp, type, relcomp: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ A) @ C))))))).
% 0.33/5.20 thf(sy_c_Relation_Otransp, type, transp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 0.33/5.20 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime, type, algebr8660921524188924756oprime: (!> [A:$tType]: ((A > (A > $o))))).
% 0.33/5.20 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Rings_Odvd__class_Odvd, type, dvd_dvd: (!> [A:$tType]: ((A > (A > $o))))).
% 0.33/5.20 thf(sy_c_Rings_Omodulo__class_Omodulo, type, modulo_modulo: (!> [A:$tType]: ((A > (A > A))))).
% 0.33/5.20 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize, type, normal6383669964737779283malize: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor, type, unit_f5069060285200089521factor: (!> [A:$tType]: ((A > A)))).
% 0.33/5.20 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))).
% 0.33/5.20 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 0.33/5.20 thf(sy_c_Set_OBex, type, bex: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 0.33/5.20 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.33/5.20 thf(sy_c_Set_OPow, type, pow2: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set_Odisjnt, type, disjnt: (!> [A:$tType]: (((set @ A) > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 0.33/5.20 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 0.33/5.20 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat, type, set_fo6178422350223883121st_nat: (!> [A:$tType]: (((nat > (A > A)) > (nat > (nat > (A > A))))))).
% 0.33/5.20 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))) > $o))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OatLeast, type, set_ord_atLeast: (!> [A:$tType]: ((A > (set @ A))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or1337092689740270186AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan, type, set_or7035219750837199246ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OgreaterThan, type, set_ord_greaterThan: (!> [A:$tType]: ((A > (set @ A))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost, type, set_or3652927894154168847AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan, type, set_or5935395276787703475ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 0.33/5.20 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 0.33/5.20 thf(sy_c_String_OLiteral, type, literal2: ($o > ($o > ($o > ($o > ($o > ($o > ($o > (literal > literal))))))))).
% 0.33/5.20 thf(sy_c_String_Ochar_OChar, type, char2: ($o > ($o > ($o > ($o > ($o > ($o > ($o > ($o > char))))))))).
% 0.33/5.20 thf(sy_c_String_Ochar_Osize__char, type, size_char: (char > nat)).
% 0.33/5.20 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char, type, comm_s6883823935334413003f_char: (!> [A:$tType]: ((char > A)))).
% 0.33/5.20 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of, type, unique5772411509450598832har_of: (!> [A:$tType]: ((A > char)))).
% 0.33/5.20 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 0.33/5.20 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 0.33/5.20 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 0.33/5.20 thf(sy_c_Sum__Type_Osum_Ocase__sum, type, sum_case_sum: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((B > C) > (((sum_sum @ A) @ B) > C)))))).
% 0.33/5.20 thf(sy_c_Transfer_Obi__total, type, bi_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 0.33/5.20 thf(sy_c_Transitive__Closure_Ontrancl, type, transitive_ntrancl: (!> [A:$tType]: ((nat > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))).
% 0.33/5.20 thf(sy_c_Transitive__Closure_Ortrancl, type, transitive_rtrancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_Transitive__Closure_Otrancl, type, transitive_trancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_Transitive__Closure_Otranclp, type, transitive_tranclp: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 0.33/5.20 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 0.33/5.20 thf(sy_c_Typerep_Otyperep_OTyperep, type, typerep3: (literal > ((list @ typerep) > typerep))).
% 0.33/5.20 thf(sy_c_Typerep_Otyperep_Osize__typerep, type, size_typerep: (typerep > nat)).
% 0.33/5.20 thf(sy_c_Typerep_Otyperep__class_Otyperep, type, typerep_typerep: (!> [A:$tType]: (((itself @ A) > typerep)))).
% 0.33/5.20 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Ofinite__psubset, type, finite_psubset: (!> [A:$tType]: ((set @ ((product_prod @ (set @ A)) @ (set @ A)))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Oless__than, type, less_than: (set @ ((product_prod @ nat) @ nat))).
% 0.33/5.20 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)))))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Omax__ext, type, max_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Omeasure, type, measure: (!> [A:$tType]: (((A > nat) > (set @ ((product_prod @ A) @ A)))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Omin__ext, type, min_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Omlex__prod, type, mlex_prod: (!> [A:$tType]: (((A > nat) > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))).
% 0.33/5.20 thf(sy_c_Wellfounded_Opred__nat, type, pred_nat: (set @ ((product_prod @ nat) @ nat))).
% 0.33/5.20 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 0.33/5.20 thf(sy_c_Zorn_Ochain__subset, type, chain_subset: (!> [A:$tType]: (((set @ (set @ A)) > $o)))).
% 0.33/5.20 thf(sy_c_Zorn_Ochains, type, chains: (!> [A:$tType]: (((set @ (set @ A)) > (set @ (set @ (set @ A))))))).
% 0.33/5.20 thf(sy_c_Zorn_Opred__on_Ochain, type, pred_chain: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 0.33/5.20 thf(sy_c_Zorn_Opred__on_Omaxchain, type, pred_maxchain: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 0.33/5.20 thf(sy_c_Zorn_Opred__on_Osuc, type, pred_suc: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > (set @ A))))))).
% 0.33/5.20 thf(sy_c_Zorn_Opred__on_Osuc__Union__closed, type, pred_s596693808085603175closed: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > (set @ (set @ A))))))).
% 0.33/5.20 thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp, type, pred_s7749564232668923593losedp: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 0.33/5.20 thf(sy_c_fChoice, type, fChoice: (!> [A:$tType]: (((A > $o) > A)))).
% 0.33/5.20 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.33/5.20 thf(sy_v_a, type, a2: (array @ a)).
% 0.33/5.20 thf(sy_v_h, type, h: (heap_ext @ product_unit)).
% 0.33/5.20 thf(sy_v_h_H, type, h2: (heap_ext @ product_unit)).
% 0.33/5.20 thf(sy_v_n, type, n: nat).
% 0.33/5.20 thf(sy_v_r, type, r: nat).
% 0.33/5.20 thf(sy_v_thesis, type, thesis: $o).
% 0.33/5.20 thf(fact_0_assms, axiom, ((((((heap_Time_effect @ nat) @ ((array_len @ a) @ a2)) @ h) @ h2) @ r) @ n)).
% 0.33/5.20 thf(fact_1_that, axiom, ((r = (((array_length @ a) @ h2) @ a2)) => ((h2 = h) => ((n = (one_one @ nat)) => thesis)))).
% 0.33/5.20 thf(fact_2_effect__lengthI, axiom, (! [A:$tType]: (((heap @ A) => (! [H:(heap_ext @ product_unit),H2:(heap_ext @ product_unit),R:nat,A3:(array @ A),N:nat]: (((H = H2) => ((R = (((array_length @ A) @ H2) @ A3)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ nat) @ ((array_len @ A) @ A3)) @ H2) @ H) @ R) @ N)))))))))).
% 0.33/5.20 thf(fact_3_one__natural_Orsp, axiom, ((one_one @ nat) = (one_one @ nat))).
% 0.33/5.20 thf(fact_4_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 0.33/5.20 thf(fact_5_length__update, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [B2:(array @ B),I:nat,V:B,H2:(heap_ext @ product_unit)]: ((((array_length @ A) @ (((((array_update @ B) @ B2) @ I) @ V) @ H2)) = ((array_length @ A) @ H2)))))))).
% 0.33/5.20 thf(fact_6_euclidean__size__1, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (((euclid6346220572633701492n_size @ A) @ (one_one @ A)) = (one_one @ nat)))))).
% 0.33/5.20 thf(fact_7_fact__1, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_8_of__nat__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_9_of__nat__1__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((one_one @ A) = ((semiring_1_of_nat @ A) @ N)) = (N = (one_one @ nat))))))))).
% 0.33/5.20 thf(fact_10_of__nat__eq__1__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: (((((semiring_1_of_nat @ A) @ N) = (one_one @ A)) = (N = (one_one @ nat))))))))).
% 0.33/5.20 thf(fact_11_dbl__dec__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_12_round__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_round @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 0.33/5.20 thf(fact_13_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 0.33/5.20 thf(fact_14_of__nat__fact, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ ((semiring_char_0_fact @ nat) @ N)) = ((semiring_char_0_fact @ A) @ N)))))))).
% 0.33/5.20 thf(fact_15_euclidean__size__of__nat, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: ((((euclid6346220572633701492n_size @ A) @ ((semiring_1_of_nat @ A) @ N)) = N))))))).
% 0.33/5.20 thf(fact_16_round__of__nat, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:nat]: ((((archimedean_round @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 0.33/5.20 thf(fact_17_one__integer_Orsp, axiom, ((one_one @ int) = (one_one @ int))).
% 0.33/5.20 thf(fact_18_update__swap__neqIndex, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,I2:nat,A3:(array @ A),V:A,V2:A,H2:(heap_ext @ product_unit)]: (((I != I2) => ((((((array_update @ A) @ A3) @ I) @ V) @ (((((array_update @ A) @ A3) @ I2) @ V2) @ H2)) = (((((array_update @ A) @ A3) @ I2) @ V2) @ (((((array_update @ A) @ A3) @ I) @ V) @ H2)))))))))).
% 0.33/5.20 thf(fact_19_int__ops_I2_J, axiom, (((semiring_1_of_nat @ int) @ (one_one @ nat)) = (one_one @ int))).
% 0.33/5.20 thf(fact_20_effect__deterministic_I3_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),A3:A,N:nat,H3:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H2) @ H) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H2) @ H3) @ B2) @ N2) => (N = N2)))))).
% 0.33/5.20 thf(fact_21_effect__deterministic_I2_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),A3:A,N:nat,H3:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H2) @ H) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H2) @ H3) @ B2) @ N2) => (H = H3)))))).
% 0.33/5.20 thf(fact_22_effect__deterministic_I1_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),A3:A,N:nat,H3:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H2) @ H) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H2) @ H3) @ B2) @ N2) => (A3 = B2)))))).
% 0.33/5.20 thf(fact_23_effect__ifE, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),E:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E)) @ H2) @ H) @ R) @ N) => ((C2 => (~ (((((((heap_Time_effect @ A) @ T2) @ H2) @ H) @ R) @ N)))) => (~ (((~ (C2)) => (~ (((((((heap_Time_effect @ A) @ E) @ H2) @ H) @ R) @ N))))))))))).
% 0.33/5.20 thf(fact_24_effect__ifI, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat,E:(heap_Time_Heap @ A)]: (((C2 => ((((((heap_Time_effect @ A) @ T2) @ H2) @ H) @ R) @ N)) => (((~ (C2)) => ((((((heap_Time_effect @ A) @ E) @ H2) @ H) @ R) @ N)) => ((((((heap_Time_effect @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E)) @ H2) @ H) @ R) @ N)))))).
% 0.33/5.20 thf(fact_25_division__segment__of__nat, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: ((((euclid7384307370059645450egment @ A) @ ((semiring_1_of_nat @ A) @ N)) = (one_one @ A)))))))).
% 0.33/5.20 thf(fact_26_present__update, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [B2:(array @ B),I:nat,V:B,H2:(heap_ext @ product_unit)]: ((((array_present @ A) @ (((((array_update @ B) @ B2) @ I) @ V) @ H2)) = ((array_present @ A) @ H2)))))))).
% 0.33/5.20 thf(fact_27_fact__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (zero_zero @ nat)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_28_effect__tapI, axiom, (! [A:$tType,H:(heap_ext @ product_unit),H2:(heap_ext @ product_unit),R:A,F:((heap_ext @ product_unit) > A)]: (((H = H2) => ((R = (F @ H2)) => ((((((heap_Time_effect @ A) @ ((heap_Time_tap @ A) @ F)) @ H2) @ H) @ R) @ (one_one @ nat))))))).
% 0.33/5.20 thf(fact_29_effect__tapE, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_tap @ A) @ F)) @ H2) @ H) @ R) @ N) => (~ (((H = H2) => ((R = (F @ H2)) => (N != (one_one @ nat)))))))))).
% 0.33/5.20 thf(fact_30_update__swap__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),A4:(array @ B),I:nat,V:A,I2:nat,V2:B,H2:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ A3) @ A4) => ((((((array_update @ A) @ A3) @ I) @ V) @ (((((array_update @ B) @ A4) @ I2) @ V2) @ H2)) = (((((array_update @ B) @ A4) @ I2) @ V2) @ (((((array_update @ A) @ A3) @ I) @ V) @ H2)))))))))).
% 0.33/5.20 thf(fact_31_unequal, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),A4:(array @ A)]: (((A3 != A4) = ((((array_noteq @ A) @ A) @ A3) @ A4)))))))).
% 0.33/5.20 thf(fact_32_division__segment__1, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (((euclid7384307370059645450egment @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_33_round__0, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_round @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_34_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 0.33/5.20 thf(fact_35_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 0.33/5.20 thf(fact_36_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_37_euclidean__size__eq__0__iff, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A]: (((((euclid6346220572633701492n_size @ A) @ B2) = (zero_zero @ nat)) = (B2 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_38_size__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (((euclid6346220572633701492n_size @ A) @ (zero_zero @ A)) = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_39_nat__int__comparison_I1_J, axiom, ((^ [Y:nat,Z2:nat]: ((Y = Z2))) = (^ [A5:nat,B3:nat]: ((((semiring_1_of_nat @ int) @ A5) = ((semiring_1_of_nat @ int) @ B3)))))).
% 0.33/5.20 thf(fact_40_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 0.33/5.20 thf(fact_41_division__segment__not__0, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A]: ((((euclid7384307370059645450egment @ A) @ A3) != (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_42_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 0.33/5.20 thf(fact_43_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A6)))) = A6)))).
% 0.33/5.20 thf(fact_44_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.33/5.20 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 0.33/5.20 thf(fact_46_int__if, axiom, (! [P:$o,A3:nat,B2:nat]: (((P => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A3) @ B2)) = ((semiring_1_of_nat @ int) @ A3))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A3) @ B2)) = ((semiring_1_of_nat @ int) @ B2))))))).
% 0.33/5.20 thf(fact_47_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_48_noteq__sym, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),B2:(array @ B)]: ((((((array_noteq @ A) @ B) @ A3) @ B2) => ((((array_noteq @ B) @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_49_noteq__irrefl, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A)]: ((~ (((((array_noteq @ A) @ A) @ R) @ R))))))))).
% 0.33/5.20 thf(fact_50_fact__nonzero, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semiri3467727345109120633visors @ A)) => (! [N:nat]: ((((semiring_char_0_fact @ A) @ N) != (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_51_division__segment__nat__def, axiom, ((euclid7384307370059645450egment @ nat) = (^ [N3:nat]: ((one_one @ nat))))).
% 0.33/5.20 thf(fact_52_division__segment__eq__iff, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,B2:A]: (((((euclid7384307370059645450egment @ A) @ A3) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ B2)) => (A3 = B2))))))))).
% 0.33/5.20 thf(fact_53_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 0.33/5.20 thf(fact_54_dbl__inc__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ (zero_zero @ A)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_55_effect__ureturnI, axiom, (! [A:$tType,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),X:A]: (((H2 = H) => ((((((heap_Time_effect @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H2) @ H) @ X) @ (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_56_effect__ureturnE, axiom, (! [A:$tType,X:A,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H2) @ H) @ R) @ N) => (~ (((R = X) => ((H = H2) => (N != (zero_zero @ nat)))))))))).
% 0.33/5.20 thf(fact_57_effect__assertE, axiom, (! [A:$tType,P:(A > $o),X:A,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H2) @ H) @ R) @ N) => (~ (((P @ X) => ((R = X) => ((H = H2) => (N != (one_one @ nat))))))))))).
% 0.33/5.20 thf(fact_58_effect__assertI, axiom, (! [A:$tType,P:(A > $o),X:A,H:(heap_ext @ product_unit),H2:(heap_ext @ product_unit),R:A,N:nat]: (((P @ X) => ((H = H2) => ((R = X) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H2) @ H) @ R) @ N)))))))).
% 0.33/5.20 thf(fact_59_dbl__dec__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.20 thf(fact_60_division__segment__euclidean__size, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((((times_times @ A) @ ((euclid7384307370059645450egment @ A) @ A3)) @ ((semiring_1_of_nat @ A) @ ((euclid6346220572633701492n_size @ A) @ A3))) = A3))))))).
% 0.33/5.20 thf(fact_61_euclidean__size__greater__0__iff, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((euclid6346220572633701492n_size @ A) @ B2)) = (B2 != (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_62_fact__Suc__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_63_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 0.33/5.20 thf(fact_64_effect__returnE, axiom, (! [A:$tType,X:A,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_return @ A) @ X)) @ H2) @ H) @ R) @ N) => (~ (((R = X) => ((H = H2) => (N != (one_one @ nat)))))))))).
% 0.33/5.20 thf(fact_65_verit__minus__simplify_I4_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: ((((uminus_uminus @ B) @ ((uminus_uminus @ B) @ B2)) = B2))))))).
% 0.33/5.20 thf(fact_66_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = ((uminus_uminus @ A) @ B2)) = (A3 = B2)))))))).
% 0.33/5.20 thf(fact_67_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A3)) = A3))))))).
% 0.33/5.20 thf(fact_68_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 0.33/5.20 thf(fact_69_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 0.33/5.20 thf(fact_70_nat_Oinject, axiom, (! [X22:nat,Y2:nat]: ((((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))).
% 0.33/5.20 thf(fact_71_mult__cancel2, axiom, (! [M:nat,K:nat,N:nat]: ((((((times_times @ nat) @ M) @ K) = (((times_times @ nat) @ N) @ K)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_72_mult__cancel1, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_73_mult__0__right, axiom, (! [M:nat]: (((((times_times @ nat) @ M) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_74_mult__is__0, axiom, (! [M:nat,N:nat]: ((((((times_times @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_75_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.33/5.20 thf(fact_76_bot__nat__0_Oextremum, axiom, (! [A3:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A3)))).
% 0.33/5.20 thf(fact_77_nat__mult__eq__1__iff, axiom, (! [M:nat,N:nat]: ((((((times_times @ nat) @ M) @ N) = (one_one @ nat)) = ((M = (one_one @ nat)) & (N = (one_one @ nat))))))).
% 0.33/5.20 thf(fact_78_nat__1__eq__mult__iff, axiom, (! [M:nat,N:nat]: ((((one_one @ nat) = (((times_times @ nat) @ M) @ N)) = ((M = (one_one @ nat)) & (N = (one_one @ nat))))))).
% 0.33/5.20 thf(fact_79_le__zero__eq, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_80_not__gr__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_81_mult__zero__left, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A3:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_82_mult__zero__right, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_83_mult__eq__0__iff, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ B2) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_84_mult__cancel__left, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A3:A,B2:A]: ((((((times_times @ A) @ C2) @ A3) = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A3 = B2))))))))).
% 0.33/5.20 thf(fact_85_mult__cancel__right, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [A3:A,C2:A,B2:A]: ((((((times_times @ A) @ A3) @ C2) = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A3 = B2))))))))).
% 0.33/5.20 thf(fact_86_neg__le__iff__le, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_87_neg__equal__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((uminus_uminus @ A) @ A3) = A3) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_88_equal__neg__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((A3 = ((uminus_uminus @ A) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_89_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((uminus_uminus @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_90_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A3)) = ((zero_zero @ A) = A3)))))))).
% 0.33/5.20 thf(fact_91_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_92_mult__1, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ (one_one @ A)) @ A3) = A3))))))).
% 0.33/5.20 thf(fact_93_mult_Oright__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 0.33/5.20 thf(fact_94_neg__less__iff__less, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_95_mult__minus__left, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = ((uminus_uminus @ A) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_96_minus__mult__minus, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2)) = (((times_times @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_97_mult__minus__right, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_98_of__nat__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 0.33/5.20 thf(fact_99_one__le__mult__iff, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((times_times @ nat) @ M) @ N)) = ((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ M) & (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ N)))))).
% 0.33/5.20 thf(fact_100_one__eq__mult__iff, axiom, (! [M:nat,N:nat]: ((((suc @ (zero_zero @ nat)) = (((times_times @ nat) @ M) @ N)) = ((M = (suc @ (zero_zero @ nat))) & (N = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_101_mult__eq__1__iff, axiom, (! [M:nat,N:nat]: ((((((times_times @ nat) @ M) @ N) = (suc @ (zero_zero @ nat))) = ((M = (suc @ (zero_zero @ nat))) & (N = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_102_of__nat__mult, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((times_times @ nat) @ M) @ N)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_103_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_104_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 0.33/5.20 thf(fact_105_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 0.33/5.20 thf(fact_106_mult__le__cancel2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_107_nat__0__less__mult__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((times_times @ nat) @ M) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.20 thf(fact_108_mult__less__cancel2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_109_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_110_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.33/5.20 thf(fact_111_bot__nat__0_Onot__eq__extremum, axiom, (! [A3:nat]: (((A3 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A3))))).
% 0.33/5.20 thf(fact_112_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.20 thf(fact_113_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_114_neg__le__0__iff__le, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_115_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_116_mult__cancel__right2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [A3:A,C2:A]: ((((((times_times @ A) @ A3) @ C2) = C2) = ((C2 = (zero_zero @ A)) | (A3 = (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_117_mult__cancel__right1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_118_mult__cancel__left2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,A3:A]: ((((((times_times @ A) @ C2) @ A3) = C2) = ((C2 = (zero_zero @ A)) | (A3 = (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_119_mult__cancel__left1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_120_neg__less__0__iff__less, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_121_neg__0__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_122_neg__less__pos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.20 thf(fact_123_less__neg__neg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_124_mult__minus1, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ Z3) = ((uminus_uminus @ A) @ Z3)))))))).
% 0.33/5.20 thf(fact_125_mult__minus1__right, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:A]: (((((times_times @ A) @ Z3) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ Z3)))))))).
% 0.33/5.20 thf(fact_126_of__nat__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 0.33/5.20 thf(fact_127_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 0.33/5.20 thf(fact_128_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_129_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_130_dbl__inc__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.20 thf(fact_131_fact__Suc, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_char_0_fact @ A) @ (suc @ N)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N))) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 0.33/5.20 thf(fact_132_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 0.33/5.20 thf(fact_133_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 0.33/5.20 thf(fact_134_one__less__mult, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ (((times_times @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_135_n__less__m__mult__n, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ N) @ (((times_times @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_136_n__less__n__mult__m, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ N) @ (((times_times @ nat) @ N) @ M))))))).
% 0.33/5.20 thf(fact_137_ex__least__nat__less, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ (suc @ K2))))))))))).
% 0.33/5.20 thf(fact_138_mult__le__cancel__iff1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z3) => ((((ord_less_eq @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ Y3) @ Z3)) = (((ord_less_eq @ A) @ X) @ Y3))))))))).
% 0.33/5.20 thf(fact_139_mult__le__cancel__iff2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z3) => ((((ord_less_eq @ A) @ (((times_times @ A) @ Z3) @ X)) @ (((times_times @ A) @ Z3) @ Y3)) = (((ord_less_eq @ A) @ X) @ Y3))))))))).
% 0.33/5.20 thf(fact_140_division__segment__int__def, axiom, ((euclid7384307370059645450egment @ int) = (^ [K3:int]: (((((if @ int) @ (((ord_less_eq @ int) @ (zero_zero @ int)) @ K3)) @ (one_one @ int)) @ ((uminus_uminus @ int) @ (one_one @ int))))))).
% 0.33/5.20 thf(fact_141_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((A3 = B2) | ((~ ((((ord_less_eq @ A) @ A3) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A3))))))))))).
% 0.33/5.20 thf(fact_142_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom, (! [A:$tType]: (((linord2810124833399127020strict @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_143_mult__less__cancel__right__disj, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A3) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_144_mult__strict__right__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_145_mult__strict__right__mono__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_146_mult__less__cancel__left__disj, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A3) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_147_mult__strict__left__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_148_mult__less__le__imp__less, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_149_mult__le__less__imp__less, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_150_mult__strict__left__mono__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_151_mult__less__cancel__left__pos, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_152_mult__less__cancel__left__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ B2) @ A3))))))))).
% 0.33/5.20 thf(fact_153_mult__right__le__imp__le, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_154_zero__less__mult__pos2, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ B2) @ A3)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))).
% 0.33/5.20 thf(fact_155_mult__left__le__imp__le, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_156_mult__le__cancel__left__pos, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_157_mult__le__cancel__left__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 0.33/5.20 thf(fact_158_zero__less__mult__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))).
% 0.33/5.20 thf(fact_159_mult__less__cancel__right, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_160_mult__strict__mono_H, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_161_mult__right__less__imp__less, axiom, (! [A:$tType]: (((linordered_semiring @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_162_mult__less__cancel__left, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_163_mult__strict__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_164_mult__left__less__imp__less, axiom, (! [A:$tType]: (((linordered_semiring @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_165_mult__le__cancel__right, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_166_mult__le__cancel__left, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_167_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom, (! [A:$tType]: (((ordere2520102378445227354miring @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_168_zero__less__mult__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_169_mult__pos__neg2, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ B2) @ A3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_170_mult__pos__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_171_mult__pos__neg, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_172_mult__neg__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_173_zero__le__mult__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_174_mult__nonneg__nonpos2, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ B2) @ A3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_175_mult__less__0__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 0.33/5.20 thf(fact_176_mult__nonpos__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_177_mult__nonneg__nonpos, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_178_mult__nonneg__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_179_split__mult__neg__le, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A3:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2))) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_180_mult__le__0__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 0.33/5.20 thf(fact_181_not__square__less__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ (((times_times @ A) @ A3) @ A3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_182_mult__neg__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_183_mult__right__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_184_mult__right__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_185_mult__left__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_186_mult__nonpos__nonpos, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_187_mult__left__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_188_split__mult__pos__le, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_189_zero__le__square, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ A3)))))))).
% 0.33/5.20 thf(fact_190_mult__mono_H, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_191_mult__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))).
% 0.33/5.20 thf(fact_192_verit__negate__coefficient_I2_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_193_verit__negate__coefficient_I3_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((A3 = B2) => (((uminus_uminus @ A) @ A3) = ((uminus_uminus @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_194_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 0.33/5.20 thf(fact_195_verit__comp__simplify1_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 0.33/5.20 thf(fact_196_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B4:B,A4:B]: (((~ ((((ord_less_eq @ B) @ B4) @ A4))) = (((ord_less @ B) @ A4) @ B4)))))))).
% 0.33/5.20 thf(fact_197_square__eq__iff, axiom, (! [A:$tType]: (((idom @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ A3) = (((times_times @ A) @ B2) @ B2)) = ((A3 = B2) | (A3 = ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_198_minus__mult__commute, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_199_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((X != Y3) => ((~ ((((ord_less @ A) @ X) @ Y3))) => (((ord_less @ A) @ Y3) @ X))))))))).
% 0.33/5.20 thf(fact_200_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V3:(A > nat),X:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y4:A]: (((((ord_less @ nat) @ (V3 @ Y4)) @ (V3 @ X3)) & (~ ((P @ Y4))))))))) => (P @ X))))).
% 0.33/5.20 thf(fact_201_transitive__stepwise__le, axiom, (! [M:nat,N:nat,R2:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M) @ N) => ((! [X3:nat]: (((R2 @ X3) @ X3))) => ((! [X3:nat,Y5:nat,Z4:nat]: ((((R2 @ X3) @ Y5) => (((R2 @ Y5) @ Z4) => ((R2 @ X3) @ Z4))))) => ((! [N4:nat]: (((R2 @ N4) @ (suc @ N4)))) => ((R2 @ M) @ N)))))))).
% 0.33/5.20 thf(fact_202_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I4:nat,J2:nat]: (((((ord_less @ nat) @ I4) @ J2) => (((ord_less @ nat) @ (F @ I4)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 0.33/5.20 thf(fact_203_Suc__mult__less__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ (suc @ K)) @ M)) @ (((times_times @ nat) @ (suc @ K)) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_204_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 0.33/5.20 thf(fact_205_nat__induct__at__least, axiom, (! [M:nat,N:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M) @ N) => ((P @ M) => ((! [N4:nat]: (((((ord_less_eq @ nat) @ M) @ N4) => ((P @ N4) => (P @ (suc @ N4)))))) => (P @ N))))))).
% 0.33/5.20 thf(fact_206_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ B) @ (F @ Y4)) @ (F @ X3)) => (P @ Y4)))) => (P @ X3)))) => (P @ A3)))))))).
% 0.33/5.20 thf(fact_207_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_208_Suc__mult__le__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ (suc @ K)) @ M)) @ (((times_times @ nat) @ (suc @ K)) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_209_strict__inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I) @ J) => ((! [I4:nat]: (((J = (suc @ I4)) => (P @ I4)))) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ J) => ((P @ (suc @ I4)) => (P @ I4))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_210_linorder__neqE__nat, axiom, (! [X:nat,Y3:nat]: (((X != Y3) => ((~ ((((ord_less @ nat) @ X) @ Y3))) => (((ord_less @ nat) @ Y3) @ X)))))).
% 0.33/5.20 thf(fact_211_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_212_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N3:nat]: (((((ord_less @ nat) @ M2) @ N3) | (M2 = N3)))))).
% 0.33/5.20 thf(fact_213_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N4:nat]: (((~ ((P @ N4))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N4) & (~ ((P @ M3))))))))) => (P @ N))))).
% 0.33/5.20 thf(fact_214_Suc__mult__cancel1, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ (suc @ K)) @ M) = (((times_times @ nat) @ (suc @ K)) @ N)) = (M = N))))).
% 0.33/5.20 thf(fact_215_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N4:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N4) => (P @ M3)))) => (P @ N4)))) => (P @ N))))).
% 0.33/5.20 thf(fact_216_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.33/5.20 thf(fact_217_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_218_less__Suc__induct, axiom, (! [I:nat,J:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I) @ J) => ((! [I4:nat]: (((P @ I4) @ (suc @ I4)))) => ((! [I4:nat,J2:nat,K2:nat]: (((((ord_less @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ K2) => (((P @ I4) @ J2) => (((P @ J2) @ K2) => ((P @ I4) @ K2))))))) => ((P @ I) @ J))))))).
% 0.33/5.20 thf(fact_219_le__imp__less__Suc, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 0.33/5.20 thf(fact_220_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N4:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N4) => (P @ M3)))) => (P @ N4)))) => (P @ N))))).
% 0.33/5.20 thf(fact_221_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 0.33/5.20 thf(fact_222_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A3:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ B) @ (F @ Y4)) @ (F @ X3)) => (P @ Y4)))) => (P @ X3)))) => (P @ A3)))))))).
% 0.33/5.20 thf(fact_223_less__trans__Suc, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (suc @ I)) @ K)))))).
% 0.33/5.20 thf(fact_224_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))).
% 0.33/5.20 thf(fact_225_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 0.33/5.20 thf(fact_226_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N3:nat]: (((ord_less_eq @ nat) @ (suc @ N3)))))).
% 0.33/5.20 thf(fact_227_less__Suc__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_228_le__less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 0.33/5.20 thf(fact_229_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 0.33/5.20 thf(fact_230_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.33/5.20 thf(fact_231_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_232_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 0.33/5.20 thf(fact_233_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N) @ M4)))))))).
% 0.33/5.20 thf(fact_234_Suc__le__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_235_All__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((! [I5:nat]: (((((ord_less @ nat) @ I5) @ (suc @ N)) => (P @ I5)))) = ((P @ N) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ N) => (P @ I5))))))))).
% 0.33/5.20 thf(fact_236_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 0.33/5.20 thf(fact_237_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 0.33/5.20 thf(fact_238_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N3:nat]: (((((ord_less_eq @ nat) @ M2) @ N3) & (M2 != N3)))))).
% 0.33/5.20 thf(fact_239_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 0.33/5.20 thf(fact_240_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 0.33/5.20 thf(fact_241_Ex__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((? [I5:nat]: (((((ord_less @ nat) @ I5) @ (suc @ N)) & (P @ I5)))) = ((P @ N) | (? [I5:nat]: (((((ord_less @ nat) @ I5) @ N) & (P @ I5))))))))).
% 0.33/5.20 thf(fact_242_inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ J) => ((! [N4:nat]: (((((ord_less_eq @ nat) @ I) @ N4) => ((((ord_less @ nat) @ N4) @ J) => ((P @ (suc @ N4)) => (P @ N4)))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_243_dec__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ I) => ((! [N4:nat]: (((((ord_less_eq @ nat) @ I) @ N4) => ((((ord_less @ nat) @ N4) @ J) => ((P @ N4) => (P @ (suc @ N4))))))) => (P @ J))))))).
% 0.33/5.20 thf(fact_244_Suc__inject, axiom, (! [X:nat,Y3:nat]: ((((suc @ X) = (suc @ Y3)) => (X = Y3))))).
% 0.33/5.20 thf(fact_245_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 0.33/5.20 thf(fact_246_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 0.33/5.20 thf(fact_247_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 0.33/5.20 thf(fact_248_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 0.33/5.20 thf(fact_249_Suc__lessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I)) @ K) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K != (suc @ J2))))))))))).
% 0.33/5.20 thf(fact_250_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_251_Suc__le__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_252_Suc__le__D, axiom, (! [N:nat,M5:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M5) => (? [M6:nat]: ((M5 = (suc @ M6)))))))).
% 0.33/5.20 thf(fact_253_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 0.33/5.20 thf(fact_254_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 0.33/5.20 thf(fact_255_Suc__leI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_256_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_257_Nat_OlessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ I) @ K) => ((K != (suc @ I)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K != (suc @ J2)))))))))))).
% 0.33/5.20 thf(fact_258_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))))))))))).
% 0.33/5.20 thf(fact_259_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ A3)))))))).
% 0.33/5.20 thf(fact_260_less__minus__iff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_261_mult_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [B2:A,A3:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)) = (((times_times @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_262_le__imp__neg__le, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_263_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A3)))))))).
% 0.33/5.20 thf(fact_264_le__minus__iff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_265_mult_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((times_times @ A) = (^ [A5:A,B3:A]: ((((times_times @ A) @ B3) @ A5)))))))).
% 0.33/5.20 thf(fact_266_less__1__mult, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ (one_one @ A)) @ M) => ((((ord_less @ A) @ (one_one @ A)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ M) @ N)))))))))).
% 0.33/5.20 thf(fact_267_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = B2) = (((uminus_uminus @ A) @ B2) = A3)))))))).
% 0.33/5.20 thf(fact_268_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((A3 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_269_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M:nat]: (((! [N4:nat]: ((((ord_less @ A) @ (F @ N4)) @ (F @ (suc @ N4))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 0.33/5.20 thf(fact_270_mult_Oassoc, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_271_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N2:nat]: (((! [N4:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N4))) @ (F @ N4)))) => ((((ord_less_eq @ nat) @ N) @ N2) => (((ord_less_eq @ A) @ (F @ N2)) @ (F @ N)))))))))).
% 0.33/5.20 thf(fact_272_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N2:nat]: (((! [N4:nat]: ((((ord_less @ A) @ (F @ N4)) @ (F @ (suc @ N4))))) => ((((ord_less @ nat) @ N) @ N2) => (((ord_less @ A) @ (F @ N)) @ (F @ N2)))))))))).
% 0.33/5.20 thf(fact_273_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N2:nat]: (((! [N4:nat]: ((((ord_less_eq @ A) @ (F @ N4)) @ (F @ (suc @ N4))))) => ((((ord_less_eq @ nat) @ N) @ N2) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N2)))))))))).
% 0.33/5.20 thf(fact_274_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: (((ab_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))))))))).
% 0.33/5.20 thf(fact_275_fold__if__return, axiom, (! [A:$tType,B2:$o,C2:A,D2:A]: (((B2 => (((heap_Time_return @ A) @ C2) = ((heap_Time_return @ A) @ ((((if @ A) @ B2) @ C2) @ D2)))) & ((~ (B2)) => (((heap_Time_return @ A) @ D2) = ((heap_Time_return @ A) @ ((((if @ A) @ B2) @ C2) @ D2)))))))).
% 0.33/5.20 thf(fact_276_mult__less__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,C2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A3)))))))))).
% 0.33/5.20 thf(fact_277_mult__less__cancel__right1, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_278_mult__less__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A3)))))))))).
% 0.33/5.20 thf(fact_279_mult__less__cancel__left1, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_280_mult__le__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,C2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A3)))))))))).
% 0.33/5.20 thf(fact_281_mult__le__cancel__right1, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_282_mult__le__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A3)))))))))).
% 0.33/5.20 thf(fact_283_mult__le__cancel__left1, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_284_mult__less__iff1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z3) => ((((ord_less @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ Y3) @ Z3)) = (((ord_less @ A) @ X) @ Y3))))))))).
% 0.33/5.20 thf(fact_285_less__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_286_less__minus__one__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_287_le__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_288_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)))))).
% 0.33/5.20 thf(fact_289_less__Suc__eq__0__disj, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((M = (zero_zero @ nat)) | (? [J3:nat]: (((M = (suc @ J3)) & (((ord_less @ nat) @ J3) @ N))))))))).
% 0.33/5.20 thf(fact_290_gr0__implies__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (? [M6:nat]: ((N = (suc @ M6)))))))).
% 0.33/5.20 thf(fact_291_All__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((! [I5:nat]: (((((ord_less @ nat) @ I5) @ (suc @ N)) => (P @ I5)))) = ((P @ (zero_zero @ nat)) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ N) => (P @ (suc @ I5)))))))))).
% 0.33/5.20 thf(fact_292_gr0__conv__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) = (? [M2:nat]: ((N = (suc @ M2)))))))).
% 0.33/5.20 thf(fact_293_Ex__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((? [I5:nat]: (((((ord_less @ nat) @ I5) @ (suc @ N)) & (P @ I5)))) = ((P @ (zero_zero @ nat)) | (? [I5:nat]: (((((ord_less @ nat) @ I5) @ N) & (P @ (suc @ I5)))))))))).
% 0.33/5.20 thf(fact_294_mult__less__mono2, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ K) @ I)) @ (((times_times @ nat) @ K) @ J))))))).
% 0.33/5.20 thf(fact_295_mult__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ K))))))).
% 0.33/5.20 thf(fact_296_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ K2)))))))))).
% 0.33/5.20 thf(fact_297_less__imp__of__nat__less, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_298_of__nat__less__imp__less, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) => (((ord_less @ nat) @ M) @ N)))))))).
% 0.33/5.20 thf(fact_299_of__nat__mono, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I)) @ ((semiring_1_of_nat @ A) @ J))))))))).
% 0.33/5.20 thf(fact_300_int__ops_I7_J, axiom, (! [A3:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((times_times @ nat) @ A3) @ B2)) = (((times_times @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 0.33/5.20 thf(fact_301_nat__int__comparison_I3_J, axiom, ((ord_less_eq @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 0.33/5.20 thf(fact_302_mult__not__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ B2) != (zero_zero @ A)) => ((A3 != (zero_zero @ A)) & (B2 != (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_303_divisors__zero, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ B2) = (zero_zero @ A)) => ((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_304_no__zero__divisors, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ B2) != (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_305_mult__left__cancel, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ C2) @ A3) = (((times_times @ A) @ C2) @ B2)) = (A3 = B2))))))))).
% 0.33/5.20 thf(fact_306_mult__right__cancel, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A3) @ C2) = (((times_times @ A) @ B2) @ C2)) = (A3 = B2))))))))).
% 0.33/5.20 thf(fact_307_fact__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ ((semiring_char_0_fact @ A) @ M)) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 0.33/5.20 thf(fact_308_euclidean__size__mult, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A]: ((((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_309_round__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_round @ A) @ Y3))))))))).
% 0.33/5.20 thf(fact_310_mult__left__le, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [C2:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ A3))))))))).
% 0.33/5.20 thf(fact_311_mult__le__one, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_312_mult__right__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ X) @ Y3)) @ X)))))))))).
% 0.33/5.20 thf(fact_313_mult__left__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Y3) @ X)) @ X)))))))))).
% 0.33/5.20 thf(fact_314_less__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_315_less__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_316_le__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_317_le__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_318_ex__less__of__nat__mult, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (? [N4:nat]: ((((ord_less @ A) @ Y3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N4)) @ X))))))))))).
% 0.33/5.20 thf(fact_319_nat__induct__non__zero, axiom, (! [N:nat,P:(nat > $o)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((P @ (one_one @ nat)) => ((! [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => ((P @ N4) => (P @ (suc @ N4)))))) => (P @ N))))))).
% 0.33/5.20 thf(fact_320_size__mult__mono_H, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((ord_less_eq @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_321_size__mult__mono, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((ord_less_eq @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_322_fact__less__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_char_0_fact @ A) @ M)) @ ((semiring_char_0_fact @ A) @ N)))))))))).
% 0.33/5.20 thf(fact_323_fact__ge__Suc__0__nat, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ ((semiring_char_0_fact @ nat) @ N))))).
% 0.33/5.20 thf(fact_324_one__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => ((one_one @ A) != ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.20 thf(fact_325_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_326_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_327_not__less__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_328_gr__zeroI, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 0.33/5.20 thf(fact_329_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_330_mult_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 0.33/5.20 thf(fact_331_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ (one_one @ A)) @ A3) = A3))))))).
% 0.33/5.20 thf(fact_332_zero__le, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 0.33/5.20 thf(fact_333_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_334_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 0.33/5.20 thf(fact_335_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X_1:nat]: ((P @ X_1))) => (? [N4:nat]: (((~ ((P @ N4))) & (P @ (suc @ N4)))))))))).
% 0.33/5.20 thf(fact_336_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M6:nat]: ((N = (suc @ M6)))))))).
% 0.33/5.20 thf(fact_337_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 0.33/5.20 thf(fact_338_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 0.33/5.20 thf(fact_339_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_340_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N4:nat]: (((P @ (suc @ N4)) => (P @ N4)))) => (P @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_341_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X3:nat]: (((P @ X3) @ (zero_zero @ nat)))) => ((! [Y5:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y5)))) => ((! [X3:nat,Y5:nat]: ((((P @ X3) @ Y5) => ((P @ (suc @ X3)) @ (suc @ Y5))))) => ((P @ M) @ N))))))).
% 0.33/5.20 thf(fact_342_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N4:nat]: (((P @ N4) => (P @ (suc @ N4))))) => (P @ N)))))).
% 0.33/5.20 thf(fact_343_old_Onat_Oexhaust, axiom, (! [Y3:nat]: (((Y3 != (zero_zero @ nat)) => (~ ((! [Nat3:nat]: ((Y3 != (suc @ Nat3)))))))))).
% 0.33/5.20 thf(fact_344_nat_OdiscI, axiom, (! [Nat:nat,X22:nat]: (((Nat = (suc @ X22)) => (Nat != (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_345_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 0.33/5.20 thf(fact_346_old_Onat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((suc @ Nat4) != (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_347_nat_Odistinct_I1_J, axiom, (! [X22:nat]: (((zero_zero @ nat) != (suc @ X22))))).
% 0.33/5.20 thf(fact_348_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_349_mult__0, axiom, (! [N:nat]: (((((times_times @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_350_mult__of__nat__commute, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X:nat,Y3:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ X)) @ Y3) = (((times_times @ A) @ Y3) @ ((semiring_1_of_nat @ A) @ X))))))))).
% 0.33/5.20 thf(fact_351_infinite__descent0__measure, axiom, (! [A:$tType,V3:(A > nat),P:(A > $o),X:A]: (((! [X3:A]: ((((V3 @ X3) = (zero_zero @ nat)) => (P @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V3 @ X3)) => ((~ ((P @ X3))) => (? [Y4:A]: (((((ord_less @ nat) @ (V3 @ Y4)) @ (V3 @ X3)) & (~ ((P @ Y4)))))))))) => (P @ X)))))).
% 0.33/5.20 thf(fact_352_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => ((~ ((P @ N4))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N4) & (~ ((P @ M3)))))))))) => (P @ N)))))).
% 0.33/5.20 thf(fact_353_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_354_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_355_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_356_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_357_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.33/5.20 thf(fact_358_bot__nat__0_Oextremum__strict, axiom, (! [A3:nat]: ((~ ((((ord_less @ nat) @ A3) @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_359_reals__Archimedean2, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [N4:nat]: ((((ord_less @ A) @ X) @ ((semiring_1_of_nat @ A) @ N4)))))))))).
% 0.33/5.20 thf(fact_360_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_361_bot__nat__0_Oextremum__uniqueI, axiom, (! [A3:nat]: (((((ord_less_eq @ nat) @ A3) @ (zero_zero @ nat)) => (A3 = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_362_bot__nat__0_Oextremum__unique, axiom, (! [A3:nat]: (((((ord_less_eq @ nat) @ A3) @ (zero_zero @ nat)) = (A3 = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_363_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.33/5.20 thf(fact_364_real__arch__simple, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [N4:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N4)))))))))).
% 0.33/5.20 thf(fact_365_nat__mult__1__right, axiom, (! [N:nat]: (((((times_times @ nat) @ N) @ (one_one @ nat)) = N)))).
% 0.33/5.20 thf(fact_366_nat__mult__1, axiom, (! [N:nat]: (((((times_times @ nat) @ (one_one @ nat)) @ N) = N)))).
% 0.33/5.20 thf(fact_367_not__one__less__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_368_zero__less__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_369_not__one__le__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_370_linordered__nonzero__semiring__class_Ozero__le__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_371_zero__less__one__class_Ozero__le__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_372_fact__mono__nat, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N)))))).
% 0.33/5.20 thf(fact_373_fact__ge__self, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ ((semiring_char_0_fact @ nat) @ N))))).
% 0.33/5.20 thf(fact_374_zero__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => ((zero_zero @ A) != ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.20 thf(fact_375_less__numeral__extra_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.33/5.20 thf(fact_376_of__nat__neq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N)) != (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_377_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_378_of__nat__0__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 0.33/5.20 thf(fact_379_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))).
% 0.33/5.20 thf(fact_380_mult__eq__self__implies__10, axiom, (! [M:nat,N:nat]: (((M = (((times_times @ nat) @ M) @ N)) => ((N = (one_one @ nat)) | (M = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_381_fact__gt__zero, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_char_0_fact @ A) @ N)))))))).
% 0.33/5.20 thf(fact_382_fact__not__neg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((~ ((((ord_less @ A) @ ((semiring_char_0_fact @ A) @ N)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_383_fact__ge__zero, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_char_0_fact @ A) @ N)))))))).
% 0.33/5.20 thf(fact_384_division__segment__mult, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (((euclid7384307370059645450egment @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((euclid7384307370059645450egment @ A) @ A3)) @ ((euclid7384307370059645450egment @ A) @ B2))))))))))).
% 0.33/5.20 thf(fact_385_fact__ge__1, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((semiring_char_0_fact @ A) @ N)))))))).
% 0.33/5.20 thf(fact_386_int__ops_I1_J, axiom, (((semiring_1_of_nat @ int) @ (zero_zero @ nat)) = (zero_zero @ int))).
% 0.33/5.20 thf(fact_387_fact__less__mono__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N))))))).
% 0.33/5.20 thf(fact_388_effect__returnI, axiom, (! [A:$tType,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),X:A]: (((H2 = H) => ((((((heap_Time_effect @ A) @ ((heap_Time_return @ A) @ X)) @ H2) @ H) @ X) @ (one_one @ nat)))))).
% 0.33/5.20 thf(fact_389_negative__eq__positive, axiom, (! [N:nat,M:nat]: (((((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ int) @ M)) = ((N = (zero_zero @ nat)) & (M = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_390_nat__mult__less__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_391_nat__mult__le__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_392_compl__less__compl__iff, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_393_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_394_field__le__mult__one__interval, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((! [Z4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z4) => ((((ord_less @ A) @ Z4) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Z4) @ X)) @ Y3))))) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_395_int__cases3, axiom, (! [K:int]: (((K != (zero_zero @ int)) => ((! [N4:nat]: (((K = ((semiring_1_of_nat @ int) @ N4)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N4)))))) => (~ ((! [N4:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N4))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N4))))))))))))).
% 0.33/5.20 thf(fact_396_not__zle__0__negative, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N))))))))).
% 0.33/5.20 thf(fact_397_int__cases4, axiom, (! [M:int]: (((! [N4:nat]: ((M != ((semiring_1_of_nat @ int) @ N4)))) => (~ ((! [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => (M != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N4)))))))))))).
% 0.33/5.20 thf(fact_398_negative__zle, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) @ ((semiring_1_of_nat @ int) @ M))))).
% 0.33/5.20 thf(fact_399_lift__collapse, axiom, (! [A:$tType,B:$tType]: ((((heap_Time_lift @ B) @ A) = (^ [F2:(B > A),X2:B]: (((heap_Time_return @ A) @ (F2 @ X2)))))))).
% 0.33/5.20 thf(fact_400_boolean__algebra__class_Oboolean__algebra_Odouble__compl, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X)) = X))))))).
% 0.33/5.20 thf(fact_401_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((uminus_uminus @ A) @ X) = ((uminus_uminus @ A) @ Y3)) = (X = Y3)))))))).
% 0.33/5.20 thf(fact_402_negative__zless, axiom, (! [N:nat,M:nat]: ((((ord_less @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N)))) @ ((semiring_1_of_nat @ int) @ M))))).
% 0.33/5.20 thf(fact_403_verit__la__generic, axiom, (! [A3:int,X:int]: (((((ord_less_eq @ int) @ A3) @ X) | ((A3 = X) | (((ord_less_eq @ int) @ X) @ A3)))))).
% 0.33/5.20 thf(fact_404_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ B2)))) => (? [X3:nat]: (((P @ X3) & (! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ X3)))))))))))).
% 0.33/5.20 thf(fact_405_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 0.33/5.20 thf(fact_406_mult__le__mono2, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ I)) @ (((times_times @ nat) @ K) @ J)))))).
% 0.33/5.20 thf(fact_407_mult__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ K)))))).
% 0.33/5.20 thf(fact_408_mult__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) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ L))))))).
% 0.33/5.20 thf(fact_409_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 0.33/5.20 thf(fact_410_le__square, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ M))))).
% 0.33/5.20 thf(fact_411_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_412_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 0.33/5.20 thf(fact_413_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 0.33/5.20 thf(fact_414_le__cube, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ (((times_times @ nat) @ M) @ M)))))).
% 0.33/5.20 thf(fact_415_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 0.33/5.20 thf(fact_416_zmult__zless__mono2, axiom, (! [I:int,J:int,K:int]: (((((ord_less @ int) @ I) @ J) => ((((ord_less @ int) @ (zero_zero @ int)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ K) @ I)) @ (((times_times @ int) @ K) @ J))))))).
% 0.33/5.20 thf(fact_417_not__int__zless__negative, axiom, (! [N:nat,M:nat]: ((~ ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M)))))))).
% 0.33/5.20 thf(fact_418_int__one__le__iff__zero__less, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (one_one @ int)) @ Z3) = (((ord_less @ int) @ (zero_zero @ int)) @ Z3))))).
% 0.33/5.20 thf(fact_419_pos__zmult__eq__1__iff, axiom, (! [M:int,N:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ M) => (((((times_times @ int) @ M) @ N) = (one_one @ int)) = ((M = (one_one @ int)) & (N = (one_one @ int)))))))).
% 0.33/5.20 thf(fact_420_zmult__zless__mono2__lemma, axiom, (! [I:int,J:int,K:nat]: (((((ord_less @ int) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ I)) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ J))))))).
% 0.33/5.20 thf(fact_421_negative__zless__0, axiom, (! [N:nat]: ((((ord_less @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N)))) @ (zero_zero @ int))))).
% 0.33/5.20 thf(fact_422_negD, axiom, (! [X:int]: (((((ord_less @ int) @ X) @ (zero_zero @ int)) => (? [N4:nat]: ((X = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N4)))))))))).
% 0.33/5.20 thf(fact_423_nat__int__comparison_I2_J, axiom, ((ord_less @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 0.33/5.20 thf(fact_424_pos__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N4:nat]: (((K = ((semiring_1_of_nat @ int) @ N4)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N4)))))))))))).
% 0.33/5.20 thf(fact_425_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) & (K = ((semiring_1_of_nat @ int) @ N4))))))))).
% 0.33/5.20 thf(fact_426_neg__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (~ ((! [N4:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N4))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N4)))))))))))).
% 0.33/5.20 thf(fact_427_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [X_12:A]: ((((ord_less @ A) @ X4) @ X_12))))))))).
% 0.33/5.20 thf(fact_428_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X4))))))))).
% 0.33/5.20 thf(fact_429_nat__mult__eq__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((K = (zero_zero @ nat)) | (M = N)))))).
% 0.33/5.20 thf(fact_430_times__int__code_I2_J, axiom, (! [L:int]: (((((times_times @ int) @ (zero_zero @ int)) @ L) = (zero_zero @ int))))).
% 0.33/5.20 thf(fact_431_times__int__code_I1_J, axiom, (! [K:int]: (((((times_times @ int) @ K) @ (zero_zero @ int)) = (zero_zero @ int))))).
% 0.33/5.20 thf(fact_432_uminus__int__code_I1_J, axiom, (((uminus_uminus @ int) @ (zero_zero @ int)) = (zero_zero @ int))).
% 0.33/5.20 thf(fact_433_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 0.33/5.20 thf(fact_434_zmult__eq__1__iff, axiom, (! [M:int,N:int]: ((((((times_times @ int) @ M) @ N) = (one_one @ int)) = (((M = (one_one @ int)) & (N = (one_one @ int))) | ((M = ((uminus_uminus @ int) @ (one_one @ int))) & (N = ((uminus_uminus @ int) @ (one_one @ int))))))))).
% 0.33/5.20 thf(fact_435_pos__zmult__eq__1__iff__lemma, axiom, (! [M:int,N:int]: ((((((times_times @ int) @ M) @ N) = (one_one @ int)) => ((M = (one_one @ int)) | (M = ((uminus_uminus @ int) @ (one_one @ int)))))))).
% 0.33/5.20 thf(fact_436_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 0.33/5.20 thf(fact_437_int__cases2, axiom, (! [Z3:int]: (((! [N4:nat]: ((Z3 != ((semiring_1_of_nat @ int) @ N4)))) => (~ ((! [N4:nat]: ((Z3 != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N4))))))))))).
% 0.33/5.20 thf(fact_438_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_439_compl__le__swap2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)))))))).
% 0.33/5.20 thf(fact_440_compl__le__swap1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.20 thf(fact_441_compl__mono, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.20 thf(fact_442_compl__less__swap2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,X:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ Y3)) @ X) => (((ord_less @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)))))))).
% 0.33/5.20 thf(fact_443_compl__less__swap1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ ((uminus_uminus @ A) @ X)) => (((ord_less @ A) @ X) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.20 thf(fact_444_nat__mult__eq__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = (M = N)))))).
% 0.33/5.20 thf(fact_445_nat__mult__le__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_446_nat__mult__less__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_447_int__of__nat__induct, axiom, (! [P:(int > $o),Z3:int]: (((! [N4:nat]: ((P @ ((semiring_1_of_nat @ int) @ N4)))) => ((! [N4:nat]: ((P @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N4)))))) => (P @ Z3)))))).
% 0.33/5.20 thf(fact_448_int__cases, axiom, (! [Z3:int]: (((! [N4:nat]: ((Z3 != ((semiring_1_of_nat @ int) @ N4)))) => (~ ((! [N4:nat]: ((Z3 != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N4)))))))))))).
% 0.33/5.20 thf(fact_449_int__zle__neg, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M))) = ((N = (zero_zero @ nat)) & (M = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_450_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N4:nat]: ((K = ((semiring_1_of_nat @ int) @ N4)))))))).
% 0.33/5.20 thf(fact_451_nonpos__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => (~ ((! [N4:nat]: ((K != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N4))))))))))).
% 0.33/5.20 thf(fact_452_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N4:nat]: ((K != ((semiring_1_of_nat @ int) @ N4)))))))))).
% 0.33/5.20 thf(fact_453_negative__zle__0, axiom, (! [N:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) @ (zero_zero @ int))))).
% 0.33/5.20 thf(fact_454_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A7:(A > B),X2:A]: (((uminus_uminus @ B) @ (A7 @ X2))))))))).
% 0.33/5.20 thf(fact_455_dual__order_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 0.33/5.20 thf(fact_456_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.33/5.20 thf(fact_457_imp__le__cong, axiom, (! [X:int,X5:int,P:$o,P2:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P = P2)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => P2))))))).
% 0.33/5.20 thf(fact_458_conj__le__cong, axiom, (! [X:int,X5:int,P:$o,P2:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P = P2)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) & P2))))))).
% 0.33/5.20 thf(fact_459_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B2:nat]: (((P @ K) => ((! [Y5:A]: (((P @ Y5) => (((ord_less @ nat) @ (F @ Y5)) @ B2)))) => (? [X3:A]: (((P @ X3) & (! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ nat) @ (F @ Y4)) @ (F @ X3))))))))))))).
% 0.33/5.20 thf(fact_460_nat__descend__induct, axiom, (! [N:nat,P:(nat > $o),M:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) => ((! [I3:nat]: (((((ord_less @ nat) @ K2) @ I3) => (P @ I3)))) => (P @ K2))))) => (P @ M)))))).
% 0.33/5.20 thf(fact_461_dependent__nat__choice, axiom, (! [A:$tType,P:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X_1:A]: (((P @ (zero_zero @ nat)) @ X_1))) => ((! [X3:A,N4:nat]: ((((P @ N4) @ X3) => (? [Y4:A]: ((((P @ (suc @ N4)) @ Y4) & (((Q @ N4) @ X3) @ Y4))))))) => (? [F3:(nat > A)]: ((! [N5:nat]: ((((P @ N5) @ (F3 @ N5)) & (((Q @ N5) @ (F3 @ N5)) @ (F3 @ (suc @ N5))))))))))))).
% 0.33/5.20 thf(fact_462_list__decode_Ocases, axiom, (! [X:nat]: (((X != (zero_zero @ nat)) => (~ ((! [N4:nat]: ((X != (suc @ N4)))))))))).
% 0.33/5.20 thf(fact_463_order__le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ X) @ Y3) | (X = Y3))))))))).
% 0.33/5.20 thf(fact_464_nle__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((~ ((((ord_less_eq @ A) @ A3) @ B2))) = ((((ord_less_eq @ A) @ B2) @ A3) & (B2 != A3))))))))).
% 0.33/5.20 thf(fact_465_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z3)))) => (((((ord_less_eq @ A) @ Y3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y3)))) => (((((ord_less_eq @ A) @ Z3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X)))) => (((((ord_less_eq @ A) @ Y3) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y3))))))))))))))))).
% 0.33/5.20 thf(fact_466_order__class_Oorder__eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X2))))))))).
% 0.33/5.20 thf(fact_467_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_468_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_469_order__antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X) => (X = Y3))))))))).
% 0.33/5.20 thf(fact_470_order_Otrans, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_471_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 0.33/5.20 thf(fact_472_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A8:A,B5:A]: (((((ord_less_eq @ A) @ A8) @ B5) => ((P @ A8) @ B5)))) => ((! [A8:A,B5:A]: ((((P @ B5) @ A8) => ((P @ A8) @ B5)))) => ((P @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_473_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A5) & (((ord_less_eq @ A) @ A5) @ B3))))))))).
% 0.33/5.20 thf(fact_474_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ A3) @ B2) => (A3 = B2))))))))).
% 0.33/5.20 thf(fact_475_dual__order_Otrans, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_476_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (A3 = B2))))))))).
% 0.33/5.20 thf(fact_477_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.33/5.20 thf(fact_478_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.33/5.20 thf(fact_479_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 0.33/5.20 thf(fact_480_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 0.33/5.20 thf(fact_481_Orderings_Oorder__eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) & (((ord_less_eq @ A) @ B3) @ A5))))))))).
% 0.33/5.20 thf(fact_482_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less_eq @ B) @ X3) @ Y5) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_483_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_484_order__eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((X = Y3) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_485_linorder__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_486_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((A3 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less_eq @ B) @ X3) @ Y5) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_487_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B2) => (((F @ B2) = C2) => ((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_488_linorder__le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_489_order__antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_less_eq @ A) @ X) @ Y3) = (X = Y3))))))))).
% 0.33/5.20 thf(fact_490_minf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F4:D]: ((? [Z4:C]: ((! [X4:C]: (((((ord_less @ C) @ X4) @ Z4) => (F4 = F4)))))))))))).
% 0.33/5.20 thf(fact_491_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (~ ((((ord_less @ A) @ T2) @ X4)))))))))))))).
% 0.33/5.20 thf(fact_492_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (((ord_less @ A) @ X4) @ T2)))))))))))).
% 0.33/5.20 thf(fact_493_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (X4 != T2)))))))))))).
% 0.33/5.20 thf(fact_494_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (X4 != T2)))))))))))).
% 0.33/5.20 thf(fact_495_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z5) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z5) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 0.33/5.20 thf(fact_496_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z5) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z5) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 0.33/5.20 thf(fact_497_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F4:D]: ((? [Z4:C]: ((! [X4:C]: (((((ord_less @ C) @ Z4) @ X4) => (F4 = F4)))))))))))).
% 0.33/5.20 thf(fact_498_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (((ord_less @ A) @ T2) @ X4)))))))))))).
% 0.33/5.20 thf(fact_499_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (~ ((((ord_less @ A) @ X4) @ T2)))))))))))))).
% 0.33/5.20 thf(fact_500_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (X4 != T2)))))))))))).
% 0.33/5.20 thf(fact_501_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (X4 != T2)))))))))))).
% 0.33/5.20 thf(fact_502_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ Z5) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ Z5) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 0.33/5.20 thf(fact_503_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ Z5) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z5:A]: ((! [X3:A]: (((((ord_less @ A) @ Z5) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 0.33/5.20 thf(fact_504_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X))))))))).
% 0.33/5.20 thf(fact_505_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_12:A]: ((((ord_less @ A) @ X) @ X_12))))))))).
% 0.33/5.20 thf(fact_506_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (? [Z4:A]: (((((ord_less @ A) @ X) @ Z4) & (((ord_less @ A) @ Z4) @ Y3))))))))))).
% 0.33/5.20 thf(fact_507_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (X != Y3)))))))).
% 0.33/5.20 thf(fact_508_order_Oasym, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_509_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_510_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((B2 = C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_511_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X3:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X3) => (P @ Y4)))) => (P @ X3)))) => (P @ A3)))))))).
% 0.33/5.20 thf(fact_512_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X:A]: (((~ ((((ord_less @ A) @ Y3) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y3))) = (X = Y3))))))))).
% 0.33/5.20 thf(fact_513_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((X != Y3) => (((ord_less @ A) @ Y3) @ X))))))))).
% 0.33/5.20 thf(fact_514_dual__order_Oasym, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_515_dual__order_Oirrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 0.33/5.20 thf(fact_516_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P3:(A > $o)]: ((? [X6:A]: ((P3 @ X6))))) = (^ [P4:(A > $o)]: ((? [N3:A]: (((P4 @ N3) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N3) => (~ ((P4 @ M2)))))))))))))))).
% 0.33/5.20 thf(fact_517_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A8:A,B5:A]: (((((ord_less @ A) @ A8) @ B5) => ((P @ A8) @ B5)))) => ((! [A8:A]: (((P @ A8) @ A8))) => ((! [A8:A,B5:A]: ((((P @ B5) @ A8) => ((P @ A8) @ B5)))) => ((P @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_518_order_Ostrict__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_519_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) = ((((ord_less @ A) @ Y3) @ X) | (X = Y3))))))))).
% 0.33/5.20 thf(fact_520_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_521_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (A3 != B2)))))))).
% 0.33/5.20 thf(fact_522_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (A3 != B2)))))))).
% 0.33/5.20 thf(fact_523_linorder__neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((X != Y3) => ((~ ((((ord_less @ A) @ X) @ Y3))) => (((ord_less @ A) @ Y3) @ X))))))))).
% 0.33/5.20 thf(fact_524_order__less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X)))))))))).
% 0.33/5.20 thf(fact_525_linorder__neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((X != Y3) = ((((ord_less @ A) @ X) @ Y3) | (((ord_less @ A) @ Y3) @ X))))))))).
% 0.33/5.20 thf(fact_526_order__less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_527_order__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less @ A) @ X) @ Y3) => ((((ord_less @ A) @ Y3) @ Z3) => (((ord_less @ A) @ X) @ Z3))))))))).
% 0.33/5.20 thf(fact_528_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((A3 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less @ B) @ X3) @ Y5) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_529_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A3) @ B2) => (((F @ B2) = C2) => ((! [X3:A,Y5:A]: (((((ord_less @ A) @ X3) @ Y5) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ B) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_530_order__less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 0.33/5.20 thf(fact_531_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less @ B) @ X3) @ Y5) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_532_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y5:A]: (((((ord_less @ A) @ X3) @ Y5) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_533_order__less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X)))))))))).
% 0.33/5.20 thf(fact_534_order__less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,P:$o]: (((((ord_less @ A) @ X) @ Y3) => ((((ord_less @ A) @ Y3) @ X) => P)))))))).
% 0.33/5.20 thf(fact_535_linorder__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) | ((X = Y3) | (((ord_less @ A) @ Y3) @ X))))))))).
% 0.33/5.20 thf(fact_536_order__less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (X != Y3)))))))).
% 0.33/5.20 thf(fact_537_order__less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (Y3 != X)))))))).
% 0.33/5.20 thf(fact_538_order__less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X)))))))))).
% 0.33/5.20 thf(fact_539_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A7:(A > B),X2:A]: (((uminus_uminus @ B) @ (A7 @ X2))))))))).
% 0.33/5.20 thf(fact_540_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X3:A]: (((P @ X3) & (! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ Y4)))))))))))).
% 0.33/5.20 thf(fact_541_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (~ ((((ord_less_eq @ A) @ T2) @ X4)))))))))))))).
% 0.33/5.20 thf(fact_542_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z4) => (((ord_less_eq @ A) @ X4) @ T2)))))))))))).
% 0.33/5.20 thf(fact_543_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (((ord_less_eq @ A) @ T2) @ X4)))))))))))).
% 0.33/5.20 thf(fact_544_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z4:A]: ((! [X4:A]: (((((ord_less @ A) @ Z4) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T2)))))))))))))).
% 0.33/5.20 thf(fact_545_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => (~ ((((ord_less @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_546_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_547_nless__le, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((~ ((((ord_less @ A) @ A3) @ B2))) = ((~ ((((ord_less_eq @ A) @ A3) @ B2))) | (A3 = B2))))))))).
% 0.33/5.20 thf(fact_548_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((((ord_less_eq @ A) @ X) @ Y3) = (X = Y3))))))))).
% 0.33/5.20 thf(fact_549_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((~ ((((ord_less @ A) @ X) @ Y3))) = (X = Y3))))))))).
% 0.33/5.20 thf(fact_550_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z3:A,Y3:A]: (((! [X3:A]: (((((ord_less @ A) @ Z3) @ X3) => (((ord_less_eq @ A) @ Y3) @ X3)))) => (((ord_less_eq @ A) @ Y3) @ Z3)))))))).
% 0.33/5.20 thf(fact_551_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y3:A,Z3:A]: (((! [X3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less_eq @ A) @ X3) @ Z3)))) => (((ord_less_eq @ A) @ Y3) @ Z3)))))))).
% 0.33/5.20 thf(fact_552_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) & (~ ((((ord_less_eq @ A) @ Y6) @ X2))))))))))).
% 0.33/5.20 thf(fact_553_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y3) @ X))) => (((ord_less @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_554_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: (((((ord_less @ A) @ A5) @ B3) | (A5 = B3))))))))).
% 0.33/5.20 thf(fact_555_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) & (A5 != B3))))))))).
% 0.33/5.20 thf(fact_556_order_Ostrict__trans1, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_557_order_Ostrict__trans2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_558_order_Ostrict__iff__not, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) & (~ ((((ord_less_eq @ A) @ B3) @ A5))))))))))).
% 0.33/5.20 thf(fact_559_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ Z3) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z3) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y3) @ W))))) => (((ord_less_eq @ A) @ Y3) @ Z3))))))))).
% 0.33/5.20 thf(fact_560_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less @ A) @ X) @ Y3) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y3) => (((ord_less_eq @ A) @ W) @ Z3))))) => (((ord_less_eq @ A) @ Y3) @ Z3))))))))).
% 0.33/5.20 thf(fact_561_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: (((((ord_less @ A) @ B3) @ A5) | (A5 = B3))))))))).
% 0.33/5.20 thf(fact_562_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B3:A,A5:A]: (((((ord_less_eq @ A) @ B3) @ A5) & (A5 != B3))))))))).
% 0.33/5.20 thf(fact_563_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_564_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_565_dual__order_Ostrict__iff__not, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [B3:A,A5:A]: (((((ord_less_eq @ A) @ B3) @ A5) & (~ ((((ord_less_eq @ A) @ A5) @ B3))))))))))).
% 0.33/5.20 thf(fact_566_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_567_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_568_order__le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y6:A]: (((((ord_less @ A) @ X2) @ Y6) | (X2 = Y6))))))))).
% 0.33/5.20 thf(fact_569_order__less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) & (X2 != Y6))))))))).
% 0.33/5.20 thf(fact_570_linorder__not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y3))) = (((ord_less @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_571_linorder__not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) = (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_572_order__less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_573_order__le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((A3 != B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_574_order__neq__le__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((A3 != B2) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_575_order__le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ Y3) @ Z3) => (((ord_less @ A) @ X) @ Z3))))))))).
% 0.33/5.20 thf(fact_576_order__less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z3) => (((ord_less @ A) @ X) @ Z3))))))))).
% 0.33/5.20 thf(fact_577_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less @ B) @ X3) @ Y5) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_578_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_579_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y5:B]: (((((ord_less_eq @ B) @ X3) @ Y5) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 0.33/5.20 thf(fact_580_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y5:A]: (((((ord_less @ A) @ X3) @ Y5) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y5))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 0.33/5.20 thf(fact_581_linorder__le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) | (((ord_less @ A) @ Y3) @ X)))))))).
% 0.33/5.20 thf(fact_582_complete__interval, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [A3:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A3) @ B2) => ((P @ A3) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A3) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A3) @ X4) & (((ord_less @ A) @ X4) @ C3)) => (P @ X4)))) & (! [D3:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A3) @ X3) & (((ord_less @ A) @ X3) @ D3)) => (P @ X3)))) => (((ord_less_eq @ A) @ D3) @ C3)))))))))))))))))).
% 0.33/5.20 thf(fact_583_fact__reduce, axiom, (! [A:$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)))))))))))).
% 0.33/5.20 thf(fact_584_fact__num__eq__if, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [M2:nat]: (((((if @ A) @ (M2 = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat)))))))))))).
% 0.33/5.20 thf(fact_585_zero__le__ceiling, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_586_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)))))))))).
% 0.33/5.20 thf(fact_587_one__less__nat__eq, axiom, (! [Z3:int]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ (nat2 @ Z3)) = (((ord_less @ int) @ (one_one @ int)) @ Z3))))).
% 0.33/5.20 thf(fact_588_split__div_H, axiom, (! [P:(nat > $o),M:nat,N:nat]: (((P @ (((divide_divide @ nat) @ M) @ N)) = (((N = (zero_zero @ nat)) & (P @ (zero_zero @ nat))) | (? [Q3:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ N) @ Q3)) @ M) & ((((ord_less @ nat) @ M) @ (((times_times @ nat) @ N) @ (suc @ Q3))) & (P @ Q3)))))))))).
% 0.33/5.20 thf(fact_589_frac__eq, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_590_euclidean__size__times__nonunit, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((~ ((((dvd_dvd @ A) @ A3) @ (one_one @ A)))) => (((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ B2)) @ ((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ A3) @ B2)))))))))))).
% 0.33/5.20 thf(fact_591_sgn__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((sgn_sgn @ A) @ A3) = ((uminus_uminus @ A) @ (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_592_Suc__times__mod__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ M) @ N))) @ M) = (one_one @ nat)))))).
% 0.33/5.20 thf(fact_593_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B6:(A > B),X2:A]: ((((minus_minus @ B) @ (A7 @ X2)) @ (B6 @ X2))))))))).
% 0.33/5.20 thf(fact_594_mod__mod__trivial, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_595_nat__dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (one_one @ nat)) = (M = (one_one @ nat)))))).
% 0.33/5.20 thf(fact_596_int__dvd__int__iff, axiom, (! [M:nat,N:nat]: (((((dvd_dvd @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((dvd_dvd @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_597_sgn__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((sgn_sgn @ A) @ ((sgn_sgn @ A) @ A3)) = ((sgn_sgn @ A) @ A3)))))))).
% 0.33/5.20 thf(fact_598_frac__frac, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_frac @ A) @ ((archimedean_frac @ A) @ X)) = ((archimedean_frac @ A) @ X)))))))).
% 0.33/5.20 thf(fact_599_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_600_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_601_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_602_diff__zero, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_603_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_604_dvd__0__left__iff, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A3) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_605_dvd__0__right, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ A3) @ (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_606_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_607_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,C2:A,B2:A]: ((((((divide_divide @ A) @ A3) @ C2) = (((divide_divide @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A3 = B2))))))))).
% 0.33/5.20 thf(fact_608_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: ((((((divide_divide @ A) @ C2) @ A3) = (((divide_divide @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A3 = B2))))))))).
% 0.33/5.20 thf(fact_609_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_610_div__by__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_611_div__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_612_times__divide__eq__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_613_divide__divide__eq__right, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)))))))).
% 0.33/5.20 thf(fact_614_divide__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_615_times__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [B2:A,C2:A,A3:A]: (((((times_times @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((divide_divide @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2)))))))).
% 0.33/5.20 thf(fact_616_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A3) @ B2)) = (((minus_minus @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_617_div__by__1, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 0.33/5.20 thf(fact_618_mod__self, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_619_mod__by__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_620_mod__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_621_minus__dvd__iff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:A,Y3:A]: (((((dvd_dvd @ A) @ ((uminus_uminus @ A) @ X)) @ Y3) = (((dvd_dvd @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_622_dvd__minus__iff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:A,Y3:A]: (((((dvd_dvd @ A) @ X) @ ((uminus_uminus @ A) @ Y3)) = (((dvd_dvd @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_623_div__minus__minus, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_624_dvd__1__left, axiom, (! [K:nat]: ((((dvd_dvd @ nat) @ (suc @ (zero_zero @ nat))) @ K)))).
% 0.33/5.20 thf(fact_625_dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (M = (suc @ (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_626_div__dvd__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (((divide_divide @ A) @ C2) @ A3)) = (((dvd_dvd @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_627_minus__mod__self2, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_628_mod__minus__minus, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_629_diff__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M)) @ (suc @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_630_Suc__diff__diff, axiom, (! [M:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ K))))).
% 0.33/5.20 thf(fact_631_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_632_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_633_nat__mult__dvd__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = ((K = (zero_zero @ nat)) | (((dvd_dvd @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_634_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 0.33/5.20 thf(fact_635_sgn__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_636_sgn__mult__dvd__iff, axiom, (! [R:int,L:int,K:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ R)) @ L)) @ K) = ((((dvd_dvd @ int) @ L) @ K) & ((R = (zero_zero @ int)) => (K = (zero_zero @ int)))))))).
% 0.33/5.20 thf(fact_637_mult__sgn__dvd__iff, axiom, (! [L:int,R:int,K:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ L) @ ((sgn_sgn @ int) @ R))) @ K) = ((((dvd_dvd @ int) @ L) @ K) & ((R = (zero_zero @ int)) => (K = (zero_zero @ int)))))))).
% 0.33/5.20 thf(fact_638_dvd__sgn__mult__iff, axiom, (! [L:int,R:int,K:int]: (((((dvd_dvd @ int) @ L) @ (((times_times @ int) @ ((sgn_sgn @ int) @ R)) @ K)) = ((((dvd_dvd @ int) @ L) @ K) | (R = (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_639_dvd__mult__sgn__iff, axiom, (! [L:int,K:int,R:int]: (((((dvd_dvd @ int) @ L) @ (((times_times @ int) @ K) @ ((sgn_sgn @ int) @ R))) = ((((dvd_dvd @ int) @ L) @ K) | (R = (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_640_sgn__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_641_sgn__minus, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((sgn_sgn @ A) @ ((uminus_uminus @ A) @ A3)) = ((uminus_uminus @ A) @ ((sgn_sgn @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_642_sgn__divide, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A3:A,B2:A]: ((((sgn_sgn @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((sgn_sgn @ A) @ A3)) @ ((sgn_sgn @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_643_mod__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((modulo_modulo @ nat) @ M) @ N) = M))))).
% 0.33/5.20 thf(fact_644_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 0.33/5.20 thf(fact_645_diff__ge__0__iff__ge, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_646_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_647_diff__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_648_dvd__mult__cancel__left, axiom, (! [A:$tType]: (((idom @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_649_dvd__mult__cancel__right, axiom, (! [A:$tType]: (((idom @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_650_dvd__times__left__cancel__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2)) = (((dvd_dvd @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_651_dvd__times__right__cancel__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ B2) @ A3)) @ (((times_times @ A) @ C2) @ A3)) = (((dvd_dvd @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_652_mult__divide__mult__cancel__left__if, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: ((((C2 = (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (zero_zero @ A))) & ((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_653_nonzero__mult__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_654_nonzero__mult__divide__mult__cancel__left2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_655_nonzero__mult__divide__mult__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_656_nonzero__mult__divide__mult__cancel__right2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_657_div__mult__mult1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_658_div__mult__mult2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_659_div__mult__mult1__if, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A3:A,B2:A]: ((((C2 = (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (zero_zero @ A))) & ((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_660_nonzero__mult__div__cancel__left, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ A3) = B2)))))))).
% 0.33/5.20 thf(fact_661_nonzero__mult__div__cancel__right, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ B2) = A3)))))))).
% 0.33/5.20 thf(fact_662_diff__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = ((uminus_uminus @ A) @ A3)))))))).
% 0.33/5.20 thf(fact_663_verit__minus__simplify_I3_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: (((((minus_minus @ B) @ (zero_zero @ B)) @ B2) = ((uminus_uminus @ B) @ B2)))))))).
% 0.33/5.20 thf(fact_664_divide__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((((divide_divide @ A) @ A3) @ B2) = (one_one @ A)) = ((B2 != (zero_zero @ A)) & (A3 = B2))))))))).
% 0.33/5.20 thf(fact_665_one__eq__divide__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((one_one @ A) = (((divide_divide @ A) @ A3) @ B2)) = ((B2 != (zero_zero @ A)) & (A3 = B2))))))))).
% 0.33/5.20 thf(fact_666_divide__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (one_one @ A))))))))).
% 0.33/5.20 thf(fact_667_divide__self__if, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_668_divide__eq__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: ((((((divide_divide @ A) @ B2) @ A3) = (one_one @ A)) = ((A3 != (zero_zero @ A)) & (A3 = B2))))))))).
% 0.33/5.20 thf(fact_669_eq__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: ((((one_one @ A) = (((divide_divide @ A) @ B2) @ A3)) = ((A3 != (zero_zero @ A)) & (A3 = B2))))))))).
% 0.33/5.20 thf(fact_670_one__divide__eq__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_671_zero__eq__1__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_672_div__self, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (one_one @ A))))))))).
% 0.33/5.20 thf(fact_673_unit__prod, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_674_mod__mult__self1__is__0, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ B2) @ A3)) @ B2) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_675_mod__mult__self2__is__0, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ B2) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_676_mod__by__1, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_677_divide__minus1, axiom, (! [A:$tType]: (((field @ A) => (! [X:A]: (((((divide_divide @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ X)))))))).
% 0.33/5.20 thf(fact_678_div__minus1__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ A3)))))))).
% 0.33/5.20 thf(fact_679_dvd__mult__div__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ A3)) = B2)))))))).
% 0.33/5.20 thf(fact_680_dvd__div__mult__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ A3) = B2)))))))).
% 0.33/5.20 thf(fact_681_unit__div, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_682_unit__div__1__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_683_unit__div__1__div__1, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_684_div__diff, axiom, (! [A:$tType]: (((idom_modulo @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))))).
% 0.33/5.20 thf(fact_685_dvd__imp__mod__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((modulo_modulo @ A) @ B2) @ A3) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_686_mod__div__trivial, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ B2) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_687_minus__mod__self1, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [B2:A,A3:A]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ B2) = (((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 0.33/5.20 thf(fact_688_sgn__less, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ ((sgn_sgn @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_689_sgn__greater, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((sgn_sgn @ A) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.20 thf(fact_690_zero__less__diff, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_691_diff__is__0__eq, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_692_diff__is__0__eq_H, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_693_divide__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ A3) @ ((sgn_sgn @ A) @ B2)) = (((times_times @ A) @ A3) @ ((sgn_sgn @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_694_diff__Suc__1, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (suc @ N)) @ (one_one @ nat)) = N)))).
% 0.33/5.20 thf(fact_695_mod__by__Suc__0, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_696_div__by__Suc__0, axiom, (! [M:nat]: (((((divide_divide @ nat) @ M) @ (suc @ (zero_zero @ nat))) = M)))).
% 0.33/5.20 thf(fact_697_div__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_698_ceiling__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_ceiling @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_699_nat__mult__div__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: ((((K = (zero_zero @ nat)) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (zero_zero @ nat))) & ((K != (zero_zero @ nat)) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((divide_divide @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_700_ceiling__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_ceiling @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 0.33/5.20 thf(fact_701_ceiling__of__nat, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:nat]: ((((archimedean_ceiling @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 0.33/5.20 thf(fact_702_divide__le__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_703_zero__le__divide__1__iff, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_704_divide__less__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_705_divide__less__eq__1__neg, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_706_divide__less__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less @ A) @ B2) @ A3))))))))).
% 0.33/5.20 thf(fact_707_less__divide__eq__1__neg, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_708_less__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_709_zero__less__divide__1__iff, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_710_nonzero__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ A3) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2))))))))).
% 0.33/5.20 thf(fact_711_nonzero__divide__mult__cancel__right, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_712_diff__numeral__special_I12_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_713_unit__mult__div__div, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_714_unit__div__mult__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ A3) = B2)))))))).
% 0.33/5.20 thf(fact_715_mod__minus1__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_716_sgn__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((sgn_sgn @ A) @ A3) = (one_one @ A))))))))).
% 0.33/5.20 thf(fact_717_Suc__pred, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) = N))))).
% 0.33/5.20 thf(fact_718_nat__1, axiom, ((nat2 @ (one_one @ int)) = (suc @ (zero_zero @ nat)))).
% 0.33/5.20 thf(fact_719_nat__le__0, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ Z3) @ (zero_zero @ int)) => ((nat2 @ Z3) = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_720_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_721_zless__nat__conj, axiom, (! [W2:int,Z3:int]: (((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z3)) = ((((ord_less @ int) @ (zero_zero @ int)) @ Z3) & (((ord_less @ int) @ W2) @ Z3)))))).
% 0.33/5.20 thf(fact_722_div__mult__self1__is__m, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((divide_divide @ nat) @ (((times_times @ nat) @ N) @ M)) @ N) = M))))).
% 0.33/5.20 thf(fact_723_div__mult__self__is__m, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((divide_divide @ nat) @ (((times_times @ nat) @ M) @ N)) @ N) = M))))).
% 0.33/5.20 thf(fact_724_ceiling__diff__one, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_725_int__nat__eq, axiom, (! [Z3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = Z3)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_726_nat__zminus__int, axiom, (! [N:nat]: (((nat2 @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_727_divide__le__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_728_divide__le__eq__1__pos, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_729_le__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 0.33/5.20 thf(fact_730_le__divide__eq__1__pos, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_731_Suc__diff__1, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) = N))))).
% 0.33/5.20 thf(fact_732_ceiling__le__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_733_zero__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (zero_zero @ A)) @ X)))))))).
% 0.33/5.20 thf(fact_734_zero__less__nat__eq, axiom, (! [Z3:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (nat2 @ Z3)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z3))))).
% 0.33/5.20 thf(fact_735_ceiling__less__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_736_one__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ (one_one @ int)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (zero_zero @ A)) @ X)))))))).
% 0.33/5.20 thf(fact_737_ceiling__le__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_738_one__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (one_one @ int)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (one_one @ A)) @ X)))))))).
% 0.33/5.20 thf(fact_739_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 0.33/5.20 thf(fact_740_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B6:(A > B),X2:A]: ((((minus_minus @ B) @ (A7 @ X2)) @ (B6 @ X2))))))))).
% 0.33/5.20 thf(fact_741_mod__eq__0__iff__dvd, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A)) = (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_742_dvd__eq__mod__eq__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => ((dvd_dvd @ A) = (^ [A5:A,B3:A]: (((((modulo_modulo @ A) @ B3) @ A5) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_743_mod__0__imp__dvd, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A)) => (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_744_mod__eq__self__iff__div__eq__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ B2) = A3) = ((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_745_dvd__div__eq__0__iff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => (((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)) = (A3 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_746_dvd__diff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:A,Y3:A,Z3:A]: (((((dvd_dvd @ A) @ X) @ Y3) => ((((dvd_dvd @ A) @ X) @ Z3) => (((dvd_dvd @ A) @ X) @ (((minus_minus @ A) @ Y3) @ Z3)))))))))).
% 0.33/5.20 thf(fact_747_dvd__refl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ A3) @ A3))))))).
% 0.33/5.20 thf(fact_748_dvd__trans, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ B2) @ C2) => (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_749_dvd__mod__iff, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ C2) @ (((modulo_modulo @ A) @ A3) @ B2)) = (((dvd_dvd @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_750_dvd__minus__mod, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [B2:A,A3:A]: ((((dvd_dvd @ A) @ B2) @ (((minus_minus @ A) @ A3) @ (((modulo_modulo @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_751_dvd__div__eq__iff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => (((((divide_divide @ A) @ A3) @ C2) = (((divide_divide @ A) @ B2) @ C2)) = (A3 = B2)))))))))).
% 0.33/5.20 thf(fact_752_dvd__mod__imp__dvd, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ (((modulo_modulo @ A) @ A3) @ B2)) => ((((dvd_dvd @ A) @ C2) @ B2) => (((dvd_dvd @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_753_dvd__div__eq__cancel, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A,C2:A,B2:A]: ((((((divide_divide @ A) @ A3) @ C2) = (((divide_divide @ A) @ B2) @ C2)) => ((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => (A3 = B2)))))))))).
% 0.33/5.20 thf(fact_754_div__div__div__same, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [D2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ D2) @ B2) => ((((dvd_dvd @ A) @ B2) @ A3) => ((((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ D2)) @ (((divide_divide @ A) @ B2) @ D2)) = (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_755_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((A3 = B2) = (C2 = D2))))))))).
% 0.33/5.20 thf(fact_756_diff__right__commute, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_757_dvd__diff__commute, axiom, (! [A:$tType]: (((euclid5891614535332579305n_ring @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((minus_minus @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_758_mod__diff__eq, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_759_dvd__mod, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [K:A,M:A,N:A]: (((((dvd_dvd @ A) @ K) @ M) => ((((dvd_dvd @ A) @ K) @ N) => (((dvd_dvd @ A) @ K) @ (((modulo_modulo @ A) @ M) @ N)))))))))).
% 0.33/5.20 thf(fact_760_mod__diff__cong, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,A4:A,B2:A,B4:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ A4) @ C2)) => (((((modulo_modulo @ A) @ B2) @ C2) = (((modulo_modulo @ A) @ B4) @ C2)) => ((((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ C2)))))))))).
% 0.33/5.20 thf(fact_761_mod__eq__dvd__iff, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ C2) @ (((minus_minus @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_762_mod__diff__left__eq, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_763_mod__diff__right__eq, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_764_mod__mod__cancel, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((modulo_modulo @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ C2) = (((modulo_modulo @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_765_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 0.33/5.20 thf(fact_766_dvd__diff__nat, axiom, (! [K:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ K) @ M) => ((((dvd_dvd @ nat) @ K) @ N) => (((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_767_field__char__0__class_Oof__nat__div, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((divide_divide @ nat) @ M) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ (((modulo_modulo @ nat) @ M) @ N)))) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_768_diff__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_769_modulo__nat__def, axiom, ((modulo_modulo @ nat) = (^ [M2:nat,N3:nat]: ((((minus_minus @ nat) @ M2) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M2) @ N3)) @ N3)))))).
% 0.33/5.20 thf(fact_770_minus__mult__div__eq__mod, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_771_minus__mod__eq__mult__div, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((modulo_modulo @ A) @ A3) @ B2)) = (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_772_minus__mod__eq__div__mult, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((modulo_modulo @ A) @ A3) @ B2)) = (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)))))))).
% 0.33/5.20 thf(fact_773_minus__div__mult__eq__mod, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_774_div__mult__div__if__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,D2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ A3) => ((((dvd_dvd @ A) @ D2) @ C2) => ((((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (((divide_divide @ A) @ C2) @ D2)) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2))))))))))).
% 0.33/5.20 thf(fact_775_dvd__mult__imp__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2) => (((dvd_dvd @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_776_dvd__div__mult2__eq, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,C2:A,A3:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ B2) @ C2)) @ A3) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2))))))))).
% 0.33/5.20 thf(fact_777_div__div__eq__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ B2) @ A3) => ((((divide_divide @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2)))))))))).
% 0.33/5.20 thf(fact_778_div__mult__swap, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2))))))))).
% 0.33/5.20 thf(fact_779_dvd__div__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((divide_divide @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2))))))))).
% 0.33/5.20 thf(fact_780_unit__div__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((((divide_divide @ A) @ B2) @ A3) = (((divide_divide @ A) @ C2) @ A3)) = (B2 = C2))))))))).
% 0.33/5.20 thf(fact_781_div__unit__dvd__iff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_782_dvd__div__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((divide_divide @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_783_of__nat__mod, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((modulo_modulo @ nat) @ M) @ N)) = (((modulo_modulo @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_784_dvd__neg__div, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_785_dvd__div__neg, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => ((((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_786_mod__if, axiom, ((modulo_modulo @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (((ord_less @ nat) @ M2) @ N3)) @ M2) @ (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M2) @ N3)) @ N3)))))).
% 0.33/5.20 thf(fact_787_le__mod__geq, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((modulo_modulo @ nat) @ M) @ N) = (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ N)))))).
% 0.33/5.20 thf(fact_788_dvd__minus__self, axiom, (! [M:nat,N:nat]: (((((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N) @ M)) = ((((ord_less @ nat) @ N) @ M) | (((dvd_dvd @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_789_unique__euclidean__semiring__with__nat__class_Oof__nat__div, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((divide_divide @ nat) @ M) @ N)) = (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_790_of__nat__dvd__iff, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N:nat]: (((((dvd_dvd @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((dvd_dvd @ nat) @ M) @ N)))))))).
% 0.33/5.20 thf(fact_791_less__eq__dvd__minus, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((dvd_dvd @ nat) @ M) @ N) = (((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N) @ M))))))).
% 0.33/5.20 thf(fact_792_dvd__diffD1, axiom, (! [K:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N)) => ((((dvd_dvd @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ N) @ M) => (((dvd_dvd @ nat) @ K) @ N))))))).
% 0.33/5.20 thf(fact_793_dvd__diffD, axiom, (! [K:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N)) => ((((dvd_dvd @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (((dvd_dvd @ nat) @ K) @ M))))))).
% 0.33/5.20 thf(fact_794_mod__nat__eqI, axiom, (! [R:nat,N:nat,M:nat]: (((((ord_less @ nat) @ R) @ N) => ((((ord_less_eq @ nat) @ R) @ M) => ((((dvd_dvd @ nat) @ N) @ (((minus_minus @ nat) @ M) @ R)) => ((((modulo_modulo @ nat) @ M) @ N) = R))))))).
% 0.33/5.20 thf(fact_795_unit__imp__mod__eq__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_796_of__nat__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R:A]: ((((ord_less_eq @ A) @ R) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((archimedean_ceiling @ A) @ R)))))))))).
% 0.33/5.20 thf(fact_797_add__divide__eq__if__simps_I4_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((minus_minus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z3)) = A3)) & ((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z3)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ A3) @ Z3)) @ B2)) @ Z3)))))))))).
% 0.33/5.20 thf(fact_798_diff__frac__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y3:A,Z3:A,X:A,W2:A]: (((Y3 != (zero_zero @ A)) => ((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ W2) @ Z3)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ W2) @ Y3))) @ (((times_times @ A) @ Y3) @ Z3))))))))))).
% 0.33/5.20 thf(fact_799_diff__divide__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ X) @ (((divide_divide @ A) @ Y3) @ Z3)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X) @ Z3)) @ Y3)) @ Z3))))))))).
% 0.33/5.20 thf(fact_800_divide__diff__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ X) @ Z3)) @ Y3) = (((divide_divide @ A) @ (((minus_minus @ A) @ X) @ (((times_times @ A) @ Y3) @ Z3))) @ Z3))))))))).
% 0.33/5.20 thf(fact_801_dvd__div__eq__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ A3) @ B2) => (((((divide_divide @ A) @ B2) @ A3) = C2) = (B2 = (((times_times @ A) @ C2) @ A3))))))))))).
% 0.33/5.20 thf(fact_802_div__dvd__iff__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ A3) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))))))))))).
% 0.33/5.20 thf(fact_803_dvd__div__iff__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A3:A]: (((C2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)))))))))).
% 0.33/5.20 thf(fact_804_dvd__div__div__eq__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((A3 != (zero_zero @ A)) => ((C2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ C2) @ D2) => (((((divide_divide @ A) @ B2) @ A3) = (((divide_divide @ A) @ D2) @ C2)) = ((((times_times @ A) @ B2) @ C2) = (((times_times @ A) @ A3) @ D2))))))))))))).
% 0.33/5.20 thf(fact_805_unit__div__eq__0__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)) = (A3 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_806_unit__eq__div1, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_807_unit__eq__div2, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((A3 = (((divide_divide @ A) @ C2) @ B2)) = ((((times_times @ A) @ A3) @ B2) = C2))))))))).
% 0.33/5.20 thf(fact_808_div__mult__unit2, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_809_unit__div__commute, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_810_unit__div__mult__swap, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_811_is__unit__div__mult2__eq, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,C2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2)))))))))).
% 0.33/5.20 thf(fact_812_div__less__mono, axiom, (! [A6:nat,B7:nat,N:nat]: (((((ord_less @ nat) @ A6) @ B7) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((((modulo_modulo @ nat) @ A6) @ N) = (zero_zero @ nat)) => (((((modulo_modulo @ nat) @ B7) @ N) = (zero_zero @ nat)) => (((ord_less @ nat) @ (((divide_divide @ nat) @ A6) @ N)) @ (((divide_divide @ nat) @ B7) @ N))))))))).
% 0.33/5.20 thf(fact_813_mod__greater__zero__iff__not__dvd, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((modulo_modulo @ nat) @ M) @ N)) = (~ ((((dvd_dvd @ nat) @ N) @ M))))))).
% 0.33/5.20 thf(fact_814_of__nat__diff, axiom, (! [A:$tType]: (((semiring_1_cancel @ A) => (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M) @ N)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 0.33/5.20 thf(fact_815_division__segment__mod, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((~ ((((dvd_dvd @ A) @ B2) @ A3))) => (((euclid7384307370059645450egment @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) = ((euclid7384307370059645450egment @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_816_mod__mult__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_817_mod__mult__cong, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,A4:A,B2:A,B4:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ A4) @ C2)) => (((((modulo_modulo @ A) @ B2) @ C2) = (((modulo_modulo @ A) @ B4) @ C2)) => ((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((times_times @ A) @ A4) @ B4)) @ C2)))))))))).
% 0.33/5.20 thf(fact_818_mod__mult__mult2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((times_times @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_819_mult__mod__right, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A3:A,B2:A]: (((((times_times @ A) @ C2) @ (((modulo_modulo @ A) @ A3) @ B2)) = (((modulo_modulo @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2))))))))).
% 0.33/5.20 thf(fact_820_mod__mult__left__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_821_mod__mult__right__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_822_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ D2) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_823_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 0.33/5.20 thf(fact_824_diff__right__mono, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_825_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ A3) @ B2) = (((ord_less_eq @ A) @ C2) @ D2))))))))).
% 0.33/5.20 thf(fact_826_mod__minus__eq, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A3) @ B2))) @ B2) = (((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 0.33/5.20 thf(fact_827_mod__minus__cong, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A,A4:A]: ((((((modulo_modulo @ A) @ A3) @ B2) = (((modulo_modulo @ A) @ A4) @ B2)) => ((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A4)) @ B2))))))))).
% 0.33/5.20 thf(fact_828_mod__minus__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ ((uminus_uminus @ A) @ A3)) @ B2))))))))).
% 0.33/5.20 thf(fact_829_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [A5:A,B3:A]: (((((minus_minus @ A) @ A5) @ B3) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_830_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ D2) @ C2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_831_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A3) @ B2) = (((ord_less @ A) @ C2) @ D2))))))))).
% 0.33/5.20 thf(fact_832_diff__strict__left__mono, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_833_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_834_right__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_835_left__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [B2:A,C2:A,A3:A]: (((((times_times @ A) @ (((minus_minus @ A) @ B2) @ C2)) @ A3) = (((minus_minus @ A) @ (((times_times @ A) @ B2) @ A3)) @ (((times_times @ A) @ C2) @ A3))))))))).
% 0.33/5.20 thf(fact_836_right__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_837_left__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_838_inf__period_I1_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [P:(A > $o),D4:A,Q:(A > $o)]: (((! [X3:A,K2:A]: (((P @ X3) = (P @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D4)))))) => ((! [X3:A,K2:A]: (((Q @ X3) = (Q @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D4)))))) => (! [X4:A,K4:A]: ((((P @ X4) & (Q @ X4)) = ((P @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))) & (Q @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))))))))))))))).
% 0.33/5.20 thf(fact_839_inf__period_I2_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [P:(A > $o),D4:A,Q:(A > $o)]: (((! [X3:A,K2:A]: (((P @ X3) = (P @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D4)))))) => ((! [X3:A,K2:A]: (((Q @ X3) = (Q @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D4)))))) => (! [X4:A,K4:A]: ((((P @ X4) | (Q @ X4)) = ((P @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))) | (Q @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))))))))))))))).
% 0.33/5.20 thf(fact_840_dvd__field__iff, axiom, (! [A:$tType]: (((field @ A) => ((dvd_dvd @ A) = (^ [A5:A,B3:A]: (((A5 = (zero_zero @ A)) => (B3 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_841_dvd__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A3) => (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_842_mod__Suc__eq, axiom, (! [M:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (((modulo_modulo @ nat) @ M) @ N))) @ N) = (((modulo_modulo @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_843_mod__Suc__Suc__eq, axiom, (! [M:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (((modulo_modulo @ nat) @ M) @ N)))) @ N) = (((modulo_modulo @ nat) @ (suc @ (suc @ M))) @ N))))).
% 0.33/5.20 thf(fact_844_minus__diff__commute, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_845_dvd__triv__right, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A]: ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_846_dvd__mult__right, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) => (((dvd_dvd @ A) @ B2) @ C2)))))))).
% 0.33/5.20 thf(fact_847_mult__dvd__mono, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ C2) @ D2) => (((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_848_dvd__triv__left, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A]: ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_849_dvd__mult__left, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) => (((dvd_dvd @ A) @ A3) @ C2)))))))).
% 0.33/5.20 thf(fact_850_dvd__mult2, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => (((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_851_dvd__mult, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ C2) => (((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_852_dvd__def, axiom, (! [A:$tType]: (((dvd @ A) => ((dvd_dvd @ A) = (^ [B3:A,A5:A]: ((? [K3:A]: ((A5 = (((times_times @ A) @ B3) @ K3))))))))))).
% 0.33/5.20 thf(fact_853_dvdI, axiom, (! [A:$tType]: (((dvd @ A) => (! [A3:A,B2:A,K:A]: (((A3 = (((times_times @ A) @ B2) @ K)) => (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_854_dvdE, axiom, (! [A:$tType]: (((dvd @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => (~ ((! [K2:A]: ((A3 != (((times_times @ A) @ B2) @ K2))))))))))))).
% 0.33/5.20 thf(fact_855_one__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ (one_one @ A)) @ A3))))))).
% 0.33/5.20 thf(fact_856_unit__imp__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_857_dvd__unit__imp__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ A3) @ (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_858_divide__divide__eq__left_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((divide_divide @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))))))))).
% 0.33/5.20 thf(fact_859_divide__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,Y3:A,Z3:A,W2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ Z3) @ W2)) = (((divide_divide @ A) @ (((times_times @ A) @ X) @ W2)) @ (((times_times @ A) @ Y3) @ Z3))))))))).
% 0.33/5.20 thf(fact_860_times__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,Y3:A,Z3:A,W2:A]: (((((times_times @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ Z3) @ W2)) = (((divide_divide @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ Y3) @ W2))))))))).
% 0.33/5.20 thf(fact_861_mod__less__eq__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ N)) @ M)))).
% 0.33/5.20 thf(fact_862_minus__divide__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 0.33/5.20 thf(fact_863_minus__divide__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_864_minus__divide__right, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_865_div__minus__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 0.33/5.20 thf(fact_866_zero__induct__lemma, axiom, (! [P:(nat > $o),K:nat,I:nat]: (((P @ K) => ((! [N4:nat]: (((P @ (suc @ N4)) => (P @ N4)))) => (P @ (((minus_minus @ nat) @ K) @ I))))))).
% 0.33/5.20 thf(fact_867_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 0.33/5.20 thf(fact_868_diffs0__imp__equal, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N) @ M) = (zero_zero @ nat)) => (M = N)))))).
% 0.33/5.20 thf(fact_869_sgn__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_870_sgn__0__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_871_diff__less__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less @ nat) @ M) @ N) => ((((ord_less @ nat) @ M) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M))))))).
% 0.33/5.20 thf(fact_872_less__imp__diff__less, axiom, (! [J:nat,K:nat,N:nat]: (((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ N)) @ K))))).
% 0.33/5.20 thf(fact_873_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 0.33/5.20 thf(fact_874_div__le__mono, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N) @ K)))))).
% 0.33/5.20 thf(fact_875_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 0.33/5.20 thf(fact_876_le__diff__iff_H, axiom, (! [A3:nat,C2:nat,B2:nat]: (((((ord_less_eq @ nat) @ A3) @ C2) => ((((ord_less_eq @ nat) @ B2) @ C2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C2) @ A3)) @ (((minus_minus @ nat) @ C2) @ B2)) = (((ord_less_eq @ nat) @ B2) @ A3))))))).
% 0.33/5.20 thf(fact_877_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 0.33/5.20 thf(fact_878_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 0.33/5.20 thf(fact_879_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_880_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_881_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 0.33/5.20 thf(fact_882_sgn__mult, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A,B2:A]: ((((sgn_sgn @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((sgn_sgn @ A) @ A3)) @ ((sgn_sgn @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_883_div__mult2__eq, axiom, (! [M:nat,N:nat,Q4:nat]: (((((divide_divide @ nat) @ M) @ (((times_times @ nat) @ N) @ Q4)) = (((divide_divide @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ Q4))))).
% 0.33/5.20 thf(fact_884_diff__mult__distrib2, axiom, (! [K:nat,M:nat,N:nat]: (((((times_times @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N)) = (((minus_minus @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)))))).
% 0.33/5.20 thf(fact_885_diff__mult__distrib, axiom, (! [M:nat,N:nat,K:nat]: (((((times_times @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ K) = (((minus_minus @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N) @ K)))))).
% 0.33/5.20 thf(fact_886_uminus__dvd__conv_I2_J, axiom, ((dvd_dvd @ int) = (^ [D5:int,T3:int]: ((((dvd_dvd @ int) @ D5) @ ((uminus_uminus @ int) @ T3)))))).
% 0.33/5.20 thf(fact_887_uminus__dvd__conv_I1_J, axiom, ((dvd_dvd @ int) = (^ [D5:int]: (((dvd_dvd @ int) @ ((uminus_uminus @ int) @ D5)))))).
% 0.33/5.20 thf(fact_888_nat__dvd__iff, axiom, (! [Z3:int,M:nat]: (((((dvd_dvd @ nat) @ (nat2 @ Z3)) @ M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((dvd_dvd @ int) @ Z3) @ ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3))) => (M = (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_889_frac__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,Z3:A,X:A,W2:A]: (((Y3 != (zero_zero @ A)) => ((Z3 != (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ W2) @ Z3)) = (((ord_less_eq @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ W2) @ Y3))) @ (((times_times @ A) @ Y3) @ Z3))) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_890_frac__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,Z3:A,X:A,W2:A]: (((Y3 != (zero_zero @ A)) => ((Z3 != (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ W2) @ Z3)) = (((ord_less @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ W2) @ Y3))) @ (((times_times @ A) @ Y3) @ Z3))) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_891_add__divide__eq__if__simps_I6_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z3))) @ B2) = ((uminus_uminus @ A) @ B2))) & ((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z3))) @ B2) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ A3)) @ (((times_times @ A) @ B2) @ Z3))) @ Z3)))))))))).
% 0.33/5.20 thf(fact_892_add__divide__eq__if__simps_I5_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ Z3)) @ B2) = ((uminus_uminus @ A) @ B2))) & ((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ Z3)) @ B2) = (((divide_divide @ A) @ (((minus_minus @ A) @ A3) @ (((times_times @ A) @ B2) @ Z3))) @ Z3)))))))))).
% 0.33/5.20 thf(fact_893_minus__divide__diff__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ X) @ Z3))) @ Y3) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ X)) @ (((times_times @ A) @ Y3) @ Z3))) @ Z3))))))))).
% 0.33/5.20 thf(fact_894_is__unitE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (~ (((A3 != (zero_zero @ A)) => (! [B5:A]: (((B5 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B5) @ (one_one @ A)) => (((((divide_divide @ A) @ (one_one @ A)) @ A3) = B5) => (((((divide_divide @ A) @ (one_one @ A)) @ B5) = A3) => (((((times_times @ A) @ A3) @ B5) = (one_one @ A)) => ((((divide_divide @ A) @ C2) @ A3) != (((times_times @ A) @ C2) @ B5))))))))))))))))))).
% 0.33/5.20 thf(fact_895_is__unit__div__mult__cancel__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ A3) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2)))))))))).
% 0.33/5.20 thf(fact_896_is__unit__div__mult__cancel__right, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_897_div__if, axiom, ((divide_divide @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ ((((ord_less @ nat) @ M2) @ N3) | (N3 = (zero_zero @ nat)))) @ (zero_zero @ nat)) @ (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M2) @ N3)) @ N3))))))).
% 0.33/5.20 thf(fact_898_le__div__geq, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => ((((divide_divide @ nat) @ M) @ N) = (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ N)))))))).
% 0.33/5.20 thf(fact_899_nat__zero__as__int, axiom, ((zero_zero @ nat) = (nat2 @ (zero_zero @ int)))).
% 0.33/5.20 thf(fact_900_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A5) @ B3)) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_901_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A5:A,B3:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A5) @ B3)) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_902_int__sgnE, axiom, (! [K:int]: ((~ ((! [N4:nat,L2:int]: ((K != (((times_times @ int) @ ((sgn_sgn @ int) @ L2)) @ ((semiring_1_of_nat @ int) @ N4)))))))))).
% 0.33/5.20 thf(fact_903_ex__nat, axiom, ((^ [P3:(nat > $o)]: ((? [X6:nat]: ((P3 @ X6))))) = (^ [P4:(nat > $o)]: ((? [X2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X2) & (P4 @ (nat2 @ X2))))))))).
% 0.33/5.20 thf(fact_904_all__nat, axiom, ((^ [P3:(nat > $o)]: ((! [X6:nat]: ((P3 @ X6))))) = (^ [P4:(nat > $o)]: ((! [X2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X2) => (P4 @ (nat2 @ X2))))))))).
% 0.33/5.20 thf(fact_905_eq__nat__nat__iff, axiom, (! [Z3:int,Z6:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z6) => (((nat2 @ Z3) = (nat2 @ Z6)) = (Z3 = Z6))))))).
% 0.33/5.20 thf(fact_906_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 0.33/5.20 thf(fact_907_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_908_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_909_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_910_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_911_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_912_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_913_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ (((divide_divide @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_914_nat__mono, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ X) @ Y3) => (((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y3)))))).
% 0.33/5.20 thf(fact_915_divide__neg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_916_divide__neg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_917_divide__pos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_918_divide__pos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_919_divide__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 0.33/5.20 thf(fact_920_divide__less__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ B2)) & (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A3)) & (C2 != (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_921_zero__less__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_922_divide__strict__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_923_divide__strict__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_924_not__is__unit__0, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (~ ((((dvd_dvd @ A) @ (zero_zero @ A)) @ (one_one @ A)))))))).
% 0.33/5.20 thf(fact_925_frac__eq__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y3:A,Z3:A,X:A,W2:A]: (((Y3 != (zero_zero @ A)) => ((Z3 != (zero_zero @ A)) => (((((divide_divide @ A) @ X) @ Y3) = (((divide_divide @ A) @ W2) @ Z3)) = ((((times_times @ A) @ X) @ Z3) = (((times_times @ A) @ W2) @ Y3))))))))))).
% 0.33/5.20 thf(fact_926_divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A3:A]: ((((((divide_divide @ A) @ B2) @ C2) = A3) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A3) @ C2))) & ((C2 = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_927_eq__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_928_divide__eq__imp, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,B2:A,A3:A]: (((C2 != (zero_zero @ A)) => ((B2 = (((times_times @ A) @ A3) @ C2)) => ((((divide_divide @ A) @ B2) @ C2) = A3))))))))).
% 0.33/5.20 thf(fact_929_eq__divide__imp, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A3) @ C2) = B2) => (A3 = (((divide_divide @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_930_nonzero__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,B2:A,A3:A]: (((C2 != (zero_zero @ A)) => (((((divide_divide @ A) @ B2) @ C2) = A3) = (B2 = (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_931_nonzero__eq__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,A3:A,B2:A]: (((C2 != (zero_zero @ A)) => ((A3 = (((divide_divide @ A) @ B2) @ C2)) = ((((times_times @ A) @ A3) @ C2) = B2))))))))).
% 0.33/5.20 thf(fact_932_right__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((((divide_divide @ A) @ A3) @ B2) = (one_one @ A)) = (A3 = B2))))))))).
% 0.33/5.20 thf(fact_933_mod__Suc, axiom, (! [M:nat,N:nat]: (((((suc @ (((modulo_modulo @ nat) @ M) @ N)) = N) => ((((modulo_modulo @ nat) @ (suc @ M)) @ N) = (zero_zero @ nat))) & (((suc @ (((modulo_modulo @ nat) @ M) @ N)) != N) => ((((modulo_modulo @ nat) @ (suc @ M)) @ N) = (suc @ (((modulo_modulo @ nat) @ M) @ N)))))))).
% 0.33/5.20 thf(fact_934_is__unit__mult__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_935_dvd__mult__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_936_mult__unit__dvd__iff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_937_dvd__mult__unit__iff_H, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 0.33/5.20 thf(fact_938_mult__unit__dvd__iff_H, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_939_unit__mult__left__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((((times_times @ A) @ A3) @ B2) = (((times_times @ A) @ A3) @ C2)) = (B2 = C2))))))))).
% 0.33/5.20 thf(fact_940_unit__mult__right__cancel, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_941_nonzero__minus__divide__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_942_nonzero__minus__divide__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_943_mod__induct, axiom, (! [P:(nat > $o),N:nat,P5:nat,M:nat]: (((P @ N) => ((((ord_less @ nat) @ N) @ P5) => ((((ord_less @ nat) @ M) @ P5) => ((! [N4:nat]: (((((ord_less @ nat) @ N4) @ P5) => ((P @ N4) => (P @ (((modulo_modulo @ nat) @ (suc @ N4)) @ P5)))))) => (P @ M)))))))).
% 0.33/5.20 thf(fact_944_nat__one__as__int, axiom, ((one_one @ nat) = (nat2 @ (one_one @ int)))).
% 0.33/5.20 thf(fact_945_mod__less__divisor, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (((modulo_modulo @ nat) @ M) @ N)) @ N))))).
% 0.33/5.20 thf(fact_946_mod__Suc__le__divisor, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ (suc @ N))) @ N)))).
% 0.33/5.20 thf(fact_947_ceiling__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ Y3)) @ ((archimedean_ceiling @ A) @ X))))))))).
% 0.33/5.20 thf(fact_948_ceiling__less__cancel, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((archimedean_ceiling @ A) @ Y3)) => (((ord_less @ A) @ X) @ Y3)))))))).
% 0.33/5.20 thf(fact_949_div__mult2__eq_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,M:nat,N:nat]: (((((divide_divide @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ ((semiring_1_of_nat @ A) @ M))) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_950_Euclidean__Division_Odiv__eq__0__iff, axiom, (! [M:nat,N:nat]: ((((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)) = ((((ord_less @ nat) @ M) @ N) | (N = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_951_Suc__diff__Suc, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => ((suc @ (((minus_minus @ nat) @ M) @ (suc @ N))) = (((minus_minus @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_952_diff__less__Suc, axiom, (! [M:nat,N:nat]: ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ (suc @ M))))).
% 0.33/5.20 thf(fact_953_nat__dvd__not__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N) => (~ ((((dvd_dvd @ nat) @ N) @ M)))))))).
% 0.33/5.20 thf(fact_954_Suc__div__le__mono, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ (((divide_divide @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_955_diff__less, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))))).
% 0.33/5.20 thf(fact_956_Suc__diff__le, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((minus_minus @ nat) @ (suc @ M)) @ N) = (suc @ (((minus_minus @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_957_less__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_958_diff__less__mono, axiom, (! [A3:nat,B2:nat,C2:nat]: (((((ord_less @ nat) @ A3) @ B2) => ((((ord_less_eq @ nat) @ C2) @ A3) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A3) @ C2)) @ (((minus_minus @ nat) @ B2) @ C2))))))).
% 0.33/5.20 thf(fact_959_zdvd__antisym__nonneg, axiom, (! [M:int,N:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ M) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => ((((dvd_dvd @ int) @ M) @ N) => ((((dvd_dvd @ int) @ N) @ M) => (M = N)))))))).
% 0.33/5.20 thf(fact_960_sgn__not__eq__imp, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A3:A]: (((((sgn_sgn @ A) @ B2) != ((sgn_sgn @ A) @ A3)) => ((((sgn_sgn @ A) @ A3) != (zero_zero @ A)) => ((((sgn_sgn @ A) @ B2) != (zero_zero @ A)) => (((sgn_sgn @ A) @ A3) = ((uminus_uminus @ A) @ ((sgn_sgn @ A) @ B2)))))))))))).
% 0.33/5.20 thf(fact_961_zdvd__not__zless, axiom, (! [M:int,N:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ M) => ((((ord_less @ int) @ M) @ N) => (~ ((((dvd_dvd @ int) @ N) @ M)))))))).
% 0.33/5.20 thf(fact_962_less__mult__imp__div__less, axiom, (! [M:nat,I:nat,N:nat]: (((((ord_less @ nat) @ M) @ (((times_times @ nat) @ I) @ N)) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ I))))).
% 0.33/5.20 thf(fact_963_times__div__less__eq__dividend, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ N) @ (((divide_divide @ nat) @ M) @ N))) @ M)))).
% 0.33/5.20 thf(fact_964_div__times__less__eq__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ N)) @ M)))).
% 0.33/5.20 thf(fact_965_zdvd__mult__cancel, axiom, (! [K:int,M:int,N:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ K) @ M)) @ (((times_times @ int) @ K) @ N)) => ((K != (zero_zero @ int)) => (((dvd_dvd @ int) @ M) @ N)))))).
% 0.33/5.20 thf(fact_966_zdvd__mono, axiom, (! [K:int,M:int,T2:int]: (((K != (zero_zero @ int)) => ((((dvd_dvd @ int) @ M) @ T2) = (((dvd_dvd @ int) @ (((times_times @ int) @ K) @ M)) @ (((times_times @ int) @ K) @ T2))))))).
% 0.33/5.20 thf(fact_967_diff__Suc__eq__diff__pred, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ M) @ (suc @ N)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))) @ N))))).
% 0.33/5.20 thf(fact_968_sgn__minus__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.20 thf(fact_969_dvd__euclidean__size__eq__imp__dvd, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ B2)) => ((((dvd_dvd @ A) @ B2) @ A3) => (((dvd_dvd @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_970_euclidean__size__unit, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_971_fact__dvd, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((dvd_dvd @ A) @ ((semiring_char_0_fact @ A) @ N)) @ ((semiring_char_0_fact @ A) @ M))))))))).
% 0.33/5.20 thf(fact_972_is__unit__division__segment, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ ((euclid7384307370059645450egment @ A) @ A3)) @ (one_one @ A)))))))).
% 0.33/5.20 thf(fact_973_division__segment__eq__sgn, axiom, (! [K:int]: (((K != (zero_zero @ int)) => (((euclid7384307370059645450egment @ int) @ K) = ((sgn_sgn @ int) @ K)))))).
% 0.33/5.20 thf(fact_974_ceiling__ge__round, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_ceiling @ A) @ X)))))))).
% 0.33/5.20 thf(fact_975_frac__ge__0, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((archimedean_frac @ A) @ X)))))))).
% 0.33/5.20 thf(fact_976_frac__lt__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less @ A) @ ((archimedean_frac @ A) @ X)) @ (one_one @ A)))))))).
% 0.33/5.20 thf(fact_977_nat__mono__iff, axiom, (! [Z3:int,W2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z3) => ((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z3)) = (((ord_less @ int) @ W2) @ Z3)))))).
% 0.33/5.20 thf(fact_978_frac__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,X:A,W2:A,Z3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less_eq @ A) @ W2) @ Z3) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Z3)) @ (((divide_divide @ A) @ Y3) @ W2)))))))))))).
% 0.33/5.20 thf(fact_979_frac__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,W2:A,Z3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less_eq @ A) @ W2) @ Z3) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z3)) @ (((divide_divide @ A) @ Y3) @ W2)))))))))))).
% 0.33/5.20 thf(fact_980_frac__less2, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,W2:A,Z3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less @ A) @ W2) @ Z3) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z3)) @ (((divide_divide @ A) @ Y3) @ W2)))))))))))).
% 0.33/5.20 thf(fact_981_divide__le__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_982_divide__nonneg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_983_divide__nonneg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_984_divide__nonpos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_985_divide__nonpos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_986_divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))))))).
% 0.33/5.20 thf(fact_987_less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))))))).
% 0.33/5.20 thf(fact_988_neg__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_989_neg__less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_990_pos__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_991_pos__less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_992_mult__imp__div__pos__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,X:A,Z3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less @ A) @ X) @ (((times_times @ A) @ Z3) @ Y3)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ Z3))))))))).
% 0.33/5.20 thf(fact_993_mult__imp__less__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,Z3:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less @ A) @ (((times_times @ A) @ Z3) @ Y3)) @ X) => (((ord_less @ A) @ Z3) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_994_divide__strict__left__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less @ A) @ (((divide_divide @ A) @ C2) @ A3)) @ (((divide_divide @ A) @ C2) @ B2))))))))))).
% 0.33/5.20 thf(fact_995_divide__strict__left__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less @ A) @ (((divide_divide @ A) @ C2) @ A3)) @ (((divide_divide @ A) @ C2) @ B2))))))))))).
% 0.33/5.20 thf(fact_996_unit__dvdE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (~ (((A3 != (zero_zero @ A)) => (! [C3:A]: ((B2 != (((times_times @ A) @ A3) @ C3)))))))))))))).
% 0.33/5.20 thf(fact_997_divide__less__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ B2) @ A3)) | (((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ A3) @ B2)) | (A3 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_998_less__divide__eq__1, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_999_zless__nat__eq__int__zless, axiom, (! [M:nat,Z3:int]: (((((ord_less @ nat) @ M) @ (nat2 @ Z3)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ M)) @ Z3))))).
% 0.33/5.20 thf(fact_1000_int__eq__iff, axiom, (! [M:nat,Z3:int]: (((((semiring_1_of_nat @ int) @ M) = Z3) = ((M = (nat2 @ Z3)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3)))))).
% 0.33/5.20 thf(fact_1001_nat__0__le, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = Z3))))).
% 0.33/5.20 thf(fact_1002_nat__le__iff, axiom, (! [X:int,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X)) @ N) = (((ord_less_eq @ int) @ X) @ ((semiring_1_of_nat @ int) @ N)))))).
% 0.33/5.20 thf(fact_1003_eq__minus__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ C2) = ((uminus_uminus @ A) @ B2))) & ((C2 = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_1004_minus__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A3:A]: (((((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2)) = A3) = (((C2 != (zero_zero @ A)) => (((uminus_uminus @ A) @ B2) = (((times_times @ A) @ A3) @ C2))) & ((C2 = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_1005_nonzero__neg__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2)) = C2) = (((uminus_uminus @ A) @ A3) = (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.20 thf(fact_1006_nonzero__neg__divide__eq__eq2, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((C2 = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ B2))) = ((((times_times @ A) @ C2) @ B2) = ((uminus_uminus @ A) @ A3)))))))))).
% 0.33/5.20 thf(fact_1007_divide__eq__minus__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((((divide_divide @ A) @ A3) @ B2) = ((uminus_uminus @ A) @ (one_one @ A))) = ((B2 != (zero_zero @ A)) & (A3 = ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1008_mod__le__divisor, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ N)) @ N))))).
% 0.33/5.20 thf(fact_1009_fact__mod, axiom, (! [A:$tType]: ((((linordered_semidom @ A) & (semidom_modulo @ A)) => (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((modulo_modulo @ A) @ ((semiring_char_0_fact @ A) @ N)) @ ((semiring_char_0_fact @ A) @ M)) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1010_mod__size__less, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ (((modulo_modulo @ A) @ A3) @ B2))) @ ((euclid6346220572633701492n_size @ A) @ B2))))))))).
% 0.33/5.20 thf(fact_1011_diff__Suc__less, axiom, (! [N:nat,I:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (((minus_minus @ nat) @ N) @ (suc @ I))) @ N))))).
% 0.33/5.20 thf(fact_1012_div__greater__zero__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ M) @ N)) = ((((ord_less_eq @ nat) @ N) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.20 thf(fact_1013_div__le__mono2, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ K) @ N)) @ (((divide_divide @ nat) @ K) @ M))))))).
% 0.33/5.20 thf(fact_1014_sgn__1__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = (one_one @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.20 thf(fact_1015_dvd__imp__le, axiom, (! [K:nat,N:nat]: (((((dvd_dvd @ nat) @ K) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ nat) @ K) @ N)))))).
% 0.33/5.20 thf(fact_1016_nat__mult__div__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((divide_divide @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1017_div__less__iff__less__mult, axiom, (! [Q4:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Q4) => ((((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ Q4)) @ N) = (((ord_less @ nat) @ M) @ (((times_times @ nat) @ N) @ Q4))))))).
% 0.33/5.20 thf(fact_1018_nat__mult__dvd__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((dvd_dvd @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1019_dvd__mult__cancel, axiom, (! [K:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((dvd_dvd @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1020_unit__iff__euclidean__size, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) = ((((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ (one_one @ A))) & (A3 != (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1021_div__less__dividend, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))))).
% 0.33/5.20 thf(fact_1022_div__eq__dividend__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((((divide_divide @ nat) @ M) @ N) = M) = (N = (one_one @ nat))))))).
% 0.33/5.20 thf(fact_1023_euclidean__size__times__unit, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1024_zdvd__imp__le, axiom, (! [Z3:int,N:int]: (((((dvd_dvd @ int) @ Z3) @ N) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ int) @ Z3) @ N)))))).
% 0.33/5.20 thf(fact_1025_dvd__proper__imp__size__less, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((~ ((((dvd_dvd @ A) @ B2) @ A3))) => ((B2 != (zero_zero @ A)) => (((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ B2))))))))))).
% 0.33/5.20 thf(fact_1026_dvd__imp__size__le, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((B2 != (zero_zero @ A)) => (((ord_less_eq @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1027_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit5016429287641298734tinuum @ A) => (! [A3:A]: ((? [B5:A]: (((((ord_less @ A) @ A3) @ B5) | (((ord_less @ A) @ B5) @ A3)))))))))).
% 0.33/5.20 thf(fact_1028_of__nat__euclidean__size, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: ((((semiring_1_of_nat @ A) @ ((euclid6346220572633701492n_size @ A) @ A3)) = (((divide_divide @ A) @ A3) @ ((euclid7384307370059645450egment @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_1029_dvd__fact, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => (((dvd_dvd @ nat) @ M) @ ((semiring_char_0_fact @ nat) @ N))))))).
% 0.33/5.20 thf(fact_1030_zsgn__def, axiom, ((sgn_sgn @ int) = (^ [I5:int]: (((((if @ int) @ (I5 = (zero_zero @ int))) @ (zero_zero @ int)) @ ((((if @ int) @ (((ord_less @ int) @ (zero_zero @ int)) @ I5)) @ (one_one @ int)) @ ((uminus_uminus @ int) @ (one_one @ int)))))))).
% 0.33/5.20 thf(fact_1031_nat__less__eq__zless, axiom, (! [W2:int,Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => ((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z3)) = (((ord_less @ int) @ W2) @ Z3)))))).
% 0.33/5.20 thf(fact_1032_nat__le__eq__zle, axiom, (! [W2:int,Z3:int]: ((((((ord_less @ int) @ (zero_zero @ int)) @ W2) | (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3)) => ((((ord_less_eq @ nat) @ (nat2 @ W2)) @ (nat2 @ Z3)) = (((ord_less_eq @ int) @ W2) @ Z3)))))).
% 0.33/5.20 thf(fact_1033_nat__eq__iff, axiom, (! [W2:int,M:nat]: ((((nat2 @ W2) = M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => (W2 = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2))) => (M = (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_1034_nat__eq__iff2, axiom, (! [M:nat,W2:int]: (((M = (nat2 @ W2)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => (W2 = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2))) => (M = (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_1035_divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))))))).
% 0.33/5.20 thf(fact_1036_le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))))))).
% 0.33/5.20 thf(fact_1037_divide__left__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ C2) @ A3)) @ (((divide_divide @ A) @ C2) @ B2))))))))))).
% 0.33/5.20 thf(fact_1038_neg__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_1039_neg__le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1040_pos__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A3) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1041_pos__le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_1042_mult__imp__div__pos__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,X:A,Z3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ X) @ (((times_times @ A) @ Z3) @ Y3)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ Z3))))))))).
% 0.33/5.20 thf(fact_1043_mult__imp__le__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,Z3:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ (((times_times @ A) @ Z3) @ Y3)) @ X) => (((ord_less_eq @ A) @ Z3) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.20 thf(fact_1044_divide__left__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ C2) @ A3)) @ (((divide_divide @ A) @ C2) @ B2))))))))))).
% 0.33/5.20 thf(fact_1045_split__nat, axiom, (! [P:(nat > $o),I:int]: (((P @ (nat2 @ I)) = ((! [N3:nat]: (((I = ((semiring_1_of_nat @ int) @ N3)) => (P @ N3)))) & ((((ord_less @ int) @ I) @ (zero_zero @ int)) => (P @ (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_1046_divide__le__eq__1, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_1047_le__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ A3) @ B2)) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_1048_le__nat__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 0.33/5.20 thf(fact_1049_nat__mult__distrib, axiom, (! [Z3:int,Z6:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((nat2 @ (((times_times @ int) @ Z3) @ Z6)) = (((times_times @ nat) @ (nat2 @ Z3)) @ (nat2 @ Z6))))))).
% 0.33/5.20 thf(fact_1050_less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))))))).
% 0.33/5.20 thf(fact_1051_minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))))))).
% 0.33/5.20 thf(fact_1052_neg__less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1053_neg__minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1054_pos__less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1055_pos__minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1056_div__nat__eqI, axiom, (! [N:nat,Q4:nat,M:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ N) @ Q4)) @ M) => ((((ord_less @ nat) @ M) @ (((times_times @ nat) @ N) @ (suc @ Q4))) => ((((divide_divide @ nat) @ M) @ N) = Q4)))))).
% 0.33/5.20 thf(fact_1057_less__eq__div__iff__mult__less__eq, axiom, (! [Q4:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Q4) => ((((ord_less_eq @ nat) @ M) @ (((divide_divide @ nat) @ N) @ Q4)) = (((ord_less_eq @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ N)))))).
% 0.33/5.20 thf(fact_1058_sgn__if, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((sgn_sgn @ A) = (^ [X2:A]: (((((if @ A) @ (X2 = (zero_zero @ A))) @ (zero_zero @ A)) @ ((((if @ A) @ (((ord_less @ A) @ (zero_zero @ A)) @ X2)) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 0.33/5.20 thf(fact_1059_sgn__1__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = ((uminus_uminus @ A) @ (one_one @ A))) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1060_Suc__pred_H, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (N = (suc @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 0.33/5.20 thf(fact_1061_Suc__diff__eq__diff__pred, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((minus_minus @ nat) @ (suc @ M)) @ N) = (((minus_minus @ nat) @ M) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 0.33/5.20 thf(fact_1062_dvd__mult__cancel1, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ M) @ N)) @ M) = (N = (one_one @ nat))))))).
% 0.33/5.20 thf(fact_1063_dvd__mult__cancel2, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ N) @ M)) @ M) = (N = (one_one @ nat))))))).
% 0.33/5.20 thf(fact_1064_unique__euclidean__semiring__class_Odiv__eq__0__iff, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A]: (((((euclid7384307370059645450egment @ A) @ A3) = ((euclid7384307370059645450egment @ A) @ B2)) => (((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)) = ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ A3)) @ ((euclid6346220572633701492n_size @ A) @ B2)) | (B2 = (zero_zero @ A))))))))))).
% 0.33/5.20 thf(fact_1065_nat__less__iff, axiom, (! [W2:int,M:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => ((((ord_less @ nat) @ (nat2 @ W2)) @ M) = (((ord_less @ int) @ W2) @ ((semiring_1_of_nat @ int) @ M))))))).
% 0.33/5.20 thf(fact_1066_le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))))))).
% 0.33/5.20 thf(fact_1067_minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))))))).
% 0.33/5.20 thf(fact_1068_neg__le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1069_neg__minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1070_pos__le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))).
% 0.33/5.20 thf(fact_1071_pos__minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A3) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.20 thf(fact_1072_nat__mult__distrib__neg, axiom, (! [Z3:int,Z6:int]: (((((ord_less_eq @ int) @ Z3) @ (zero_zero @ int)) => ((nat2 @ (((times_times @ int) @ Z3) @ Z6)) = (((times_times @ nat) @ (nat2 @ ((uminus_uminus @ int) @ Z3))) @ (nat2 @ ((uminus_uminus @ int) @ Z6)))))))).
% 0.33/5.20 thf(fact_1073_nat__approx__posE, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) => (~ ((! [N4:nat]: ((~ ((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ (suc @ N4)))) @ E)))))))))))))).
% 0.33/5.20 thf(fact_1074_mult__ceiling__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ (((times_times @ A) @ A3) @ B2))) @ (((times_times @ int) @ ((archimedean_ceiling @ A) @ A3)) @ ((archimedean_ceiling @ A) @ B2))))))))))).
% 0.33/5.20 thf(fact_1075_bits__mod__div__trivial, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,B2:A]: (((((divide_divide @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ B2) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1076_bits__mod__by__1, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1077_choose__dvd, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((dvd_dvd @ A) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N) @ K)))) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 0.33/5.20 thf(fact_1078_div__geq, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((~ ((((ord_less @ nat) @ M) @ N))) => ((((divide_divide @ nat) @ M) @ N) = (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ N)))))))).
% 0.33/5.20 thf(fact_1079_bits__mod__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1080_bits__div__by__1, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 0.33/5.20 thf(fact_1081_bits__div__by__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1082_bits__div__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1083_unique__euclidean__semiring__numeral__class_Omod__less, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ B2) => ((((modulo_modulo @ A) @ A3) @ B2) = A3))))))))).
% 0.33/5.20 thf(fact_1084_unique__euclidean__semiring__numeral__class_Opos__mod__sign, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((modulo_modulo @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_1085_mod__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((modulo_modulo @ int) @ K) @ L) = K)))))).
% 0.33/5.20 thf(fact_1086_mod__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((modulo_modulo @ int) @ K) @ L) = K)))))).
% 0.33/5.20 thf(fact_1087_div__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1088_div__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1089_zle__diff1__eq, axiom, (! [W2:int,Z3:int]: (((((ord_less_eq @ int) @ W2) @ (((minus_minus @ int) @ Z3) @ (one_one @ int))) = (((ord_less @ int) @ W2) @ Z3))))).
% 0.33/5.20 thf(fact_1090_dvd__antisym, axiom, (! [M:nat,N:nat]: (((((dvd_dvd @ nat) @ M) @ N) => ((((dvd_dvd @ nat) @ N) @ M) => (M = N)))))).
% 0.33/5.20 thf(fact_1091_zdvd__zdiffD, axiom, (! [K:int,M:int,N:int]: (((((dvd_dvd @ int) @ K) @ (((minus_minus @ int) @ M) @ N)) => ((((dvd_dvd @ int) @ K) @ N) => (((dvd_dvd @ int) @ K) @ M)))))).
% 0.33/5.20 thf(fact_1092_zmod__zminus1__eq__if, axiom, (! [A3:int,B2:int]: (((((((modulo_modulo @ int) @ A3) @ B2) = (zero_zero @ int)) => ((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ A3)) @ B2) = (zero_zero @ int))) & (((((modulo_modulo @ int) @ A3) @ B2) != (zero_zero @ int)) => ((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ A3)) @ B2) = (((minus_minus @ int) @ B2) @ (((modulo_modulo @ int) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1093_zmod__zminus2__eq__if, axiom, (! [A3:int,B2:int]: (((((((modulo_modulo @ int) @ A3) @ B2) = (zero_zero @ int)) => ((((modulo_modulo @ int) @ A3) @ ((uminus_uminus @ int) @ B2)) = (zero_zero @ int))) & (((((modulo_modulo @ int) @ A3) @ B2) != (zero_zero @ int)) => ((((modulo_modulo @ int) @ A3) @ ((uminus_uminus @ int) @ B2)) = (((minus_minus @ int) @ (((modulo_modulo @ int) @ A3) @ B2)) @ B2))))))).
% 0.33/5.20 thf(fact_1094_zdiv__zminus1__eq__if, axiom, (! [B2:int,A3:int]: (((B2 != (zero_zero @ int)) => ((((((modulo_modulo @ int) @ A3) @ B2) = (zero_zero @ int)) => ((((divide_divide @ int) @ ((uminus_uminus @ int) @ A3)) @ B2) = ((uminus_uminus @ int) @ (((divide_divide @ int) @ A3) @ B2)))) & (((((modulo_modulo @ int) @ A3) @ B2) != (zero_zero @ int)) => ((((divide_divide @ int) @ ((uminus_uminus @ int) @ A3)) @ B2) = (((minus_minus @ int) @ ((uminus_uminus @ int) @ (((divide_divide @ int) @ A3) @ B2))) @ (one_one @ int))))))))).
% 0.33/5.20 thf(fact_1095_zdiv__zminus2__eq__if, axiom, (! [B2:int,A3:int]: (((B2 != (zero_zero @ int)) => ((((((modulo_modulo @ int) @ A3) @ B2) = (zero_zero @ int)) => ((((divide_divide @ int) @ A3) @ ((uminus_uminus @ int) @ B2)) = ((uminus_uminus @ int) @ (((divide_divide @ int) @ A3) @ B2)))) & (((((modulo_modulo @ int) @ A3) @ B2) != (zero_zero @ int)) => ((((divide_divide @ int) @ A3) @ ((uminus_uminus @ int) @ B2)) = (((minus_minus @ int) @ ((uminus_uminus @ int) @ (((divide_divide @ int) @ A3) @ B2))) @ (one_one @ int))))))))).
% 0.33/5.20 thf(fact_1096_zdiv__mono__strict, axiom, (! [A6:int,B7:int,N:int]: (((((ord_less @ int) @ A6) @ B7) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((((modulo_modulo @ int) @ A6) @ N) = (zero_zero @ int)) => (((((modulo_modulo @ int) @ B7) @ N) = (zero_zero @ int)) => (((ord_less @ int) @ (((divide_divide @ int) @ A6) @ N)) @ (((divide_divide @ int) @ B7) @ N))))))))).
% 0.33/5.20 thf(fact_1097_mod__pos__geq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ L) @ K) => ((((modulo_modulo @ int) @ K) @ L) = (((modulo_modulo @ int) @ (((minus_minus @ int) @ K) @ L)) @ L))))))).
% 0.33/5.20 thf(fact_1098_sgn__mod, axiom, (! [L:int,K:int]: (((L != (zero_zero @ int)) => ((~ ((((dvd_dvd @ int) @ L) @ K))) => (((sgn_sgn @ int) @ (((modulo_modulo @ int) @ K) @ L)) = ((sgn_sgn @ int) @ L))))))).
% 0.33/5.20 thf(fact_1099_minus__int__code_I1_J, axiom, (! [K:int]: (((((minus_minus @ int) @ K) @ (zero_zero @ int)) = K)))).
% 0.33/5.20 thf(fact_1100_int__distrib_I3_J, axiom, (! [Z1:int,Z22:int,W2:int]: (((((times_times @ int) @ (((minus_minus @ int) @ Z1) @ Z22)) @ W2) = (((minus_minus @ int) @ (((times_times @ int) @ Z1) @ W2)) @ (((times_times @ int) @ Z22) @ W2)))))).
% 0.33/5.20 thf(fact_1101_int__distrib_I4_J, axiom, (! [W2:int,Z1:int,Z22:int]: (((((times_times @ int) @ W2) @ (((minus_minus @ int) @ Z1) @ Z22)) = (((minus_minus @ int) @ (((times_times @ int) @ W2) @ Z1)) @ (((times_times @ int) @ W2) @ Z22)))))).
% 0.33/5.20 thf(fact_1102_int__diff__cases, axiom, (! [Z3:int]: ((~ ((! [M6:nat,N4:nat]: ((Z3 != (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ M6)) @ ((semiring_1_of_nat @ int) @ N4)))))))))).
% 0.33/5.20 thf(fact_1103_zmod__le__nonneg__dividend, axiom, (! [M:int,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ M) => (((ord_less_eq @ int) @ (((modulo_modulo @ int) @ M) @ K)) @ M))))).
% 0.33/5.20 thf(fact_1104_zmod__eq__0__iff, axiom, (! [M:int,D2:int]: ((((((modulo_modulo @ int) @ M) @ D2) = (zero_zero @ int)) = (? [Q3:int]: ((M = (((times_times @ int) @ D2) @ Q3)))))))).
% 0.33/5.20 thf(fact_1105_zmod__eq__0D, axiom, (! [M:int,D2:int]: ((((((modulo_modulo @ int) @ M) @ D2) = (zero_zero @ int)) => (? [Q5:int]: ((M = (((times_times @ int) @ D2) @ Q5)))))))).
% 0.33/5.20 thf(fact_1106_zmod__zminus2__not__zero, axiom, (! [K:int,L:int]: ((((((modulo_modulo @ int) @ K) @ ((uminus_uminus @ int) @ L)) != (zero_zero @ int)) => ((((modulo_modulo @ int) @ K) @ L) != (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_1107_zmod__zminus1__not__zero, axiom, (! [K:int,L:int]: ((((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ K)) @ L) != (zero_zero @ int)) => ((((modulo_modulo @ int) @ K) @ L) != (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_1108_div__neg__pos__less0, axiom, (! [A3:int,B2:int]: (((((ord_less @ int) @ A3) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1109_neg__imp__zdiv__neg__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ (zero_zero @ int)) @ A3)))))).
% 0.33/5.20 thf(fact_1110_pos__imp__zdiv__neg__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ A3) @ (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1111_zmod__int, axiom, (! [A3:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ A3) @ B2)) = (((modulo_modulo @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 0.33/5.20 thf(fact_1112_zdiv__int, axiom, (! [A3:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ A3) @ B2)) = (((divide_divide @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 0.33/5.20 thf(fact_1113_minus__mod__int__eq, axiom, (! [L:int,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ L) => ((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ K)) @ L) = (((minus_minus @ int) @ (((minus_minus @ int) @ L) @ (one_one @ int))) @ (((modulo_modulo @ int) @ (((minus_minus @ int) @ K) @ (one_one @ int))) @ L))))))).
% 0.33/5.20 thf(fact_1114_zmod__minus1, axiom, (! [B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ B2) = (((minus_minus @ int) @ B2) @ (one_one @ int))))))).
% 0.33/5.20 thf(fact_1115_neg__mod__conj, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (((modulo_modulo @ int) @ A3) @ B2)) @ (zero_zero @ int)) & (((ord_less @ int) @ B2) @ (((modulo_modulo @ int) @ A3) @ B2))))))).
% 0.33/5.20 thf(fact_1116_pos__mod__conj, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ A3) @ B2)) & (((ord_less @ int) @ (((modulo_modulo @ int) @ A3) @ B2)) @ B2)))))).
% 0.33/5.20 thf(fact_1117_zmod__trivial__iff, axiom, (! [I:int,K:int]: ((((((modulo_modulo @ int) @ I) @ K) = I) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) & (((ord_less @ int) @ I) @ K)) | ((((ord_less_eq @ int) @ I) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I)))))))).
% 0.33/5.20 thf(fact_1118_mod__int__pos__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ K) @ L)) = ((((dvd_dvd @ int) @ L) @ K) | (((L = (zero_zero @ int)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ K)) | (((ord_less @ int) @ (zero_zero @ int)) @ L))))))).
% 0.33/5.20 thf(fact_1119_neg__mod__bound, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less @ int) @ L) @ (((modulo_modulo @ int) @ K) @ L)))))).
% 0.33/5.20 thf(fact_1120_Euclidean__Division_Opos__mod__bound, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less @ int) @ (((modulo_modulo @ int) @ K) @ L)) @ L))))).
% 0.33/5.20 thf(fact_1121_zdiv__mono1, axiom, (! [A3:int,A4:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ A4) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (((divide_divide @ int) @ A4) @ B2))))))).
% 0.33/5.20 thf(fact_1122_zdiv__mono2, axiom, (! [A3:int,B4:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => ((((ord_less_eq @ int) @ B4) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (((divide_divide @ int) @ A3) @ B4)))))))).
% 0.33/5.20 thf(fact_1123_zdiv__eq__0__iff, axiom, (! [I:int,K:int]: ((((((divide_divide @ int) @ I) @ K) = (zero_zero @ int)) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) & (((ord_less @ int) @ I) @ K)) | ((((ord_less_eq @ int) @ I) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I)))))))).
% 0.33/5.20 thf(fact_1124_zdiv__mono1__neg, axiom, (! [A3:int,A4:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ A4) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A4) @ B2)) @ (((divide_divide @ int) @ A3) @ B2))))))).
% 0.33/5.20 thf(fact_1125_zdiv__mono2__neg, axiom, (! [A3:int,B4:int,B2:int]: (((((ord_less @ int) @ A3) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => ((((ord_less_eq @ int) @ B4) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B4)) @ (((divide_divide @ int) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1126_div__int__pos__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L)) = ((K = (zero_zero @ int)) | ((L = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)) | ((((ord_less @ int) @ K) @ (zero_zero @ int)) & (((ord_less @ int) @ L) @ (zero_zero @ int)))))))))).
% 0.33/5.20 thf(fact_1127_div__positive__int, axiom, (! [L:int,K:int]: (((((ord_less_eq @ int) @ L) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L))))))).
% 0.33/5.20 thf(fact_1128_div__nonneg__neg__le0, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A3) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1129_div__nonpos__pos__le0, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1130_pos__imp__zdiv__pos__iff, axiom, (! [K:int,I:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ I) @ K)) = (((ord_less_eq @ int) @ K) @ I)))))).
% 0.33/5.20 thf(fact_1131_neg__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A3) @ B2)) = (((ord_less_eq @ int) @ A3) @ (zero_zero @ int))))))).
% 0.33/5.20 thf(fact_1132_pos__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A3) @ B2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A3)))))).
% 0.33/5.20 thf(fact_1133_nonneg1__imp__zdiv__pos__iff, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A3) @ B2)) = ((((ord_less_eq @ int) @ B2) @ A3) & (((ord_less @ int) @ (zero_zero @ int)) @ B2))))))).
% 0.33/5.20 thf(fact_1134_int__div__less__self, axiom, (! [X:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ X) => ((((ord_less @ int) @ (one_one @ int)) @ K) => (((ord_less @ int) @ (((divide_divide @ int) @ X) @ K)) @ X)))))).
% 0.33/5.20 thf(fact_1135_minus__int__code_I2_J, axiom, (! [L:int]: (((((minus_minus @ int) @ (zero_zero @ int)) @ L) = ((uminus_uminus @ int) @ L))))).
% 0.33/5.20 thf(fact_1136_int__le__induct, axiom, (! [I:int,K:int,P:(int > $o)]: (((((ord_less_eq @ int) @ I) @ K) => ((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_1137_int__less__induct, axiom, (! [I:int,K:int,P:(int > $o)]: (((((ord_less @ int) @ I) @ K) => ((P @ (((minus_minus @ int) @ K) @ (one_one @ int))) => ((! [I4:int]: (((((ord_less @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_1138_div__eq__minus1, axiom, (! [B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((divide_divide @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ B2) = ((uminus_uminus @ int) @ (one_one @ int))))))).
% 0.33/5.20 thf(fact_1139_neg__mod__sign, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((modulo_modulo @ int) @ K) @ L)) @ (zero_zero @ int)))))).
% 0.33/5.20 thf(fact_1140_Euclidean__Division_Opos__mod__sign, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ K) @ L)))))).
% 0.33/5.20 thf(fact_1141_plusinfinity, axiom, (! [D2:int,P2:(int > $o),P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X3:int,K2:int]: (((P2 @ X3) = (P2 @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [Z5:int]: ((! [X3:int]: (((((ord_less @ int) @ Z5) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [X_1:int]: ((P2 @ X_1))) => (? [X_12:int]: ((P @ X_12)))))))))).
% 0.33/5.20 thf(fact_1142_minusinfinity, axiom, (! [D2:int,P1:(int > $o),P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X3:int,K2:int]: (((P1 @ X3) = (P1 @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [Z5:int]: ((! [X3:int]: (((((ord_less @ int) @ X3) @ Z5) => ((P @ X3) = (P1 @ X3))))))) => ((? [X_1:int]: ((P1 @ X_1))) => (? [X_12:int]: ((P @ X_12)))))))))).
% 0.33/5.20 thf(fact_1143_zdiv__zmult2__eq, axiom, (! [C2:int,A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ C2) => ((((divide_divide @ int) @ A3) @ (((times_times @ int) @ B2) @ C2)) = (((divide_divide @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ C2)))))).
% 0.33/5.20 thf(fact_1144_int__minus, axiom, (! [N:nat,M:nat]: ((((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ N) @ M)) = ((semiring_1_of_nat @ int) @ (nat2 @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((semiring_1_of_nat @ int) @ M)))))))).
% 0.33/5.20 thf(fact_1145_nat__mod__distrib, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((nat2 @ (((modulo_modulo @ int) @ X) @ Y3)) = (((modulo_modulo @ nat) @ (nat2 @ X)) @ (nat2 @ Y3)))))))).
% 0.33/5.20 thf(fact_1146_decr__mult__lemma, axiom, (! [D2:int,P:(int > $o),K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X3:int]: (((P @ X3) => (P @ (((minus_minus @ int) @ X3) @ D2))))) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (! [X4:int]: (((P @ X4) => (P @ (((minus_minus @ int) @ X4) @ (((times_times @ int) @ K) @ D2)))))))))))).
% 0.33/5.20 thf(fact_1147_int__ops_I6_J, axiom, (! [A3:nat,B2:nat]: ((((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)) => (((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ A3) @ B2)) = (zero_zero @ int))) & ((~ ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))) => (((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ A3) @ B2)) = (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))))).
% 0.33/5.20 thf(fact_1148_nat__diff__distrib, axiom, (! [Z6:int,Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z6) => ((((ord_less_eq @ int) @ Z6) @ Z3) => ((nat2 @ (((minus_minus @ int) @ Z3) @ Z6)) = (((minus_minus @ nat) @ (nat2 @ Z3)) @ (nat2 @ Z6)))))))).
% 0.33/5.20 thf(fact_1149_nat__diff__distrib_H, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((nat2 @ (((minus_minus @ int) @ X) @ Y3)) = (((minus_minus @ nat) @ (nat2 @ X)) @ (nat2 @ Y3)))))))).
% 0.33/5.20 thf(fact_1150_verit__less__mono__div__int2, axiom, (! [A6:int,B7:int,N:int]: (((((ord_less_eq @ int) @ A6) @ B7) => ((((ord_less @ int) @ (zero_zero @ int)) @ ((uminus_uminus @ int) @ N)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ B7) @ N)) @ (((divide_divide @ int) @ A6) @ N))))))).
% 0.33/5.20 thf(fact_1151_nat__div__distrib_H, axiom, (! [Y3:int,X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((nat2 @ (((divide_divide @ int) @ X) @ Y3)) = (((divide_divide @ nat) @ (nat2 @ X)) @ (nat2 @ Y3))))))).
% 0.33/5.20 thf(fact_1152_nat__div__distrib, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((nat2 @ (((divide_divide @ int) @ X) @ Y3)) = (((divide_divide @ nat) @ (nat2 @ X)) @ (nat2 @ Y3))))))).
% 0.33/5.20 thf(fact_1153_zdiff__int__split, axiom, (! [P:(int > $o),X:nat,Y3:nat]: (((P @ ((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ X) @ Y3))) = (((((ord_less_eq @ nat) @ Y3) @ X) => (P @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y3)))) & ((((ord_less @ nat) @ X) @ Y3) => (P @ (zero_zero @ int)))))))).
% 0.33/5.20 thf(fact_1154_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ A3)))))))).
% 0.33/5.20 thf(fact_1155_unique__euclidean__semiring__numeral__class_Opos__mod__bound, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ B2)))))))).
% 0.33/5.20 thf(fact_1156_mod__eq__0D, axiom, (! [M:nat,D2:nat]: ((((((modulo_modulo @ nat) @ M) @ D2) = (zero_zero @ nat)) => (? [Q5:nat]: ((M = (((times_times @ nat) @ D2) @ Q5)))))))).
% 0.33/5.20 thf(fact_1157_mod__geq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) => ((((modulo_modulo @ nat) @ M) @ N) = (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ N)))))).
% 0.33/5.20 thf(fact_1158_mod__eq__dvd__iff__nat, axiom, (! [N:nat,M:nat,Q4:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N) @ Q4)) = (((dvd_dvd @ nat) @ Q4) @ (((minus_minus @ nat) @ M) @ N))))))).
% 0.33/5.20 thf(fact_1159_div__positive, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1160_unique__euclidean__semiring__numeral__class_Odiv__less, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ B2) => ((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1161_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2))))))))).
% 0.33/5.20 thf(fact_1162_bezout1__nat, axiom, (! [A3:nat,B2:nat]: ((? [D6:nat,X3:nat,Y5:nat]: (((((dvd_dvd @ nat) @ D6) @ A3) & ((((dvd_dvd @ nat) @ D6) @ B2) & (((((minus_minus @ nat) @ (((times_times @ nat) @ A3) @ X3)) @ (((times_times @ nat) @ B2) @ Y5)) = D6) | ((((minus_minus @ nat) @ (((times_times @ nat) @ B2) @ X3)) @ (((times_times @ nat) @ A3) @ Y5)) = D6))))))))).
% 0.33/5.20 thf(fact_1163_gcd__nat__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [M6:nat]: (((P @ M6) @ (zero_zero @ nat)))) => ((! [M6:nat,N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => (((P @ N4) @ (((modulo_modulo @ nat) @ M6) @ N4)) => ((P @ M6) @ N4))))) => ((P @ M) @ N)))))).
% 0.33/5.20 thf(fact_1164_dvd__pos__nat, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((dvd_dvd @ nat) @ M) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ M)))))).
% 0.33/5.20 thf(fact_1165_gbinomial__absorption_H, axiom, (! [A:$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)))))))))))).
% 0.33/5.20 thf(fact_1166_modulo__int__unfold, axiom, (! [L:int,K:int,N:nat,M:nat]: (((((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N = (zero_zero @ nat)))) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M)))) & ((~ (((((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)) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ M) @ N))))) & ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ (((times_times @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ N) @ M))))))) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ M) @ N)))))))))))).
% 0.33/5.20 thf(fact_1167_divmod__cases, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,A3:A]: ((((B2 != (zero_zero @ A)) => (((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A)) => (A3 != (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)))) => (((B2 != (zero_zero @ A)) => (! [Q5:A,R3:A]: (((((euclid7384307370059645450egment @ A) @ R3) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R3)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => ((R3 != (zero_zero @ A)) => (((((divide_divide @ A) @ A3) @ B2) = Q5) => (((((modulo_modulo @ A) @ A3) @ B2) = R3) => (A3 != (((plus_plus @ A) @ (((times_times @ A) @ Q5) @ B2)) @ R3)))))))))) => (B2 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1168_gcd__nat_Oextremum, axiom, (! [A3:nat]: ((((dvd_dvd @ nat) @ A3) @ (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_1169_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) = (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1170_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) = (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1171_add__Suc__right, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ M) @ (suc @ N)) = (suc @ (((plus_plus @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1172_add__is__0, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 0.33/5.20 thf(fact_1173_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 0.33/5.20 thf(fact_1174_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1175_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1176_diff__diff__left, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)))))).
% 0.33/5.20 thf(fact_1177_add__le__cancel__left, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1178_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1179_double__eq__0__iff, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: ((((((plus_plus @ A) @ A3) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1180_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_1181_double__zero__sym, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A3) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1182_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [B2:A,A3:A]: ((((((plus_plus @ A) @ B2) @ A3) = A3) = (B2 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1183_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = A3) = (B2 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1184_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ B2) @ A3)) = (B2 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1185_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ A3) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1186_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A,Y3:A]: ((((((plus_plus @ A) @ X) @ Y3) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1187_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A,Y3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X) @ Y3)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1188_add__0, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.33/5.20 thf(fact_1189_add__less__cancel__left, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1190_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1191_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 0.33/5.20 thf(fact_1192_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 0.33/5.20 thf(fact_1193_add__diff__cancel__left, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1194_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ A3) = B2))))))).
% 0.33/5.20 thf(fact_1195_add__diff__cancel__right, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1196_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 0.33/5.20 thf(fact_1197_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)) = B2))))))).
% 0.33/5.20 thf(fact_1198_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ (((plus_plus @ A) @ A3) @ B2)) = B2))))))).
% 0.33/5.20 thf(fact_1199_minus__add__distrib, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1200_dvd__add__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1201_dvd__add__triv__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ A3)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1202_of__nat__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M) @ N)) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.20 thf(fact_1203_mod__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1204_mod__add__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1205_add__gr__0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.20 thf(fact_1206_mult__Suc__right, axiom, (! [M:nat,N:nat]: (((((times_times @ nat) @ M) @ (suc @ N)) = (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1207_Nat_Odiff__diff__right, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))).
% 0.33/5.20 thf(fact_1208_Nat_Oadd__diff__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K)))))).
% 0.33/5.20 thf(fact_1209_Nat_Oadd__diff__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K)))))).
% 0.33/5.20 thf(fact_1210_of__bool__less__eq__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [P:$o,Q:$o]: (((((ord_less_eq @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q)) = (P => Q)))))))).
% 0.33/5.20 thf(fact_1211_of__bool__eq__0__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:$o]: (((((zero_neq_one_of_bool @ A) @ P) = (zero_zero @ A)) = (~ (P))))))))).
% 0.33/5.20 thf(fact_1212_of__bool__eq_I1_J, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (((zero_neq_one_of_bool @ A) @ $false) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_1213_of__bool__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [P:$o,Q:$o]: (((((ord_less @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q)) = ((~ (P)) & Q)))))))).
% 0.33/5.20 thf(fact_1214_of__bool__eq_I2_J, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (((zero_neq_one_of_bool @ A) @ $true) = (one_one @ A)))))).
% 0.33/5.20 thf(fact_1215_of__bool__eq__1__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:$o]: (((((zero_neq_one_of_bool @ A) @ P) = (one_one @ A)) = P))))))).
% 0.33/5.20 thf(fact_1216_gbinomial__1, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A]: (((((gbinomial @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 0.33/5.20 thf(fact_1217_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1218_add__le__same__cancel2, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1219_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 0.33/5.20 thf(fact_1220_le__add__same__cancel2, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1221_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1222_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1223_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1224_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1225_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 0.33/5.20 thf(fact_1226_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ B2) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 0.33/5.20 thf(fact_1227_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1228_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.20 thf(fact_1229_le__add__diff__inverse2, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ B2) = A3)))))))).
% 0.33/5.20 thf(fact_1230_le__add__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A3) @ B2)) = A3)))))))).
% 0.33/5.20 thf(fact_1231_diff__add__zero, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1232_add_Oright__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1233_ab__left__minus, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1234_dvd__add__times__triv__right__iff, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1235_dvd__add__times__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ A3)) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1236_diff__minus__eq__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((plus_plus @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1237_uminus__add__conv__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((minus_minus @ A) @ B2) @ A3)))))))).
% 0.33/5.20 thf(fact_1238_div__add, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_1239_mod__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1240_mod__mult__self2, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1241_mod__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A3:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A3)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1242_mod__mult__self4, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1243_zero__less__of__bool__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((zero_neq_one_of_bool @ A) @ P)) = P))))))).
% 0.33/5.20 thf(fact_1244_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))))))).
% 0.33/5.20 thf(fact_1245_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))))))).
% 0.33/5.20 thf(fact_1246_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))))))))).
% 0.33/5.20 thf(fact_1247_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))))))))).
% 0.33/5.20 thf(fact_1248_gbinomial__0_I2_J, axiom, (! [B:$tType]: ((((semiring_char_0 @ B) & (semidom_divide @ B)) => (! [K:nat]: (((((gbinomial @ B) @ (zero_zero @ B)) @ (suc @ K)) = (zero_zero @ B)))))))).
% 0.33/5.20 thf(fact_1249_zle__add1__eq__le, axiom, (! [W2:int,Z3:int]: (((((ord_less @ int) @ W2) @ (((plus_plus @ int) @ Z3) @ (one_one @ int))) = (((ord_less_eq @ int) @ W2) @ Z3))))).
% 0.33/5.20 thf(fact_1250_gbinomial__0_I1_J, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A]: (((((gbinomial @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 0.33/5.20 thf(fact_1251_Suc__mod__mult__self1, axiom, (! [M:nat,K:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ K) @ N)))) @ N) = (((modulo_modulo @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_1252_Suc__mod__mult__self2, axiom, (! [M:nat,N:nat,K:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ N) @ K)))) @ N) = (((modulo_modulo @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_1253_Suc__mod__mult__self3, axiom, (! [K:nat,N:nat,M:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ N)) @ M))) @ N) = (((modulo_modulo @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_1254_Suc__mod__mult__self4, axiom, (! [N:nat,K:nat,M:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ K)) @ M))) @ N) = (((modulo_modulo @ nat) @ (suc @ M)) @ N))))).
% 0.33/5.20 thf(fact_1255_gbinomial__Suc0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A]: (((((gbinomial @ A) @ A3) @ (suc @ (zero_zero @ nat))) = A3))))))).
% 0.33/5.20 thf(fact_1256_of__nat__of__bool, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [P:$o]: ((((semiring_1_of_nat @ A) @ ((zero_neq_one_of_bool @ nat) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))).
% 0.33/5.20 thf(fact_1257_add__neg__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_1258_add__neg__numeral__special_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 0.33/5.20 thf(fact_1259_div__mult__self4, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1260_div__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A3)) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1261_div__mult__self2, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1262_div__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1263_of__nat__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat]: ((((semiring_1_of_nat @ A) @ (suc @ M)) = (((plus_plus @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ M))))))))).
% 0.33/5.20 thf(fact_1264_sgn__mult__self__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ A3)) @ ((sgn_sgn @ A) @ A3)) = ((zero_neq_one_of_bool @ A) @ (A3 != (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1265_ceiling__add__one, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1266_Suc__0__mod__eq, axiom, (! [N:nat]: (((((modulo_modulo @ nat) @ (suc @ (zero_zero @ nat))) @ N) = ((zero_neq_one_of_bool @ nat) @ (N != (suc @ (zero_zero @ nat)))))))).
% 0.33/5.20 thf(fact_1267_sgn__of__nat, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat]: ((((sgn_sgn @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.20 thf(fact_1268_of__nat__gbinomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat,K:nat]: ((((semiring_1_of_nat @ A) @ (((gbinomial @ nat) @ N) @ K)) = (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N)) @ K)))))))).
% 0.33/5.20 thf(fact_1269_int__plus, axiom, (! [N:nat,M:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ N) @ M)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((semiring_1_of_nat @ int) @ M)))))).
% 0.33/5.20 thf(fact_1270_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)))))).
% 0.33/5.20 thf(fact_1271_zadd__int__left, axiom, (! [M:nat,N:nat,Z3:int]: (((((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ M)) @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ Z3)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ M) @ N))) @ Z3))))).
% 0.33/5.20 thf(fact_1272_Euclid__induct, axiom, (! [P:(nat > (nat > $o)),A3:nat,B2:nat]: (((! [A8:nat,B5:nat]: ((((P @ A8) @ B5) = ((P @ B5) @ A8)))) => ((! [A8:nat]: (((P @ A8) @ (zero_zero @ nat)))) => ((! [A8:nat,B5:nat]: ((((P @ A8) @ B5) => ((P @ A8) @ (((plus_plus @ nat) @ A8) @ B5))))) => ((P @ A3) @ B2))))))).
% 0.33/5.20 thf(fact_1273_of__bool__eq__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P5:$o,Q4:$o]: (((((zero_neq_one_of_bool @ A) @ P5) = ((zero_neq_one_of_bool @ A) @ Q4)) = (P5 = Q4)))))))).
% 0.33/5.20 thf(fact_1274_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1275_is__num__normalize_I1_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1276_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 0.33/5.20 thf(fact_1277_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A6:A,K:A,A3:A,B2:A]: (((A6 = (((plus_plus @ A) @ K) @ A3)) => ((((plus_plus @ A) @ A6) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1278_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B7:A,K:A,B2:A,A3:A]: (((B7 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A3) @ B7) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1279_add_Oassoc, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1280_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) = (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1281_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) = (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1282_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B3:A]: ((((plus_plus @ A) @ B3) @ A5)))))))).
% 0.33/5.20 thf(fact_1283_add_Oleft__commute, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1284_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) => (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1285_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) => (B2 = C2)))))))).
% 0.33/5.20 thf(fact_1286_gbinomial__Suc__Suc, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ A3) @ K)) @ (((gbinomial @ A) @ A3) @ (suc @ K)))))))))).
% 0.33/5.20 thf(fact_1287_nat__int__add, axiom, (! [A3:nat,B2:nat]: (((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2))) = (((plus_plus @ nat) @ A3) @ B2))))).
% 0.33/5.20 thf(fact_1288_ceiling__add__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ Y3))) @ (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((archimedean_ceiling @ A) @ Y3))))))))).
% 0.33/5.20 thf(fact_1289_of__bool__conj, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [P:$o,Q:$o]: ((((zero_neq_one_of_bool @ A) @ (P & Q)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q))))))))).
% 0.33/5.20 thf(fact_1290_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1291_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 0.33/5.20 thf(fact_1292_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1293_add__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_1294_add__left__mono, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1295_less__eqE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (~ ((! [C3:A]: ((B2 != (((plus_plus @ A) @ A3) @ C3))))))))))))).
% 0.33/5.20 thf(fact_1296_add__right__mono, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1297_le__iff__add, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: ((? [C4:A]: ((B3 = (((plus_plus @ A) @ A5) @ C4))))))))))).
% 0.33/5.20 thf(fact_1298_add__le__imp__le__left, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1299_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1300_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.33/5.20 thf(fact_1301_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_1302_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.33/5.20 thf(fact_1303_verit__sum__simplify, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.20 thf(fact_1304_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1305_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 0.33/5.20 thf(fact_1306_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1307_add__strict__mono, axiom, (! [A:$tType]: (((strict9044650504122735259up_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_1308_add__strict__left__mono, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1309_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1310_add__less__imp__less__left, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1311_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.20 thf(fact_1312_combine__common__factor, axiom, (! [A:$tType]: (((semiring @ A) => (! [A3:A,E:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ E)) @ C2)))))))).
% 0.33/5.20 thf(fact_1313_distrib__right, axiom, (! [A:$tType]: (((semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1314_distrib__left, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1315_comm__semiring__class_Odistrib, axiom, (! [A:$tType]: (((comm_semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1316_ring__class_Oring__distribs_I1_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1317_ring__class_Oring__distribs_I2_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1318_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A6:A,K:A,A3:A,B2:A]: (((A6 = (((plus_plus @ A) @ K) @ A3)) => ((((minus_minus @ A) @ A6) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1319_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A3) @ B2) = C2) = (A3 = (((plus_plus @ A) @ C2) @ B2))))))))).
% 0.33/5.20 thf(fact_1320_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,C2:A,B2:A]: (((A3 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A3) @ B2) = C2)))))))).
% 0.33/5.20 thf(fact_1321_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_1322_diff__diff__eq2, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1323_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2)))))))).
% 0.33/5.20 thf(fact_1324_diff__add__eq__diff__diff__swap, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1325_add__implies__diff, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [C2:A,B2:A,A3:A]: ((((((plus_plus @ A) @ C2) @ B2) = A3) => (C2 = (((minus_minus @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_1326_diff__diff__eq, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1327_is__num__normalize_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_1328_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A6:A,K:A,A3:A]: (((A6 = (((plus_plus @ A) @ K) @ A3)) => (((uminus_uminus @ A) @ A6) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A3)))))))))).
% 0.33/5.20 thf(fact_1329_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_1330_dvd__add, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1331_dvd__add__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.20 thf(fact_1332_dvd__add__right__iff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1333_add__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 0.33/5.20 thf(fact_1334_nat__arith_Osuc1, axiom, (! [A6:nat,K:nat,A3:nat]: (((A6 = (((plus_plus @ nat) @ K) @ A3)) => ((suc @ A6) = (((plus_plus @ nat) @ K) @ (suc @ A3))))))).
% 0.33/5.20 thf(fact_1335_add__Suc, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N) = (suc @ (((plus_plus @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1336_add__Suc__shift, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N) = (((plus_plus @ nat) @ M) @ (suc @ N)))))).
% 0.33/5.20 thf(fact_1337_plus__nat_Oadd__0, axiom, (! [N:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 0.33/5.20 thf(fact_1338_add__eq__self__zero, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = M) => (N = (zero_zero @ nat)))))).
% 0.33/5.20 thf(fact_1339_mod__add__eq, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1340_mod__add__cong, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,A4:A,B2:A,B4:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ A4) @ C2)) => (((((modulo_modulo @ A) @ B2) @ C2) = (((modulo_modulo @ A) @ B4) @ C2)) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A4) @ B4)) @ C2)))))))))).
% 0.33/5.20 thf(fact_1341_mod__add__left__eq, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1342_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)))))))).
% 0.33/5.20 thf(fact_1343_plus__int__code_I2_J, axiom, (! [L:int]: (((((plus_plus @ int) @ (zero_zero @ int)) @ L) = L)))).
% 0.33/5.20 thf(fact_1344_plus__int__code_I1_J, axiom, (! [K:int]: (((((plus_plus @ int) @ K) @ (zero_zero @ int)) = K)))).
% 0.33/5.20 thf(fact_1345_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))).
% 0.33/5.20 thf(fact_1346_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))))))).
% 0.33/5.20 thf(fact_1347_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))).
% 0.33/5.20 thf(fact_1348_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))).
% 0.33/5.20 thf(fact_1349_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)))))).
% 0.33/5.20 thf(fact_1350_trans__less__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 0.33/5.20 thf(fact_1351_trans__less__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 0.33/5.20 thf(fact_1352_less__add__eq__less, axiom, (! [K:nat,L:nat,M:nat,N:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1353_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N3:nat]: ((? [K3:nat]: ((N3 = (((plus_plus @ nat) @ M2) @ K3)))))))).
% 0.33/5.20 thf(fact_1354_trans__le__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 0.33/5.20 thf(fact_1355_trans__le__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 0.33/5.20 thf(fact_1356_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)))))).
% 0.33/5.20 thf(fact_1357_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))))))).
% 0.33/5.20 thf(fact_1358_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N4:nat]: ((L = (((plus_plus @ nat) @ K) @ N4)))))))).
% 0.33/5.20 thf(fact_1359_add__leD2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 0.33/5.20 thf(fact_1360_add__leD1, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1361_le__add2, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1362_le__add1, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M))))).
% 0.33/5.20 thf(fact_1363_add__leE, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 0.33/5.20 thf(fact_1364_Nat_Odiff__cancel, axiom, (! [K:nat,M:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1365_diff__cancel2, axiom, (! [M:nat,K:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))).
% 0.33/5.20 thf(fact_1366_diff__add__inverse, axiom, (! [N:nat,M:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ M)) @ N) = M)))).
% 0.33/5.20 thf(fact_1367_diff__add__inverse2, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ N) = M)))).
% 0.33/5.20 thf(fact_1368_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))))).
% 0.33/5.20 thf(fact_1369_add__mult__distrib2, axiom, (! [K:nat,M:nat,N:nat]: (((((times_times @ nat) @ K) @ (((plus_plus @ nat) @ M) @ N)) = (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)))))).
% 0.33/5.20 thf(fact_1370_add__mult__distrib, axiom, (! [M:nat,N:nat,K:nat]: (((((times_times @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ K) = (((plus_plus @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N) @ K)))))).
% 0.33/5.20 thf(fact_1371_int__distrib_I2_J, axiom, (! [W2:int,Z1:int,Z22:int]: (((((times_times @ int) @ W2) @ (((plus_plus @ int) @ Z1) @ Z22)) = (((plus_plus @ int) @ (((times_times @ int) @ W2) @ Z1)) @ (((times_times @ int) @ W2) @ Z22)))))).
% 0.33/5.20 thf(fact_1372_int__distrib_I1_J, axiom, (! [Z1:int,Z22:int,W2:int]: (((((times_times @ int) @ (((plus_plus @ int) @ Z1) @ Z22)) @ W2) = (((plus_plus @ int) @ (((times_times @ int) @ Z1) @ W2)) @ (((times_times @ int) @ Z22) @ W2)))))).
% 0.33/5.20 thf(fact_1373_div__mod__decomp__int, axiom, (! [A6:int,N:int]: ((A6 = (((plus_plus @ int) @ (((times_times @ int) @ (((divide_divide @ int) @ A6) @ N)) @ N)) @ (((modulo_modulo @ int) @ A6) @ N)))))).
% 0.33/5.20 thf(fact_1374_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))))))))).
% 0.33/5.20 thf(fact_1375_gbinomial__addition__formula, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ A3) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ (suc @ K))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 0.33/5.20 thf(fact_1376_bezout__lemma__nat, axiom, (! [D2:nat,A3:nat,B2:nat,X:nat,Y3:nat]: (((((dvd_dvd @ nat) @ D2) @ A3) => ((((dvd_dvd @ nat) @ D2) @ B2) => ((((((times_times @ nat) @ A3) @ X) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y3)) @ D2)) | ((((times_times @ nat) @ B2) @ X) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y3)) @ D2))) => (? [X3:nat,Y5:nat]: (((((dvd_dvd @ nat) @ D2) @ A3) & ((((dvd_dvd @ nat) @ D2) @ (((plus_plus @ nat) @ A3) @ B2)) & (((((times_times @ nat) @ A3) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ Y5)) @ D2)) | ((((times_times @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y5)) @ D2))))))))))))).
% 0.33/5.20 thf(fact_1377_bezout__add__nat, axiom, (! [A3:nat,B2:nat]: ((? [D6:nat,X3:nat,Y5:nat]: (((((dvd_dvd @ nat) @ D6) @ A3) & ((((dvd_dvd @ nat) @ D6) @ B2) & (((((times_times @ nat) @ A3) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y5)) @ D6)) | ((((times_times @ nat) @ B2) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y5)) @ D6)))))))))).
% 0.33/5.20 thf(fact_1378_gbinomial__mult__1_H, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_1379_gbinomial__mult__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ A3) @ (((gbinomial @ A) @ A3) @ K)) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ K)) @ (((gbinomial @ A) @ A3) @ K))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A3) @ (suc @ K))))))))))).
% 0.33/5.20 thf(fact_1380_nat__add__distrib, axiom, (! [Z3:int,Z6:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z6) => ((nat2 @ (((plus_plus @ int) @ Z3) @ Z6)) = (((plus_plus @ nat) @ (nat2 @ Z3)) @ (nat2 @ Z6)))))))).
% 0.33/5.20 thf(fact_1381_Suc__times__gbinomial, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1382_zero__less__eq__of__bool, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((zero_neq_one_of_bool @ A) @ P)))))))).
% 0.33/5.20 thf(fact_1383_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)))))))).
% 0.33/5.20 thf(fact_1384_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))))))))).
% 0.33/5.20 thf(fact_1385_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))))))))))).
% 0.33/5.20 thf(fact_1386_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))))))))))))))).
% 0.33/5.20 thf(fact_1387_add__decreasing, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_1388_add__increasing, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1389_add__decreasing2, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2))))))))).
% 0.33/5.20 thf(fact_1390_add__increasing2, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1391_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1392_add__nonpos__nonpos, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1393_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((((plus_plus @ A) @ X) @ Y3) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))))).
% 0.33/5.20 thf(fact_1394_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y3) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X) @ Y3) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))))).
% 0.33/5.20 thf(fact_1395_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 0.33/5.20 thf(fact_1396_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1397_add__le__less__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_1398_add__less__le__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))).
% 0.33/5.20 thf(fact_1399_add__neg__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1400_add__pos__pos, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1401_canonically__ordered__monoid__add__class_OlessE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((! [C3:A]: (((B2 = (((plus_plus @ A) @ A3) @ C3)) => (C3 = (zero_zero @ A)))))))))))))).
% 0.33/5.20 thf(fact_1402_pos__add__strict, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1403_add__less__zeroD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (zero_zero @ A)) => ((((ord_less @ A) @ X) @ (zero_zero @ A)) | (((ord_less @ A) @ Y3) @ (zero_zero @ A)))))))))).
% 0.33/5.20 thf(fact_1404_add__le__imp__le__diff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1405_add__le__add__imp__diff__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N:A,J:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ N) @ K)) @ J))))))))))).
% 0.33/5.20 thf(fact_1406_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_1407_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ A3) @ (((minus_minus @ A) @ B2) @ A3)) = B2)))))))).
% 0.33/5.20 thf(fact_1408_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1409_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A3) = (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ C2))))))))).
% 0.33/5.20 thf(fact_1410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1411_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A3) = (((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)))))))))).
% 0.33/5.20 thf(fact_1412_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1413_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ B2))))))))).
% 0.33/5.20 thf(fact_1414_le__add__diff, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1415_diff__add, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ A3) = B2)))))))).
% 0.33/5.20 thf(fact_1416_le__diff__eq, axiom, (! [A:$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)))))))).
% 0.33/5.20 thf(fact_1417_diff__le__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((ord_less_eq @ A) @ A3) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 0.33/5.20 thf(fact_1418_less__add__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A]: ((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_1419_add__mono1, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (((plus_plus @ A) @ B2) @ (one_one @ A)))))))))).
% 0.33/5.20 thf(fact_1420_diff__less__eq, axiom, (! [A:$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))))))))).
% 0.33/5.20 thf(fact_1421_less__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ A3) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.20 thf(fact_1422_linordered__semidom__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,B2:A]: (((~ ((((ord_less @ A) @ A3) @ B2))) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A3) @ B2)) = A3)))))))).
% 0.33/5.20 thf(fact_1423_add__eq__0__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A)) = (B2 = ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.20 thf(fact_1424_ab__group__add__class_Oab__left__minus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.20 thf(fact_1425_add_Oinverse__unique, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A)) => (((uminus_uminus @ A) @ A3) = B2)))))))).
% 0.33/5.20 thf(fact_1426_eq__neg__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((A3 = ((uminus_uminus @ A) @ B2)) = ((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1427_neg__eq__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = B2) = ((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 0.33/5.20 thf(fact_1428_square__diff__square__factored, axiom, (! [A:$tType]: (((comm_ring @ A) => (! [X:A,Y3:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3)) = (((times_times @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (((minus_minus @ A) @ X) @ Y3))))))))).
% 0.33/5.20 thf(fact_1429_eq__add__iff2, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = (C2 = (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E)) @ D2))))))))).
% 0.33/5.20 thf(fact_1430_eq__add__iff1, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = ((((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E)) @ C2) = D2)))))))).
% 0.33/5.20 thf(fact_1431_pinf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z4:B]: ((! [X4:B]: (((((ord_less @ B) @ Z4) @ X4) => ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)) = (((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))))))))))))).
% 0.33/5.20 thf(fact_1432_pinf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z4:B]: ((! [X4:B]: (((((ord_less @ B) @ Z4) @ X4) => ((~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))) = (~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))))))))))))))).
% 0.33/5.20 thf(fact_1433_minf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z4:B]: ((! [X4:B]: (((((ord_less @ B) @ X4) @ Z4) => ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)) = (((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))))))))))))).
% 0.33/5.20 thf(fact_1434_minf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z4:B]: ((! [X4:B]: (((((ord_less @ B) @ X4) @ Z4) => ((~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))) = (~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X4) @ S)))))))))))))))).
% 0.33/5.20 thf(fact_1435_group__cancel_Osub2, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B7:A,K:A,B2:A,A3:A]: (((B7 = (((plus_plus @ A) @ K) @ B2)) => ((((minus_minus @ A) @ A3) @ B7) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ (((minus_minus @ A) @ A3) @ B2)))))))))).
% 0.33/5.20 thf(fact_1436_diff__conv__add__uminus, axiom, (! [A:$tType]: (((group_add @ A) => ((minus_minus @ A) = (^ [A5:A,B3:A]: ((((plus_plus @ A) @ A5) @ ((uminus_uminus @ A) @ B3))))))))).
% 0.33/5.20 thf(fact_1437_ab__group__add__class_Oab__diff__conv__add__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => ((minus_minus @ A) = (^ [A5:A,B3:A]: ((((plus_plus @ A) @ A5) @ ((uminus_uminus @ A) @ B3))))))))).
% 0.33/5.20 thf(fact_1438_div__plus__div__distrib__dvd__left, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1439_div__plus__div__distrib__dvd__right, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 0.33/5.20 thf(fact_1440_mod__eqE, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ B2) @ C2)) => (~ ((! [D6:A]: ((B2 != (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ D6)))))))))))))).
% 0.33/5.20 thf(fact_1441_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))))))))).
% 0.33/5.20 thf(fact_1442_add__is__1, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (suc @ (zero_zero @ nat))) = (((M = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 0.33/5.20 thf(fact_1443_one__is__add, axiom, (! [M:nat,N:nat]: ((((suc @ (zero_zero @ nat)) = (((plus_plus @ nat) @ M) @ N)) = (((M = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 0.33/5.20 thf(fact_1444_less__natE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (~ ((! [Q5:nat]: ((N != (suc @ (((plus_plus @ nat) @ M) @ Q5))))))))))).
% 0.33/5.20 thf(fact_1445_less__add__Suc1, axiom, (! [I:nat,M:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ I) @ M)))))).
% 0.33/5.20 thf(fact_1446_less__add__Suc2, axiom, (! [I:nat,M:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ M) @ I)))))).
% 0.33/5.20 thf(fact_1447_less__iff__Suc__add, axiom, ((ord_less @ nat) = (^ [M2:nat,N3:nat]: ((? [K3:nat]: ((N3 = (suc @ (((plus_plus @ nat) @ M2) @ K3))))))))).
% 0.33/5.20 thf(fact_1448_less__imp__Suc__add, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (? [K2:nat]: ((N = (suc @ (((plus_plus @ nat) @ M) @ K2))))))))).
% 0.33/5.20 thf(fact_1449_less__imp__add__positive, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (? [K2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K2) & ((((plus_plus @ nat) @ I) @ K2) = J)))))))).
% 0.33/5.20 thf(fact_1450_mono__nat__linear__lb, axiom, (! [F:(nat > nat),M:nat,K:nat]: (((! [M6:nat,N4:nat]: (((((ord_less @ nat) @ M6) @ N4) => (((ord_less @ nat) @ (F @ M6)) @ (F @ N4))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F @ M)) @ K)) @ (F @ (((plus_plus @ nat) @ M) @ K))))))).
% 0.33/5.20 thf(fact_1451_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),N:nat]: (((P @ K) => ((! [X3:A]: (((P @ X3) => (? [Y4:A]: (((P @ Y4) & (~ ((((ord_less_eq @ nat) @ (F @ Y4)) @ (F @ X3)))))))))) => (? [Y5:A]: (((P @ Y5) & (~ ((((ord_less @ nat) @ (F @ Y5)) @ (((plus_plus @ nat) @ (F @ K)) @ N)))))))))))).
% 0.33/5.20 thf(fact_1452_bezout__add__strong__nat, axiom, (! [A3:nat,B2:nat]: (((A3 != (zero_zero @ nat)) => (? [D6:nat,X3:nat,Y5:nat]: (((((dvd_dvd @ nat) @ D6) @ A3) & ((((dvd_dvd @ nat) @ D6) @ B2) & ((((times_times @ nat) @ A3) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y5)) @ D6)))))))))).
% 0.33/5.20 thf(fact_1453_diff__add__0, axiom, (! [N:nat,M:nat]: (((((minus_minus @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M)) = (zero_zero @ nat))))).
% 0.33/5.20 thf(fact_1454_odd__nonzero, axiom, (! [Z3:int]: (((((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z3)) @ Z3) != (zero_zero @ int))))).
% 0.33/5.20 thf(fact_1455_mult__Suc, axiom, (! [M:nat,N:nat]: (((((times_times @ nat) @ (suc @ M)) @ N) = (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ M) @ N)))))).
% 0.33/5.20 thf(fact_1456_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))))).
% 0.33/5.20 thf(fact_1457_add__diff__inverse__nat, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) => ((((plus_plus @ nat) @ N) @ (((minus_minus @ nat) @ M) @ N)) = M))))).
% 0.33/5.20 thf(fact_1458_Suc__eq__plus1__left, axiom, (suc = ((plus_plus @ nat) @ (one_one @ nat)))).
% 0.33/5.20 thf(fact_1459_plus__1__eq__Suc, axiom, (((plus_plus @ nat) @ (one_one @ nat)) = suc)).
% 0.33/5.20 thf(fact_1460_Suc__eq__plus1, axiom, (suc = (^ [N3:nat]: ((((plus_plus @ nat) @ N3) @ (one_one @ nat)))))).
% 0.33/5.20 thf(fact_1461_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))))))).
% 0.33/5.20 thf(fact_1462_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)))))).
% 0.33/5.20 thf(fact_1463_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))))))).
% 0.33/5.20 thf(fact_1464_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)))))).
% 0.33/5.20 thf(fact_1465_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)))))).
% 0.33/5.20 thf(fact_1466_int__ge__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less_eq @ int) @ K) @ I) => ((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_1467_zless__add1__eq, axiom, (! [W2:int,Z3:int]: (((((ord_less @ int) @ W2) @ (((plus_plus @ int) @ Z3) @ (one_one @ int))) = ((((ord_less @ int) @ W2) @ Z3) | (W2 = Z3)))))).
% 0.33/5.20 thf(fact_1468_int__gr__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less @ int) @ K) @ I) => ((P @ (((plus_plus @ int) @ K) @ (one_one @ int))) => ((! [I4:int]: (((((ord_less @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))).
% 0.33/5.20 thf(fact_1469_zle__iff__zadd, axiom, ((ord_less_eq @ int) = (^ [W3:int,Z7:int]: ((? [N3:nat]: ((Z7 = (((plus_plus @ int) @ W3) @ ((semiring_1_of_nat @ int) @ N3))))))))).
% 0.33/5.20 thf(fact_1470_zdvd__period, axiom, (! [A3:int,D2:int,X:int,T2:int,C2:int]: (((((dvd_dvd @ int) @ A3) @ D2) => ((((dvd_dvd @ int) @ A3) @ (((plus_plus @ int) @ X) @ T2)) = (((dvd_dvd @ int) @ A3) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X) @ (((times_times @ int) @ C2) @ D2))) @ T2))))))).
% 0.33/5.20 thf(fact_1471_zdvd__reduce, axiom, (! [K:int,N:int,M:int]: (((((dvd_dvd @ int) @ K) @ (((plus_plus @ int) @ N) @ (((times_times @ int) @ K) @ M))) = (((dvd_dvd @ int) @ K) @ N))))).
% 0.33/5.20 thf(fact_1472_nat__mod__eq__iff, axiom, (! [X:nat,N:nat,Y3:nat]: ((((((modulo_modulo @ nat) @ X) @ N) = (((modulo_modulo @ nat) @ Y3) @ N)) = (? [Q1:nat,Q22:nat]: (((((plus_plus @ nat) @ X) @ (((times_times @ nat) @ N) @ Q1)) = (((plus_plus @ nat) @ Y3) @ (((times_times @ nat) @ N) @ Q22))))))))).
% 0.33/5.20 thf(fact_1473_frac__1__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A))) = ((archimedean_frac @ A) @ X)))))))).
% 0.33/5.20 thf(fact_1474_gbinomial__factors, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((semiring_1_of_nat @ A) @ (suc @ K)))) @ (((gbinomial @ A) @ A3) @ K))))))))).
% 0.33/5.20 thf(fact_1475_gbinomial__rec, axiom, (! [A:$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))))))))))).
% 0.33/5.20 thf(fact_1476_dbl__inc__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_inc @ A) = (^ [X2:A]: ((((plus_plus @ A) @ (((plus_plus @ A) @ X2) @ X2)) @ (one_one @ A))))))))).
% 0.33/5.20 thf(fact_1477_gbinomial__reduce__nat, axiom, (! [A:$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)))))))))).
% 0.33/5.20 thf(fact_1478_field__le__epsilon, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((! [E2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E2) => (((ord_less_eq @ A) @ X) @ (((plus_plus @ A) @ Y3) @ E2))))) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_1479_add__strict__increasing2, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1480_add__strict__increasing, axiom, (! [A:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 0.33/5.21 thf(fact_1481_add__pos__nonneg, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1482_add__nonpos__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1483_add__nonneg__pos, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1484_add__neg__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1485_sum__squares__ge__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3))))))))).
% 0.33/5.21 thf(fact_1486_discrete, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((ord_less @ A) = (^ [A5:A]: (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_1487_not__sum__squares__lt__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y3:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3))) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1488_zero__less__two, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_1489_le__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = (((ord_less_eq @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E)) @ D2))))))))).
% 0.33/5.21 thf(fact_1490_le__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E)) @ C2)) @ D2)))))))).
% 0.33/5.21 thf(fact_1491_less__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = (((ord_less @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E)) @ D2))))))))).
% 0.33/5.21 thf(fact_1492_less__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E)) @ D2)) = (((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E)) @ C2)) @ D2)))))))).
% 0.33/5.21 thf(fact_1493_unity__coeff__ex, axiom, (! [A:$tType]: ((((dvd @ A) & (semiring_0 @ A)) => (! [P:(A > $o),L:A]: (((? [X2:A]: ((P @ (((times_times @ A) @ L) @ X2)))) = (? [X2:A]: (((((dvd_dvd @ A) @ L) @ (((plus_plus @ A) @ X2) @ (zero_zero @ A))) & (P @ X2))))))))))).
% 0.33/5.21 thf(fact_1494_add__divide__eq__if__simps_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ Z3)) @ B2) = B2)) & ((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ Z3)) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ B2) @ Z3))) @ Z3)))))))))).
% 0.33/5.21 thf(fact_1495_add__divide__eq__if__simps_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((plus_plus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z3)) = A3)) & ((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z3)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ Z3)) @ B2)) @ Z3)))))))))).
% 0.33/5.21 thf(fact_1496_add__frac__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y3:A,Z3:A,X:A,W2:A]: (((Y3 != (zero_zero @ A)) => ((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (((divide_divide @ A) @ W2) @ Z3)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ Z3)) @ (((times_times @ A) @ W2) @ Y3))) @ (((times_times @ A) @ Y3) @ Z3))))))))))).
% 0.33/5.21 thf(fact_1497_add__frac__num, axiom, (! [A:$tType]: (((field @ A) => (! [Y3:A,X:A,Z3:A]: (((Y3 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ Z3) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Z3) @ Y3))) @ Y3))))))))).
% 0.33/5.21 thf(fact_1498_add__num__frac, axiom, (! [A:$tType]: (((field @ A) => (! [Y3:A,Z3:A,X:A]: (((Y3 != (zero_zero @ A)) => ((((plus_plus @ A) @ Z3) @ (((divide_divide @ A) @ X) @ Y3)) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Z3) @ Y3))) @ Y3))))))))).
% 0.33/5.21 thf(fact_1499_add__divide__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ X) @ (((divide_divide @ A) @ Y3) @ Z3)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ Z3)) @ Y3)) @ Z3))))))))).
% 0.33/5.21 thf(fact_1500_divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Z3)) @ Y3) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Y3) @ Z3))) @ Z3))))))))).
% 0.33/5.21 thf(fact_1501_div__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_1502_div__add__self2, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1503_square__diff__one__factored, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X) @ X)) @ (one_one @ A)) = (((times_times @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A))) @ (((minus_minus @ A) @ X) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_1504_gt__half__sum, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_1505_less__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_1506_inf__period_I4_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D2:A,D4:A,T2:A]: (((((dvd_dvd @ A) @ D2) @ D4) => (! [X4:A,K4:A]: (((~ ((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ X4) @ T2)))) = (~ ((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))) @ T2)))))))))))))).
% 0.33/5.21 thf(fact_1507_inf__period_I3_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D2:A,D4:A,T2:A]: (((((dvd_dvd @ A) @ D2) @ D4) => (! [X4:A,K4:A]: (((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ X4) @ T2)) = (((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K4) @ D4))) @ T2)))))))))))).
% 0.33/5.21 thf(fact_1508_gbinomial__of__nat__symmetric, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => ((((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N)) @ K) = (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((minus_minus @ nat) @ N) @ K)))))))))).
% 0.33/5.21 thf(fact_1509_div__mult1__eq, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2))) @ (((divide_divide @ A) @ (((times_times @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2))))))))).
% 0.33/5.21 thf(fact_1510_mult__div__mod__eq, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_1511_mod__mult__div__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))) = A3))))))).
% 0.33/5.21 thf(fact_1512_mod__div__mult__eq, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_1513_div__mult__mod__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A3) @ B2)) = A3))))))).
% 0.33/5.21 thf(fact_1514_mod__div__decomp, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1515_cancel__div__mod__rules_I1_J, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A3) @ B2))) @ C2) = (((plus_plus @ A) @ A3) @ C2)))))))).
% 0.33/5.21 thf(fact_1516_cancel__div__mod__rules_I2_J, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_1517_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))))))))))))).
% 0.33/5.21 thf(fact_1518_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))))))))).
% 0.33/5.21 thf(fact_1519_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))))))).
% 0.33/5.21 thf(fact_1520_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))))))).
% 0.33/5.21 thf(fact_1521_odd__less__0__iff, axiom, (! [Z3:int]: (((((ord_less @ int) @ (((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z3)) @ Z3)) @ (zero_zero @ int)) = (((ord_less @ int) @ Z3) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_1522_zless__iff__Suc__zadd, axiom, ((ord_less @ int) = (^ [W3:int,Z7:int]: ((? [N3:nat]: ((Z7 = (((plus_plus @ int) @ W3) @ ((semiring_1_of_nat @ int) @ (suc @ N3)))))))))).
% 0.33/5.21 thf(fact_1523_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)))))).
% 0.33/5.21 thf(fact_1524_int__Suc, axiom, (! [N:nat]: ((((semiring_1_of_nat @ int) @ (suc @ N)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_1525_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))))))).
% 0.33/5.21 thf(fact_1526_zless__imp__add1__zle, axiom, (! [W2:int,Z3:int]: (((((ord_less @ int) @ W2) @ Z3) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ W2) @ (one_one @ int))) @ Z3))))).
% 0.33/5.21 thf(fact_1527_add1__zle__eq, axiom, (! [W2:int,Z3:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ W2) @ (one_one @ int))) @ Z3) = (((ord_less @ int) @ W2) @ Z3))))).
% 0.33/5.21 thf(fact_1528_nat__eq__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = ((((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M) = N)))))).
% 0.33/5.21 thf(fact_1529_nat__eq__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (M = (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 0.33/5.21 thf(fact_1530_nat__le__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((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)) @ M)) @ N)))))).
% 0.33/5.21 thf(fact_1531_nat__le__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 0.33/5.21 thf(fact_1532_nat__diff__add__eq1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M)) @ N)))))).
% 0.33/5.21 thf(fact_1533_nat__diff__add__eq2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((minus_minus @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 0.33/5.21 thf(fact_1534_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))))))))).
% 0.33/5.21 thf(fact_1535_int__induct, axiom, (! [P:(int > $o),K:int,I:int]: (((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => ((! [I4:int]: (((((ord_less_eq @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))).
% 0.33/5.21 thf(fact_1536_mod__eq__nat1E, axiom, (! [M:nat,Q4:nat,N:nat]: ((((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N) @ Q4)) => ((((ord_less_eq @ nat) @ N) @ M) => (~ ((! [S2:nat]: ((M != (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ Q4) @ S2)))))))))))).
% 0.33/5.21 thf(fact_1537_mod__eq__nat2E, axiom, (! [M:nat,Q4:nat,N:nat]: ((((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N) @ Q4)) => ((((ord_less_eq @ nat) @ M) @ N) => (~ ((! [S2:nat]: ((N != (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ Q4) @ S2)))))))))))).
% 0.33/5.21 thf(fact_1538_nat__mod__eq__lemma, axiom, (! [X:nat,N:nat,Y3:nat]: ((((((modulo_modulo @ nat) @ X) @ N) = (((modulo_modulo @ nat) @ Y3) @ N)) => ((((ord_less_eq @ nat) @ Y3) @ X) => (? [Q5:nat]: ((X = (((plus_plus @ nat) @ Y3) @ (((times_times @ nat) @ N) @ Q5)))))))))).
% 0.33/5.21 thf(fact_1539_mod__mult2__eq, axiom, (! [M:nat,N:nat,Q4:nat]: (((((modulo_modulo @ nat) @ M) @ (((times_times @ nat) @ N) @ Q4)) = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ (((modulo_modulo @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ Q4))) @ (((modulo_modulo @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_1540_div__mod__decomp, axiom, (! [A6:nat,N:nat]: ((A6 = (((plus_plus @ nat) @ (((times_times @ nat) @ (((divide_divide @ nat) @ A6) @ N)) @ N)) @ (((modulo_modulo @ nat) @ A6) @ N)))))).
% 0.33/5.21 thf(fact_1541_dbl__dec__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_dec @ A) = (^ [X2:A]: ((((minus_minus @ A) @ (((plus_plus @ A) @ X2) @ X2)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_1542_gbinomial__absorb__comp, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (((gbinomial @ A) @ A3) @ K)) = (((times_times @ A) @ A3) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 0.33/5.21 thf(fact_1543_convex__bound__le, axiom, (! [A:$tType]: (((linord6961819062388156250ring_1 @ A) => (! [X:A,A3:A,Y3:A,U:A,V:A]: (((((ord_less_eq @ A) @ X) @ A3) => ((((ord_less_eq @ A) @ Y3) @ 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) @ Y3))) @ A3)))))))))))).
% 0.33/5.21 thf(fact_1544_divide__int__unfold, axiom, (! [L:int,K:int,N:nat,M: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) @ M))) @ (((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) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ M) @ N)))) & ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((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) @ M) @ N)) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ N) @ M))))))))))))))).
% 0.33/5.21 thf(fact_1545_add__divide__eq__if__simps_I3_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,A3:A,B2:A]: ((((Z3 = (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z3))) @ B2) = B2)) & ((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z3))) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ (((times_times @ A) @ B2) @ Z3))) @ Z3)))))))))).
% 0.33/5.21 thf(fact_1546_minus__divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z3:A,X:A,Y3:A]: (((Z3 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ X) @ Z3))) @ Y3) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ X)) @ (((times_times @ A) @ Y3) @ Z3))) @ Z3))))))))).
% 0.33/5.21 thf(fact_1547_mod__mult2__eq_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,M:nat,N:nat]: (((((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (((modulo_modulo @ A) @ (((divide_divide @ A) @ A3) @ ((semiring_1_of_nat @ A) @ M))) @ ((semiring_1_of_nat @ A) @ N)))) @ (((modulo_modulo @ A) @ A3) @ ((semiring_1_of_nat @ A) @ M)))))))))).
% 0.33/5.21 thf(fact_1548_add__eq__if, axiom, ((plus_plus @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N3) @ (suc @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))) @ N3))))))).
% 0.33/5.21 thf(fact_1549_le__imp__0__less, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((ord_less @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (one_one @ int)) @ Z3)))))).
% 0.33/5.21 thf(fact_1550_verit__le__mono__div__int, axiom, (! [A6:int,B7:int,N:int]: (((((ord_less @ int) @ A6) @ B7) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ (((divide_divide @ int) @ A6) @ N)) @ ((((if @ int) @ ((((modulo_modulo @ int) @ B7) @ N) = (zero_zero @ int))) @ (one_one @ int)) @ (zero_zero @ int)))) @ (((divide_divide @ int) @ B7) @ N))))))).
% 0.33/5.21 thf(fact_1551_nat__less__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M)) @ N)))))).
% 0.33/5.21 thf(fact_1552_nat__less__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 0.33/5.21 thf(fact_1553_incr__mult__lemma, axiom, (! [D2:int,P:(int > $o),K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X3:int]: (((P @ X3) => (P @ (((plus_plus @ int) @ X3) @ D2))))) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (! [X4:int]: (((P @ X4) => (P @ (((plus_plus @ int) @ X4) @ (((times_times @ int) @ K) @ D2)))))))))))).
% 0.33/5.21 thf(fact_1554_unique__quotient__lemma__neg, axiom, (! [B2:int,Q6:int,R4:int,Q4:int,R:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q6)) @ R4)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((ord_less @ int) @ B2) @ R4) => (((ord_less_eq @ int) @ Q4) @ Q6)))))))).
% 0.33/5.21 thf(fact_1555_unique__quotient__lemma, axiom, (! [B2:int,Q6:int,R4:int,Q4:int,R:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q6)) @ R4)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R4) => ((((ord_less @ int) @ R4) @ B2) => ((((ord_less @ int) @ R) @ B2) => (((ord_less_eq @ int) @ Q6) @ Q4)))))))).
% 0.33/5.21 thf(fact_1556_zdiv__mono2__neg__lemma, axiom, (! [B2:int,Q4:int,R:int,B4:int,Q6:int,R4:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R) = (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q6)) @ R4)) => ((((ord_less @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q6)) @ 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) @ Q6) @ Q4)))))))))).
% 0.33/5.21 thf(fact_1557_zdiv__mono2__lemma, axiom, (! [B2:int,Q4:int,R:int,B4:int,Q6:int,R4:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R) = (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q6)) @ R4)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q6)) @ 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) @ Q4) @ Q6)))))))))).
% 0.33/5.21 thf(fact_1558_q__pos__lemma, axiom, (! [B4:int,Q6:int,R4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q6)) @ R4)) => ((((ord_less @ int) @ R4) @ B4) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ Q6))))))).
% 0.33/5.21 thf(fact_1559_int__div__pos__eq, axiom, (! [A3:int,B2:int,Q4:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) => ((((ord_less @ int) @ R) @ B2) => ((((divide_divide @ int) @ A3) @ B2) = Q4))))))).
% 0.33/5.21 thf(fact_1560_int__div__neg__eq, axiom, (! [A3:int,B2:int,Q4:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((divide_divide @ int) @ A3) @ B2) = Q4))))))).
% 0.33/5.21 thf(fact_1561_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) => (! [I5:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => (P @ I5))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I5:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => (P @ I5))))))))))).
% 0.33/5.21 thf(fact_1562_int__mod__pos__eq, axiom, (! [A3:int,B2:int,Q4:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) => ((((ord_less @ int) @ R) @ B2) => ((((modulo_modulo @ int) @ A3) @ B2) = R))))))).
% 0.33/5.21 thf(fact_1563_int__mod__neg__eq, axiom, (! [A3:int,B2:int,Q4:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((modulo_modulo @ int) @ A3) @ B2) = R))))))).
% 0.33/5.21 thf(fact_1564_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) => (! [I5:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => (P @ J3))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I5:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => (P @ J3))))))))))).
% 0.33/5.21 thf(fact_1565_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)) = (! [I5:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => ((P @ I5) @ J3))))))))).
% 0.33/5.21 thf(fact_1566_split__neg__lemma, axiom, (! [K:int,P:(int > (int > $o)),N:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (((P @ (((divide_divide @ int) @ N) @ K)) @ (((modulo_modulo @ int) @ N) @ K)) = (! [I5:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I5)) @ J3)))) => ((P @ I5) @ J3))))))))).
% 0.33/5.21 thf(fact_1567_frac__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3))) @ (one_one @ A)) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ Y3)) = (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3))) @ (one_one @ A)))) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ Y3)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3))) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_1568_mult__eq__if, axiom, ((times_times @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N3) @ (((times_times @ nat) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))) @ N3))))))).
% 0.33/5.21 thf(fact_1569_Suc__as__int, axiom, (suc = (^ [A5:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_1570_split__mod, axiom, (! [P:(nat > $o),M:nat,N:nat]: (((P @ (((modulo_modulo @ nat) @ M) @ N)) = (((N = (zero_zero @ nat)) => (P @ M)) & ((N != (zero_zero @ nat)) => (! [I5:nat,J3:nat]: (((((ord_less @ nat) @ J3) @ N) => ((M = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ I5)) @ J3)) => (P @ J3))))))))))).
% 0.33/5.21 thf(fact_1571_dividend__less__times__div, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ N) @ (((divide_divide @ nat) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_1572_dividend__less__div__times, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ N))))))).
% 0.33/5.21 thf(fact_1573_split__div, axiom, (! [P:(nat > $o),M:nat,N:nat]: (((P @ (((divide_divide @ nat) @ M) @ N)) = (((N = (zero_zero @ nat)) => (P @ (zero_zero @ nat))) & ((N != (zero_zero @ nat)) => (! [I5:nat,J3:nat]: (((((ord_less @ nat) @ J3) @ N) => ((M = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ I5)) @ J3)) => (P @ I5))))))))))).
% 0.33/5.21 thf(fact_1574_dvd__minus__add, axiom, (! [Q4:nat,N:nat,R:nat,M:nat]: (((((ord_less_eq @ nat) @ Q4) @ N) => ((((ord_less_eq @ nat) @ Q4) @ (((times_times @ nat) @ R) @ M)) => ((((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N) @ Q4)) = (((dvd_dvd @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (((minus_minus @ nat) @ (((times_times @ nat) @ R) @ M)) @ Q4))))))))).
% 0.33/5.21 thf(fact_1575_convex__bound__lt, axiom, (! [A:$tType]: (((linord715952674999750819strict @ A) => (! [X:A,A3:A,Y3:A,U:A,V:A]: (((((ord_less @ A) @ X) @ A3) => ((((ord_less @ A) @ Y3) @ 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) @ Y3))) @ A3)))))))))))).
% 0.33/5.21 thf(fact_1576_gbinomial__absorption, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A3) @ (suc @ K))) = (((times_times @ A) @ A3) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 0.33/5.21 thf(fact_1577_scaling__mono, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1578_gbinomial__trinomial__revision, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,M:nat,A3:A]: (((((ord_less_eq @ nat) @ K) @ M) => ((((times_times @ A) @ (((gbinomial @ A) @ A3) @ M)) @ (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ M)) @ K)) = (((times_times @ A) @ (((gbinomial @ A) @ A3) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (((minus_minus @ nat) @ M) @ K))))))))))).
% 0.33/5.21 thf(fact_1579_unique__euclidean__semiring__numeral__class_Omod__mult2__eq, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1580_Suc__nat__eq__nat__zadd1, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((suc @ (nat2 @ Z3)) = (nat2 @ (((plus_plus @ int) @ (one_one @ int)) @ Z3))))))).
% 0.33/5.21 thf(fact_1581_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)))))))).
% 0.33/5.21 thf(fact_1582_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)))))))).
% 0.33/5.21 thf(fact_1583_div__eqI, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q4:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((euclid7384307370059645450egment @ A) @ R) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => (((((plus_plus @ A) @ (((times_times @ A) @ Q4) @ B2)) @ R) = A3) => ((((divide_divide @ A) @ A3) @ B2) = Q4))))))))))).
% 0.33/5.21 thf(fact_1584_div__bounded, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q4: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) @ Q4) @ B2)) @ R)) @ B2) = Q4)))))))))).
% 0.33/5.21 thf(fact_1585_mod__eqI, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q4:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((euclid7384307370059645450egment @ A) @ R) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => (((((plus_plus @ A) @ (((times_times @ A) @ Q4) @ B2)) @ R) = A3) => ((((modulo_modulo @ A) @ A3) @ B2) = R))))))))))).
% 0.33/5.21 thf(fact_1586_verit__le__mono__div, axiom, (! [A6:nat,B7:nat,N:nat]: (((((ord_less @ nat) @ A6) @ B7) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ A6) @ N)) @ ((((if @ nat) @ ((((modulo_modulo @ nat) @ B7) @ N) = (zero_zero @ nat))) @ (one_one @ nat)) @ (zero_zero @ nat)))) @ (((divide_divide @ nat) @ B7) @ N))))))).
% 0.33/5.21 thf(fact_1587_division__decomp, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) => (? [B8:A,C5:A]: (((A3 = (((times_times @ A) @ B8) @ C5)) & ((((dvd_dvd @ A) @ B8) @ B2) & (((dvd_dvd @ A) @ C5) @ C2)))))))))))).
% 0.33/5.21 thf(fact_1588_dvd__productE, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [P5:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ P5) @ (((times_times @ A) @ A3) @ B2)) => (~ ((! [X3:A,Y5:A]: (((P5 = (((times_times @ A) @ X3) @ Y5)) => ((((dvd_dvd @ A) @ X3) @ A3) => (~ ((((dvd_dvd @ A) @ Y5) @ B2)))))))))))))))).
% 0.33/5.21 thf(fact_1589_gcd__nat_Oextremum__uniqueI, axiom, (! [A3:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A3) => (A3 = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_1590_gcd__nat_Onot__eq__extremum, axiom, (! [A3:nat]: (((A3 != (zero_zero @ nat)) = ((((dvd_dvd @ nat) @ A3) @ (zero_zero @ nat)) & (A3 != (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_1591_gcd__nat_Oextremum__unique, axiom, (! [A3:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A3) = (A3 = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_1592_gcd__nat_Oextremum__strict, axiom, (! [A3:nat]: ((~ (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A3) & ((zero_zero @ nat) != A3))))))).
% 0.33/5.21 thf(fact_1593_sum__squares__gt__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3))) = ((X != (zero_zero @ A)) | (Y3 != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1594_sum__squares__le__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3))) @ (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1595_sum__squares__eq__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y3:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y3) @ Y3)) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1596_add__scale__eq__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [R:A,A3:A,B2:A,C2:A,D2:A]: (((R != (zero_zero @ A)) => (((A3 = B2) & (C2 != D2)) => ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ R) @ C2)) != (((plus_plus @ A) @ B2) @ (((times_times @ A) @ R) @ D2))))))))))).
% 0.33/5.21 thf(fact_1597_divide__int__def, axiom, ((divide_divide @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ (L3 = (zero_zero @ int))) @ (zero_zero @ int)) @ ((((if @ int) @ (((sgn_sgn @ int) @ K3) = ((sgn_sgn @ int) @ L3))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L3))))) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L3)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ int) @ L3) @ K3))))))))))))).
% 0.33/5.21 thf(fact_1598_gbinomial__pochhammer_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: ((((divide_divide @ A) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ K3))) @ (one_one @ A))) @ K3)) @ ((semiring_char_0_fact @ A) @ K3))))))))).
% 0.33/5.21 thf(fact_1599_abs__idempotent, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1600_abs__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1601_abs__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_1602_abs__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_1603_abs__eq__0, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((abs_abs @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1604_abs__0__eq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1605_abs__mult__self__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((times_times @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ A3)) = (((times_times @ A) @ A3) @ A3)))))))).
% 0.33/5.21 thf(fact_1606_abs__add__abs, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1607_abs__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_1608_abs__minus, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1609_abs__minus__cancel, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1610_dvd__abs__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:A,K:A]: (((((dvd_dvd @ A) @ M) @ ((abs_abs @ A) @ K)) = (((dvd_dvd @ A) @ M) @ K)))))))).
% 0.33/5.21 thf(fact_1611_abs__dvd__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:A,K:A]: (((((dvd_dvd @ A) @ ((abs_abs @ A) @ M)) @ K) = (((dvd_dvd @ A) @ M) @ K)))))))).
% 0.33/5.21 thf(fact_1612_abs__divide, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A3:A,B2:A]: ((((abs_abs @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 0.33/5.21 thf(fact_1613_abs__of__nat, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat]: ((((abs_abs @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1614_abs__bool__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P:$o]: ((((abs_abs @ A) @ ((zero_neq_one_of_bool @ A) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))).
% 0.33/5.21 thf(fact_1615_pochhammer__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 0.33/5.21 thf(fact_1616_nat__of__bool, axiom, (! [P:$o]: (((nat2 @ ((zero_neq_one_of_bool @ int) @ P)) = ((zero_neq_one_of_bool @ nat) @ P))))).
% 0.33/5.21 thf(fact_1617_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1618_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_1619_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((abs_abs @ A) @ A3) = A3)))))))).
% 0.33/5.21 thf(fact_1620_zero__less__abs__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A3)) = (A3 != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1621_abs__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_1622_pochhammer__0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_1623_pochhammer__Suc0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ (zero_zero @ nat))) = A3))))))).
% 0.33/5.21 thf(fact_1624_zdvd1__eq, axiom, (! [X:int]: (((((dvd_dvd @ int) @ X) @ (one_one @ int)) = (((abs_abs @ int) @ X) = (one_one @ int)))))).
% 0.33/5.21 thf(fact_1625_abs__division__segment, axiom, (! [K:int]: ((((abs_abs @ int) @ ((euclid7384307370059645450egment @ int) @ K)) = (one_one @ int))))).
% 0.33/5.21 thf(fact_1626_abs__of__nonpos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => (((abs_abs @ A) @ A3) = ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_1627_zero__le__divide__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ ((abs_abs @ A) @ B2))) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1628_divide__le__0__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A3) @ ((abs_abs @ A) @ B2))) @ (zero_zero @ A)) = ((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1629_abs__sgn__eq__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_1630_zabs__less__one__iff, axiom, (! [Z3:int]: (((((ord_less @ int) @ ((abs_abs @ int) @ Z3)) @ (one_one @ int)) = (Z3 = (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_1631_idom__abs__sgn__class_Oabs__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((sgn_sgn @ A) @ ((abs_abs @ A) @ A3)) = ((zero_neq_one_of_bool @ A) @ (A3 != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1632_sgn__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = ((zero_neq_one_of_bool @ A) @ (A3 != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1633_nat__abs__dvd__iff, axiom, (! [K:int,N:nat]: (((((dvd_dvd @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N) = (((dvd_dvd @ int) @ K) @ ((semiring_1_of_nat @ int) @ N)))))).
% 0.33/5.21 thf(fact_1634_dvd__nat__abs__iff, axiom, (! [N:nat,K:int]: (((((dvd_dvd @ nat) @ N) @ (nat2 @ ((abs_abs @ int) @ K))) = (((dvd_dvd @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K))))).
% 0.33/5.21 thf(fact_1635_pochhammer__of__nat, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:nat,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((semiring_1_of_nat @ A) @ X)) @ N) = ((semiring_1_of_nat @ A) @ (((comm_s3205402744901411588hammer @ nat) @ X) @ N))))))))).
% 0.33/5.21 thf(fact_1636_abs__ge__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1637_abs__le__D1, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_1638_abs__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: (((((abs_abs @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1639_abs__mult, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A,B2:A]: ((((abs_abs @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 0.33/5.21 thf(fact_1640_abs__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_1641_abs__minus__commute, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2)) = ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ A3))))))))).
% 0.33/5.21 thf(fact_1642_abs__eq__iff, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y3:A]: (((((abs_abs @ A) @ X) = ((abs_abs @ A) @ Y3)) = ((X = Y3) | (X = ((uminus_uminus @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_1643_dvd__if__abs__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [L:A,K:A]: (((((abs_abs @ A) @ L) = ((abs_abs @ A) @ K)) => (((dvd_dvd @ A) @ L) @ K)))))))).
% 0.33/5.21 thf(fact_1644_zdvd__antisym__abs, axiom, (! [A3:int,B2:int]: (((((dvd_dvd @ int) @ A3) @ B2) => ((((dvd_dvd @ int) @ B2) @ A3) => (((abs_abs @ int) @ A3) = ((abs_abs @ int) @ B2))))))).
% 0.33/5.21 thf(fact_1645_abs__ge__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1646_abs__not__less__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ ((abs_abs @ A) @ A3)) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1647_abs__of__pos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((abs_abs @ A) @ A3) = A3)))))))).
% 0.33/5.21 thf(fact_1648_abs__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((plus_plus @ A) @ A3) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 0.33/5.21 thf(fact_1649_abs__triangle__ineq2, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1650_abs__triangle__ineq3, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_1651_abs__triangle__ineq2__sym, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1652_abs__mult__less, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ A3)) @ C2) => ((((ord_less @ A) @ ((abs_abs @ A) @ B2)) @ D2) => (((ord_less @ A) @ (((times_times @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))) @ (((times_times @ A) @ C2) @ D2)))))))))).
% 0.33/5.21 thf(fact_1653_abs__ge__minus__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ ((abs_abs @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_1654_abs__le__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2) = ((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2))))))))).
% 0.33/5.21 thf(fact_1655_abs__le__D2, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_1656_abs__leI, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2))))))))).
% 0.33/5.21 thf(fact_1657_abs__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ A3)) @ B2) = ((((ord_less @ A) @ A3) @ B2) & (((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ B2))))))))).
% 0.33/5.21 thf(fact_1658_nonzero__abs__divide, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((abs_abs @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)))))))))).
% 0.33/5.21 thf(fact_1659_linordered__idom__class_Oabs__sgn, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((abs_abs @ A) = (^ [K3:A]: ((((times_times @ A) @ K3) @ ((sgn_sgn @ A) @ K3))))))))).
% 0.33/5.21 thf(fact_1660_abs__mult__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: (((((times_times @ A) @ ((abs_abs @ A) @ A3)) @ ((sgn_sgn @ A) @ A3)) = A3))))))).
% 0.33/5.21 thf(fact_1661_sgn__mult__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A3:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ A3)) @ ((abs_abs @ A) @ A3)) = A3))))))).
% 0.33/5.21 thf(fact_1662_mult__sgn__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ X)) @ ((abs_abs @ A) @ X)) = X))))))).
% 0.33/5.21 thf(fact_1663_same__sgn__abs__add, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1664_abs__zmult__eq__1, axiom, (! [M:int,N:int]: (((((abs_abs @ int) @ (((times_times @ int) @ M) @ N)) = (one_one @ int)) => (((abs_abs @ int) @ M) = (one_one @ int)))))).
% 0.33/5.21 thf(fact_1665_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))))))).
% 0.33/5.21 thf(fact_1666_pochhammer__pos, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (((ord_less @ A) @ (zero_zero @ A)) @ (((comm_s3205402744901411588hammer @ A) @ X) @ N))))))))).
% 0.33/5.21 thf(fact_1667_pochhammer__neq__0__mono, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,M:nat,N:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A3) @ M) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N) @ M) => ((((comm_s3205402744901411588hammer @ A) @ A3) @ N) != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1668_pochhammer__eq__0__mono, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,N:nat,M:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A3) @ N) = (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N) @ M) => ((((comm_s3205402744901411588hammer @ A) @ A3) @ M) = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1669_pochhammer__fact, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (comm_semiring_1 @ A)) => ((semiring_char_0_fact @ A) = ((comm_s3205402744901411588hammer @ A) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_1670_dense__eq0__I, axiom, (! [A:$tType]: ((((ordere166539214618696060dd_abs @ A) & (dense_linorder @ A)) => (! [X:A]: (((! [E2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E2) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ E2)))) => (X = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1671_abs__eq__mult, axiom, (! [A:$tType]: (((ordered_ring_abs @ A) => (! [A3:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) | (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))) & ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) | (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((abs_abs @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)))))))))).
% 0.33/5.21 thf(fact_1672_abs__mult__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((times_times @ A) @ ((abs_abs @ A) @ Y3)) @ X) = ((abs_abs @ A) @ (((times_times @ A) @ Y3) @ X)))))))))).
% 0.33/5.21 thf(fact_1673_abs__minus__le__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((abs_abs @ A) @ A3))) @ (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_1674_eq__abs__iff_H, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A3:A,B2:A]: (((A3 = ((abs_abs @ A) @ B2)) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & ((B2 = A3) | (B2 = ((uminus_uminus @ A) @ A3))))))))))).
% 0.33/5.21 thf(fact_1675_abs__eq__iff_H, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A3:A,B2:A]: (((((abs_abs @ A) @ A3) = B2) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) & ((A3 = B2) | (A3 = ((uminus_uminus @ A) @ B2))))))))))).
% 0.33/5.21 thf(fact_1676_abs__of__neg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((abs_abs @ A) @ A3) = ((uminus_uminus @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_1677_abs__if__raw, axiom, (! [A:$tType]: (((abs_if @ A) => ((abs_abs @ A) = (^ [A5:A]: (((((if @ A) @ (((ord_less @ A) @ A5) @ (zero_zero @ A))) @ ((uminus_uminus @ A) @ A5)) @ A5)))))))).
% 0.33/5.21 thf(fact_1678_abs__if, axiom, (! [A:$tType]: (((abs_if @ A) => ((abs_abs @ A) = (^ [A5:A]: (((((if @ A) @ (((ord_less @ A) @ A5) @ (zero_zero @ A))) @ ((uminus_uminus @ A) @ A5)) @ A5)))))))).
% 0.33/5.21 thf(fact_1679_abs__diff__le__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1680_abs__diff__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (((plus_plus @ A) @ C2) @ D2)))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ C2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_1681_abs__triangle__ineq4, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1682_abs__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => ((((divide_divide @ A) @ ((abs_abs @ A) @ X)) @ Y3) = ((abs_abs @ A) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 0.33/5.21 thf(fact_1683_abs__diff__less__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1684_div__noneq__sgn__abs, axiom, (! [L:int,K:int]: (((L != (zero_zero @ int)) => ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ K) @ L) = (((minus_minus @ int) @ ((uminus_uminus @ int) @ (((divide_divide @ int) @ ((abs_abs @ int) @ K)) @ ((abs_abs @ int) @ L)))) @ ((zero_neq_one_of_bool @ int) @ (~ ((((dvd_dvd @ int) @ L) @ K))))))))))).
% 0.33/5.21 thf(fact_1685_abs__sgn__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: ((((A3 = (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (zero_zero @ A))) & ((A3 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_1686_zabs__def, axiom, ((abs_abs @ int) = (^ [I5:int]: (((((if @ int) @ (((ord_less @ int) @ I5) @ (zero_zero @ int))) @ ((uminus_uminus @ int) @ I5)) @ I5))))).
% 0.33/5.21 thf(fact_1687_dvd__imp__le__int, axiom, (! [I:int,D2:int]: (((I != (zero_zero @ int)) => ((((dvd_dvd @ int) @ D2) @ I) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ D2)) @ ((abs_abs @ int) @ I))))))).
% 0.33/5.21 thf(fact_1688_nat__abs__mult__distrib, axiom, (! [W2:int,Z3:int]: (((nat2 @ ((abs_abs @ int) @ (((times_times @ int) @ W2) @ Z3))) = (((times_times @ nat) @ (nat2 @ ((abs_abs @ int) @ W2))) @ (nat2 @ ((abs_abs @ int) @ Z3))))))).
% 0.33/5.21 thf(fact_1689_abs__mod__less, axiom, (! [L:int,K:int]: (((L != (zero_zero @ int)) => (((ord_less @ int) @ ((abs_abs @ int) @ (((modulo_modulo @ int) @ K) @ L))) @ ((abs_abs @ int) @ L)))))).
% 0.33/5.21 thf(fact_1690_pochhammer__nonneg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((comm_s3205402744901411588hammer @ A) @ X) @ N))))))))).
% 0.33/5.21 thf(fact_1691_pochhammer__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N) = (one_one @ A))) & ((N != (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1692_pochhammer__product_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z3:A,N:nat,M:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z3) @ (((plus_plus @ nat) @ N) @ M)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z3) @ N)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z3) @ ((semiring_1_of_nat @ A) @ N))) @ M))))))))).
% 0.33/5.21 thf(fact_1693_abs__add__one__gt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: ((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X))))))))).
% 0.33/5.21 thf(fact_1694_zdvd__mult__cancel1, axiom, (! [M:int,N:int]: (((M != (zero_zero @ int)) => ((((dvd_dvd @ int) @ (((times_times @ int) @ M) @ N)) @ M) = (((abs_abs @ int) @ N) = (one_one @ int))))))).
% 0.33/5.21 thf(fact_1695_div__sgn__abs__cancel, axiom, (! [V:int,K:int,L:int]: (((V != (zero_zero @ int)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ V)) @ ((abs_abs @ int) @ K))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ V)) @ ((abs_abs @ int) @ L))) = (((divide_divide @ int) @ ((abs_abs @ int) @ K)) @ ((abs_abs @ int) @ L))))))).
% 0.33/5.21 thf(fact_1696_div__abs__eq__div__nat, axiom, (! [K:int,L:int]: (((((divide_divide @ int) @ ((abs_abs @ int) @ K)) @ ((abs_abs @ int) @ L)) = ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ (nat2 @ ((abs_abs @ int) @ L)))))))).
% 0.33/5.21 thf(fact_1697_mod__abs__eq__div__nat, axiom, (! [K:int,L:int]: (((((modulo_modulo @ int) @ ((abs_abs @ int) @ K)) @ ((abs_abs @ int) @ L)) = ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ (nat2 @ ((abs_abs @ int) @ L)))))))).
% 0.33/5.21 thf(fact_1698_pochhammer__rec, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1699_pochhammer__Suc, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ A3) @ N)) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 0.33/5.21 thf(fact_1700_pochhammer__rec_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z3) @ (suc @ N)) = (((times_times @ A) @ (((plus_plus @ A) @ Z3) @ ((semiring_1_of_nat @ A) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ Z3) @ N))))))))).
% 0.33/5.21 thf(fact_1701_div__dvd__sgn__abs, axiom, (! [L:int,K:int]: (((((dvd_dvd @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (((times_times @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((sgn_sgn @ int) @ L))) @ (((divide_divide @ int) @ ((abs_abs @ int) @ K)) @ ((abs_abs @ int) @ L)))))))).
% 0.33/5.21 thf(fact_1702_pochhammer__of__nat__eq__0__lemma, axiom, (! [A:$tType]: (((idom @ A) => (! [N:nat,K:nat]: (((((ord_less @ nat) @ N) @ K) => ((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ K) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1703_pochhammer__of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & (idom @ A)) => (! [N:nat,K:nat]: ((((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ K) = (zero_zero @ A)) = (((ord_less @ nat) @ N) @ K)))))))).
% 0.33/5.21 thf(fact_1704_pochhammer__eq__0__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,N:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A3) @ N) = (zero_zero @ A)) = (? [K3:nat]: (((((ord_less @ nat) @ K3) @ N) & (A3 = ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ K3))))))))))))).
% 0.33/5.21 thf(fact_1705_pochhammer__of__nat__eq__0__lemma_H, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & (idom @ A)) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => ((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ K) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1706_modulo__int__def, axiom, ((modulo_modulo @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ (L3 = (zero_zero @ int))) @ K3) @ ((((if @ int) @ (((sgn_sgn @ int) @ K3) = ((sgn_sgn @ int) @ L3))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L3)) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L3)))))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L3)) @ (((minus_minus @ int) @ (((times_times @ int) @ ((abs_abs @ int) @ L3)) @ ((zero_neq_one_of_bool @ int) @ (~ ((((dvd_dvd @ int) @ L3) @ K3)))))) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L3)))))))))))).
% 0.33/5.21 thf(fact_1707_incr__lemma, axiom, (! [D2:int,Z3:int,X:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => (((ord_less @ int) @ Z3) @ (((plus_plus @ int) @ X) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X) @ Z3))) @ (one_one @ int))) @ D2))))))).
% 0.33/5.21 thf(fact_1708_decr__lemma, axiom, (! [D2:int,X:int,Z3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => (((ord_less @ int) @ (((minus_minus @ int) @ X) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X) @ Z3))) @ (one_one @ int))) @ D2))) @ Z3))))).
% 0.33/5.21 thf(fact_1709_nat__abs__int__diff, axiom, (! [A3:nat,B2:nat]: ((((((ord_less_eq @ nat) @ A3) @ B2) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ B2) @ A3))) & ((~ ((((ord_less_eq @ nat) @ A3) @ B2))) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ A3) @ B2))))))).
% 0.33/5.21 thf(fact_1710_pochhammer__product, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [M:nat,N:nat,Z3:A]: (((((ord_less_eq @ nat) @ M) @ N) => ((((comm_s3205402744901411588hammer @ A) @ Z3) @ N) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z3) @ M)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z3) @ ((semiring_1_of_nat @ A) @ M))) @ (((minus_minus @ nat) @ N) @ M))))))))))).
% 0.33/5.21 thf(fact_1711_nat__intermed__int__val, axiom, (! [M:nat,N:nat,F:(nat > int),K:int]: (((! [I4:nat]: ((((((ord_less_eq @ nat) @ M) @ I4) & (((ord_less @ nat) @ I4) @ N)) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F @ (suc @ I4))) @ (F @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ int) @ (F @ M)) @ K) => ((((ord_less_eq @ int) @ K) @ (F @ N)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ M) @ I4) & ((((ord_less_eq @ nat) @ I4) @ N) & ((F @ I4) = K)))))))))))).
% 0.33/5.21 thf(fact_1712_pochhammer__absorb__comp, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1713_add__0__iff, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [B2:A,A3:A]: (((B2 = (((plus_plus @ A) @ B2) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1714_crossproduct__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((A3 != B2) & (C2 != D2)) = ((((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D2)) != (((plus_plus @ A) @ (((times_times @ A) @ A3) @ D2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.21 thf(fact_1715_crossproduct__eq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [W2:A,Y3:A,X:A,Z3:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ W2) @ Y3)) @ (((times_times @ A) @ X) @ Z3)) = (((plus_plus @ A) @ (((times_times @ A) @ W2) @ Z3)) @ (((times_times @ A) @ X) @ Y3))) = ((W2 = X) | (Y3 = Z3))))))))).
% 0.33/5.21 thf(fact_1716_nat__ivt__aux, axiom, (! [N:nat,F:(nat > int),K:int]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F @ (suc @ I4))) @ (F @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ int) @ (F @ (zero_zero @ nat))) @ K) => ((((ord_less_eq @ int) @ K) @ (F @ N)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ N) & ((F @ I4) = K)))))))))).
% 0.33/5.21 thf(fact_1717_nat0__intermed__int__val, axiom, (! [N:nat,F:(nat > int),K:int]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F @ (((plus_plus @ nat) @ I4) @ (one_one @ nat)))) @ (F @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ int) @ (F @ (zero_zero @ nat))) @ K) => ((((ord_less_eq @ int) @ K) @ (F @ N)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ N) & ((F @ I4) = K)))))))))).
% 0.33/5.21 thf(fact_1718_gbinomial__pochhammer, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K3)) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ A5)) @ K3))) @ ((semiring_char_0_fact @ A) @ K3))))))))).
% 0.33/5.21 thf(fact_1719_int__power__div__base, axiom, (! [M:nat,K:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((divide_divide @ int) @ (((power_power @ int) @ K) @ M)) @ K) = (((power_power @ int) @ K) @ (((minus_minus @ nat) @ M) @ (suc @ (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_1720_pochhammer__minus, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [B2:A,K:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ B2)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))).
% 0.33/5.21 thf(fact_1721_pochhammer__minus_H, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1722_gbinomial__minus, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ ((uminus_uminus @ A) @ A3)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))).
% 0.33/5.21 thf(fact_1723_triangle__Suc, axiom, (! [N:nat]: (((nat_triangle @ (suc @ N)) = (((plus_plus @ nat) @ (nat_triangle @ N)) @ (suc @ N)))))).
% 0.33/5.21 thf(fact_1724_frac__unique__iff, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_1725_power__Suc__0, axiom, (! [N:nat]: (((((power_power @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_1726_nat__power__eq__Suc__0__iff, axiom, (! [X:nat,M:nat]: ((((((power_power @ nat) @ X) @ M) = (suc @ (zero_zero @ nat))) = ((M = (zero_zero @ nat)) | (X = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.21 thf(fact_1727_power__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat]: (((((power_power @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_1728_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_1729_of__nat__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((power_power @ nat) @ M) @ N)) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N)))))))).
% 0.33/5.21 thf(fact_1730_of__nat__eq__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [B2:nat,W2:nat,X:nat]: ((((((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2) = ((semiring_1_of_nat @ A) @ X)) = ((((power_power @ nat) @ B2) @ W2) = X)))))))).
% 0.33/5.21 thf(fact_1731_of__nat__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:nat,B2:nat,W2:nat]: (((((semiring_1_of_nat @ A) @ X) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) = (X = (((power_power @ nat) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1732_power__one__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 0.33/5.21 thf(fact_1733_frac__in__Ints__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((member @ A) @ ((archimedean_frac @ A) @ X)) @ (ring_1_Ints @ A)) = (((member @ A) @ X) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1734_triangle__0, axiom, ((nat_triangle @ (zero_zero @ nat)) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_1735_power__inject__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((((power_power @ A) @ A3) @ M) = (((power_power @ A) @ A3) @ N)) = (M = N))))))))).
% 0.33/5.21 thf(fact_1736_power__0__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ (zero_zero @ A)) @ (suc @ N)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_1737_power__Suc0__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ (suc @ (zero_zero @ nat))) = A3))))))).
% 0.33/5.21 thf(fact_1738_abs__power__minus, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: ((((abs_abs @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N)) = ((abs_abs @ A) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1739_frac__eq__0__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((archimedean_frac @ A) @ X) = (zero_zero @ A)) = (((member @ A) @ X) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1740_minus__one__mult__self, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_1741_left__minus__one__mult__self, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat,A3:A]: (((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ A3)) = A3))))))).
% 0.33/5.21 thf(fact_1742_power__strict__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X:nat,Y3:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ X)) @ (((power_power @ A) @ B2) @ Y3)) = (((ord_less @ nat) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_1743_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A3:A,N:nat]: ((((((power_power @ A) @ A3) @ N) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.21 thf(fact_1744_of__nat__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B2:nat,W2:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) = (((ord_less @ nat) @ X) @ (((power_power @ nat) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1745_of__nat__less__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W2:nat,X:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less @ nat) @ (((power_power @ nat) @ B2) @ W2)) @ X)))))))).
% 0.33/5.21 thf(fact_1746_of__nat__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B2:nat,W2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1747_of__nat__le__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W2:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ W2)) @ X)))))))).
% 0.33/5.21 thf(fact_1748_pow__divides__pow__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [N:nat,A3:A,B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)) = (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_1749_frac__gt__0__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((archimedean_frac @ A) @ X)) = (~ ((((member @ A) @ X) @ (ring_1_Ints @ A))))))))))).
% 0.33/5.21 thf(fact_1750_power__strict__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ B2) @ (one_one @ A)) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ M)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less @ nat) @ N) @ M)))))))))).
% 0.33/5.21 thf(fact_1751_power__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X:nat,Y3:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B2) @ X)) @ (((power_power @ A) @ B2) @ Y3)) = (((ord_less_eq @ nat) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_1752_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,B2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less_eq @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_1753_zero__less__power__abs__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A3)) @ N)) = ((A3 != (zero_zero @ A)) | (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_1754_power__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M: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) @ M)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less_eq @ nat) @ N) @ M)))))))))).
% 0.33/5.21 thf(fact_1755_of__nat__zero__less__power__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ X)) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_1756_Ints__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,N:nat]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => (((member @ A) @ (((power_power @ A) @ A3) @ N)) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1757_power__not__zero, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A3:A,N:nat]: (((A3 != (zero_zero @ A)) => ((((power_power @ A) @ A3) @ N) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1758_power__commutes, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,N:nat]: (((((times_times @ A) @ (((power_power @ A) @ A3) @ N)) @ A3) = (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1759_power__mult__distrib, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A,N:nat]: (((((power_power @ A) @ (((times_times @ A) @ A3) @ B2)) @ N) = (((times_times @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N))))))))).
% 0.33/5.21 thf(fact_1760_power__commuting__commutes, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A,Y3:A,N:nat]: ((((((times_times @ A) @ X) @ Y3) = (((times_times @ A) @ Y3) @ X)) => ((((times_times @ A) @ (((power_power @ A) @ X) @ N)) @ Y3) = (((times_times @ A) @ Y3) @ (((power_power @ A) @ X) @ N)))))))))).
% 0.33/5.21 thf(fact_1761_power__mod, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A,N:nat]: (((((modulo_modulo @ A) @ (((power_power @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ N)) @ B2) = (((modulo_modulo @ A) @ (((power_power @ A) @ A3) @ N)) @ B2)))))))).
% 0.33/5.21 thf(fact_1762_power__mult, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M:nat,N:nat]: (((((power_power @ A) @ A3) @ (((times_times @ nat) @ M) @ N)) = (((power_power @ A) @ (((power_power @ A) @ A3) @ M)) @ N)))))))).
% 0.33/5.21 thf(fact_1763_Ints__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (ring_1_Ints @ A)))))).
% 0.33/5.21 thf(fact_1764_Ints__mult, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((times_times @ A) @ A3) @ B2)) @ (ring_1_Ints @ A)))))))))).
% 0.33/5.21 thf(fact_1765_Ints__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (ring_1_Ints @ A)))))))))).
% 0.33/5.21 thf(fact_1766_Ints__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (ring_1_Ints @ A)))))).
% 0.33/5.21 thf(fact_1767_Ints__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ (ring_1_Ints @ A)))))))))).
% 0.33/5.21 thf(fact_1768_minus__in__Ints__iff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A]: (((((member @ A) @ ((uminus_uminus @ A) @ X)) @ (ring_1_Ints @ A)) = (((member @ A) @ X) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1769_Ints__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => (((member @ A) @ ((uminus_uminus @ A) @ A3)) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1770_Ints__of__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: ((((member @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (ring_1_Ints @ A)))))))).
% 0.33/5.21 thf(fact_1771_Ints__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => (((member @ A) @ ((abs_abs @ A) @ A3)) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_1772_fact__in__Ints, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (ring_1 @ A)) => (! [N:nat]: ((((member @ A) @ ((semiring_char_0_fact @ A) @ N)) @ (ring_1_Ints @ A)))))))).
% 0.33/5.21 thf(fact_1773_nat__power__eq, axiom, (! [Z3:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((nat2 @ (((power_power @ int) @ Z3) @ N)) = (((power_power @ nat) @ (nat2 @ Z3)) @ N)))))).
% 0.33/5.21 thf(fact_1774_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,B2:A,N:nat]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)))))))))).
% 0.33/5.21 thf(fact_1775_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1776_zero__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1777_one__le__power, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1778_left__right__inverse__power, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A,Y3:A,N:nat]: ((((((times_times @ A) @ X) @ Y3) = (one_one @ A)) => ((((times_times @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ Y3) @ N)) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_1779_power__Suc2, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ A3) @ (suc @ N)) = (((times_times @ A) @ (((power_power @ A) @ A3) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_1780_power__Suc, axiom, (! [A:$tType]: (((power @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ A3) @ (suc @ N)) = (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1781_power__0, axiom, (! [A:$tType]: (((power @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_1782_power__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ N) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1783_power__add, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M:nat,N:nat]: (((((power_power @ A) @ A3) @ (((plus_plus @ nat) @ M) @ N)) = (((times_times @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1784_le__imp__power__dvd, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [M:nat,N:nat,A3:A]: (((((ord_less_eq @ nat) @ M) @ N) => (((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1785_power__le__dvd, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat,B2:A,M:nat]: (((((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ N)) @ B2) => ((((ord_less_eq @ nat) @ M) @ N) => (((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ M)) @ B2))))))))).
% 0.33/5.21 thf(fact_1786_dvd__power__le, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:A,Y3:A,N:nat,M:nat]: (((((dvd_dvd @ A) @ X) @ Y3) => ((((ord_less_eq @ nat) @ N) @ M) => (((dvd_dvd @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ Y3) @ M)))))))))).
% 0.33/5.21 thf(fact_1787_nat__power__less__imp__less, axiom, (! [I:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_1788_Ints__double__eq__0__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1789_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat,B2:A]: (((((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_1790_power__le__one, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1791_power__inject__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat,B2:A]: ((((((power_power @ A) @ A3) @ (suc @ N)) = (((power_power @ A) @ B2) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (A3 = B2)))))))))).
% 0.33/5.21 thf(fact_1792_power__le__imp__le__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat,B2:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ (suc @ N))) @ (((power_power @ A) @ B2) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_1793_power__gt1__lemma, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N)))))))))).
% 0.33/5.21 thf(fact_1794_power__less__power__Suc, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1795_power__0__left, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (one_one @ A))) & ((N != (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1796_power__gt1, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1797_power__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1798_power__less__imp__less__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N)) => (((ord_less @ nat) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1799_power__strict__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less @ nat) @ N) @ N6) => ((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ A3) @ N6)))))))))).
% 0.33/5.21 thf(fact_1800_power__increasing, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1801_zero__le__power__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A3)) @ N)))))))).
% 0.33/5.21 thf(fact_1802_zero__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_1803_is__unit__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ N)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) | (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_1804_power__gt__expt, axiom, (! [N:nat,K:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N) => (((ord_less @ nat) @ K) @ (((power_power @ nat) @ N) @ K)))))).
% 0.33/5.21 thf(fact_1805_nat__one__le__power, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ I) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((power_power @ nat) @ I) @ N)))))).
% 0.33/5.21 thf(fact_1806_fact__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ ((semiring_char_0_fact @ A) @ N)) @ ((semiring_1_of_nat @ A) @ (((power_power @ nat) @ N) @ N))))))))).
% 0.33/5.21 thf(fact_1807_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))))))))).
% 0.33/5.21 thf(fact_1808_bounded__Max__nat, axiom, (! [P:(nat > $o),X:nat,M7:nat]: (((P @ X) => ((! [X3:nat]: (((P @ X3) => (((ord_less_eq @ nat) @ X3) @ M7)))) => (~ ((! [M6:nat]: (((P @ M6) => (~ ((! [X4:nat]: (((P @ X4) => (((ord_less_eq @ nat) @ X4) @ M6)))))))))))))))).
% 0.33/5.21 thf(fact_1809_power__Suc__less, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1810_power__Suc__le__self, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ (suc @ N))) @ A3))))))))).
% 0.33/5.21 thf(fact_1811_power__Suc__less__one, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1812_power__strict__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less @ nat) @ N) @ N6) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N6)) @ (((power_power @ A) @ A3) @ N))))))))))).
% 0.33/5.21 thf(fact_1813_power__decreasing, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_1814_power__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N)) => (((ord_less_eq @ nat) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1815_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat,B2:A]: ((((((power_power @ A) @ A3) @ N) = (((power_power @ A) @ B2) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (A3 = B2))))))))))).
% 0.33/5.21 thf(fact_1816_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,A3:A,B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((((power_power @ A) @ A3) @ N) = (((power_power @ A) @ B2) @ N)) = (A3 = B2))))))))))).
% 0.33/5.21 thf(fact_1817_self__le__power, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1818_one__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ N)))))))))).
% 0.33/5.21 thf(fact_1819_dvd__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [X:A,M:nat,N:nat]: (((X != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((power_power @ A) @ X) @ M)) @ (((power_power @ A) @ X) @ N)) = ((((dvd_dvd @ A) @ X) @ (one_one @ A)) | (((ord_less_eq @ nat) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_1820_dvd__power, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat,X:A]: ((((((ord_less @ nat) @ (zero_zero @ nat)) @ N) | (X = (one_one @ A))) => (((dvd_dvd @ A) @ X) @ (((power_power @ A) @ X) @ N))))))))).
% 0.33/5.21 thf(fact_1821_power__diff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A,N:nat,M:nat]: (((A3 != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N) @ M) => ((((power_power @ A) @ A3) @ (((minus_minus @ nat) @ M) @ N)) = (((divide_divide @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N))))))))))).
% 0.33/5.21 thf(fact_1822_gbinomial__ge__n__over__k__pow__k, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [K:nat,A3:A]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ K)) @ A3) => (((ord_less_eq @ A) @ (((power_power @ A) @ (((divide_divide @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ K)) @ (((gbinomial @ A) @ A3) @ K))))))))).
% 0.33/5.21 thf(fact_1823_power__dvd__imp__le, axiom, (! [I:nat,M:nat,N:nat]: (((((dvd_dvd @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N)) => ((((ord_less @ nat) @ (one_one @ nat)) @ I) => (((ord_less_eq @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_1824_Ints__odd__less__0, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1825_fact__div__fact__le__pow, axiom, (! [R:nat,N:nat]: (((((ord_less_eq @ nat) @ R) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ N)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N) @ R)))) @ (((power_power @ nat) @ N) @ R)))))).
% 0.33/5.21 thf(fact_1826_Ints__nonzero__abs__ge1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((X != (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X)))))))))).
% 0.33/5.21 thf(fact_1827_Ints__nonzero__abs__less1, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1828_Ints__eq__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((((member @ A) @ Y3) @ (ring_1_Ints @ A)) => ((X = Y3) = (((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ Y3))) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_1829_power__strict__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,B2:A,N:nat]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N))))))))))).
% 0.33/5.21 thf(fact_1830_power__diff__power__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,N:nat,M:nat]: (((A3 != (zero_zero @ A)) => (((((ord_less_eq @ nat) @ N) @ M) => ((((divide_divide @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N)) = (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ M) @ N)))) & ((~ ((((ord_less_eq @ nat) @ N) @ M))) => ((((divide_divide @ A) @ (((power_power @ A) @ A3) @ M)) @ (((power_power @ A) @ A3) @ N)) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ N) @ M))))))))))))).
% 0.33/5.21 thf(fact_1831_power__eq__if, axiom, (! [A:$tType]: (((power @ A) => ((power_power @ A) = (^ [P6:A,M2:nat]: (((((if @ A) @ (M2 = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ P6) @ (((power_power @ A) @ P6) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat)))))))))))).
% 0.33/5.21 thf(fact_1832_power__minus__mult, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat,A3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((times_times @ A) @ (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))) @ A3) = (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_1833_frac__neg, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_1834_gbinomial__negated__upper, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: ((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K3)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ K3)) @ A5)) @ (one_one @ A))) @ K3))))))))).
% 0.33/5.21 thf(fact_1835_gbinomial__index__swap, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_1836_pochhammer__same, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & ((comm_ring_1 @ A) & (semiri3467727345109120633visors @ A))) => (! [N:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 0.33/5.21 thf(fact_1837_neg__one__power__add__eq__neg__one__power__diff, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1838_power__minus_H, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A,N:nat]: ((((((nO_MATCH @ A) @ A) @ (one_one @ A)) @ X) => ((((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_power @ A) @ X) @ N)))))))))).
% 0.33/5.21 thf(fact_1839_ceiling__divide__lower, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ (((divide_divide @ A) @ P5) @ Q4)))) @ (one_one @ A))) @ Q4)) @ P5)))))))).
% 0.33/5.21 thf(fact_1840_power_Opower__eq__if, axiom, (! [A:$tType]: (((power2 @ A) = (^ [One:A,Times:(A > (A > A)),P6:A,M2:nat]: (((((if @ A) @ (M2 = (zero_zero @ nat))) @ One) @ ((Times @ P6) @ (((((power2 @ A) @ One) @ Times) @ P6) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))))))))))).
% 0.33/5.21 thf(fact_1841_le__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_1842_divide__le__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))))))).
% 0.33/5.21 thf(fact_1843_mult__ceiling__le__Ints, axiom, (! [A:$tType,B:$tType]: ((((archim2362893244070406136eiling @ B) & (linordered_idom @ A)) => (! [A3:B,B2:B]: (((((ord_less_eq @ B) @ (zero_zero @ B)) @ A3) => ((((member @ B) @ A3) @ (ring_1_Ints @ B)) => (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ B) @ (((times_times @ B) @ A3) @ B2)))) @ ((ring_1_of_int @ A) @ (((times_times @ int) @ ((archimedean_ceiling @ B) @ A3)) @ ((archimedean_ceiling @ B) @ B2)))))))))))).
% 0.33/5.21 thf(fact_1844_numeral__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:num,N:num]: (((((numeral_numeral @ A) @ M) = ((numeral_numeral @ A) @ N)) = (M = N)))))))).
% 0.33/5.21 thf(fact_1845_int__eq__iff__numeral, axiom, (! [M:nat,V:num]: (((((semiring_1_of_nat @ int) @ M) = ((numeral_numeral @ int) @ V)) = (M = ((numeral_numeral @ nat) @ V)))))).
% 0.33/5.21 thf(fact_1846_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W2:int,Z3:int]: (((((ring_1_of_int @ A) @ W2) = ((ring_1_of_int @ A) @ Z3)) = (W2 = Z3)))))))).
% 0.33/5.21 thf(fact_1847_nat__numeral, axiom, (! [K:num]: (((nat2 @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ nat) @ K))))).
% 0.33/5.21 thf(fact_1848_numeral__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_1849_numeral__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:num,N:num]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ num) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_1850_mult__numeral__left__semiring__numeral, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [V:num,W2:num,Z3:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ Z3)) = (((times_times @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ V) @ W2))) @ Z3)))))))).
% 0.33/5.21 thf(fact_1851_numeral__times__numeral, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [M:num,N:num]: (((((times_times @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((times_times @ num) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1852_add__numeral__left, axiom, (! [A:$tType]: (((numeral @ A) => (! [V:num,W2:num,Z3:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W2)) @ Z3)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V) @ W2))) @ Z3)))))))).
% 0.33/5.21 thf(fact_1853_numeral__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [M:num,N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1854_neg__numeral__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (M = N)))))))).
% 0.33/5.21 thf(fact_1855_power__zero__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [K:num]: (((((power_power @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ nat) @ K)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_1856_power__add__numeral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M:num,N:num]: (((((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ M))) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ N))) = (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_1857_power__add__numeral2, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M:num,N:num,B2:A]: (((((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ M))) @ (((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ N))) @ B2)) = (((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ M) @ N)))) @ B2)))))))).
% 0.33/5.21 thf(fact_1858_of__nat__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:num]: ((((semiring_1_of_nat @ A) @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1859_abs__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((abs_abs @ A) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1860_of__int__eq__numeral__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z3:int,N:num]: (((((ring_1_of_int @ A) @ Z3) = ((numeral_numeral @ A) @ N)) = (Z3 = ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_1861_of__int__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ A) @ K)))))))).
% 0.33/5.21 thf(fact_1862_ceiling__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num]: ((((archimedean_ceiling @ A) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ int) @ V)))))))).
% 0.33/5.21 thf(fact_1863_euclidean__size__numeral, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [K:num]: ((((euclid6346220572633701492n_size @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ nat) @ K)))))))).
% 0.33/5.21 thf(fact_1864_of__int__fact, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (ring_1 @ A)) => (! [N:nat]: ((((ring_1_of_int @ A) @ ((semiring_char_0_fact @ int) @ N)) = ((semiring_char_0_fact @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1865_ceiling__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int]: ((((archimedean_ceiling @ A) @ ((ring_1_of_int @ A) @ Z3)) = Z3))))))).
% 0.33/5.21 thf(fact_1866_round__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:num]: ((((archimedean_round @ A) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_1867_round__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:int]: ((((archimedean_round @ A) @ ((ring_1_of_int @ A) @ N)) = N))))))).
% 0.33/5.21 thf(fact_1868_neg__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((ord_less_eq @ num) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_1869_distrib__left__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (semiring @ A)) => (! [V:num,B2:A,C2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ B2)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ C2))))))))).
% 0.33/5.21 thf(fact_1870_distrib__right__numeral, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1871_neg__numeral__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((ord_less @ num) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_1872_right__diff__distrib__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (ring @ A)) => (! [V:num,B2:A,C2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ B2)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ C2))))))))).
% 0.33/5.21 thf(fact_1873_left__diff__distrib__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (ring @ A)) => (! [A3:A,B2:A,V:num]: (((((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ ((numeral_numeral @ A) @ V)) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ V))) @ (((times_times @ A) @ B2) @ ((numeral_numeral @ A) @ V)))))))))).
% 0.33/5.21 thf(fact_1874_mult__neg__numeral__simps_I1_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [M:num,N:num]: (((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ (((times_times @ num) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1875_mult__neg__numeral__simps_I2_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [M:num,N:num]: (((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_1876_mult__neg__numeral__simps_I3_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [M:num,N:num]: (((((times_times @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_1877_semiring__norm_I172_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [V:num,W2:num,Y3:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y3)) = (((times_times @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ V) @ W2))) @ Y3)))))))).
% 0.33/5.21 thf(fact_1878_semiring__norm_I171_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [V:num,W2:num,Y3:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y3)) = (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ V) @ W2)))) @ Y3)))))))).
% 0.33/5.21 thf(fact_1879_semiring__norm_I170_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [V:num,W2:num,Y3:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ Y3)) = (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((times_times @ num) @ V) @ W2)))) @ Y3)))))))).
% 0.33/5.21 thf(fact_1880_semiring__norm_I168_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W2:num,Y3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y3)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V) @ W2)))) @ Y3)))))))).
% 0.33/5.21 thf(fact_1881_add__neg__numeral__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((uminus_uminus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)))))))))).
% 0.33/5.21 thf(fact_1882_diff__numeral__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_1883_diff__numeral__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_1884_abs__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1885_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_1886_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z3:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z3)) = (Z3 = (zero_zero @ int))))))))).
% 0.33/5.21 thf(fact_1887_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z3:int]: (((((ring_1_of_int @ A) @ Z3) = (zero_zero @ A)) = (Z3 = (zero_zero @ int))))))))).
% 0.33/5.21 thf(fact_1888_of__int__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W2:int,Z3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less_eq @ int) @ W2) @ Z3)))))))).
% 0.33/5.21 thf(fact_1889_of__int__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W2:int,Z3:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less @ int) @ W2) @ Z3)))))))).
% 0.33/5.21 thf(fact_1890_of__int__eq__1__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z3:int]: (((((ring_1_of_int @ A) @ Z3) = (one_one @ A)) = (Z3 = (one_one @ int))))))))).
% 0.33/5.21 thf(fact_1891_of__int__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (one_one @ int)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_1892_of__int__mult, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z3:int]: ((((ring_1_of_int @ A) @ (((times_times @ int) @ W2) @ Z3)) = (((times_times @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_1893_of__int__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z3:int]: ((((ring_1_of_int @ A) @ (((plus_plus @ int) @ W2) @ Z3)) = (((plus_plus @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_1894_of__int__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:int]: ((((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ Z3)) = ((uminus_uminus @ A) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_1895_nat__neg__numeral, axiom, (! [K:num]: (((nat2 @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_1896_of__int__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z3:int]: ((((ring_1_of_int @ A) @ (((minus_minus @ int) @ W2) @ Z3)) = (((minus_minus @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_1897_division__segment__numeral, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [K:num]: ((((euclid7384307370059645450egment @ A) @ ((numeral_numeral @ A) @ K)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_1898_of__int__of__nat__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: ((((ring_1_of_int @ A) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1899_diff__nat__numeral, axiom, (! [V:num,V2:num]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ ((numeral_numeral @ nat) @ V2)) = (nat2 @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ V)) @ ((numeral_numeral @ int) @ V2))))))).
% 0.33/5.21 thf(fact_1900_of__int__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int]: ((((ring_1_of_int @ A) @ ((abs_abs @ int) @ X)) = ((abs_abs @ A) @ ((ring_1_of_int @ A) @ X))))))))).
% 0.33/5.21 thf(fact_1901_of__int__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:int,B2:int,W2:nat]: (((((ring_1_of_int @ A) @ X) = (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (X = (((power_power @ int) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1902_of__int__eq__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [B2:int,W2:nat,X:int]: ((((((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2) = ((ring_1_of_int @ A) @ X)) = ((((power_power @ int) @ B2) @ W2) = X)))))))).
% 0.33/5.21 thf(fact_1903_of__int__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:int,N:nat]: ((((ring_1_of_int @ A) @ (((power_power @ int) @ Z3) @ N)) = (((power_power @ A) @ ((ring_1_of_int @ A) @ Z3)) @ N)))))))).
% 0.33/5.21 thf(fact_1904_frac__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int]: ((((archimedean_frac @ A) @ ((ring_1_of_int @ A) @ Z3)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_1905_nat__eq__numeral__power__cancel__iff, axiom, (! [Y3:int,X:num,N:nat]: ((((nat2 @ Y3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) = (Y3 = (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))).
% 0.33/5.21 thf(fact_1906_numeral__power__eq__nat__cancel__iff, axiom, (! [X:num,N:nat,Y3:int]: ((((((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N) = (nat2 @ Y3)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N) = Y3))))).
% 0.33/5.21 thf(fact_1907_of__int__of__bool, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [P:$o]: ((((ring_1_of_int @ A) @ ((zero_neq_one_of_bool @ int) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))).
% 0.33/5.21 thf(fact_1908_le__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,W2:num]: (((((ord_less_eq @ A) @ A3) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2))) @ B2)))))))).
% 0.33/5.21 thf(fact_1909_divide__le__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) @ A3) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2)))))))))).
% 0.33/5.21 thf(fact_1910_divide__eq__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,W2:num,A3:A]: ((((((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2)) = A3) = (((((numeral_numeral @ A) @ W2) != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2)))) & ((((numeral_numeral @ A) @ W2) = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_1911_eq__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,W2:num]: (((A3 = (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((((numeral_numeral @ A) @ W2) != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2)) = B2)) & ((((numeral_numeral @ A) @ W2) = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_1912_less__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,W2:num]: (((((ord_less @ A) @ A3) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2))) @ B2)))))))).
% 0.33/5.21 thf(fact_1913_divide__less__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) @ A3) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ W2)))))))))).
% 0.33/5.21 thf(fact_1914_of__int__le__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int,N:num]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ int) @ Z3) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_1915_of__int__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,Z3:int]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ N)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1916_of__int__numeral__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,Z3:int]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less @ int) @ ((numeral_numeral @ int) @ N)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1917_of__int__less__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int,N:num]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z3)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ int) @ Z3) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_1918_ceiling__le__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less_eq @ A) @ X) @ ((numeral_numeral @ A) @ V))))))))).
% 0.33/5.21 thf(fact_1919_numeral__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((numeral_numeral @ int) @ V)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ ((numeral_numeral @ A) @ V)) @ X)))))))).
% 0.33/5.21 thf(fact_1920_real__of__nat__eq__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [Y3:nat,X:num,N:nat]: (((((semiring_1_of_nat @ A) @ Y3) = (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (Y3 = (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N))))))))).
% 0.33/5.21 thf(fact_1921_numeral__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:num,N:nat,Y3:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N) = ((semiring_1_of_nat @ A) @ Y3)) = ((((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N) = Y3)))))))).
% 0.33/5.21 thf(fact_1922_ceiling__add__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ ((numeral_numeral @ A) @ V))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V))))))))).
% 0.33/5.21 thf(fact_1923_ceiling__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num]: ((((archimedean_ceiling @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))))))))).
% 0.33/5.21 thf(fact_1924_ceiling__diff__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: ((((archimedean_ceiling @ A) @ (((minus_minus @ A) @ X) @ ((numeral_numeral @ A) @ V))) = (((minus_minus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V))))))))).
% 0.33/5.21 thf(fact_1925_Suc__times__numeral__mod__eq, axiom, (! [K:num,N:nat]: (((((numeral_numeral @ nat) @ K) != (one_one @ nat)) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ K)) @ N))) @ ((numeral_numeral @ nat) @ K)) = (one_one @ nat)))))).
% 0.33/5.21 thf(fact_1926_numeral__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:num,N:nat,Y3:int]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N) = ((ring_1_of_int @ A) @ Y3)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N) = Y3)))))))).
% 0.33/5.21 thf(fact_1927_of__int__eq__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Y3:int,X:num,N:nat]: (((((ring_1_of_int @ A) @ Y3) = (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (Y3 = (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N))))))))).
% 0.33/5.21 thf(fact_1928_ceiling__add__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ ((ring_1_of_int @ A) @ Z3))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1929_ceiling__diff__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: ((((archimedean_ceiling @ A) @ (((minus_minus @ A) @ X) @ ((ring_1_of_int @ A) @ Z3))) = (((minus_minus @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1930_round__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:num]: ((((archimedean_round @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_1931_le__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,W2:num]: (((((ord_less_eq @ A) @ A3) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))))))))).
% 0.33/5.21 thf(fact_1932_divide__le__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ A3) = (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ B2)))))))).
% 0.33/5.21 thf(fact_1933_divide__eq__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,W2:num,A3:A]: ((((((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = A3) = (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))) & ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_1934_eq__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,W2:num]: (((A3 = (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = B2)) & ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_1935_less__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,W2:num]: (((((ord_less @ A) @ A3) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))))))))).
% 0.33/5.21 thf(fact_1936_divide__less__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ A3) = (((ord_less @ A) @ (((times_times @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ B2)))))))).
% 0.33/5.21 thf(fact_1937_dbl__dec__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl_dec @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = ((uminus_uminus @ A) @ ((neg_numeral_dbl_inc @ A) @ ((numeral_numeral @ A) @ K)))))))))).
% 0.33/5.21 thf(fact_1938_dbl__inc__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl_inc @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = ((uminus_uminus @ A) @ ((neg_numeral_dbl_dec @ A) @ ((numeral_numeral @ A) @ K)))))))))).
% 0.33/5.21 thf(fact_1939_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z3) @ (zero_zero @ int))))))))).
% 0.33/5.21 thf(fact_1940_of__int__0__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1941_of__int__less__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (zero_zero @ A)) = (((ord_less @ int) @ Z3) @ (zero_zero @ int))))))))).
% 0.33/5.21 thf(fact_1942_of__int__0__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1943_of__int__1__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less_eq @ int) @ (one_one @ int)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1944_of__int__le__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A)) = (((ord_less_eq @ int) @ Z3) @ (one_one @ int))))))))).
% 0.33/5.21 thf(fact_1945_of__int__1__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z3)) = (((ord_less @ int) @ (one_one @ int)) @ Z3)))))))).
% 0.33/5.21 thf(fact_1946_of__int__less__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A)) = (((ord_less @ int) @ Z3) @ (one_one @ int))))))))).
% 0.33/5.21 thf(fact_1947_of__nat__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((semiring_1_of_nat @ A) @ (nat2 @ Z3)) = ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_1948_of__int__le__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:int,W2:nat,X:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less_eq @ int) @ (((power_power @ int) @ B2) @ W2)) @ X)))))))).
% 0.33/5.21 thf(fact_1949_of__int__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B2:int,W2:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (((ord_less_eq @ int) @ X) @ (((power_power @ int) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1950_of__int__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B2:int,W2:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (((ord_less @ int) @ X) @ (((power_power @ int) @ B2) @ W2))))))))).
% 0.33/5.21 thf(fact_1951_of__int__less__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:int,W2:nat,X:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less @ int) @ (((power_power @ int) @ B2) @ W2)) @ X)))))))).
% 0.33/5.21 thf(fact_1952_nat__numeral__diff__1, axiom, (! [V:num]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat)) = (nat2 @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ V)) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_1953_nat__less__numeral__power__cancel__iff, axiom, (! [A3:int,X:num,N:nat]: (((((ord_less @ nat) @ (nat2 @ A3)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) = (((ord_less @ int) @ A3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))).
% 0.33/5.21 thf(fact_1954_numeral__power__less__nat__cancel__iff, axiom, (! [X:num,N:nat,A3:int]: (((((ord_less @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) @ (nat2 @ A3)) = (((ord_less @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A3))))).
% 0.33/5.21 thf(fact_1955_numeral__power__le__nat__cancel__iff, axiom, (! [X:num,N:nat,A3:int]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) @ (nat2 @ A3)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A3))))).
% 0.33/5.21 thf(fact_1956_nat__le__numeral__power__cancel__iff, axiom, (! [A3:int,X:num,N:nat]: (((((ord_less_eq @ nat) @ (nat2 @ A3)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X)) @ N)) = (((ord_less_eq @ int) @ A3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)))))).
% 0.33/5.21 thf(fact_1957_ceiling__less__numeral, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1958_numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_1959_numeral__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:num,N:nat,X:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N)) @ X)))))))).
% 0.33/5.21 thf(fact_1960_of__nat__less__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,I:num,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N)) = (((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N))))))))).
% 0.33/5.21 thf(fact_1961_of__nat__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,I:num,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N))))))))).
% 0.33/5.21 thf(fact_1962_numeral__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:num,N:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N)) @ X)))))))).
% 0.33/5.21 thf(fact_1963_ceiling__le__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) = (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V)))))))))).
% 0.33/5.21 thf(fact_1964_neg__numeral__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ X)))))))).
% 0.33/5.21 thf(fact_1965_of__int__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:int,X:num,N:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (((ord_less_eq @ int) @ A3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N))))))))).
% 0.33/5.21 thf(fact_1966_numeral__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:num,N:nat,A3:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) @ ((ring_1_of_int @ A) @ A3)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_1967_numeral__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:num,N:nat,A3:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) @ ((ring_1_of_int @ A) @ A3)) = (((ord_less @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_1968_of__int__less__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:int,X:num,N:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ A3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X)) @ N)) = (((ord_less @ int) @ A3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X)) @ N))))))))).
% 0.33/5.21 thf(fact_1969_of__int__eq__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Y3:int,X:num,N:nat]: (((((ring_1_of_int @ A) @ Y3) = (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N)) = (Y3 = (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N))))))))).
% 0.33/5.21 thf(fact_1970_neg__numeral__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:num,N:nat,Y3:int]: ((((((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N) = ((ring_1_of_int @ A) @ Y3)) = ((((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N) = Y3)))))))).
% 0.33/5.21 thf(fact_1971_ceiling__less__neg__numeral, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_1972_neg__numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_1973_neg__numeral__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:num,N:nat,A3:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N)) @ ((ring_1_of_int @ A) @ A3)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_1974_of__int__le__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:int,X:num,N:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A3)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N)) = (((ord_less_eq @ int) @ A3) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N))))))))).
% 0.33/5.21 thf(fact_1975_neg__numeral__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:num,N:nat,A3:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N)) @ ((ring_1_of_int @ A) @ A3)) = (((ord_less @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_1976_of__int__less__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:int,X:num,N:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ A3)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X))) @ N)) = (((ord_less @ int) @ A3) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X))) @ N))))))))).
% 0.33/5.21 thf(fact_1977_of__int__neg__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))))))))).
% 0.33/5.21 thf(fact_1978_nat__numeral__as__int, axiom, ((numeral_numeral @ nat) = (^ [I5:num]: ((nat2 @ ((numeral_numeral @ int) @ I5)))))).
% 0.33/5.21 thf(fact_1979_int__ops_I3_J, axiom, (! [N:num]: ((((semiring_1_of_nat @ int) @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ int) @ N))))).
% 0.33/5.21 thf(fact_1980_ex__le__of__int, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [Z4:int]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z4)))))))))).
% 0.33/5.21 thf(fact_1981_ex__of__int__less, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [Z4:int]: ((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z4)) @ X))))))))).
% 0.33/5.21 thf(fact_1982_ex__less__of__int, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [Z4:int]: ((((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ Z4)))))))))).
% 0.33/5.21 thf(fact_1983_mult__of__int__commute, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:int,Y3:A]: (((((times_times @ A) @ ((ring_1_of_int @ A) @ X)) @ Y3) = (((times_times @ A) @ Y3) @ ((ring_1_of_int @ A) @ X))))))))).
% 0.33/5.21 thf(fact_1984_zero__neq__numeral, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((zero_zero @ A) != ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1985_numeral__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: ((((numeral_numeral @ A) @ M) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_1986_neg__numeral__neq__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [M:num,N:num]: ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M)) != ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1987_div__mult2__numeral__eq, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,K:num,L:num]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ L)) = (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (((times_times @ num) @ K) @ L)))))))))).
% 0.33/5.21 thf(fact_1988_pochhammer__of__int, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:int,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((ring_1_of_int @ A) @ X)) @ N) = ((ring_1_of_int @ A) @ (((comm_s3205402744901411588hammer @ int) @ X) @ N))))))))).
% 0.33/5.21 thf(fact_1989_Ints__of__int, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z3:int]: ((((member @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (ring_1_Ints @ A)))))))).
% 0.33/5.21 thf(fact_1990_Ints__induct, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q4:A,P:(A > $o)]: (((((member @ A) @ Q4) @ (ring_1_Ints @ A)) => ((! [Z4:int]: ((P @ ((ring_1_of_int @ A) @ Z4)))) => (P @ Q4))))))))).
% 0.33/5.21 thf(fact_1991_Ints__cases, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q4:A]: (((((member @ A) @ Q4) @ (ring_1_Ints @ A)) => (~ ((! [Z4:int]: ((Q4 != ((ring_1_of_int @ A) @ Z4))))))))))))).
% 0.33/5.21 thf(fact_1992_Ints__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:num]: ((((member @ A) @ ((numeral_numeral @ A) @ N)) @ (ring_1_Ints @ A)))))))).
% 0.33/5.21 thf(fact_1993_not__numeral__le__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1994_zero__le__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1995_not__numeral__less__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_1996_zero__less__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((((ord_less @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1997_le__of__int__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X))))))))).
% 0.33/5.21 thf(fact_1998_one__le__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_1999_not__numeral__less__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2000_not__numeral__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 0.33/5.21 thf(fact_2001_neg__numeral__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2002_zero__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((zero_zero @ A) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2003_not__numeral__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 0.33/5.21 thf(fact_2004_neg__numeral__less__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2005_one__plus__numeral__commute, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2006_numeral__times__minus__swap, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [W2:num,X:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ ((uminus_uminus @ A) @ X)) = (((times_times @ A) @ X) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))).
% 0.33/5.21 thf(fact_2007_numeral__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: ((((numeral_numeral @ A) @ N) != ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2008_one__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((one_one @ A) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2009_power_Opower_Opower__Suc, axiom, (! [A:$tType,One2:A,Times2:(A > (A > A)),A3:A,N:nat]: (((((((power2 @ A) @ One2) @ Times2) @ A3) @ (suc @ N)) = ((Times2 @ A3) @ (((((power2 @ A) @ One2) @ Times2) @ A3) @ N)))))).
% 0.33/5.21 thf(fact_2010_power_Opower_Opower__0, axiom, (! [A:$tType,One2:A,Times2:(A > (A > A)),A3:A]: (((((((power2 @ A) @ One2) @ Times2) @ A3) @ (zero_zero @ nat)) = One2)))).
% 0.33/5.21 thf(fact_2011_ceiling__le__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z3) = (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_2012_less__ceiling__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less @ int) @ Z3) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X)))))))).
% 0.33/5.21 thf(fact_2013_neg__numeral__le__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) @ (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2014_not__zero__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 0.33/5.21 thf(fact_2015_neg__numeral__less__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) @ (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2016_not__zero__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num]: ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))))).
% 0.33/5.21 thf(fact_2017_not__one__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))).
% 0.33/5.21 thf(fact_2018_not__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_2019_neg__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2020_neg__one__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M)))))))).
% 0.33/5.21 thf(fact_2021_neg__numeral__le__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2022_divide__eq__eq__numeral_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,W2:num]: ((((((divide_divide @ A) @ B2) @ C2) = ((numeral_numeral @ A) @ W2)) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((C2 = (zero_zero @ A)) => (((numeral_numeral @ A) @ W2) = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_2023_eq__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num,B2:A,C2:A]: (((((numeral_numeral @ A) @ W2) = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (((numeral_numeral @ A) @ W2) = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_2024_not__neg__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))).
% 0.33/5.21 thf(fact_2025_not__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))).
% 0.33/5.21 thf(fact_2026_not__numeral__less__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_2027_neg__one__less__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M)))))))).
% 0.33/5.21 thf(fact_2028_neg__numeral__less__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2029_of__int__divide__in__Ints, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:int,A3:int]: (((((dvd_dvd @ int) @ B2) @ A3) => (((member @ A) @ (((divide_divide @ A) @ ((ring_1_of_int @ A) @ A3)) @ ((ring_1_of_int @ A) @ B2))) @ (ring_1_Ints @ A))))))))).
% 0.33/5.21 thf(fact_2030_of__int__nonneg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_2031_of__int__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z3) => (((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_2032_of__int__leD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:int,X:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N))) @ X) => ((N = (zero_zero @ int)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X))))))))).
% 0.33/5.21 thf(fact_2033_of__int__lessD, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2034_floor__exists1, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [X3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X3)) @ X) & ((((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ X3) @ (one_one @ int)))) & (! [Y4:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y4)) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Y4) @ (one_one @ int))))) => (Y4 = X3)))))))))))))).
% 0.33/5.21 thf(fact_2035_floor__exists, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [Z4:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z4)) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Z4) @ (one_one @ int))))))))))))).
% 0.33/5.21 thf(fact_2036_of__nat__less__of__int__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat,X:int]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ N)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ N)) @ X)))))))).
% 0.33/5.21 thf(fact_2037_divide__less__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((numeral_numeral @ A) @ W2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ W2))))))))))))).
% 0.33/5.21 thf(fact_2038_less__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ W2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ ((numeral_numeral @ A) @ W2)) @ (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_2039_divide__eq__eq__numeral_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,W2:num]: ((((((divide_divide @ A) @ B2) @ C2) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((C2 = (zero_zero @ A)) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_2040_eq__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num,B2:A,C2:A]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A))))))))))).
% 0.33/5.21 thf(fact_2041_round__diff__minimal, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:A,M:int]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ Z3) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ Z3))))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ Z3) @ ((ring_1_of_int @ A) @ M)))))))))).
% 0.33/5.21 thf(fact_2042_ceiling__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X))) @ (one_one @ A))) @ X) & (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X)))))))))).
% 0.33/5.21 thf(fact_2043_ceiling__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A))) @ X) => ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z3)) => (((archimedean_ceiling @ A) @ X) = Z3))))))))).
% 0.33/5.21 thf(fact_2044_ceiling__eq__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,A3:int]: (((((archimedean_ceiling @ A) @ X) = A3) = ((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ A3)) @ (one_one @ A))) @ X) & (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_2045_ceiling__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archimedean_ceiling @ A) @ T2)) = (! [I5:int]: ((((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ I5)) @ (one_one @ A))) @ T2) & (((ord_less_eq @ A) @ T2) @ ((ring_1_of_int @ A) @ I5))) => (P @ I5))))))))))).
% 0.33/5.21 thf(fact_2046_ceiling__less__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z3) = (((ord_less_eq @ A) @ X) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2047_le__ceiling__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less_eq @ int) @ Z3) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_2048_divide__le__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((numeral_numeral @ A) @ W2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ W2))))))))))))).
% 0.33/5.21 thf(fact_2049_le__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ W2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ W2)) @ (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_2050_divide__less__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))))))).
% 0.33/5.21 thf(fact_2051_less__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_2052_distrib__left__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X:B,Y3:B,A3:A,B2:A,C2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y3)) @ A3) => ((((times_times @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.21 thf(fact_2053_distrib__right__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X:B,Y3:B,C2:A,A3:A,B2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y3)) @ C2) => ((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.21 thf(fact_2054_left__diff__distrib__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((ring @ A) => (! [X:B,Y3:B,C2:A,A3:A,B2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y3)) @ C2) => ((((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 0.33/5.21 thf(fact_2055_right__diff__distrib__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((ring @ A) => (! [X:B,Y3:B,A3:A,B2:A,C2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y3)) @ A3) => ((((times_times @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 0.33/5.21 thf(fact_2056_ceiling__divide__upper, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less_eq @ A) @ P5) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ (((divide_divide @ A) @ P5) @ Q4)))) @ Q4))))))))).
% 0.33/5.21 thf(fact_2057_of__int__of__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [K3:int]: (((((if @ A) @ (((ord_less @ int) @ K3) @ (zero_zero @ int))) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((uminus_uminus @ int) @ K3))))) @ ((semiring_1_of_nat @ A) @ (nat2 @ K3)))))))))).
% 0.33/5.21 thf(fact_2058_div__add__self2__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X:B,B2:A,A3:A]: ((((((nO_MATCH @ B) @ A) @ X) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_2059_div__add__self1__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X:B,B2:A,A3:A]: ((((((nO_MATCH @ B) @ A) @ X) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_2060_floor__le__neg__numeral, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2061_neg__numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_2062_diff__numeral__special_I5_J, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2063_diff__numeral__special_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((numeral_numeral @ A) @ (inc @ M))))))))).
% 0.33/5.21 thf(fact_2064_add__neg__numeral__special_I5_J, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2065_floor__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int]: ((((archim6421214686448440834_floor @ A) @ ((ring_1_of_int @ A) @ Z3)) = Z3))))))).
% 0.33/5.21 thf(fact_2066_floor__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archim6421214686448440834_floor @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2067_floor__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num]: ((((archim6421214686448440834_floor @ A) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ int) @ V)))))))).
% 0.33/5.21 thf(fact_2068_floor__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archim6421214686448440834_floor @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 0.33/5.21 thf(fact_2069_floor__of__nat, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N:nat]: ((((archim6421214686448440834_floor @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 0.33/5.21 thf(fact_2070_floor__uminus__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int]: ((((archim6421214686448440834_floor @ A) @ ((uminus_uminus @ A) @ ((ring_1_of_int @ A) @ Z3))) = ((uminus_uminus @ int) @ Z3)))))))).
% 0.33/5.21 thf(fact_2071_floor__diff__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: ((((archim6421214686448440834_floor @ A) @ (((minus_minus @ A) @ X) @ ((ring_1_of_int @ A) @ Z3))) = (((minus_minus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z3)))))))).
% 0.33/5.21 thf(fact_2072_floor__add2, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: ((((((member @ A) @ X) @ (ring_1_Ints @ A)) | (((member @ A) @ Y3) @ (ring_1_Ints @ A))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y3)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2073_zero__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ X)))))))).
% 0.33/5.21 thf(fact_2074_floor__less__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (zero_zero @ int)) = (((ord_less @ A) @ X) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2075_numeral__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ V)) @ X)))))))).
% 0.33/5.21 thf(fact_2076_zero__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X)))))))).
% 0.33/5.21 thf(fact_2077_floor__le__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (zero_zero @ int)) = (((ord_less @ A) @ X) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2078_floor__less__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less @ A) @ X) @ ((numeral_numeral @ A) @ V))))))))).
% 0.33/5.21 thf(fact_2079_one__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ (one_one @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X)))))))).
% 0.33/5.21 thf(fact_2080_floor__less__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int)) = (((ord_less @ A) @ X) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2081_floor__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num]: ((((archim6421214686448440834_floor @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))))))))).
% 0.33/5.21 thf(fact_2082_floor__diff__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: ((((archim6421214686448440834_floor @ A) @ (((minus_minus @ A) @ X) @ ((numeral_numeral @ A) @ V))) = (((minus_minus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((numeral_numeral @ int) @ V))))))))).
% 0.33/5.21 thf(fact_2083_add__neg__numeral__special_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ M)))))))))).
% 0.33/5.21 thf(fact_2084_floor__diff__one, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2085_numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((numeral_numeral @ int) @ V)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_2086_floor__le__numeral, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2087_neg__numeral__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ X)))))))).
% 0.33/5.21 thf(fact_2088_floor__less__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) = (((ord_less @ A) @ X) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V)))))))))).
% 0.33/5.21 thf(fact_2089_add__inc, axiom, (! [X:num,Y3:num]: (((((plus_plus @ num) @ X) @ (inc @ Y3)) = (inc @ (((plus_plus @ num) @ X) @ Y3)))))).
% 0.33/5.21 thf(fact_2090_mult__inc, axiom, (! [X:num,Y3:num]: (((((times_times @ num) @ X) @ (inc @ Y3)) = (((plus_plus @ num) @ (((times_times @ num) @ X) @ Y3)) @ X))))).
% 0.33/5.21 thf(fact_2091_floor__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_2092_floor__less__cancel, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3)) => (((ord_less @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_2093_of__int__floor__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X))) @ X))))))).
% 0.33/5.21 thf(fact_2094_floor__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archimedean_ceiling @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2095_floor__le__round, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archimedean_round @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2096_le__floor__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less_eq @ int) @ Z3) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X)))))))).
% 0.33/5.21 thf(fact_2097_floor__less__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z3) = (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ Z3))))))))).
% 0.33/5.21 thf(fact_2098_le__floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3))) @ ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_2099_int__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((plus_plus @ int) @ Z3) @ ((archim6421214686448440834_floor @ A) @ X)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X))))))))).
% 0.33/5.21 thf(fact_2100_floor__add__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z3) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ ((ring_1_of_int @ A) @ Z3)))))))))).
% 0.33/5.21 thf(fact_2101_floor__divide__of__int__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [K:int,L:int]: ((((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ ((ring_1_of_int @ A) @ K)) @ ((ring_1_of_int @ A) @ L))) = (((divide_divide @ int) @ K) @ L)))))))).
% 0.33/5.21 thf(fact_2102_ceiling__def, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_ceiling @ A) = (^ [X2:A]: (((uminus_uminus @ int) @ ((archim6421214686448440834_floor @ A) @ ((uminus_uminus @ A) @ X2)))))))))).
% 0.33/5.21 thf(fact_2103_floor__minus, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archim6421214686448440834_floor @ A) @ ((uminus_uminus @ A) @ X)) = ((uminus_uminus @ int) @ ((archimedean_ceiling @ A) @ X))))))))).
% 0.33/5.21 thf(fact_2104_ceiling__minus, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_ceiling @ A) @ ((uminus_uminus @ A) @ X)) = ((uminus_uminus @ int) @ ((archim6421214686448440834_floor @ A) @ X))))))))).
% 0.33/5.21 thf(fact_2105_frac__def, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_frac @ A) = (^ [X2:A]: ((((minus_minus @ A) @ X2) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X2)))))))))).
% 0.33/5.21 thf(fact_2106_of__nat__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ R) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((archim6421214686448440834_floor @ A) @ R)))) @ R)))))))).
% 0.33/5.21 thf(fact_2107_one__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2108_numeral__inc, axiom, (! [A:$tType]: (((numeral @ A) => (! [X:num]: ((((numeral_numeral @ A) @ (inc @ X)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2109_floor__divide__of__nat__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [M:nat,N:nat]: ((((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))) = ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_2110_ceiling__diff__floor__le__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ int) @ (((minus_minus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ X))) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_2111_ceiling__altdef, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_ceiling @ A) = (^ [X2:A]: (((((if @ int) @ (X2 = ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X2)))) @ ((archim6421214686448440834_floor @ A) @ X2)) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X2)) @ (one_one @ int)))))))))).
% 0.33/5.21 thf(fact_2112_floor__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archim6421214686448440834_floor @ A) @ T2)) = (! [I5:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ I5)) @ T2) & (((ord_less @ A) @ T2) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ I5)) @ (one_one @ A)))) => (P @ I5))))))))))).
% 0.33/5.21 thf(fact_2113_floor__eq__iff, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2114_floor__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X) => ((((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A))) => (((archim6421214686448440834_floor @ A) @ X) = Z3))))))))).
% 0.33/5.21 thf(fact_2115_le__mult__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ int) @ (((times_times @ int) @ ((archim6421214686448440834_floor @ A) @ A3)) @ ((archim6421214686448440834_floor @ A) @ B2))) @ ((archim6421214686448440834_floor @ A) @ (((times_times @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_2116_less__floor__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z3:int,X:A]: (((((ord_less @ int) @ Z3) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A))) @ X)))))))).
% 0.33/5.21 thf(fact_2117_floor__le__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z3:int]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z3) = (((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z3)) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2118_floor__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X))) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int))))))))))).
% 0.33/5.21 thf(fact_2119_floor__divide__lower, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ P5) @ Q4)))) @ Q4)) @ P5)))))))).
% 0.33/5.21 thf(fact_2120_le__mult__floor__Ints, axiom, (! [A:$tType,B:$tType]: ((((archim2362893244070406136eiling @ B) & (linordered_idom @ A)) => (! [A3:B,B2:B]: (((((ord_less_eq @ B) @ (zero_zero @ B)) @ A3) => ((((member @ B) @ A3) @ (ring_1_Ints @ B)) => (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ (((times_times @ int) @ ((archim6421214686448440834_floor @ B) @ A3)) @ ((archim6421214686448440834_floor @ B) @ B2)))) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ B) @ (((times_times @ B) @ A3) @ B2)))))))))))).
% 0.33/5.21 thf(fact_2121_floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3))) @ (one_one @ A)) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y3)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y3))) @ (one_one @ A)))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y3)) = (((plus_plus @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y3))) @ (one_one @ int))))))))))).
% 0.33/5.21 thf(fact_2122_floor__divide__upper, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less @ A) @ P5) @ (((times_times @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ P5) @ Q4)))) @ (one_one @ A))) @ Q4))))))))).
% 0.33/5.21 thf(fact_2123_diff__numeral__special_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ one2)))))))))).
% 0.33/5.21 thf(fact_2124_diff__numeral__special_I3_J, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2125_cpmi, axiom, (! [D4:int,P:(int > $o),P2:(int > $o),B7:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((? [Z5:int]: ((! [X3:int]: (((((ord_less @ int) @ X3) @ Z5) => ((P @ X3) = (P2 @ X3))))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ B7) => (X3 != (((plus_plus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((minus_minus @ int) @ X3) @ D4)))))) => ((! [X3:int,K2:int]: (((P2 @ X3) = (P2 @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ K2) @ D4)))))) => ((? [X7:int]: ((P @ X7))) = ((? [X2:int]: (((((member @ int) @ X2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (P2 @ X2)))) | (? [X2:int]: (((((member @ int) @ X2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (? [Y6:int]: (((((member @ int) @ Y6) @ B7) & (P @ (((plus_plus @ int) @ Y6) @ X2))))))))))))))))).
% 0.33/5.21 thf(fact_2126_cppi, axiom, (! [D4:int,P:(int > $o),P2:(int > $o),A6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((? [Z5:int]: ((! [X3:int]: (((((ord_less @ int) @ Z5) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ A6) => (X3 != (((minus_minus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((plus_plus @ int) @ X3) @ D4)))))) => ((! [X3:int,K2:int]: (((P2 @ X3) = (P2 @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ K2) @ D4)))))) => ((? [X7:int]: ((P @ X7))) = ((? [X2:int]: (((((member @ int) @ X2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (P2 @ X2)))) | (? [X2:int]: (((((member @ int) @ X2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (? [Y6:int]: (((((member @ int) @ Y6) @ A6) & (P @ (((minus_minus @ int) @ Y6) @ X2))))))))))))))))).
% 0.33/5.21 thf(fact_2127_aset_I8_J, axiom, (! [D4:int,A6:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less_eq @ int) @ T2) @ X4) => (((ord_less_eq @ int) @ T2) @ (((plus_plus @ int) @ X4) @ D4)))))))))).
% 0.33/5.21 thf(fact_2128_aset_I6_J, axiom, (! [D4:int,T2:int,A6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A6) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less_eq @ int) @ X4) @ T2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ X4) @ D4)) @ T2)))))))))).
% 0.33/5.21 thf(fact_2129_bset_I8_J, axiom, (! [D4:int,T2:int,B7:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B7) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less_eq @ int) @ T2) @ X4) => (((ord_less_eq @ int) @ T2) @ (((minus_minus @ int) @ X4) @ D4))))))))))).
% 0.33/5.21 thf(fact_2130_atLeastatMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((~ ((((ord_less_eq @ A) @ A3) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_2131_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))).
% 0.33/5.21 thf(fact_2132_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H2:A,L4:A,H:A]: ((((((set_or1337092689740270186AtMost @ A) @ L) @ H2) = (((set_or1337092689740270186AtMost @ A) @ L4) @ H)) = (((L = L4) & (H2 = H)) | ((~ ((((ord_less_eq @ A) @ L) @ H2))) & (~ ((((ord_less_eq @ A) @ L4) @ H)))))))))))).
% 0.33/5.21 thf(fact_2133_semiring__norm_I12_J, axiom, (! [N:num]: (((((times_times @ num) @ one2) @ N) = N)))).
% 0.33/5.21 thf(fact_2134_semiring__norm_I11_J, axiom, (! [M:num]: (((((times_times @ num) @ M) @ one2) = M)))).
% 0.33/5.21 thf(fact_2135_semiring__norm_I68_J, axiom, (! [N:num]: ((((ord_less_eq @ num) @ one2) @ N)))).
% 0.33/5.21 thf(fact_2136_semiring__norm_I75_J, axiom, (! [M:num]: ((~ ((((ord_less @ num) @ M) @ one2)))))).
% 0.33/5.21 thf(fact_2137_one__eq__numeral__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ N)) = (one2 = N)))))))).
% 0.33/5.21 thf(fact_2138_numeral__eq__one__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((((numeral_numeral @ A) @ N) = (one_one @ A)) = (N = one2)))))))).
% 0.33/5.21 thf(fact_2139_numeral__eq__neg__one__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ (one_one @ A))) = (N = one2)))))))).
% 0.33/5.21 thf(fact_2140_neg__one__eq__numeral__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((((uminus_uminus @ A) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (N = one2)))))))).
% 0.33/5.21 thf(fact_2141_Suc__numeral, axiom, (! [N:num]: (((suc @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ nat) @ (((plus_plus @ num) @ N) @ one2)))))).
% 0.33/5.21 thf(fact_2142_not__neg__one__le__neg__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: (((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))) = (M != one2)))))))).
% 0.33/5.21 thf(fact_2143_neg__numeral__less__neg__one__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (M != one2)))))))).
% 0.33/5.21 thf(fact_2144_one__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ N))))))))).
% 0.33/5.21 thf(fact_2145_numeral__plus__one, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ N) @ one2))))))))).
% 0.33/5.21 thf(fact_2146_numeral__le__one__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = (((ord_less_eq @ num) @ N) @ one2)))))))).
% 0.33/5.21 thf(fact_2147_one__less__numeral__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: (((((ord_less @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ num) @ one2) @ N)))))))).
% 0.33/5.21 thf(fact_2148_add__One__commute, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ N) = (((plus_plus @ num) @ N) @ one2))))).
% 0.33/5.21 thf(fact_2149_num__induct, axiom, (! [P:(num > $o),X:num]: (((P @ one2) => ((! [X3:num]: (((P @ X3) => (P @ (inc @ X3))))) => (P @ X)))))).
% 0.33/5.21 thf(fact_2150_le__num__One__iff, axiom, (! [X:num]: (((((ord_less_eq @ num) @ X) @ one2) = (X = one2))))).
% 0.33/5.21 thf(fact_2151_atLeastatMost__psubset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = (((~ ((((ord_less_eq @ A) @ A3) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A3) & ((((ord_less_eq @ A) @ B2) @ D2) & ((((ord_less @ A) @ C2) @ A3) | (((ord_less @ A) @ B2) @ D2))))) & (((ord_less_eq @ A) @ C2) @ D2))))))))).
% 0.33/5.21 thf(fact_2152_mult__numeral__1, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [A3:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ one2)) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_2153_mult__numeral__1__right, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ one2)) = A3))))))).
% 0.33/5.21 thf(fact_2154_numeral__One, axiom, (! [A:$tType]: (((numeral @ A) => (((numeral_numeral @ A) @ one2) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2155_divide__numeral__1, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ one2)) = A3))))))).
% 0.33/5.21 thf(fact_2156_numerals_I1_J, axiom, (((numeral_numeral @ nat) @ one2) = (one_one @ nat))).
% 0.33/5.21 thf(fact_2157_add__One, axiom, (! [X:num]: (((((plus_plus @ num) @ X) @ one2) = (inc @ X))))).
% 0.33/5.21 thf(fact_2158_bset_I1_J, axiom, (! [D4:int,B7:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ B7) => (X3 != (((plus_plus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((minus_minus @ int) @ X3) @ D4)))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ B7) => (X3 != (((plus_plus @ int) @ Xb) @ Xa)))))))) => ((Q @ X3) => (Q @ (((minus_minus @ int) @ X3) @ D4)))))) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => (((P @ X4) & (Q @ X4)) => ((P @ (((minus_minus @ int) @ X4) @ D4)) & (Q @ (((minus_minus @ int) @ X4) @ D4)))))))))))).
% 0.33/5.21 thf(fact_2159_bset_I2_J, axiom, (! [D4:int,B7:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ B7) => (X3 != (((plus_plus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((minus_minus @ int) @ X3) @ D4)))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ B7) => (X3 != (((plus_plus @ int) @ Xb) @ Xa)))))))) => ((Q @ X3) => (Q @ (((minus_minus @ int) @ X3) @ D4)))))) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => (((P @ X4) | (Q @ X4)) => ((P @ (((minus_minus @ int) @ X4) @ D4)) | (Q @ (((minus_minus @ int) @ X4) @ D4)))))))))))).
% 0.33/5.21 thf(fact_2160_aset_I1_J, axiom, (! [D4:int,A6:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ A6) => (X3 != (((minus_minus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((plus_plus @ int) @ X3) @ D4)))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ A6) => (X3 != (((minus_minus @ int) @ Xb) @ Xa)))))))) => ((Q @ X3) => (Q @ (((plus_plus @ int) @ X3) @ D4)))))) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => (((P @ X4) & (Q @ X4)) => ((P @ (((plus_plus @ int) @ X4) @ D4)) & (Q @ (((plus_plus @ int) @ X4) @ D4)))))))))))).
% 0.33/5.21 thf(fact_2161_aset_I2_J, axiom, (! [D4:int,A6:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ A6) => (X3 != (((minus_minus @ int) @ Xb) @ Xa)))))))) => ((P @ X3) => (P @ (((plus_plus @ int) @ X3) @ D4)))))) => ((! [X3:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb:int]: (((((member @ int) @ Xb) @ A6) => (X3 != (((minus_minus @ int) @ Xb) @ Xa)))))))) => ((Q @ X3) => (Q @ (((plus_plus @ int) @ X3) @ D4)))))) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => (((P @ X4) | (Q @ X4)) => ((P @ (((plus_plus @ int) @ X4) @ D4)) | (Q @ (((plus_plus @ int) @ X4) @ D4)))))))))))).
% 0.33/5.21 thf(fact_2162_mult__1s__ring__1_I1_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ one2))) @ B2) = ((uminus_uminus @ A) @ B2)))))))).
% 0.33/5.21 thf(fact_2163_mult__1s__ring__1_I2_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [B2:A]: (((((times_times @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ one2))) = ((uminus_uminus @ A) @ B2)))))))).
% 0.33/5.21 thf(fact_2164_uminus__numeral__One, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ one2)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_2165_cong__exp__iff__simps_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ one2)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2166_numeral__1__eq__Suc__0, axiom, (((numeral_numeral @ nat) @ one2) = (suc @ (zero_zero @ nat)))).
% 0.33/5.21 thf(fact_2167_bset_I9_J, axiom, (! [D2:int,D4:int,B7:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D2) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ X4) @ T2)) => (((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ (((minus_minus @ int) @ X4) @ D4)) @ T2)))))))))).
% 0.33/5.21 thf(fact_2168_bset_I10_J, axiom, (! [D2:int,D4:int,B7:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D2) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((~ ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ X4) @ T2)))) => (~ ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ (((minus_minus @ int) @ X4) @ D4)) @ T2)))))))))))).
% 0.33/5.21 thf(fact_2169_aset_I9_J, axiom, (! [D2:int,D4:int,A6:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D2) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ X4) @ T2)) => (((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X4) @ D4)) @ T2)))))))))).
% 0.33/5.21 thf(fact_2170_aset_I10_J, axiom, (! [D2:int,D4:int,A6:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D2) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((~ ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ X4) @ T2)))) => (~ ((((dvd_dvd @ int) @ D2) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X4) @ D4)) @ T2)))))))))))).
% 0.33/5.21 thf(fact_2171_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))))).
% 0.33/5.21 thf(fact_2172_periodic__finite__ex, axiom, (! [D2:int,P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X3:int,K2:int]: (((P @ X3) = (P @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [X7:int]: ((P @ X7))) = (? [X2:int]: (((((member @ int) @ X2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D2)) & (P @ X2)))))))))).
% 0.33/5.21 thf(fact_2173_aset_I7_J, axiom, (! [D4:int,A6:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less @ int) @ T2) @ X4) => (((ord_less @ int) @ T2) @ (((plus_plus @ int) @ X4) @ D4)))))))))).
% 0.33/5.21 thf(fact_2174_aset_I5_J, axiom, (! [D4:int,T2:int,A6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ A6) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less @ int) @ X4) @ T2) => (((ord_less @ int) @ (((plus_plus @ int) @ X4) @ D4)) @ T2)))))))))).
% 0.33/5.21 thf(fact_2175_aset_I4_J, axiom, (! [D4:int,T2:int,A6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ A6) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((X4 != T2) => ((((plus_plus @ int) @ X4) @ D4) != T2)))))))))).
% 0.33/5.21 thf(fact_2176_aset_I3_J, axiom, (! [D4:int,T2:int,A6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A6) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A6) => (X4 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((X4 = T2) => ((((plus_plus @ int) @ X4) @ D4) = T2)))))))))).
% 0.33/5.21 thf(fact_2177_bset_I7_J, axiom, (! [D4:int,T2:int,B7:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ B7) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less @ int) @ T2) @ X4) => (((ord_less @ int) @ T2) @ (((minus_minus @ int) @ X4) @ D4))))))))))).
% 0.33/5.21 thf(fact_2178_bset_I5_J, axiom, (! [D4:int,B7:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less @ int) @ X4) @ T2) => (((ord_less @ int) @ (((minus_minus @ int) @ X4) @ D4)) @ T2))))))))).
% 0.33/5.21 thf(fact_2179_bset_I4_J, axiom, (! [D4:int,T2:int,B7:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ B7) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((X4 != T2) => ((((minus_minus @ int) @ X4) @ D4) != T2)))))))))).
% 0.33/5.21 thf(fact_2180_bset_I3_J, axiom, (! [D4:int,T2:int,B7:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B7) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((X4 = T2) => ((((minus_minus @ int) @ X4) @ D4) = T2)))))))))).
% 0.33/5.21 thf(fact_2181_bset_I6_J, axiom, (! [D4:int,B7:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X4:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B7) => (X4 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((((ord_less_eq @ int) @ X4) @ T2) => (((ord_less_eq @ int) @ (((minus_minus @ int) @ X4) @ D4)) @ T2))))))))).
% 0.33/5.21 thf(fact_2182_pochhammer__double, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z3)) @ (((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) @ Z3) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ N))))))))).
% 0.33/5.21 thf(fact_2183_even__succ__mod__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((plus_plus @ A) @ (one_one @ A)) @ (((modulo_modulo @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)))))))))))).
% 0.33/5.21 thf(fact_2184_even__succ__div__exp, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2185_fact__double, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat]: ((((semiring_char_0_fact @ A) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N))) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2186_round__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y3:int]: (((((ord_less @ A) @ (((minus_minus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ Y3)) => ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y3)) @ (((plus_plus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) => (((archimedean_round @ A) @ X) = Y3))))))))).
% 0.33/5.21 thf(fact_2187_even__mult__exp__div__exp__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,M:nat,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))) = ((((ord_less @ nat) @ N) @ M) | (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) = (zero_zero @ A)) | ((((ord_less_eq @ nat) @ M) @ N) & (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ M)))))))))))))).
% 0.33/5.21 thf(fact_2188_divmod__digit__1_I1_J, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2189_verit__eq__simplify_I8_J, axiom, (! [X22:num,Y2:num]: ((((bit0 @ X22) = (bit0 @ Y2)) = (X22 = Y2))))).
% 0.33/5.21 thf(fact_2190_semiring__norm_I87_J, axiom, (! [M:num,N:num]: ((((bit0 @ M) = (bit0 @ N)) = (M = N))))).
% 0.33/5.21 thf(fact_2191_semiring__norm_I85_J, axiom, (! [M:num]: (((bit0 @ M) != one2)))).
% 0.33/5.21 thf(fact_2192_semiring__norm_I83_J, axiom, (! [N:num]: ((one2 != (bit0 @ N))))).
% 0.33/5.21 thf(fact_2193_semiring__norm_I6_J, axiom, (! [M:num,N:num]: (((((plus_plus @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (bit0 @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.21 thf(fact_2194_semiring__norm_I13_J, axiom, (! [M:num,N:num]: (((((times_times @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (bit0 @ (bit0 @ (((times_times @ num) @ M) @ N))))))).
% 0.33/5.21 thf(fact_2195_semiring__norm_I78_J, axiom, (! [M:num,N:num]: (((((ord_less @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (((ord_less @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2196_semiring__norm_I71_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2197_semiring__norm_I2_J, axiom, ((((plus_plus @ num) @ one2) @ one2) = (bit0 @ one2))).
% 0.33/5.21 thf(fact_2198_num__double, axiom, (! [N:num]: (((((times_times @ num) @ (bit0 @ one2)) @ N) = (bit0 @ N))))).
% 0.33/5.21 thf(fact_2199_semiring__norm_I69_J, axiom, (! [M:num]: ((~ ((((ord_less_eq @ num) @ (bit0 @ M)) @ one2)))))).
% 0.33/5.21 thf(fact_2200_semiring__norm_I76_J, axiom, (! [N:num]: ((((ord_less @ num) @ one2) @ (bit0 @ N))))).
% 0.33/5.21 thf(fact_2201_one__add__one, axiom, (! [A:$tType]: (((numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2202_even__mult__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ A) @ A3) @ B2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) | (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))))))))).
% 0.33/5.21 thf(fact_2203_bits__one__mod__two__eq__one, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2204_one__mod__two__eq__one, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => ((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2205_zero__eq__power2, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A3:A]: ((((((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2206_power2__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A]: (((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2207_add__2__eq__Suc_H, axiom, (! [N:nat]: (((((plus_plus @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (suc @ N)))))).
% 0.33/5.21 thf(fact_2208_add__2__eq__Suc, axiom, (! [N:nat]: (((((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) = (suc @ (suc @ N)))))).
% 0.33/5.21 thf(fact_2209_Suc__1, axiom, ((suc @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))).
% 0.33/5.21 thf(fact_2210_even__Suc, axiom, (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N)) = (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2211_even__Suc__Suc__iff, axiom, (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ N))) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2212_mod2__Suc__Suc, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ M))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2213_div2__Suc__Suc, axiom, (! [M:nat]: (((((divide_divide @ nat) @ (suc @ (suc @ M))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2214_fact__2, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2215_add__self__div__2, axiom, (! [M:nat]: (((((divide_divide @ nat) @ (((plus_plus @ nat) @ M) @ M)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = M)))).
% 0.33/5.21 thf(fact_2216_zmod__numeral__Bit0, axiom, (! [V:num,W2:num]: (((((modulo_modulo @ int) @ ((numeral_numeral @ int) @ (bit0 @ V))) @ ((numeral_numeral @ int) @ (bit0 @ W2))) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ ((numeral_numeral @ int) @ V)) @ ((numeral_numeral @ int) @ W2))))))).
% 0.33/5.21 thf(fact_2217_one__div__two__eq__zero, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2218_bits__1__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2219_add__neg__numeral__special_I9_J, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2220_diff__numeral__special_I11_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2221_diff__numeral__special_I10_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2222_even__plus__one__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) = (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 0.33/5.21 thf(fact_2223_not__mod__2__eq__0__eq__1, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2224_not__mod__2__eq__1__eq__0, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (one_one @ A)) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2225_minus__1__div__2__eq, axiom, (! [A:$tType]: (((euclid8789492081693882211th_nat @ A) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_2226_power2__less__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2227_power2__eq__iff__nonneg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (X = Y3)))))))))).
% 0.33/5.21 thf(fact_2228_zero__less__power2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (A3 != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2229_bits__minus__1__mod__2__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2230_minus__1__mod__2__eq, axiom, (! [A:$tType]: (((euclid8789492081693882211th_nat @ A) => ((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2231_of__bool__half__eq__0, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [B2:$o]: (((((divide_divide @ A) @ ((zero_neq_one_of_bool @ A) @ B2)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2232_Power_Oring__1__class_Opower__minus__even, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (((power_power @ A) @ A3) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.21 thf(fact_2233_power__minus__odd, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat,A3:A]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((uminus_uminus @ A) @ (((power_power @ A) @ A3) @ N)))))))))).
% 0.33/5.21 thf(fact_2234_Parity_Oring__1__class_Opower__minus__even, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat,A3:A]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_2235_not__mod2__eq__Suc__0__eq__0, axiom, (! [N:nat]: ((((((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) != (suc @ (zero_zero @ nat))) = ((((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_2236_add__self__mod__2, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M) @ M)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2237_half__nonnegative__int__iff, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2238_half__negative__int__iff, axiom, (! [K:int]: (((((ord_less @ int) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2239_even__Suc__div__two, axiom, (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((divide_divide @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2240_odd__Suc__div__two, axiom, (! [N:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((divide_divide @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))).
% 0.33/5.21 thf(fact_2241_even__succ__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2242_even__succ__div__two, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2243_odd__succ__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2244_even__power, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((power_power @ A) @ A3) @ N)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.21 thf(fact_2245_zero__le__power__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,W2:num]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ W2))) = ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))))).
% 0.33/5.21 thf(fact_2246_power__less__zero__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,W2:num]: (((((ord_less @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ W2))) @ (zero_zero @ A)) = ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less @ A) @ A3) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2247_power__less__zero__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (zero_zero @ A)) = ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) & (((ord_less @ A) @ A3) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2248_power__minus1__even, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2249_neg__one__even__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2250_neg__one__odd__power, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2251_even__of__nat, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ N)) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))))).
% 0.33/5.21 thf(fact_2252_odd__Suc__minus__one, axiom, (! [N:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((suc @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) = N))))).
% 0.33/5.21 thf(fact_2253_one__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (one_one @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X)))))))).
% 0.33/5.21 thf(fact_2254_floor__le__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int)) = (((ord_less @ A) @ X) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2255_even__diff__nat, axiom, (! [M:nat,N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ M) @ N)) = ((((ord_less @ nat) @ M) @ N) | (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N))))))).
% 0.33/5.21 thf(fact_2256_mod2__gr__0, axiom, (! [M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ nat)))))).
% 0.33/5.21 thf(fact_2257_odd__two__times__div__two__succ, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2258_semiring__parity__class_Oeven__mask__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ (one_one @ A))) = (N = (zero_zero @ nat))))))))).
% 0.33/5.21 thf(fact_2259_zero__less__power__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,W2:num]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ W2))) = ((((numeral_numeral @ nat) @ W2) = (zero_zero @ nat)) | (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) & (A3 != (zero_zero @ A))) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less @ A) @ (zero_zero @ A)) @ A3))))))))))).
% 0.33/5.21 thf(fact_2260_bits__1__div__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: (((((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = ((zero_neq_one_of_bool @ A) @ (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2261_one__div__2__pow__eq, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2262_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))))))).
% 0.33/5.21 thf(fact_2263_power__le__zero__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,W2:num]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ W2))) @ (zero_zero @ A)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ W2)) & (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))) | ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) & (A3 = (zero_zero @ A)))))))))))).
% 0.33/5.21 thf(fact_2264_one__mod__2__pow__eq, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2265_verit__eq__simplify_I10_J, axiom, (! [X22:num]: ((one2 != (bit0 @ X22))))).
% 0.33/5.21 thf(fact_2266_ex__nat__less, axiom, (! [N:nat,P:(nat > $o)]: (((? [M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N) & (P @ M2)))) = (? [X2:nat]: (((((member @ nat) @ X2) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) & (P @ X2)))))))).
% 0.33/5.21 thf(fact_2267_all__nat__less, axiom, (! [N:nat,P:(nat > $o)]: (((! [M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (P @ M2)))) = (! [X2:nat]: (((((member @ nat) @ X2) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) => (P @ X2)))))))).
% 0.33/5.21 thf(fact_2268_numeral__Bit0, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit0 @ N)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2269_inc_Osimps_I1_J, axiom, ((inc @ one2) = (bit0 @ one2))).
% 0.33/5.21 thf(fact_2270_cong__exp__iff__simps_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num,Q4:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit0 @ N))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (zero_zero @ A)) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2271_numeral__Bit0__div__2, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:num]: (((((divide_divide @ A) @ ((numeral_numeral @ A) @ (bit0 @ N))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2272_power__minus__Bit0, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A,K:num]: (((((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ ((numeral_numeral @ nat) @ (bit0 @ K))) = (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ K)))))))))).
% 0.33/5.21 thf(fact_2273_mult__2, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z3:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z3) = (((plus_plus @ A) @ Z3) @ Z3)))))))).
% 0.33/5.21 thf(fact_2274_mult__2__right, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z3:A]: (((((times_times @ A) @ Z3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ Z3) @ Z3)))))))).
% 0.33/5.21 thf(fact_2275_left__add__twice, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2276_even__zero, axiom, (! [A:$tType]: (((semiring_parity @ A) => (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2277_evenE, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => (~ ((! [B5:A]: ((A3 != (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B5))))))))))))).
% 0.33/5.21 thf(fact_2278_odd__one, axiom, (! [A:$tType]: (((semiring_parity @ A) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2279_even__minus, axiom, (! [A:$tType]: (((ring_parity @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((uminus_uminus @ A) @ A3)) = (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))).
% 0.33/5.21 thf(fact_2280_zero__power2, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((power_power @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2281_power2__eq__square, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((times_times @ A) @ A3) @ A3)))))))).
% 0.33/5.21 thf(fact_2282_power4__eq__xxxx, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A]: (((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = (((times_times @ A) @ (((times_times @ A) @ (((times_times @ A) @ X) @ X)) @ X)) @ X)))))))).
% 0.33/5.21 thf(fact_2283_numeral__2__eq__2, axiom, (((numeral_numeral @ nat) @ (bit0 @ one2)) = (suc @ (suc @ (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2284_one__power2, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((power_power @ A) @ (one_one @ A)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2285_power2__eq__iff, axiom, (! [A:$tType]: (((idom @ A) => (! [X:A,Y3:A]: ((((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((X = Y3) | (X = ((uminus_uminus @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2286_double__not__eq__Suc__double, axiom, (! [M:nat,N:nat]: (((((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M) != (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2287_Suc__double__not__eq__double, axiom, (! [M:nat,N:nat]: (((suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) != (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2288_power__even__eq, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ A3) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (((power_power @ A) @ (((power_power @ A) @ A3) @ N)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2289_nat__1__add__1, axiom, ((((plus_plus @ nat) @ (one_one @ nat)) @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))).
% 0.33/5.21 thf(fact_2290_less__exp, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2291_self__le__ge2__pow, axiom, (! [K:nat,M:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((ord_less_eq @ nat) @ M) @ (((power_power @ nat) @ K) @ M)))))).
% 0.33/5.21 thf(fact_2292_power2__nat__le__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.21 thf(fact_2293_power2__nat__le__imp__le, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.21 thf(fact_2294_even__even__mod__4__iff, axiom, (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2))))))))).
% 0.33/5.21 thf(fact_2295_div2__even__ext__nat, axiom, (! [X:nat,Y3:nat]: ((((((divide_divide @ nat) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((divide_divide @ nat) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ X) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Y3)) => (X = Y3)))))).
% 0.33/5.21 thf(fact_2296_half__gt__zero__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_2297_half__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2298_even__two__times__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = A3)))))))).
% 0.33/5.21 thf(fact_2299_zero__le__power2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2300_power2__eq__imp__eq, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,Y3:A]: ((((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (X = Y3)))))))))).
% 0.33/5.21 thf(fact_2301_power2__le__imp__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_2302_even__iff__mod__2__eq__zero, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2303_power2__less__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2304_exp__add__not__zero__imp__left, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2305_exp__add__not__zero__imp__right, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2306_odd__iff__mod__2__eq__one, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2307_sum__power2__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: ((((((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2308_exp__not__zero__imp__exp__diff__not__zero, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat,M:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ M)) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2309_power2__eq__1__iff, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2310_less__2__cases__iff, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = ((N = (zero_zero @ nat)) | (N = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.21 thf(fact_2311_less__2__cases, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((N = (zero_zero @ nat)) | (N = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.21 thf(fact_2312_abs__le__square__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ ((abs_abs @ A) @ Y3)) = (((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2313_power__mono__odd, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat,A3:A,B2:A]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)))))))))).
% 0.33/5.21 thf(fact_2314_abs__square__eq__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: ((((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)) = (((abs_abs @ A) @ X) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2315_odd__pos, axiom, (! [N:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.33/5.21 thf(fact_2316_uminus__power__if, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat,A3:A]: ((((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((power_power @ A) @ A3) @ N))) & ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((uminus_uminus @ A) @ (((power_power @ A) @ A3) @ N))))))))))).
% 0.33/5.21 thf(fact_2317_nat__2, axiom, ((nat2 @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (suc @ (suc @ (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2318_nat__induct2, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((P @ (one_one @ nat)) => ((! [N4:nat]: (((P @ N4) => (P @ (((plus_plus @ nat) @ N4) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) => (P @ N))))))).
% 0.33/5.21 thf(fact_2319_not__exp__less__eq__0__int, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2320_diff__le__diff__pow, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ (((minus_minus @ nat) @ (((power_power @ nat) @ K) @ M)) @ (((power_power @ nat) @ K) @ N))))))).
% 0.33/5.21 thf(fact_2321_dvd__power__iff__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => ((((dvd_dvd @ nat) @ (((power_power @ nat) @ K) @ M)) @ (((power_power @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_2322_divmod__digit__0_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ (((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) => ((((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)) = (((modulo_modulo @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_2323_oddE, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => (~ ((! [B5:A]: ((A3 != (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B5)) @ (one_one @ A)))))))))))))).
% 0.33/5.21 thf(fact_2324_power2__less__imp__less, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less @ A) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_2325_sum__power2__le__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2326_sum__power2__ge__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2327_mod2__eq__if, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))) & ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2328_parity__cases, axiom, (! [A:$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)))))))))))).
% 0.33/5.21 thf(fact_2329_bits__stable__imp__add__self, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: ((((((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A3) => ((((plus_plus @ A) @ A3) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2330_not__sum__power2__lt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2331_sum__power2__gt__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) = ((X != (zero_zero @ A)) | (Y3 != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2332_power2__sum, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:A,Y3:A]: (((((power_power @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X)) @ Y3))))))))).
% 0.33/5.21 thf(fact_2333_square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) => ((((ord_less_eq @ A) @ X) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2334_zero__le__even__power_H, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.21 thf(fact_2335_power2__le__iff__abs__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ Y3))))))))).
% 0.33/5.21 thf(fact_2336_zero__le__power__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N)) = ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))))).
% 0.33/5.21 thf(fact_2337_zero__le__odd__power, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat,A3:A]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3))))))))).
% 0.33/5.21 thf(fact_2338_zero__le__even__power, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat,A3:A]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_2339_abs__square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)) = (((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2340_abs__square__less__1, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2341_bits__induct, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [P:(A > $o),A3:A]: (((! [A8:A]: ((((((divide_divide @ A) @ A8) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A8) => (P @ A8)))) => ((! [A8:A,B5:$o]: (((P @ A8) => (((((divide_divide @ A) @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B5)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A8))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A8) => (P @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B5)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A8))))))) => (P @ A3))))))))).
% 0.33/5.21 thf(fact_2342_minus__power__mult__self, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [A3:A,N:nat]: (((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N)) = (((power_power @ A) @ A3) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.21 thf(fact_2343_power__odd__eq, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ A3) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) = (((times_times @ A) @ A3) @ (((power_power @ A) @ (((power_power @ A) @ A3) @ N)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2344_nat__bit__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N4:nat]: (((P @ N4) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => (P @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N4)))))) => ((! [N4:nat]: (((P @ N4) => (P @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N4)))))) => (P @ N))))))).
% 0.33/5.21 thf(fact_2345_minus__one__power__iff, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2346_power__mono__even, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:nat,A3:A,B2:A]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ B2) @ N)))))))))).
% 0.33/5.21 thf(fact_2347_Suc__n__div__2__gt__zero, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2348_div__2__gt__zero, axiom, (! [N:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2349_minus__1__div__exp__eq__int, axiom, (! [N:nat]: (((((divide_divide @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) = ((uminus_uminus @ int) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2350_triangle__def, axiom, (nat_triangle = (^ [N3:nat]: ((((divide_divide @ nat) @ (((times_times @ nat) @ N3) @ (suc @ N3))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2351_divmod__digit__0_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ (((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) = (((divide_divide @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_2352_power2__diff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:A,Y3:A]: (((((power_power @ A) @ (((minus_minus @ A) @ X) @ Y3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((minus_minus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X)) @ Y3))))))))).
% 0.33/5.21 thf(fact_2353_odd__0__le__power__imp__0__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_2354_odd__power__less__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2355_zero__less__power__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A3) @ N)) = ((N = (zero_zero @ nat)) | (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) & (A3 != (zero_zero @ A))) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) & (((ord_less @ A) @ (zero_zero @ A)) @ A3))))))))))).
% 0.33/5.21 thf(fact_2356_mult__exp__mod__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat,A3:A]: (((((ord_less_eq @ nat) @ M) @ N) => ((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ (((modulo_modulo @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ M)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)))))))))).
% 0.33/5.21 thf(fact_2357_power__minus1__odd, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2358_exp__mod__exp, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N:nat]: (((((modulo_modulo @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ M) @ N))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))))))))).
% 0.33/5.21 thf(fact_2359_int__bit__induct, axiom, (! [P:(int > $o),K:int]: (((P @ (zero_zero @ int)) => ((P @ ((uminus_uminus @ int) @ (one_one @ int))) => ((! [K2:int]: (((P @ K2) => ((K2 != (zero_zero @ int)) => (P @ (((times_times @ int) @ K2) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))) => ((! [K2:int]: (((P @ K2) => ((K2 != ((uminus_uminus @ int) @ (one_one @ int))) => (P @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ K2) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))) => (P @ K)))))))).
% 0.33/5.21 thf(fact_2360_ex__power__ivl1, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ K) => (? [N4:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ N4)) @ K) & (((ord_less @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N4) @ (one_one @ nat)))))))))))).
% 0.33/5.21 thf(fact_2361_ex__power__ivl2, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (? [N4:nat]: (((((ord_less @ nat) @ (((power_power @ nat) @ B2) @ N4)) @ K) & (((ord_less_eq @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N4) @ (one_one @ nat)))))))))))).
% 0.33/5.21 thf(fact_2362_round__def, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X2:A]: (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X2) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2363_mod__double__modulus, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ M) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => (((((modulo_modulo @ A) @ X) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) = (((modulo_modulo @ A) @ X) @ M)) | ((((modulo_modulo @ A) @ X) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ X) @ M)) @ M))))))))))).
% 0.33/5.21 thf(fact_2364_divmod__digit__1_I2_J, axiom, (! [A:$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))) => ((((minus_minus @ A) @ (((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) = (((modulo_modulo @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_2365_even__mask__div__iff_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_2366_power__le__zero__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (zero_zero @ A)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) & (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))) | ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) & (A3 = (zero_zero @ A)))))))))))).
% 0.33/5.21 thf(fact_2367_of__int__round__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X))) @ (((plus_plus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2368_of__int__round__ge, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2369_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))))))))).
% 0.33/5.21 thf(fact_2370_even__mod__4__div__2, axiom, (! [N:nat]: ((((((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = (suc @ (zero_zero @ nat))) => (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2371_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)))))).
% 0.33/5.21 thf(fact_2372_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)))))).
% 0.33/5.21 thf(fact_2373_even__nat__iff, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (nat2 @ K)) = (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K)))))).
% 0.33/5.21 thf(fact_2374_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)))))))).
% 0.33/5.21 thf(fact_2375_even__mask__div__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M: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))) @ M)) @ (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) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_2376_exp__div__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat]: (((((divide_divide @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M) != (zero_zero @ A)) & (((ord_less_eq @ nat) @ N) @ M)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ M) @ N)))))))))).
% 0.33/5.21 thf(fact_2377_of__int__round__abs__le, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2378_round__unique_H, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,N:int]: (((((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ ((ring_1_of_int @ A) @ N)))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) => (((archimedean_round @ A) @ X) = N)))))))).
% 0.33/5.21 thf(fact_2379_neg__zmod__mult__2, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ (zero_zero @ int)) => ((((modulo_modulo @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3)) = (((minus_minus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ (((plus_plus @ int) @ B2) @ (one_one @ int))) @ A3))) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_2380_round__altdef, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X2:A]: (((((if @ int) @ (((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((archimedean_frac @ A) @ X2))) @ ((archimedean_ceiling @ A) @ X2)) @ ((archim6421214686448440834_floor @ A) @ X2))))))))).
% 0.33/5.21 thf(fact_2381_flip__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se8732182000553998342ip_bit @ A) @ (zero_zero @ nat)) @ A3) = (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2382_set__bit__0, axiom, (! [A:$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)))))))))))).
% 0.33/5.21 thf(fact_2383_psubsetI, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != B7) => (((ord_less @ (set @ A)) @ A6) @ B7)))))).
% 0.33/5.21 thf(fact_2384_unset__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se2638667681897837118et_bit @ A) @ (zero_zero @ nat)) @ A3) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2385_unset__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se2638667681897837118et_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2638667681897837118et_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 0.33/5.21 thf(fact_2386_flip__bit__Suc, axiom, (! [A:$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))))))))))))).
% 0.33/5.21 thf(fact_2387_set__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se5668285175392031749et_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5668285175392031749et_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 0.33/5.21 thf(fact_2388_Compl__subset__Compl__iff, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ ((uminus_uminus @ (set @ A)) @ B7)) = (((ord_less_eq @ (set @ A)) @ B7) @ A6))))).
% 0.33/5.21 thf(fact_2389_Compl__anti__mono, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B7)) @ ((uminus_uminus @ (set @ A)) @ A6)))))).
% 0.33/5.21 thf(fact_2390_unset__bit__nonnegative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se2638667681897837118et_bit @ int) @ N) @ K)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2391_set__bit__nonnegative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se5668285175392031749et_bit @ int) @ N) @ K)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2392_flip__bit__nonnegative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se8732182000553998342ip_bit @ int) @ N) @ K)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2393_unset__bit__negative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se2638667681897837118et_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2394_set__bit__negative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se5668285175392031749et_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2395_flip__bit__negative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se8732182000553998342ip_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2396_unset__bit__nat__def, axiom, ((bit_se2638667681897837118et_bit @ nat) = (^ [M2:nat,N3:nat]: ((nat2 @ (((bit_se2638667681897837118et_bit @ int) @ M2) @ ((semiring_1_of_nat @ int) @ N3))))))).
% 0.33/5.21 thf(fact_2397_unset__bit__less__eq, axiom, (! [N:nat,K:int]: ((((ord_less_eq @ int) @ (((bit_se2638667681897837118et_bit @ int) @ N) @ K)) @ K)))).
% 0.33/5.21 thf(fact_2398_set__bit__greater__eq, axiom, (! [K:int,N:nat]: ((((ord_less_eq @ int) @ K) @ (((bit_se5668285175392031749et_bit @ int) @ N) @ K))))).
% 0.33/5.21 thf(fact_2399_psubset__imp__ex__mem, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) => (? [B5:A]: ((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B7) @ A6)))))))).
% 0.33/5.21 thf(fact_2400_psubset__trans, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A),C6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) => ((((ord_less @ (set @ A)) @ B7) @ C6) => (((ord_less @ (set @ A)) @ A6) @ C6)))))).
% 0.33/5.21 thf(fact_2401_psubsetD, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A6) @ B7) => ((((member @ A) @ C2) @ A6) => (((member @ A) @ C2) @ B7)))))).
% 0.33/5.21 thf(fact_2402_psubsetE, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) => (~ (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 0.33/5.21 thf(fact_2403_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (A7 != B6)))))))).
% 0.33/5.21 thf(fact_2404_psubset__imp__subset, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ (set @ A)) @ A6) @ B7))))).
% 0.33/5.21 thf(fact_2405_psubset__subset__trans, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A),C6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) => ((((ord_less_eq @ (set @ A)) @ B7) @ C6) => (((ord_less @ (set @ A)) @ A6) @ C6)))))).
% 0.33/5.21 thf(fact_2406_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A7)))))))))).
% 0.33/5.21 thf(fact_2407_subset__psubset__trans, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A),C6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((((ord_less @ (set @ A)) @ B7) @ C6) => (((ord_less @ (set @ A)) @ A6) @ C6)))))).
% 0.33/5.21 thf(fact_2408_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) | (A7 = B6)))))))).
% 0.33/5.21 thf(fact_2409_even__set__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A3)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) & (M != (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2410_even__flip__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A3)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) != (M = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2411_even__unset__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A3)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) | (M = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2412_signed__take__bit__rec, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N3:nat,A5:A]: (((((if @ A) @ (N3 = (zero_zero @ nat))) @ ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_ri4674362597316999326ke_bit @ A) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) @ (((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))))).
% 0.33/5.21 thf(fact_2413_dbl__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2414_dbl__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2415_concat__bit__Suc, axiom, (! [N:nat,K:int,L:int]: (((((bit_concat_bit @ (suc @ N)) @ K) @ L) = (((plus_plus @ int) @ (((modulo_modulo @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_concat_bit @ N) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ L))))))).
% 0.33/5.21 thf(fact_2416_set__decode__Suc, axiom, (! [N:nat,X:nat]: (((((member @ nat) @ (suc @ N)) @ (nat_set_decode @ X)) = (((member @ nat) @ N) @ (nat_set_decode @ (((divide_divide @ nat) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))).
% 0.33/5.21 thf(fact_2417_signed__take__bit__Suc, axiom, (! [A:$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))))))))))))).
% 0.33/5.21 thf(fact_2418_Compl__eq__Compl__iff, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A6) = ((uminus_uminus @ (set @ A)) @ B7)) = (A6 = B7))))).
% 0.33/5.21 thf(fact_2419_Compl__iff, axiom, (! [A:$tType,C2:A,A6:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A6)) = (~ ((((member @ A) @ C2) @ A6))))))).
% 0.33/5.21 thf(fact_2420_ComplI, axiom, (! [A:$tType,C2:A,A6:(set @ A)]: (((~ ((((member @ A) @ C2) @ A6))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A6)))))).
% 0.33/5.21 thf(fact_2421_signed__take__bit__of__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2422_concat__bit__0, axiom, (! [K:int,L:int]: (((((bit_concat_bit @ (zero_zero @ nat)) @ K) @ L) = L)))).
% 0.33/5.21 thf(fact_2423_dbl__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2424_signed__take__bit__of__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2425_signed__take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ (suc @ N)) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2426_signed__take__bit__numeral__of__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [K:num]: (((((bit_ri4674362597316999326ke_bit @ A) @ ((numeral_numeral @ nat) @ K)) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2427_concat__bit__nonnegative__iff, axiom, (! [N:nat,K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_concat_bit @ N) @ K) @ L)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ L))))).
% 0.33/5.21 thf(fact_2428_concat__bit__negative__iff, axiom, (! [N:nat,K:int,L:int]: (((((ord_less @ int) @ (((bit_concat_bit @ N) @ K) @ L)) @ (zero_zero @ int)) = (((ord_less @ int) @ L) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2429_dbl__simps_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ A) @ (bit0 @ K))))))))).
% 0.33/5.21 thf(fact_2430_dbl__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = ((uminus_uminus @ A) @ ((neg_numeral_dbl @ A) @ ((numeral_numeral @ A) @ K)))))))))).
% 0.33/5.21 thf(fact_2431_signed__take__bit__Suc__bit0, axiom, (! [N:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N)) @ ((numeral_numeral @ int) @ (bit0 @ K))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2432_signed__take__bit__Suc__minus__bit0, axiom, (! [N:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2433_set__decode__0, axiom, (! [X:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ (nat_set_decode @ X)) = (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ X))))))).
% 0.33/5.21 thf(fact_2434_signed__take__bit__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ (zero_zero @ nat)) @ A3) = ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2435_double__complement, axiom, (! [A:$tType,A6:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) = A6)))).
% 0.33/5.21 thf(fact_2436_ComplD, axiom, (! [A:$tType,C2:A,A6:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A6)) => (~ ((((member @ A) @ C2) @ A6))))))).
% 0.33/5.21 thf(fact_2437_signed__take__bit__mult, axiom, (! [N:nat,K:int,L:int]: (((((bit_ri4674362597316999326ke_bit @ int) @ N) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ L))) = (((bit_ri4674362597316999326ke_bit @ int) @ N) @ (((times_times @ int) @ K) @ L)))))).
% 0.33/5.21 thf(fact_2438_signed__take__bit__minus, axiom, (! [N:nat,K:int]: (((((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K))) = (((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ K)))))).
% 0.33/5.21 thf(fact_2439_subset__decode__imp__le, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ (set @ nat)) @ (nat_set_decode @ M)) @ (nat_set_decode @ N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.21 thf(fact_2440_dbl__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl @ A) = (^ [X2:A]: ((((plus_plus @ A) @ X2) @ X2)))))))).
% 0.33/5.21 thf(fact_2441_signed__take__bit__int__less__exp, axiom, (! [N:nat,K:int]: ((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2442_signed__take__bit__int__greater__eq__self__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ K) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) = (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2443_signed__take__bit__int__less__self__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ K) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ K))))).
% 0.33/5.21 thf(fact_2444_signed__take__bit__int__greater__eq__minus__exp, axiom, (! [N:nat,K:int]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K))))).
% 0.33/5.21 thf(fact_2445_signed__take__bit__int__less__eq__self__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ K) = (((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) @ K))))).
% 0.33/5.21 thf(fact_2446_signed__take__bit__int__greater__self__iff, axiom, (! [K:int,N:nat]: (((((ord_less @ int) @ K) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) = (((ord_less @ int) @ K) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2447_signed__take__bit__int__less__eq, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ K) => (((ord_less_eq @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ (((minus_minus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N)))))))).
% 0.33/5.21 thf(fact_2448_signed__take__bit__int__eq__self, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) @ K) => ((((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => ((((bit_ri4674362597316999326ke_bit @ int) @ N) @ K) = K)))))).
% 0.33/5.21 thf(fact_2449_signed__take__bit__int__eq__self__iff, axiom, (! [N:nat,K:int]: ((((((bit_ri4674362597316999326ke_bit @ int) @ N) @ K) = K) = ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) @ K) & (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2450_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)))))).
% 0.33/5.21 thf(fact_2451_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)))))).
% 0.33/5.21 thf(fact_2452_fact__code, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N3:nat]: (((semiring_1_of_nat @ A) @ (((((set_fo6178422350223883121st_nat @ nat) @ (times_times @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3) @ (one_one @ nat)))))))))).
% 0.33/5.21 thf(fact_2453_take__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se2584673776208193580ke_bit @ A) = (^ [N3:nat,A5:A]: (((((if @ A) @ (N3 = (zero_zero @ nat))) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) @ (((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((modulo_modulo @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2454_num_Osize__gen_I2_J, axiom, (! [X22:num]: (((size_num @ (bit0 @ X22)) = (((plus_plus @ nat) @ (size_num @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2455_and__int__unfold, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ ((K3 = (zero_zero @ int)) | (L3 = (zero_zero @ int)))) @ (zero_zero @ int)) @ ((((if @ int) @ (K3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ L3) @ ((((if @ int) @ (L3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ K3) @ (((plus_plus @ int) @ (((times_times @ int) @ (((modulo_modulo @ int) @ K3) @ ((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) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 0.33/5.21 thf(fact_2456_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))))))))).
% 0.33/5.21 thf(fact_2457_semiring__norm_I90_J, axiom, (! [M:num,N:num]: ((((bit1 @ M) = (bit1 @ N)) = (M = N))))).
% 0.33/5.21 thf(fact_2458_verit__eq__simplify_I9_J, axiom, (! [X32:num,Y32:num]: ((((bit1 @ X32) = (bit1 @ Y32)) = (X32 = Y32))))).
% 0.33/5.21 thf(fact_2459_take__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2460_and__zero__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2461_zero__and__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2462_bit_Oconj__zero__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ (zero_zero @ A)) @ X) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2463_bit_Oconj__zero__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ X) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2464_semiring__norm_I88_J, axiom, (! [M:num,N:num]: (((bit0 @ M) != (bit1 @ N))))).
% 0.33/5.21 thf(fact_2465_semiring__norm_I89_J, axiom, (! [M:num,N:num]: (((bit1 @ M) != (bit0 @ N))))).
% 0.33/5.21 thf(fact_2466_semiring__norm_I84_J, axiom, (! [N:num]: ((one2 != (bit1 @ N))))).
% 0.33/5.21 thf(fact_2467_semiring__norm_I86_J, axiom, (! [M:num]: (((bit1 @ M) != one2)))).
% 0.33/5.21 thf(fact_2468_bit_Oxor__self, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X) @ X) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2469_xor__self__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2470_xor_Oleft__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_2471_xor_Oright__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.21 thf(fact_2472_concat__bit__of__zero__2, axiom, (! [N:nat,K:int]: (((((bit_concat_bit @ N) @ K) @ (zero_zero @ int)) = (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))))).
% 0.33/5.21 thf(fact_2473_semiring__norm_I73_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit1 @ M)) @ (bit1 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2474_semiring__norm_I80_J, axiom, (! [M:num,N:num]: (((((ord_less @ num) @ (bit1 @ M)) @ (bit1 @ N)) = (((ord_less @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2475_take__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (zero_zero @ nat)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2476_take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2477_bit_Oconj__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = X))))))).
% 0.33/5.21 thf(fact_2478_and_Oright__neutral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = A3))))))).
% 0.33/5.21 thf(fact_2479_and_Oleft__neutral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_2480_take__bit__numeral__1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [L:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2481_of__nat__nat__take__bit__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat,K:int]: ((((semiring_1_of_nat @ A) @ (nat2 @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))) = ((ring_1_of_int @ A) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))))))))).
% 0.33/5.21 thf(fact_2482_and__nonnegative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se5824344872417868541ns_and @ int) @ K) @ L)) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) | (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)))))).
% 0.33/5.21 thf(fact_2483_and__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) & (((ord_less @ int) @ L) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2484_semiring__norm_I9_J, axiom, (! [M:num,N:num]: (((((plus_plus @ num) @ (bit1 @ M)) @ (bit0 @ N)) = (bit1 @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.21 thf(fact_2485_semiring__norm_I7_J, axiom, (! [M:num,N:num]: (((((plus_plus @ num) @ (bit0 @ M)) @ (bit1 @ N)) = (bit1 @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.21 thf(fact_2486_semiring__norm_I14_J, axiom, (! [M:num,N:num]: (((((times_times @ num) @ (bit0 @ M)) @ (bit1 @ N)) = (bit0 @ (((times_times @ num) @ M) @ (bit1 @ N))))))).
% 0.33/5.21 thf(fact_2487_semiring__norm_I15_J, axiom, (! [M:num,N:num]: (((((times_times @ num) @ (bit1 @ M)) @ (bit0 @ N)) = (bit0 @ (((times_times @ num) @ (bit1 @ M)) @ N)))))).
% 0.33/5.21 thf(fact_2488_semiring__norm_I81_J, axiom, (! [M:num,N:num]: (((((ord_less @ num) @ (bit1 @ M)) @ (bit0 @ N)) = (((ord_less @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2489_semiring__norm_I72_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit0 @ M)) @ (bit1 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2490_semiring__norm_I77_J, axiom, (! [N:num]: ((((ord_less @ num) @ one2) @ (bit1 @ N))))).
% 0.33/5.21 thf(fact_2491_semiring__norm_I70_J, axiom, (! [M:num]: ((~ ((((ord_less_eq @ num) @ (bit1 @ M)) @ one2)))))).
% 0.33/5.21 thf(fact_2492_dbl__inc__simps_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl_inc @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ A) @ (bit1 @ K))))))))).
% 0.33/5.21 thf(fact_2493_take__bit__of__1__eq__0__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: ((((((bit_se2584673776208193580ke_bit @ A) @ N) @ (one_one @ A)) = (zero_zero @ A)) = (N = (zero_zero @ nat))))))))).
% 0.33/5.21 thf(fact_2494_and__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2495_and__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_2496_semiring__norm_I10_J, axiom, (! [M:num,N:num]: (((((plus_plus @ num) @ (bit1 @ M)) @ (bit1 @ N)) = (bit0 @ (((plus_plus @ num) @ (((plus_plus @ num) @ M) @ N)) @ one2)))))).
% 0.33/5.21 thf(fact_2497_semiring__norm_I8_J, axiom, (! [M:num]: (((((plus_plus @ num) @ (bit1 @ M)) @ one2) = (bit0 @ (((plus_plus @ num) @ M) @ one2)))))).
% 0.33/5.21 thf(fact_2498_semiring__norm_I5_J, axiom, (! [M:num]: (((((plus_plus @ num) @ (bit0 @ M)) @ one2) = (bit1 @ M))))).
% 0.33/5.21 thf(fact_2499_semiring__norm_I4_J, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bit1 @ N)) = (bit0 @ (((plus_plus @ num) @ N) @ one2)))))).
% 0.33/5.21 thf(fact_2500_semiring__norm_I3_J, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bit0 @ N)) = (bit1 @ N))))).
% 0.33/5.21 thf(fact_2501_take__bit__of__Suc__0, axiom, (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ nat) @ N) @ (suc @ (zero_zero @ nat))) = ((zero_neq_one_of_bool @ nat) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.21 thf(fact_2502_semiring__norm_I16_J, axiom, (! [M:num,N:num]: (((((times_times @ num) @ (bit1 @ M)) @ (bit1 @ N)) = (bit1 @ (((plus_plus @ num) @ (((plus_plus @ num) @ M) @ N)) @ (bit0 @ (((times_times @ num) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_2503_semiring__norm_I74_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit1 @ M)) @ (bit0 @ N)) = (((ord_less @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2504_semiring__norm_I79_J, axiom, (! [M:num,N:num]: (((((ord_less @ num) @ (bit0 @ M)) @ (bit1 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 0.33/5.21 thf(fact_2505_and__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2506_and__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2507_and__numerals_I3_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2508_xor__numerals_I3_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2509_xor__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ X))))))))).
% 0.33/5.21 thf(fact_2510_xor__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 0.33/5.21 thf(fact_2511_xor__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = ((numeral_numeral @ A) @ (bit0 @ Y3))))))))).
% 0.33/5.21 thf(fact_2512_xor__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = ((numeral_numeral @ A) @ (bit1 @ Y3))))))))).
% 0.33/5.21 thf(fact_2513_take__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.21 thf(fact_2514_and__minus__numerals_I2_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (one_one @ int))))).
% 0.33/5.21 thf(fact_2515_and__minus__numerals_I6_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) @ (one_one @ int)) = (one_one @ int))))).
% 0.33/5.21 thf(fact_2516_dbl__inc__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ one2))))))).
% 0.33/5.21 thf(fact_2517_even__take__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)) = ((N = (zero_zero @ nat)) | (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))))))))).
% 0.33/5.21 thf(fact_2518_and__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2519_and__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2520_xor__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2521_mod__Suc__eq__mod__add3, axiom, (! [M:nat,N:nat]: (((((modulo_modulo @ nat) @ M) @ (suc @ (suc @ (suc @ N)))) = (((modulo_modulo @ nat) @ M) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2522_Suc__mod__eq__add3__mod__numeral, axiom, (! [M:nat,V:num]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M)))) @ ((numeral_numeral @ nat) @ V)) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ ((numeral_numeral @ nat) @ V)))))).
% 0.33/5.21 thf(fact_2523_div__Suc__eq__div__add3, axiom, (! [M:nat,N:nat]: (((((divide_divide @ nat) @ M) @ (suc @ (suc @ (suc @ N)))) = (((divide_divide @ nat) @ M) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2524_Suc__div__eq__add3__div__numeral, axiom, (! [M:nat,V:num]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M)))) @ ((numeral_numeral @ nat) @ V)) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ ((numeral_numeral @ nat) @ V)))))).
% 0.33/5.21 thf(fact_2525_and__minus__numerals_I5_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) @ (one_one @ int)) = (zero_zero @ int))))).
% 0.33/5.21 thf(fact_2526_and__minus__numerals_I1_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (zero_zero @ int))))).
% 0.33/5.21 thf(fact_2527_xor__nat__numerals_I4_J, axiom, (! [X:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit0 @ X)))))).
% 0.33/5.21 thf(fact_2528_xor__nat__numerals_I3_J, axiom, (! [X:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X)))))).
% 0.33/5.21 thf(fact_2529_xor__nat__numerals_I2_J, axiom, (! [Y3:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y3))) = ((numeral_numeral @ nat) @ (bit0 @ Y3)))))).
% 0.33/5.21 thf(fact_2530_xor__nat__numerals_I1_J, axiom, (! [Y3:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y3))) = ((numeral_numeral @ nat) @ (bit1 @ Y3)))))).
% 0.33/5.21 thf(fact_2531_dbl__dec__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit1 @ one2)))))))).
% 0.33/5.21 thf(fact_2532_take__bit__Suc__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ (zero_zero @ nat))) @ A3) = (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2533_and__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2534_xor__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((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) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2535_xor__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2536_take__bit__of__exp, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:nat,N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ M) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ N) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.21 thf(fact_2537_take__bit__of__2, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2538_zmod__numeral__Bit1, axiom, (! [V:num,W2:num]: (((((modulo_modulo @ int) @ ((numeral_numeral @ int) @ (bit1 @ V))) @ ((numeral_numeral @ int) @ (bit0 @ W2))) = (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ ((numeral_numeral @ int) @ V)) @ ((numeral_numeral @ int) @ W2)))) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2539_signed__take__bit__Suc__bit1, axiom, (! [N:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N)) @ ((numeral_numeral @ int) @ (bit1 @ K))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2540_take__bit__of__nat, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,M:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ ((semiring_1_of_nat @ A) @ M)) = ((semiring_1_of_nat @ A) @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_2541_of__nat__xor__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((bit_se5824344971392196577ns_xor @ nat) @ M) @ N)) = (((bit_se5824344971392196577ns_xor @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2542_of__nat__and__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((bit_se5824344872417868541ns_and @ nat) @ M) @ N)) = (((bit_se5824344872417868541ns_and @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2543_take__bit__tightened, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A,B2:A,M:nat]: ((((((bit_se2584673776208193580ke_bit @ A) @ N) @ A3) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ B2)) => ((((ord_less_eq @ nat) @ M) @ N) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ A3) = (((bit_se2584673776208193580ke_bit @ A) @ M) @ B2)))))))))).
% 0.33/5.21 thf(fact_2544_take__bit__tightened__less__eq__nat, axiom, (! [M:nat,N:nat,Q4:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ M) @ Q4)) @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ Q4)))))).
% 0.33/5.21 thf(fact_2545_take__bit__nat__less__eq__self, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ M)) @ M)))).
% 0.33/5.21 thf(fact_2546_verit__eq__simplify_I14_J, axiom, (! [X22:num,X32:num]: (((bit0 @ X22) != (bit1 @ X32))))).
% 0.33/5.21 thf(fact_2547_verit__eq__simplify_I12_J, axiom, (! [X32:num]: ((one2 != (bit1 @ X32))))).
% 0.33/5.21 thf(fact_2548_take__bit__mult, axiom, (! [N:nat,K:int,L:int]: (((((bit_se2584673776208193580ke_bit @ int) @ N) @ (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ L))) = (((bit_se2584673776208193580ke_bit @ int) @ N) @ (((times_times @ int) @ K) @ L)))))).
% 0.33/5.21 thf(fact_2549_take__bit__minus, axiom, (! [N:nat,K:int]: (((((bit_se2584673776208193580ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))) = (((bit_se2584673776208193580ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ K)))))).
% 0.33/5.21 thf(fact_2550_take__bit__nat__eq, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((bit_se2584673776208193580ke_bit @ nat) @ N) @ (nat2 @ K)) = (nat2 @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))))))).
% 0.33/5.21 thf(fact_2551_nat__take__bit__eq, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((nat2 @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) = (((bit_se2584673776208193580ke_bit @ nat) @ N) @ (nat2 @ K))))))).
% 0.33/5.21 thf(fact_2552_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))))))))))).
% 0.33/5.21 thf(fact_2553_num_Oexhaust, axiom, (! [Y3:num]: (((Y3 != one2) => ((! [X23:num]: ((Y3 != (bit0 @ X23)))) => (~ ((! [X33:num]: ((Y3 != (bit1 @ X33))))))))))).
% 0.33/5.21 thf(fact_2554_AND__upper2_H, axiom, (! [Y3:int,Z3:int,X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((((ord_less_eq @ int) @ Y3) @ Z3) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y3)) @ Z3)))))).
% 0.33/5.21 thf(fact_2555_AND__upper1_H, axiom, (! [Y3:int,Z3:int,Ya:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((((ord_less_eq @ int) @ Y3) @ Z3) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ Y3) @ Ya)) @ Z3)))))).
% 0.33/5.21 thf(fact_2556_AND__upper2, axiom, (! [Y3:int,X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y3)) @ Y3))))).
% 0.33/5.21 thf(fact_2557_AND__upper1, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y3)) @ X))))).
% 0.33/5.21 thf(fact_2558_AND__lower, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y3)))))).
% 0.33/5.21 thf(fact_2559_take__bit__int__less__eq__self__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ K) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2560_take__bit__nonnegative, axiom, (! [N:nat,K:int]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))))).
% 0.33/5.21 thf(fact_2561_not__take__bit__negative, axiom, (! [N:nat,K:int]: ((~ ((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2562_take__bit__int__greater__self__iff, axiom, (! [K:int,N:nat]: (((((ord_less @ int) @ K) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2563_take__bit__tightened__less__eq__int, axiom, (! [M:nat,N:nat,K:int]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ M) @ K)) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)))))).
% 0.33/5.21 thf(fact_2564_signed__take__bit__eq__iff__take__bit__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat,A3:A,B2:A]: ((((((bit_ri4674362597316999326ke_bit @ A) @ N) @ A3) = (((bit_ri4674362597316999326ke_bit @ A) @ N) @ B2)) = ((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ A3) = (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ B2))))))))).
% 0.33/5.21 thf(fact_2565_signed__take__bit__take__bit, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N:nat,A3:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)) = (((((if @ (A > A)) @ (((ord_less_eq @ nat) @ N) @ M)) @ ((bit_se2584673776208193580ke_bit @ A) @ N)) @ ((bit_ri4674362597316999326ke_bit @ A) @ M)) @ A3)))))))).
% 0.33/5.21 thf(fact_2566_num_Osize__gen_I3_J, axiom, (! [X32:num]: (((size_num @ (bit1 @ X32)) = (((plus_plus @ nat) @ (size_num @ X32)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2567_take__bit__unset__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,M:nat,A3:A]: ((((((ord_less_eq @ nat) @ N) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A3)) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))) & ((~ ((((ord_less_eq @ nat) @ N) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A3)) = (((bit_se2638667681897837118et_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))))))))))).
% 0.33/5.21 thf(fact_2568_take__bit__set__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,M:nat,A3:A]: ((((((ord_less_eq @ nat) @ N) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A3)) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))) & ((~ ((((ord_less_eq @ nat) @ N) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A3)) = (((bit_se5668285175392031749et_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))))))))))).
% 0.33/5.21 thf(fact_2569_take__bit__flip__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,M:nat,A3:A]: ((((((ord_less_eq @ nat) @ N) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A3)) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))) & ((~ ((((ord_less_eq @ nat) @ N) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A3)) = (((bit_se8732182000553998342ip_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3))))))))))).
% 0.33/5.21 thf(fact_2570_inc_Osimps_I3_J, axiom, (! [X:num]: (((inc @ (bit1 @ X)) = (bit0 @ (inc @ X)))))).
% 0.33/5.21 thf(fact_2571_inc_Osimps_I2_J, axiom, (! [X:num]: (((inc @ (bit0 @ X)) = (bit1 @ X))))).
% 0.33/5.21 thf(fact_2572_numeral__Bit1, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_2573_take__bit__signed__take__bit, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N:nat,A3:A]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ (((bit_ri4674362597316999326ke_bit @ A) @ N) @ A3)) = (((bit_se2584673776208193580ke_bit @ A) @ M) @ A3))))))))).
% 0.33/5.21 thf(fact_2574_and__less__eq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ K) @ L)) @ K))))).
% 0.33/5.21 thf(fact_2575_AND__upper1_H_H, axiom, (! [Y3:int,Z3:int,Ya:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((((ord_less @ int) @ Y3) @ Z3) => (((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ Y3) @ Ya)) @ Z3)))))).
% 0.33/5.21 thf(fact_2576_AND__upper2_H_H, axiom, (! [Y3:int,Z3:int,X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => ((((ord_less @ int) @ Y3) @ Z3) => (((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y3)) @ Z3)))))).
% 0.33/5.21 thf(fact_2577_eval__nat__numeral_I3_J, axiom, (! [N:num]: ((((numeral_numeral @ nat) @ (bit1 @ N)) = (suc @ ((numeral_numeral @ nat) @ (bit0 @ N))))))).
% 0.33/5.21 thf(fact_2578_power__minus__Bit1, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A,K:num]: (((((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ ((numeral_numeral @ nat) @ (bit1 @ K))) = ((uminus_uminus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit1 @ K))))))))))).
% 0.33/5.21 thf(fact_2579_take__bit__decr__eq, axiom, (! [N:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) != (zero_zero @ int)) => ((((bit_se2584673776208193580ke_bit @ int) @ N) @ (((minus_minus @ int) @ K) @ (one_one @ int))) = (((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_2580_take__bit__Suc__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2581_cong__exp__iff__simps_I3_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num,Q4:num]: (((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ N))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) != (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2582_numeral__Bit1__div__2, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:num]: (((((divide_divide @ A) @ ((numeral_numeral @ A) @ (bit1 @ N))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2583_power3__eq__cube, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) = (((times_times @ A) @ (((times_times @ A) @ A3) @ A3)) @ A3)))))))).
% 0.33/5.21 thf(fact_2584_numeral__3__eq__3, axiom, (((numeral_numeral @ nat) @ (bit1 @ one2)) = (suc @ (suc @ (suc @ (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_2585_Suc3__eq__add__3, axiom, (! [N:nat]: (((suc @ (suc @ (suc @ N))) = (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2586_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)))))).
% 0.33/5.21 thf(fact_2587_take__bit__Suc__bit0, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ (bit0 @ K))) = (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2588_and__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ A3) @ (one_one @ A)) = (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2589_one__and__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ A3) = (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2590_take__bit__nat__eq__self, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) => ((((bit_se2584673776208193580ke_bit @ nat) @ N) @ M) = M))))).
% 0.33/5.21 thf(fact_2591_take__bit__nat__less__exp, axiom, (! [N:nat,M:nat]: ((((ord_less @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ M)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2592_take__bit__nat__eq__self__iff, axiom, (! [N:nat,M:nat]: ((((((bit_se2584673776208193580ke_bit @ nat) @ N) @ M) = M) = (((ord_less @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2593_cong__exp__iff__simps_I7_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [Q4:num,N:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ one2)) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ N))) @ ((numeral_numeral @ A) @ (bit0 @ Q4)))) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2594_cong__exp__iff__simps_I11_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,Q4:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ M))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ one2)) @ ((numeral_numeral @ A) @ (bit0 @ Q4)))) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2595_take__bit__int__less__exp, axiom, (! [N:nat,K:int]: ((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2596_fold__atLeastAtMost__nat_Osimps, axiom, (! [A:$tType]: (((set_fo6178422350223883121st_nat @ A) = (^ [F2:(nat > (A > A)),A5:nat,B3:nat,Acc:A]: (((((if @ A) @ (((ord_less @ nat) @ B3) @ A5)) @ Acc) @ (((((set_fo6178422350223883121st_nat @ A) @ F2) @ (((plus_plus @ nat) @ A5) @ (one_one @ nat))) @ B3) @ ((F2 @ A5) @ Acc))))))))).
% 0.33/5.21 thf(fact_2597_fold__atLeastAtMost__nat_Oelims, axiom, (! [A:$tType,X:(nat > (A > A)),Xa3:nat,Xb3:nat,Xc:A,Y3:A]: ((((((((set_fo6178422350223883121st_nat @ A) @ X) @ Xa3) @ Xb3) @ Xc) = Y3) => (((((ord_less @ nat) @ Xb3) @ Xa3) => (Y3 = Xc)) & ((~ ((((ord_less @ nat) @ Xb3) @ Xa3))) => (Y3 = (((((set_fo6178422350223883121st_nat @ A) @ X) @ (((plus_plus @ nat) @ Xa3) @ (one_one @ nat))) @ Xb3) @ ((X @ Xa3) @ Xc))))))))).
% 0.33/5.21 thf(fact_2598_Suc__mod__eq__add3__mod, axiom, (! [M:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M)))) @ N) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ N))))).
% 0.33/5.21 thf(fact_2599_Suc__div__eq__add3__div, axiom, (! [M:nat,N:nat]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M)))) @ N) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ N))))).
% 0.33/5.21 thf(fact_2600_num_Osize__gen_I1_J, axiom, ((size_num @ one2) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_2601_take__bit__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: ((((((bit_se2584673776208193580ke_bit @ A) @ N) @ A3) = (zero_zero @ A)) = (((dvd_dvd @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_2602_take__bit__nat__less__self__iff, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ M)) @ M) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ M))))).
% 0.33/5.21 thf(fact_2603_take__bit__Suc__minus__bit0, axiom, (! [N:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2604_take__bit__int__less__self__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ K) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ K))))).
% 0.33/5.21 thf(fact_2605_take__bit__int__greater__eq__self__iff, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ K) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) = (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2606_take__bit__int__eq__self, axiom, (! [K:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => ((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) = K)))))).
% 0.33/5.21 thf(fact_2607_take__bit__int__eq__self__iff, axiom, (! [N:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) = K) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2608_signed__take__bit__eq__take__bit__shift, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N3:nat,K3:int]: ((((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (suc @ N3)) @ (((plus_plus @ int) @ K3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3)))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3)))))).
% 0.33/5.21 thf(fact_2609_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))))))).
% 0.33/5.21 thf(fact_2610_take__bit__Suc__minus__1__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (suc @ N))) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2611_take__bit__Suc, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2612_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))))))))).
% 0.33/5.21 thf(fact_2613_take__bit__int__less__eq, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ K) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)) @ (((minus_minus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))))).
% 0.33/5.21 thf(fact_2614_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)))))).
% 0.33/5.21 thf(fact_2615_xor__nat__unfold, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N3) @ ((((if @ nat) @ (N3 = (zero_zero @ nat))) @ M2) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N3) @ ((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) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2616_and__int__rec, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K3:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2617_xor__nat__rec, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M2:nat,N3:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) != (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2618_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)))))))))))).
% 0.33/5.21 thf(fact_2619_one__xor__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ A3) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))) @ ((zero_neq_one_of_bool @ A) @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))))).
% 0.33/5.21 thf(fact_2620_stable__imp__take__bit__eq, axiom, (! [A:$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)))))))))))).
% 0.33/5.21 thf(fact_2621_take__bit__minus__small__eq, axiom, (! [K:int,N:nat]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => ((((bit_se2584673776208193580ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ K)) = (((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ K))))))).
% 0.33/5.21 thf(fact_2622_odd__mod__4__div__2, axiom, (! [N:nat]: ((((((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = ((numeral_numeral @ nat) @ (bit1 @ one2))) => (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2623_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))))))))).
% 0.33/5.21 thf(fact_2624_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)))))).
% 0.33/5.21 thf(fact_2625_binomial__code, axiom, (binomial = (^ [N3:nat,K3:nat]: (((((if @ nat) @ (((ord_less @ nat) @ N3) @ K3)) @ (zero_zero @ nat)) @ ((((if @ nat) @ (((ord_less @ nat) @ N3) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K3))) @ ((binomial @ N3) @ (((minus_minus @ nat) @ N3) @ K3))) @ (((divide_divide @ nat) @ (((((set_fo6178422350223883121st_nat @ nat) @ (times_times @ nat)) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ N3) @ K3)) @ (one_one @ nat))) @ N3) @ (one_one @ nat))) @ ((semiring_char_0_fact @ nat) @ K3)))))))).
% 0.33/5.21 thf(fact_2626_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)))))).
% 0.33/5.21 thf(fact_2627_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)))))).
% 0.33/5.21 thf(fact_2628_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))))))))).
% 0.33/5.21 thf(fact_2629_signed__take__bit__numeral__minus__bit0, axiom, (! [L:num,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ (pred_numeral @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2630_xor__nonnegative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se5824344971392196577ns_xor @ int) @ K) @ L)) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)))))).
% 0.33/5.21 thf(fact_2631_xor__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) != (((ord_less @ int) @ L) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2632_binomial__Suc__n, axiom, (! [N:nat]: ((((binomial @ (suc @ N)) @ N) = (suc @ N))))).
% 0.33/5.21 thf(fact_2633_binomial__n__n, axiom, (! [N:nat]: ((((binomial @ N) @ N) = (one_one @ nat))))).
% 0.33/5.21 thf(fact_2634_binomial__1, axiom, (! [N:nat]: ((((binomial @ N) @ (suc @ (zero_zero @ nat))) = N)))).
% 0.33/5.21 thf(fact_2635_binomial__0__Suc, axiom, (! [K:nat]: ((((binomial @ (zero_zero @ nat)) @ (suc @ K)) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2636_binomial__eq__0__iff, axiom, (! [N:nat,K:nat]: (((((binomial @ N) @ K) = (zero_zero @ nat)) = (((ord_less @ nat) @ N) @ K))))).
% 0.33/5.21 thf(fact_2637_binomial__Suc__Suc, axiom, (! [N:nat,K:nat]: ((((binomial @ (suc @ N)) @ (suc @ K)) = (((plus_plus @ nat) @ ((binomial @ N) @ K)) @ ((binomial @ N) @ (suc @ K))))))).
% 0.33/5.21 thf(fact_2638_pred__numeral__simps_I1_J, axiom, ((pred_numeral @ one2) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_2639_binomial__n__0, axiom, (! [N:nat]: ((((binomial @ N) @ (zero_zero @ nat)) = (one_one @ nat))))).
% 0.33/5.21 thf(fact_2640_eq__numeral__Suc, axiom, (! [K:num,N:nat]: (((((numeral_numeral @ nat) @ K) = (suc @ N)) = ((pred_numeral @ K) = N))))).
% 0.33/5.21 thf(fact_2641_Suc__eq__numeral, axiom, (! [N:nat,K:num]: ((((suc @ N) = ((numeral_numeral @ nat) @ K)) = (N = (pred_numeral @ K)))))).
% 0.33/5.21 thf(fact_2642_pred__numeral__inc, axiom, (! [K:num]: (((pred_numeral @ (inc @ K)) = ((numeral_numeral @ nat) @ K))))).
% 0.33/5.21 thf(fact_2643_zero__less__binomial__iff, axiom, (! [N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((binomial @ N) @ K)) = (((ord_less_eq @ nat) @ K) @ N))))).
% 0.33/5.21 thf(fact_2644_less__Suc__numeral, axiom, (! [N:nat,K:num]: (((((ord_less @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ K)) = (((ord_less @ nat) @ N) @ (pred_numeral @ K)))))).
% 0.33/5.21 thf(fact_2645_less__numeral__Suc, axiom, (! [K:num,N:nat]: (((((ord_less @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N)) = (((ord_less @ nat) @ (pred_numeral @ K)) @ N))))).
% 0.33/5.21 thf(fact_2646_le__Suc__numeral, axiom, (! [N:nat,K:num]: (((((ord_less_eq @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ K)) = (((ord_less_eq @ nat) @ N) @ (pred_numeral @ K)))))).
% 0.33/5.21 thf(fact_2647_le__numeral__Suc, axiom, (! [K:num,N:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N)) = (((ord_less_eq @ nat) @ (pred_numeral @ K)) @ N))))).
% 0.33/5.21 thf(fact_2648_diff__numeral__Suc, axiom, (! [K:num,N:nat]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N)) = (((minus_minus @ nat) @ (pred_numeral @ K)) @ N))))).
% 0.33/5.21 thf(fact_2649_diff__Suc__numeral, axiom, (! [N:nat,K:num]: (((((minus_minus @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ K)) = (((minus_minus @ nat) @ N) @ (pred_numeral @ K)))))).
% 0.33/5.21 thf(fact_2650_pred__numeral__simps_I3_J, axiom, (! [K:num]: (((pred_numeral @ (bit1 @ K)) = ((numeral_numeral @ nat) @ (bit0 @ K)))))).
% 0.33/5.21 thf(fact_2651_and__nat__numerals_I3_J, axiom, (! [X:num]: (((((bit_se5824344872417868541ns_and @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X))) @ (suc @ (zero_zero @ nat))) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2652_and__nat__numerals_I1_J, axiom, (! [Y3:num]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y3))) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2653_and__nat__numerals_I2_J, axiom, (! [Y3:num]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y3))) = (one_one @ nat))))).
% 0.33/5.21 thf(fact_2654_and__nat__numerals_I4_J, axiom, (! [X:num]: (((((bit_se5824344872417868541ns_and @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X))) @ (suc @ (zero_zero @ nat))) = (one_one @ nat))))).
% 0.33/5.21 thf(fact_2655_Suc__0__and__eq, axiom, (! [N:nat]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2656_and__Suc__0__eq, axiom, (! [N:nat]: (((((bit_se5824344872417868541ns_and @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2657_signed__take__bit__numeral__bit0, axiom, (! [L:num,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((numeral_numeral @ int) @ (bit0 @ K))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ (pred_numeral @ L)) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2658_choose__one, axiom, (! [N:nat]: ((((binomial @ N) @ (one_one @ nat)) = N)))).
% 0.33/5.21 thf(fact_2659_binomial__eq__0, axiom, (! [N:nat,K:nat]: (((((ord_less @ nat) @ N) @ K) => (((binomial @ N) @ K) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_2660_Suc__times__binomial__eq, axiom, (! [N:nat,K:nat]: (((((times_times @ nat) @ (suc @ N)) @ ((binomial @ N) @ K)) = (((times_times @ nat) @ ((binomial @ (suc @ N)) @ (suc @ K))) @ (suc @ K)))))).
% 0.33/5.21 thf(fact_2661_Suc__times__binomial, axiom, (! [K:nat,N:nat]: (((((times_times @ nat) @ (suc @ K)) @ ((binomial @ (suc @ N)) @ (suc @ K))) = (((times_times @ nat) @ (suc @ N)) @ ((binomial @ N) @ K)))))).
% 0.33/5.21 thf(fact_2662_XOR__lower, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se5824344971392196577ns_xor @ int) @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_2663_binomial__symmetric, axiom, (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((binomial @ N) @ K) = ((binomial @ N) @ (((minus_minus @ nat) @ N) @ K))))))).
% 0.33/5.21 thf(fact_2664_choose__mult__lemma, axiom, (! [M:nat,R:nat,K:nat]: (((((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M) @ R)) @ K)) @ (((plus_plus @ nat) @ M) @ K))) @ ((binomial @ (((plus_plus @ nat) @ M) @ K)) @ K)) = (((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M) @ R)) @ K)) @ K)) @ ((binomial @ (((plus_plus @ nat) @ M) @ R)) @ M)))))).
% 0.33/5.21 thf(fact_2665_binomial__le__pow, axiom, (! [R:nat,N:nat]: (((((ord_less_eq @ nat) @ R) @ N) => (((ord_less_eq @ nat) @ ((binomial @ N) @ R)) @ (((power_power @ nat) @ N) @ R)))))).
% 0.33/5.21 thf(fact_2666_binomial__gbinomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat,K:nat]: ((((semiring_1_of_nat @ A) @ ((binomial @ N) @ K)) = (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N)) @ K)))))))).
% 0.33/5.21 thf(fact_2667_numeral__eq__Suc, axiom, ((numeral_numeral @ nat) = (^ [K3:num]: ((suc @ (pred_numeral @ K3)))))).
% 0.33/5.21 thf(fact_2668_zero__less__binomial, axiom, (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((binomial @ N) @ K)))))).
% 0.33/5.21 thf(fact_2669_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)))))).
% 0.33/5.21 thf(fact_2670_choose__mult, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => ((((times_times @ nat) @ ((binomial @ N) @ M)) @ ((binomial @ M) @ K)) = (((times_times @ nat) @ ((binomial @ N) @ K)) @ ((binomial @ (((minus_minus @ nat) @ N) @ K)) @ (((minus_minus @ nat) @ M) @ K))))))))).
% 0.33/5.21 thf(fact_2671_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)))))).
% 0.33/5.21 thf(fact_2672_binomial__Suc__Suc__eq__times, axiom, (! [N:nat,K:nat]: ((((binomial @ (suc @ N)) @ (suc @ K)) = (((divide_divide @ nat) @ (((times_times @ nat) @ (suc @ N)) @ ((binomial @ N) @ K))) @ (suc @ K)))))).
% 0.33/5.21 thf(fact_2673_and__nat__def, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M2:nat,N3:nat]: ((nat2 @ (((bit_se5824344872417868541ns_and @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N3))))))).
% 0.33/5.21 thf(fact_2674_xor__nat__def, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M2:nat,N3:nat]: ((nat2 @ (((bit_se5824344971392196577ns_xor @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N3))))))).
% 0.33/5.21 thf(fact_2675_pred__numeral__def, axiom, (pred_numeral = (^ [K3:num]: ((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ K3)) @ (one_one @ nat)))))).
% 0.33/5.21 thf(fact_2676_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)))))).
% 0.33/5.21 thf(fact_2677_binomial__fact__lemma, axiom, (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => ((((times_times @ nat) @ (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ K)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N) @ K)))) @ ((binomial @ N) @ K)) = ((semiring_char_0_fact @ nat) @ N)))))).
% 0.33/5.21 thf(fact_2678_fact__numeral, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [K:num]: ((((semiring_char_0_fact @ A) @ ((numeral_numeral @ nat) @ K)) = (((times_times @ A) @ ((numeral_numeral @ A) @ K)) @ ((semiring_char_0_fact @ A) @ (pred_numeral @ K)))))))))).
% 0.33/5.21 thf(fact_2679_binomial__ge__n__over__k__pow__k, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((ord_less_eq @ A) @ (((power_power @ A) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ N)) @ ((semiring_1_of_nat @ A) @ K))) @ K)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K)))))))))).
% 0.33/5.21 thf(fact_2680_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)))))))).
% 0.33/5.21 thf(fact_2681_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))))))))).
% 0.33/5.21 thf(fact_2682_binomial__le__pow2, axiom, (! [N:nat,K:nat]: ((((ord_less_eq @ nat) @ ((binomial @ N) @ K)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2683_binomial__altdef__nat, axiom, (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((binomial @ N) @ K) = (((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ N)) @ (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ K)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N) @ K))))))))).
% 0.33/5.21 thf(fact_2684_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))))))).
% 0.33/5.21 thf(fact_2685_fact__binomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => ((((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K))) = (((divide_divide @ A) @ ((semiring_char_0_fact @ A) @ N)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N) @ K))))))))))).
% 0.33/5.21 thf(fact_2686_binomial__fact, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((semiring_1_of_nat @ A) @ ((binomial @ N) @ K)) = (((divide_divide @ A) @ ((semiring_char_0_fact @ A) @ N)) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N) @ K)))))))))))).
% 0.33/5.21 thf(fact_2687_take__bit__numeral__bit0, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [L:num,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ ((numeral_numeral @ A) @ (bit0 @ K))) = (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (pred_numeral @ L)) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2688_take__bit__numeral__minus__bit0, axiom, (! [L:num,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (pred_numeral @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))).
% 0.33/5.21 thf(fact_2689_XOR__upper, axiom, (! [X:int,N:nat,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less @ int) @ X) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => ((((ord_less @ int) @ Y3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => (((ord_less @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ X) @ Y3)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))))).
% 0.33/5.21 thf(fact_2690_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))))))).
% 0.33/5.21 thf(fact_2691_and__nat__unfold, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ ((M2 = (zero_zero @ nat)) | (N3 = (zero_zero @ nat)))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2692_and__nat__rec, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M2:nat,N3:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) & (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2693_xor__int__rec, axiom, ((bit_se5824344971392196577ns_xor @ int) = (^ [K3:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) != (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2694_xor__int__unfold, axiom, ((bit_se5824344971392196577ns_xor @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ (K3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ ((bit_ri4277139882892585799ns_not @ int) @ L3)) @ ((((if @ int) @ (L3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ ((bit_ri4277139882892585799ns_not @ int) @ K3)) @ ((((if @ int) @ (K3 = (zero_zero @ int))) @ L3) @ ((((if @ int) @ (L3 = (zero_zero @ int))) @ K3) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (((modulo_modulo @ int) @ K3) @ ((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) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))))))).
% 0.33/5.21 thf(fact_2695_signed__take__bit__eq__take__bit__minus, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N3:nat,K3:int]: ((((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (suc @ N3)) @ K3)) @ (((times_times @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N3))) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K3) @ N3)))))))).
% 0.33/5.21 thf(fact_2696_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))))))))))).
% 0.33/5.21 thf(fact_2697_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))))))).
% 0.33/5.21 thf(fact_2698_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)))))).
% 0.33/5.21 thf(fact_2699_push__bit__numeral__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: (((((bit_se4730199178511100633sh_bit @ A) @ ((numeral_numeral @ nat) @ N)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ N)))))))))).
% 0.33/5.21 thf(fact_2700_mask__nat__positive__iff, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((bit_se2239418461657761734s_mask @ nat) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.33/5.21 thf(fact_2701_push__bit__nonnegative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se4730199178511100633sh_bit @ int) @ N) @ K)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2702_push__bit__negative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se4730199178511100633sh_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2703_push__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2704_push__bit__eq__0__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,A3:A]: ((((((bit_se4730199178511100633sh_bit @ A) @ N) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2705_concat__bit__of__zero__1, axiom, (! [N:nat,L:int]: (((((bit_concat_bit @ N) @ (zero_zero @ int)) @ L) = (((bit_se4730199178511100633sh_bit @ int) @ N) @ L))))).
% 0.33/5.21 thf(fact_2706_mask__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se2239418461657761734s_mask @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2707_mask__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se2239418461657761734s_mask @ A) @ N) = (zero_zero @ A)) = (N = (zero_zero @ nat))))))))).
% 0.33/5.21 thf(fact_2708_bit_Oconj__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ X) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2709_bit_Oconj__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) @ X) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2710_bit_Ocompl__zero, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_2711_bit_Ocompl__one, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2712_bit__numeral__Bit0__Suc__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ (bit0 @ M))) @ (suc @ N)) = (((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ M)) @ N)))))))).
% 0.33/5.21 thf(fact_2713_push__bit__Suc__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,K:num]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ K)) = (((bit_se4730199178511100633sh_bit @ A) @ N) @ ((numeral_numeral @ A) @ (bit0 @ K)))))))))).
% 0.33/5.21 thf(fact_2714_bit__numeral__Bit1__Suc__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ (bit1 @ M))) @ (suc @ N)) = (((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ M)) @ N)))))))).
% 0.33/5.21 thf(fact_2715_mask__Suc__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se2239418461657761734s_mask @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2716_bit_Oxor__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2717_bit_Oxor__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2718_bit_Oxor__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_ri4277139882892585799ns_not @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2719_bit_Oxor__one__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) = ((bit_ri4277139882892585799ns_not @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2720_take__bit__minus__one__eq__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_se2239418461657761734s_mask @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2721_not__negative__int__iff, axiom, (! [K:int]: (((((ord_less @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ K)) @ (zero_zero @ int)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2722_not__nonnegative__int__iff, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ K)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2723_minus__not__numeral__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((uminus_uminus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ (inc @ N))))))))).
% 0.33/5.21 thf(fact_2724_signed__take__bit__nonnegative__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N))))))).
% 0.33/5.21 thf(fact_2725_signed__take__bit__negative__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((bit_se5641148757651400278ts_bit @ int) @ K) @ N))))).
% 0.33/5.21 thf(fact_2726_push__bit__Suc__minus__numeral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat,K:num]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = (((bit_se4730199178511100633sh_bit @ A) @ N) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ K))))))))))).
% 0.33/5.21 thf(fact_2727_bit__minus__numeral__Bit0__Suc__iff, axiom, (! [W2:num,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ W2)))) @ (suc @ N)) = (((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ W2))) @ N))))).
% 0.33/5.21 thf(fact_2728_bit__minus__numeral__Bit1__Suc__iff, axiom, (! [W2:num,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ W2)))) @ (suc @ N)) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ ((numeral_numeral @ int) @ W2)) @ N))))))).
% 0.33/5.21 thf(fact_2729_push__bit__minus__one__eq__not__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2730_not__one__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 0.33/5.21 thf(fact_2731_bit__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (zero_zero @ nat)) = (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 0.33/5.21 thf(fact_2732_push__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N)) @ A3) = (((bit_se4730199178511100633sh_bit @ A) @ N) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2733_push__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (one_one @ A)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)))))))).
% 0.33/5.21 thf(fact_2734_even__push__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ A3)) = ((N != (zero_zero @ nat)) | (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))))))))).
% 0.33/5.21 thf(fact_2735_push__bit__of__Suc__0, axiom, (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2736_push__bit__minus__numeral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [L:num,K:num]: (((((bit_se4730199178511100633sh_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = (((bit_se4730199178511100633sh_bit @ A) @ (pred_numeral @ L)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ K))))))))))).
% 0.33/5.21 thf(fact_2737_bit__minus__numeral__int_I1_J, axiom, (! [W2:num,N:num]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ W2)))) @ ((numeral_numeral @ nat) @ N)) = (((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ W2))) @ (pred_numeral @ N)))))).
% 0.33/5.21 thf(fact_2738_bit__minus__numeral__int_I2_J, axiom, (! [W2:num,N:num]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ W2)))) @ ((numeral_numeral @ nat) @ N)) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ ((numeral_numeral @ int) @ W2)) @ (pred_numeral @ N)))))))).
% 0.33/5.21 thf(fact_2739_bit__mod__2__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N) = ((N = (zero_zero @ nat)) & (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))))))))))).
% 0.33/5.21 thf(fact_2740_of__nat__mask__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ ((bit_se2239418461657761734s_mask @ nat) @ N)) = ((bit_se2239418461657761734s_mask @ A) @ N)))))))).
% 0.33/5.21 thf(fact_2741_bit__of__nat__iff__bit, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N) = (((bit_se5641148757651400278ts_bit @ nat) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_2742_push__bit__of__nat, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,M:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ ((semiring_1_of_nat @ A) @ M)) = ((semiring_1_of_nat @ A) @ (((bit_se4730199178511100633sh_bit @ nat) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_2743_of__nat__push__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((bit_se4730199178511100633sh_bit @ nat) @ M) @ N)) = (((bit_se4730199178511100633sh_bit @ A) @ M) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2744_bit__push__bit__iff__int, axiom, (! [M:nat,K:int,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ (((bit_se4730199178511100633sh_bit @ int) @ M) @ K)) @ N) = ((((ord_less_eq @ nat) @ M) @ N) & (((bit_se5641148757651400278ts_bit @ int) @ K) @ (((minus_minus @ nat) @ N) @ M))))))).
% 0.33/5.21 thf(fact_2745_push__bit__minus, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ ((uminus_uminus @ A) @ A3)) = ((uminus_uminus @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ A3))))))))).
% 0.33/5.21 thf(fact_2746_less__eq__mask, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ ((bit_se2239418461657761734s_mask @ nat) @ N))))).
% 0.33/5.21 thf(fact_2747_bit__minus__int__iff, axiom, (! [K:int,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ K)) @ N) = (((bit_se5641148757651400278ts_bit @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ (((minus_minus @ int) @ K) @ (one_one @ int)))) @ N))))).
% 0.33/5.21 thf(fact_2748_bit__iff__and__push__bit__not__eq__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A5:A,N3:nat]: (((((bit_se5824344872417868541ns_and @ A) @ A5) @ (((bit_se4730199178511100633sh_bit @ A) @ N3) @ (one_one @ A))) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2749_unset__bit__eq__and__not, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_se2638667681897837118et_bit @ A) = (^ [N3:nat,A5:A]: ((((bit_se5824344872417868541ns_and @ A) @ A5) @ ((bit_ri4277139882892585799ns_not @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N3) @ (one_one @ A))))))))))).
% 0.33/5.21 thf(fact_2750_unset__bit__int__def, axiom, ((bit_se2638667681897837118et_bit @ int) = (^ [N3:nat,K3:int]: ((((bit_se5824344872417868541ns_and @ int) @ K3) @ ((bit_ri4277139882892585799ns_not @ int) @ (((bit_se4730199178511100633sh_bit @ int) @ N3) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_2751_take__bit__not__mask__eq__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N))) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2752_not__bit__1__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ (suc @ N)))))))))).
% 0.33/5.21 thf(fact_2753_bit__1__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ N) = (N = (zero_zero @ nat))))))))).
% 0.33/5.21 thf(fact_2754_bit__numeral__simps_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:num]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ ((numeral_numeral @ nat) @ N)))))))))).
% 0.33/5.21 thf(fact_2755_bit__take__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ M) @ A3)) @ N) = ((((ord_less @ nat) @ N) @ M) & (((bit_se5641148757651400278ts_bit @ A) @ A3) @ N))))))))).
% 0.33/5.21 thf(fact_2756_bit__of__bool__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [B2:$o,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((zero_neq_one_of_bool @ A) @ B2)) @ N) = (B2 & (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2757_bit__not__iff__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A3)) @ N) = (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) != (zero_zero @ A)) & (~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N))))))))))).
% 0.33/5.21 thf(fact_2758_mask__nonnegative__int, axiom, (! [N:nat]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((bit_se2239418461657761734s_mask @ int) @ N))))).
% 0.33/5.21 thf(fact_2759_not__mask__negative__int, axiom, (! [N:nat]: ((~ ((((ord_less @ int) @ ((bit_se2239418461657761734s_mask @ int) @ N)) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2760_flip__bit__nat__def, axiom, ((bit_se8732182000553998342ip_bit @ nat) = (^ [M2:nat,N3:nat]: ((((bit_se5824344971392196577ns_xor @ nat) @ N3) @ (((bit_se4730199178511100633sh_bit @ nat) @ M2) @ (one_one @ nat))))))).
% 0.33/5.21 thf(fact_2761_flip__bit__int__def, axiom, ((bit_se8732182000553998342ip_bit @ int) = (^ [N3:nat,K3:int]: ((((bit_se5824344971392196577ns_xor @ int) @ K3) @ (((bit_se4730199178511100633sh_bit @ int) @ N3) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_2762_minus__exp__eq__not__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: ((((uminus_uminus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2763_minus__eq__not__plus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((uminus_uminus @ A) = (^ [A5:A]: ((((plus_plus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A5)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2764_minus__eq__not__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((uminus_uminus @ A) = (^ [A5:A]: (((bit_ri4277139882892585799ns_not @ A) @ (((minus_minus @ A) @ A5) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2765_not__eq__complement, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4277139882892585799ns_not @ A) = (^ [A5:A]: ((((minus_minus @ A) @ ((uminus_uminus @ A) @ A5)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2766_not__int__def, axiom, ((bit_ri4277139882892585799ns_not @ int) = (^ [K3:int]: ((((minus_minus @ int) @ ((uminus_uminus @ int) @ K3)) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2767_bit__not__int__iff_H, axiom, (! [K:int,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ K)) @ (one_one @ int))) @ N) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N))))))).
% 0.33/5.21 thf(fact_2768_and__not__numerals_I1_J, axiom, ((((bit_se5824344872417868541ns_and @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = (zero_zero @ int))).
% 0.33/5.21 thf(fact_2769_minus__numeral__inc__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ N))) = ((bit_ri4277139882892585799ns_not @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2770_less__mask, axiom, (! [N:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N) => (((ord_less @ nat) @ N) @ ((bit_se2239418461657761734s_mask @ nat) @ N)))))).
% 0.33/5.21 thf(fact_2771_flip__bit__eq__xor, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se8732182000553998342ip_bit @ A) = (^ [N3:nat,A5:A]: ((((bit_se5824344971392196577ns_xor @ A) @ A5) @ (((bit_se4730199178511100633sh_bit @ A) @ N3) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2772_and__not__numerals_I4_J, axiom, (! [M:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = ((numeral_numeral @ int) @ (bit0 @ M)))))).
% 0.33/5.21 thf(fact_2773_and__not__numerals_I2_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (one_one @ int))))).
% 0.33/5.21 thf(fact_2774_bit__imp__take__bit__positive, axiom, (! [N:nat,M:nat,K:int]: (((((ord_less @ nat) @ N) @ M) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N) => (((ord_less @ int) @ (zero_zero @ int)) @ (((bit_se2584673776208193580ke_bit @ int) @ M) @ K))))))).
% 0.33/5.21 thf(fact_2775_not__numeral__Bit0__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((bit_ri4277139882892585799ns_not @ A) @ ((numeral_numeral @ A) @ (bit0 @ N))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit1 @ N)))))))))).
% 0.33/5.21 thf(fact_2776_bit__concat__bit__iff, axiom, (! [M:nat,K:int,L:int,N:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ (((bit_concat_bit @ M) @ K) @ L)) @ N) = (((((ord_less @ nat) @ N) @ M) & (((bit_se5641148757651400278ts_bit @ int) @ K) @ N)) | ((((ord_less_eq @ nat) @ M) @ N) & (((bit_se5641148757651400278ts_bit @ int) @ L) @ (((minus_minus @ nat) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_2777_push__bit__double, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (((times_times @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ A3)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2778_signed__take__bit__eq__concat__bit, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N3:nat,K3:int]: ((((bit_concat_bit @ N3) @ K3) @ ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K3) @ N3)))))))).
% 0.33/5.21 thf(fact_2779_exp__eq__0__imp__not__bit, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat,A3:A]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) = (zero_zero @ A)) => (~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N)))))))))).
% 0.33/5.21 thf(fact_2780_bit__Suc, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (suc @ N)) = (((bit_se5641148757651400278ts_bit @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N)))))))).
% 0.33/5.21 thf(fact_2781_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)))))).
% 0.33/5.21 thf(fact_2782_int__bit__bound, axiom, (! [K:int]: ((~ ((! [N4:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ N4) @ M3) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ M3) = (((bit_se5641148757651400278ts_bit @ int) @ K) @ N4))))) => (~ (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ (((minus_minus @ nat) @ N4) @ (one_one @ nat))) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N4))))))))))))))).
% 0.33/5.21 thf(fact_2783_and__not__numerals_I5_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_2784_and__not__numerals_I7_J, axiom, (! [M:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = ((numeral_numeral @ int) @ (bit0 @ M)))))).
% 0.33/5.21 thf(fact_2785_and__not__numerals_I3_J, axiom, (! [N:num]: (((((bit_se5824344872417868541ns_and @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (zero_zero @ int))))).
% 0.33/5.21 thf(fact_2786_push__bit__int__def, axiom, ((bit_se4730199178511100633sh_bit @ int) = (^ [N3:nat,K3:int]: ((((times_times @ int) @ K3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3)))))).
% 0.33/5.21 thf(fact_2787_push__bit__nat__def, axiom, ((bit_se4730199178511100633sh_bit @ nat) = (^ [N3:nat,M2:nat]: ((((times_times @ nat) @ M2) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))).
% 0.33/5.21 thf(fact_2788_push__bit__eq__mult, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se4730199178511100633sh_bit @ A) = (^ [N3:nat,A5:A]: ((((times_times @ A) @ A5) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N3))))))))).
% 0.33/5.21 thf(fact_2789_and__exp__eq__0__iff__not__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,N:nat]: ((((((bit_se5824344872417868541ns_and @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (zero_zero @ A)) = (~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N)))))))))).
% 0.33/5.21 thf(fact_2790_Suc__mask__eq__exp, axiom, (! [N:nat]: (((suc @ ((bit_se2239418461657761734s_mask @ nat) @ N)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2791_mask__nat__less__exp, axiom, (! [N:nat]: ((((ord_less @ nat) @ ((bit_se2239418461657761734s_mask @ nat) @ N)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))).
% 0.33/5.21 thf(fact_2792_and__not__numerals_I9_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_2793_and__not__numerals_I6_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_2794_push__bit__minus__one, axiom, (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ int) @ N) @ ((uminus_uminus @ int) @ (one_one @ int))) = ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))).
% 0.33/5.21 thf(fact_2795_semiring__bit__operations__class_Oeven__mask__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ N)) = (N = (zero_zero @ nat))))))))).
% 0.33/5.21 thf(fact_2796_even__bit__succ__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N) | (N = (zero_zero @ nat))))))))))).
% 0.33/5.21 thf(fact_2797_odd__bit__iff__bit__pred, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_2798_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))))))).
% 0.33/5.21 thf(fact_2799_mask__int__def, axiom, ((bit_se2239418461657761734s_mask @ int) = (^ [N3:nat]: ((((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3)) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2800_mask__nat__def, axiom, ((bit_se2239418461657761734s_mask @ nat) = (^ [N3:nat]: ((((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)) @ (one_one @ nat)))))).
% 0.33/5.21 thf(fact_2801_bit__sum__mult__2__cases, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,B2:A,N:nat]: (((! [J2:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (suc @ J2)))))) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ N) = (((N = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))) & ((N != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)) @ N))))))))))).
% 0.33/5.21 thf(fact_2802_bit__rec, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A5:A,N3:nat]: ((((N3 = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A5)))) & ((N3 != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat)))))))))))).
% 0.33/5.21 thf(fact_2803_mask__eq__exp__minus__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se2239418461657761734s_mask @ A) = (^ [N3:nat]: ((((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N3)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2804_and__not__numerals_I8_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((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) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_2805_not__int__rec, axiom, ((bit_ri4277139882892585799ns_not @ int) = (^ [K3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((bit_ri4277139882892585799ns_not @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2806_set__bit__eq, axiom, ((bit_se5668285175392031749et_bit @ int) = (^ [N3:nat,K3:int]: ((((plus_plus @ int) @ K3) @ (((times_times @ int) @ ((zero_neq_one_of_bool @ int) @ (~ ((((bit_se5641148757651400278ts_bit @ int) @ K3) @ N3))))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3))))))).
% 0.33/5.21 thf(fact_2807_unset__bit__eq, axiom, ((bit_se2638667681897837118et_bit @ int) = (^ [N3:nat,K3:int]: ((((minus_minus @ int) @ K3) @ (((times_times @ int) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K3) @ N3))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N3))))))).
% 0.33/5.21 thf(fact_2808_num_Osize_I6_J, axiom, (! [X32:num]: ((((size_size @ num) @ (bit1 @ X32)) = (((plus_plus @ nat) @ ((size_size @ num) @ X32)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2809_num_Osize_I5_J, axiom, (! [X22:num]: ((((size_size @ num) @ (bit0 @ X22)) = (((plus_plus @ nat) @ ((size_size @ num) @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2810_or__int__rec, axiom, ((bit_se1065995026697491101ons_or @ int) = (^ [K3:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) | (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2811_drop__bit__numeral__minus__bit1, axiom, (! [L:num,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ (pred_numeral @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K)))))))).
% 0.33/5.21 thf(fact_2812_or__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((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) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2813_or__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2814_or_Oright__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se1065995026697491101ons_or @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.33/5.21 thf(fact_2815_or_Oleft__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se1065995026697491101ons_or @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_2816_drop__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_2817_bit_Odisj__one__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2818_bit_Odisj__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2819_drop__bit__of__bool, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,B2:$o]: (((((bit_se4197421643247451524op_bit @ A) @ N) @ ((zero_neq_one_of_bool @ A) @ B2)) = ((zero_neq_one_of_bool @ A) @ ((N = (zero_zero @ nat)) & B2))))))))).
% 0.33/5.21 thf(fact_2820_or__nonnegative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se1065995026697491101ons_or @ int) @ K) @ L)) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)))))).
% 0.33/5.21 thf(fact_2821_or__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se1065995026697491101ons_or @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) | (((ord_less @ int) @ L) @ (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_2822_drop__bit__nonnegative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se4197421643247451524op_bit @ int) @ N) @ K)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_2823_drop__bit__negative__int__iff, axiom, (! [N:nat,K:int]: (((((ord_less @ int) @ (((bit_se4197421643247451524op_bit @ int) @ N) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2824_drop__bit__minus__one, axiom, (! [N:nat]: (((((bit_se4197421643247451524op_bit @ int) @ N) @ ((uminus_uminus @ int) @ (one_one @ int))) = ((uminus_uminus @ int) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2825_or__nat__numerals_I4_J, axiom, (! [X:num]: (((((bit_se1065995026697491101ons_or @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X)))))).
% 0.33/5.21 thf(fact_2826_or__nat__numerals_I2_J, axiom, (! [Y3:num]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y3))) = ((numeral_numeral @ nat) @ (bit1 @ Y3)))))).
% 0.33/5.21 thf(fact_2827_drop__bit__Suc__bit0, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,K:num]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ (bit0 @ K))) = (((bit_se4197421643247451524op_bit @ A) @ N) @ ((numeral_numeral @ A) @ K))))))))).
% 0.33/5.21 thf(fact_2828_or__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = ((numeral_numeral @ A) @ (bit1 @ Y3))))))))).
% 0.33/5.21 thf(fact_2829_or__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 0.33/5.21 thf(fact_2830_bit_Odisj__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2831_bit_Odisj__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ X) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2832_drop__bit__Suc__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,K:num]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((bit_se4197421643247451524op_bit @ A) @ N) @ ((numeral_numeral @ A) @ K))))))))).
% 0.33/5.21 thf(fact_2833_drop__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (N = (zero_zero @ nat)))))))))).
% 0.33/5.21 thf(fact_2834_or__numerals_I3_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_2835_or__nat__numerals_I1_J, axiom, (! [Y3:num]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y3))) = ((numeral_numeral @ nat) @ (bit1 @ Y3)))))).
% 0.33/5.21 thf(fact_2836_or__nat__numerals_I3_J, axiom, (! [X:num]: (((((bit_se1065995026697491101ons_or @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X)))))).
% 0.33/5.21 thf(fact_2837_or__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y3))) = ((numeral_numeral @ A) @ (bit1 @ Y3))))))))).
% 0.33/5.21 thf(fact_2838_or__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 0.33/5.21 thf(fact_2839_or__minus__numerals_I6_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) @ (one_one @ int)) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N))))))).
% 0.33/5.21 thf(fact_2840_or__minus__numerals_I2_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N))))))).
% 0.33/5.21 thf(fact_2841_drop__bit__Suc__minus__bit0, axiom, (! [N:nat,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ N) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))))))).
% 0.33/5.21 thf(fact_2842_drop__bit__numeral__minus__bit0, axiom, (! [L:num,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ (pred_numeral @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))))))).
% 0.33/5.21 thf(fact_2843_drop__bit__Suc__minus__bit1, axiom, (! [N:nat,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ N) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K)))))))).
% 0.33/5.21 thf(fact_2844_and__minus__minus__numerals, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((bit_ri4277139882892585799ns_not @ int) @ (((bit_se1065995026697491101ons_or @ int) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ N)) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_2845_or__minus__minus__numerals, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((bit_ri4277139882892585799ns_not @ int) @ (((bit_se5824344872417868541ns_and @ int) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ N)) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_2846_or__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y3:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y3))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y3))))))))))).
% 0.33/5.21 thf(fact_2847_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X:A,Y3:A]: (((((size_size @ A) @ X) != ((size_size @ A) @ Y3)) => (X != Y3)))))))).
% 0.33/5.21 thf(fact_2848_or__nat__def, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M2:nat,N3:nat]: ((nat2 @ (((bit_se1065995026697491101ons_or @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N3))))))).
% 0.33/5.21 thf(fact_2849_of__nat__or__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((bit_se1065995026697491101ons_or @ nat) @ M) @ N)) = (((bit_se1065995026697491101ons_or @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2850_drop__bit__of__nat, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,M:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N) @ ((semiring_1_of_nat @ A) @ M)) = ((semiring_1_of_nat @ A) @ (((bit_se4197421643247451524op_bit @ nat) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_2851_of__nat__drop__bit, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((bit_se4197421643247451524op_bit @ nat) @ M) @ N)) = (((bit_se4197421643247451524op_bit @ A) @ M) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2852_bit_Odisj__zero__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ X) @ (zero_zero @ A)) = X))))))).
% 0.33/5.21 thf(fact_2853_or__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,B2:A]: ((((((bit_se1065995026697491101ons_or @ A) @ A3) @ B2) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) & (B2 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_2854_set__bit__nat__def, axiom, ((bit_se5668285175392031749et_bit @ nat) = (^ [M2:nat,N3:nat]: ((((bit_se1065995026697491101ons_or @ nat) @ N3) @ (((bit_se4730199178511100633sh_bit @ nat) @ M2) @ (one_one @ nat))))))).
% 0.33/5.21 thf(fact_2855_bit__Suc__0__iff, axiom, (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (N = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_2856_not__bit__Suc__0__Suc, axiom, (! [N:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ (suc @ N))))))).
% 0.33/5.21 thf(fact_2857_bit__push__bit__iff__nat, axiom, (! [M:nat,Q4:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ nat) @ (((bit_se4730199178511100633sh_bit @ nat) @ M) @ Q4)) @ N) = ((((ord_less_eq @ nat) @ M) @ N) & (((bit_se5641148757651400278ts_bit @ nat) @ Q4) @ (((minus_minus @ nat) @ N) @ M))))))).
% 0.33/5.21 thf(fact_2858_OR__lower, axiom, (! [X:int,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((bit_se1065995026697491101ons_or @ int) @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_2859_or__greater__eq, axiom, (! [L:int,K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ L) => (((ord_less_eq @ int) @ K) @ (((bit_se1065995026697491101ons_or @ int) @ K) @ L)))))).
% 0.33/5.21 thf(fact_2860_set__bit__int__def, axiom, ((bit_se5668285175392031749et_bit @ int) = (^ [N3:nat,K3:int]: ((((bit_se1065995026697491101ons_or @ int) @ K3) @ (((bit_se4730199178511100633sh_bit @ int) @ N3) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_2861_take__bit__eq__self__iff__drop__bit__eq__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: ((((((bit_se2584673776208193580ke_bit @ A) @ N) @ A3) = A3) = ((((bit_se4197421643247451524op_bit @ A) @ N) @ A3) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_2862_not__bit__Suc__0__numeral, axiom, (! [N:num]: ((~ ((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ N))))))).
% 0.33/5.21 thf(fact_2863_or__not__numerals_I1_J, axiom, ((((bit_se1065995026697491101ons_or @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = ((bit_ri4277139882892585799ns_not @ int) @ (zero_zero @ int)))).
% 0.33/5.21 thf(fact_2864_set__bit__eq__or, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5668285175392031749et_bit @ A) = (^ [N3:nat,A5:A]: ((((bit_se1065995026697491101ons_or @ A) @ A5) @ (((bit_se4730199178511100633sh_bit @ A) @ N3) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_2865_div__push__bit__of__1__eq__drop__bit, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2866_bit__iff__and__drop__bit__eq__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A5:A,N3:nat]: (((((bit_se5824344872417868541ns_and @ A) @ (((bit_se4197421643247451524op_bit @ A) @ N3) @ A5)) @ (one_one @ A)) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_2867_bit_Ocomplement__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,X:A,Y3: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) @ Y3) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ A3) @ Y3) = ((uminus_uminus @ A) @ (one_one @ A))) => (X = Y3))))))))))).
% 0.33/5.21 thf(fact_2868_or__not__numerals_I2_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N))))))).
% 0.33/5.21 thf(fact_2869_or__not__numerals_I4_J, axiom, (! [M:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_2870_bit__nat__iff, axiom, (! [K:int,N:nat]: (((((bit_se5641148757651400278ts_bit @ nat) @ (nat2 @ K)) @ N) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((bit_se5641148757651400278ts_bit @ int) @ K) @ N)))))).
% 0.33/5.21 thf(fact_2871_num_Osize_I4_J, axiom, (((size_size @ num) @ one2) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_2872_bit_Ocompl__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A,Y3:A]: ((((((bit_se5824344872417868541ns_and @ A) @ X) @ Y3) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ X) @ Y3) = ((uminus_uminus @ A) @ (one_one @ A))) => (((bit_ri4277139882892585799ns_not @ A) @ X) = Y3))))))))).
% 0.33/5.21 thf(fact_2873_or__not__numerals_I3_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ (one_one @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N))))))).
% 0.33/5.21 thf(fact_2874_or__not__numerals_I7_J, axiom, (! [M:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ (one_one @ int))) = ((bit_ri4277139882892585799ns_not @ int) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_2875_drop__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N)) @ A3) = (((bit_se4197421643247451524op_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_2876_mask__Suc__exp, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: ((((bit_se2239418461657761734s_mask @ A) @ (suc @ N)) = (((bit_se1065995026697491101ons_or @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ ((bit_se2239418461657761734s_mask @ A) @ N))))))))).
% 0.33/5.21 thf(fact_2877_or__not__numerals_I6_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_2878_one__or__eq, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2879_or__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se1065995026697491101ons_or @ A) @ A3) @ (one_one @ A)) = (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 0.33/5.21 thf(fact_2880_mask__Suc__double, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_2881_OR__upper, axiom, (! [X:int,N:nat,Y3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less @ int) @ X) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => ((((ord_less @ int) @ Y3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) => (((ord_less @ int) @ (((bit_se1065995026697491101ons_or @ int) @ X) @ Y3)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))))).
% 0.33/5.21 thf(fact_2882_or__not__numerals_I5_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit0 @ M))) @ ((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) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_2883_signed__take__bit__def, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N3:nat,A5:A]: ((((bit_se1065995026697491101ons_or @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N3) @ A5)) @ (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ A5) @ N3))) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N3))))))))))).
% 0.33/5.21 thf(fact_2884_or__Suc__0__eq, axiom, (! [N:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2885_Suc__0__or__eq, axiom, (! [N:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))).
% 0.33/5.21 thf(fact_2886_drop__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se4197421643247451524op_bit @ A) = (^ [N3:nat,A5:A]: (((((if @ A) @ (N3 = (zero_zero @ nat))) @ A5) @ (((bit_se4197421643247451524op_bit @ A) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) @ (((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2887_or__not__numerals_I8_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((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) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_2888_or__not__numerals_I9_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit1 @ M))) @ ((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) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 0.33/5.21 thf(fact_2889_or__nat__rec, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M2:nat,N3:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) | (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 0.33/5.21 thf(fact_2890_or__int__unfold, axiom, ((bit_se1065995026697491101ons_or @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ ((K3 = ((uminus_uminus @ int) @ (one_one @ int))) | (L3 = ((uminus_uminus @ int) @ (one_one @ int))))) @ ((uminus_uminus @ int) @ (one_one @ int))) @ ((((if @ int) @ (K3 = (zero_zero @ int))) @ L3) @ ((((if @ int) @ (L3 = (zero_zero @ int))) @ K3) @ (((plus_plus @ int) @ (((ord_max @ int) @ (((modulo_modulo @ int) @ K3) @ ((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) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 0.33/5.21 thf(fact_2891_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)))))).
% 0.33/5.21 thf(fact_2892_or__minus__numerals_I4_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ (bit0 @ N)))))))).
% 0.33/5.21 thf(fact_2893_or__minus__numerals_I8_J, axiom, (! [N:num,M:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) @ ((numeral_numeral @ int) @ M)) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ (bit0 @ N)))))))).
% 0.33/5.21 thf(fact_2894_or__nat__unfold, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N3) @ ((((if @ nat) @ (N3 = (zero_zero @ nat))) @ M2) @ (((plus_plus @ nat) @ (((ord_max @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))).
% 0.33/5.21 thf(fact_2895_xor__minus__numerals_I1_J, axiom, (! [N:num,K:int]: (((((bit_se5824344971392196577ns_xor @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) @ K) = ((bit_ri4277139882892585799ns_not @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ (((neg_numeral_sub @ int) @ N) @ one2)) @ K)))))).
% 0.33/5.21 thf(fact_2896_max_Oright__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_max @ A) @ (((ord_max @ A) @ A3) @ B2)) @ B2) = (((ord_max @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_2897_max_Oleft__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_max @ A) @ A3) @ (((ord_max @ A) @ A3) @ B2)) = (((ord_max @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_2898_max_Oidem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A]: (((((ord_max @ A) @ A3) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_2899_max_Oabsorb1, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_max @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_2900_max_Oabsorb2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_max @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_2901_max_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A3) = ((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 0.33/5.21 thf(fact_2902_max_Oabsorb3, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_max @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_2903_max_Oabsorb4, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_max @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_2904_max__less__iff__conj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less @ A) @ (((ord_max @ A) @ X) @ Y3)) @ Z3) = ((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_2905_of__bool__or__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o,Q:$o]: ((((zero_neq_one_of_bool @ A) @ (P | Q)) = (((ord_max @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q))))))))).
% 0.33/5.21 thf(fact_2906_max__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((ord_max @ nat) @ (suc @ M)) @ (suc @ N)) = (suc @ (((ord_max @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_2907_max__0R, axiom, (! [N:nat]: (((((ord_max @ nat) @ N) @ (zero_zero @ nat)) = N)))).
% 0.33/5.21 thf(fact_2908_max__0L, axiom, (! [N:nat]: (((((ord_max @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 0.33/5.21 thf(fact_2909_max__nat_Oright__neutral, axiom, (! [A3:nat]: (((((ord_max @ nat) @ A3) @ (zero_zero @ nat)) = A3)))).
% 0.33/5.21 thf(fact_2910_max__nat_Oneutr__eq__iff, axiom, (! [A3:nat,B2:nat]: ((((zero_zero @ nat) = (((ord_max @ nat) @ A3) @ B2)) = ((A3 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2911_max__nat_Oleft__neutral, axiom, (! [A3:nat]: (((((ord_max @ nat) @ (zero_zero @ nat)) @ A3) = A3)))).
% 0.33/5.21 thf(fact_2912_max__nat_Oeq__neutr__iff, axiom, (! [A3:nat,B2:nat]: ((((((ord_max @ nat) @ A3) @ B2) = (zero_zero @ nat)) = ((A3 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2913_max__number__of_I1_J, axiom, (! [A:$tType]: ((((numeral @ A) & (ord @ A)) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ V))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ U)))))))))).
% 0.33/5.21 thf(fact_2914_max__0__1_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ ((numeral_numeral @ A) @ X)) @ (zero_zero @ A)) = ((numeral_numeral @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2915_max__0__1_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ X)) = ((numeral_numeral @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2916_max__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2917_max__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_2918_max__0__1_I5_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X)) = ((numeral_numeral @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2919_max__0__1_I6_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A)) = ((numeral_numeral @ A) @ X)))))))).
% 0.33/5.21 thf(fact_2920_sub__num__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((neg_numeral_sub @ A) @ one2) @ one2) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_2921_diff__numeral__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_2922_sub__num__simps_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num,L:num]: (((((neg_numeral_sub @ A) @ (bit0 @ K)) @ (bit0 @ L)) = ((neg_numeral_dbl @ A) @ (((neg_numeral_sub @ A) @ K) @ L))))))))).
% 0.33/5.21 thf(fact_2923_sub__num__simps_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num,L:num]: (((((neg_numeral_sub @ A) @ (bit1 @ K)) @ (bit1 @ L)) = ((neg_numeral_dbl @ A) @ (((neg_numeral_sub @ A) @ K) @ L))))))))).
% 0.33/5.21 thf(fact_2924_max__number__of_I2_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V)))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((numeral_numeral @ A) @ U)))))))))).
% 0.33/5.21 thf(fact_2925_max__number__of_I3_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ V))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))))))))))).
% 0.33/5.21 thf(fact_2926_max__number__of_I4_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))))))))))).
% 0.33/5.21 thf(fact_2927_add__neg__numeral__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_2928_add__neg__numeral__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((neg_numeral_sub @ A) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_2929_semiring__norm_I166_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W2:num,Y3:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y3)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ V) @ W2)) @ Y3)))))))).
% 0.33/5.21 thf(fact_2930_semiring__norm_I167_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W2:num,Y3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W2)) @ Y3)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ W2) @ V)) @ Y3)))))))).
% 0.33/5.21 thf(fact_2931_diff__numeral__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((neg_numeral_sub @ A) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_2932_max__numeral__Suc, axiom, (! [K:num,N:nat]: (((((ord_max @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N)) = (suc @ (((ord_max @ nat) @ (pred_numeral @ K)) @ N)))))).
% 0.33/5.21 thf(fact_2933_max__Suc__numeral, axiom, (! [N:nat,K:num]: (((((ord_max @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ K)) = (suc @ (((ord_max @ nat) @ N) @ (pred_numeral @ K))))))).
% 0.33/5.21 thf(fact_2934_sub__num__simps_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num,L:num]: (((((neg_numeral_sub @ A) @ (bit0 @ K)) @ (bit1 @ L)) = ((neg_numeral_dbl_dec @ A) @ (((neg_numeral_sub @ A) @ K) @ L))))))))).
% 0.33/5.21 thf(fact_2935_sub__num__simps_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num,L:num]: (((((neg_numeral_sub @ A) @ (bit1 @ K)) @ (bit0 @ L)) = ((neg_numeral_dbl_inc @ A) @ (((neg_numeral_sub @ A) @ K) @ L))))))))).
% 0.33/5.21 thf(fact_2936_drop__bit__of__Suc__0, axiom, (! [N:nat]: (((((bit_se4197421643247451524op_bit @ nat) @ N) @ (suc @ (zero_zero @ nat))) = ((zero_neq_one_of_bool @ nat) @ (N = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_2937_diff__numeral__special_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M)) @ (one_one @ A)) = (((neg_numeral_sub @ A) @ M) @ one2)))))))).
% 0.33/5.21 thf(fact_2938_diff__numeral__special_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((minus_minus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ one2) @ N)))))))).
% 0.33/5.21 thf(fact_2939_sub__num__simps_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: (((((neg_numeral_sub @ A) @ (bit1 @ K)) @ one2) = ((numeral_numeral @ A) @ (bit0 @ K))))))))).
% 0.33/5.21 thf(fact_2940_not__minus__numeral__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((bit_ri4277139882892585799ns_not @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((neg_numeral_sub @ A) @ N) @ one2)))))))).
% 0.33/5.21 thf(fact_2941_add__neg__numeral__special_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) = (((neg_numeral_sub @ A) @ one2) @ M)))))))).
% 0.33/5.21 thf(fact_2942_add__neg__numeral__special_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)) = (((neg_numeral_sub @ A) @ one2) @ M)))))))).
% 0.33/5.21 thf(fact_2943_add__neg__numeral__special_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((neg_numeral_sub @ A) @ M) @ one2)))))))).
% 0.33/5.21 thf(fact_2944_add__neg__numeral__special_I4_J, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2945_minus__sub__one__diff__one, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [M:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((neg_numeral_sub @ A) @ M) @ one2))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))).
% 0.33/5.21 thf(fact_2946_diff__numeral__special_I7_J, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_2947_diff__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((neg_numeral_sub @ A) @ one2) @ M)))))))).
% 0.33/5.21 thf(fact_2948_sub__num__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [L:num]: (((((neg_numeral_sub @ A) @ one2) @ (bit1 @ L)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ L)))))))))).
% 0.33/5.21 thf(fact_2949_xor__minus__numerals_I2_J, axiom, (! [K:int,N:num]: (((((bit_se5824344971392196577ns_xor @ int) @ K) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((bit_ri4277139882892585799ns_not @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ K) @ (((neg_numeral_sub @ int) @ N) @ one2))))))).
% 0.33/5.21 thf(fact_2950_max_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ (((ord_max @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_2951_max_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ A3) => (((ord_less @ A) @ C2) @ (((ord_max @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_2952_max_Ostrict__order__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [B3:A,A5:A]: (((A5 = (((ord_max @ A) @ A5) @ B3)) & (A5 != B3))))))))).
% 0.33/5.21 thf(fact_2953_max_Ostrict__boundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A3) => (~ (((((ord_less @ A) @ B2) @ A3) => (~ ((((ord_less @ A) @ C2) @ A3))))))))))))).
% 0.33/5.21 thf(fact_2954_less__max__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ Z3) @ (((ord_max @ A) @ X) @ Y3)) = ((((ord_less @ A) @ Z3) @ X) | (((ord_less @ A) @ Z3) @ Y3))))))))).
% 0.33/5.21 thf(fact_2955_max__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y3:A,Z3:A]: (((((plus_plus @ A) @ (((ord_max @ A) @ X) @ Y3)) @ Z3) = (((ord_max @ A) @ (((plus_plus @ A) @ X) @ Z3)) @ (((plus_plus @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_2956_max__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y3:A,Z3:A]: (((((plus_plus @ A) @ X) @ (((ord_max @ A) @ Y3) @ Z3)) = (((ord_max @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (((plus_plus @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_2957_max__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A,Z3:A]: (((((minus_minus @ A) @ (((ord_max @ A) @ X) @ Y3)) @ Z3) = (((ord_max @ A) @ (((minus_minus @ A) @ X) @ Z3)) @ (((minus_minus @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_2958_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),Y3:A,Z3:A,X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Conj @ ((Xor @ Y3) @ Z3)) @ X) = ((Xor @ ((Conj @ Y3) @ X)) @ ((Conj @ Z3) @ X))))))).
% 0.33/5.21 thf(fact_2959_abstract__boolean__algebra__sym__diff_Oxor__cancel__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ (Compl @ X)) = One2))))).
% 0.33/5.21 thf(fact_2960_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A,Z3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Conj @ X) @ ((Xor @ Y3) @ Z3)) = ((Xor @ ((Conj @ X) @ Y3)) @ ((Conj @ X) @ Z3))))))).
% 0.33/5.21 thf(fact_2961_abstract__boolean__algebra__sym__diff_Oxor__compl__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ (Compl @ Y3)) = (Compl @ ((Xor @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_2962_abstract__boolean__algebra__sym__diff_Oxor__cancel__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ (Compl @ X)) @ X) = One2))))).
% 0.33/5.21 thf(fact_2963_abstract__boolean__algebra__sym__diff_Oxor__compl__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ (Compl @ X)) @ Y3) = (Compl @ ((Xor @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_2964_abstract__boolean__algebra__sym__diff_Oxor__one__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ One2) = (Compl @ X)))))).
% 0.33/5.21 thf(fact_2965_abstract__boolean__algebra__sym__diff_Oxor__left__self, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ ((Xor @ X) @ Y3)) = Y3))))).
% 0.33/5.21 thf(fact_2966_abstract__boolean__algebra__sym__diff_Oxor__one__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ One2) @ X) = (Compl @ X)))))).
% 0.33/5.21 thf(fact_2967_abstract__boolean__algebra__sym__diff_Oxor__self, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ X) = Zero))))).
% 0.33/5.21 thf(fact_2968_abstract__boolean__algebra__sym__diff_Oxor__def2, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ Y3) = ((Conj @ ((Disj @ X) @ Y3)) @ ((Disj @ (Compl @ X)) @ (Compl @ Y3)))))))).
% 0.33/5.21 thf(fact_2969_abstract__boolean__algebra__sym__diff_Oxor__def, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A)),X:A,Y3:A]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((Xor @ X) @ Y3) = ((Disj @ ((Conj @ X) @ (Compl @ Y3))) @ ((Conj @ (Compl @ X)) @ Y3))))))).
% 0.33/5.21 thf(fact_2970_of__nat__max, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:nat,Y3:nat]: ((((semiring_1_of_nat @ A) @ (((ord_max @ nat) @ X) @ Y3)) = (((ord_max @ A) @ ((semiring_1_of_nat @ A) @ X)) @ ((semiring_1_of_nat @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_2971_max_Oleft__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_max @ A) @ B2) @ (((ord_max @ A) @ A3) @ C2)) = (((ord_max @ A) @ A3) @ (((ord_max @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_2972_max_Ocommute, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_max @ A) = (^ [A5:A,B3:A]: ((((ord_max @ A) @ B3) @ A5)))))))).
% 0.33/5.21 thf(fact_2973_max_Oassoc, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_max @ A) @ (((ord_max @ A) @ A3) @ B2)) @ C2) = (((ord_max @ A) @ A3) @ (((ord_max @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_2974_of__int__max, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,Y3:int]: ((((ring_1_of_int @ A) @ (((ord_max @ int) @ X) @ Y3)) = (((ord_max @ A) @ ((ring_1_of_int @ A) @ X)) @ ((ring_1_of_int @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_2975_max__absorb2, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_max @ A) @ X) @ Y3) = Y3)))))))).
% 0.33/5.21 thf(fact_2976_max__absorb1, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_max @ A) @ X) @ Y3) = X)))))))).
% 0.33/5.21 thf(fact_2977_max_OcoboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((ord_max @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_2978_max_OcoboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ C2) @ (((ord_max @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_2979_max_Oabsorb__iff2, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: (((((ord_max @ A) @ A5) @ B3) = B3)))))))).
% 0.33/5.21 thf(fact_2980_max_Oabsorb__iff1, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: (((((ord_max @ A) @ A5) @ B3) = A5)))))))).
% 0.33/5.21 thf(fact_2981_le__max__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less_eq @ A) @ Z3) @ (((ord_max @ A) @ X) @ Y3)) = ((((ord_less_eq @ A) @ Z3) @ X) | (((ord_less_eq @ A) @ Z3) @ Y3))))))))).
% 0.33/5.21 thf(fact_2982_max_Ocobounded2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: ((((ord_less_eq @ A) @ B2) @ (((ord_max @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_2983_max_Ocobounded1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ A3) @ (((ord_max @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_2984_max_Oorder__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: ((A5 = (((ord_max @ A) @ A5) @ B3))))))))).
% 0.33/5.21 thf(fact_2985_max_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A3))))))))).
% 0.33/5.21 thf(fact_2986_max_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A3) => (~ (((((ord_less_eq @ A) @ B2) @ A3) => (~ ((((ord_less_eq @ A) @ C2) @ A3))))))))))))).
% 0.33/5.21 thf(fact_2987_max_OorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((A3 = (((ord_max @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 0.33/5.21 thf(fact_2988_max_OorderE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => (A3 = (((ord_max @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_2989_max_Omono, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A3:A,D2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A3) => ((((ord_less_eq @ A) @ D2) @ B2) => (((ord_less_eq @ A) @ (((ord_max @ A) @ C2) @ D2)) @ (((ord_max @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_2990_max__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_max @ A) = (^ [A5:A,B3:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A5) @ B3)) @ B3) @ A5)))))))).
% 0.33/5.21 thf(fact_2991_nat__add__max__left, axiom, (! [M:nat,N:nat,Q4:nat]: (((((plus_plus @ nat) @ (((ord_max @ nat) @ M) @ N)) @ Q4) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M) @ Q4)) @ (((plus_plus @ nat) @ N) @ Q4)))))).
% 0.33/5.21 thf(fact_2992_nat__add__max__right, axiom, (! [M:nat,N:nat,Q4:nat]: (((((plus_plus @ nat) @ M) @ (((ord_max @ nat) @ N) @ Q4)) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ (((plus_plus @ nat) @ M) @ Q4)))))).
% 0.33/5.21 thf(fact_2993_nat__mult__max__left, axiom, (! [M:nat,N:nat,Q4:nat]: (((((times_times @ nat) @ (((ord_max @ nat) @ M) @ N)) @ Q4) = (((ord_max @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ (((times_times @ nat) @ N) @ Q4)))))).
% 0.33/5.21 thf(fact_2994_nat__mult__max__right, axiom, (! [M:nat,N:nat,Q4:nat]: (((((times_times @ nat) @ M) @ (((ord_max @ nat) @ N) @ Q4)) = (((ord_max @ nat) @ (((times_times @ nat) @ M) @ N)) @ (((times_times @ nat) @ M) @ Q4)))))).
% 0.33/5.21 thf(fact_2995_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys)))) => (P @ Xs2)))) => (P @ Xs))))).
% 0.33/5.21 thf(fact_2996_nat__minus__add__max, axiom, (! [N:nat,M:nat]: (((((plus_plus @ nat) @ (((minus_minus @ nat) @ N) @ M)) @ M) = (((ord_max @ nat) @ N) @ M))))).
% 0.33/5.21 thf(fact_2997_neg__numeral__class_Osub__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_sub @ A) = (^ [K3:num,L3:num]: ((((minus_minus @ A) @ ((numeral_numeral @ A) @ K3)) @ ((numeral_numeral @ A) @ L3))))))))).
% 0.33/5.21 thf(fact_2998_Heap_Osize__neq, axiom, (! [A:$tType,X:(heap_Time_Heap @ A)]: ((((size_size @ (heap_Time_Heap @ A)) @ X) != (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_2999_sub__non__negative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((neg_numeral_sub @ A) @ N) @ M)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_3000_sub__non__positive, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less_eq @ A) @ (((neg_numeral_sub @ A) @ N) @ M)) @ (zero_zero @ A)) = (((ord_less_eq @ num) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_3001_sub__negative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less @ A) @ (((neg_numeral_sub @ A) @ N) @ M)) @ (zero_zero @ A)) = (((ord_less @ num) @ N) @ M)))))))).
% 0.33/5.21 thf(fact_3002_sub__positive, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:num,M:num]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((neg_numeral_sub @ A) @ N) @ M)) = (((ord_less @ num) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_3003_sub__inc__One__eq, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((neg_numeral_sub @ A) @ (inc @ N)) @ one2) = ((numeral_numeral @ A) @ N)))))))).
% 0.33/5.21 thf(fact_3004_minus__numeral__eq__not__sub__one, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N)) = ((bit_ri4277139882892585799ns_not @ A) @ (((neg_numeral_sub @ A) @ N) @ one2))))))))).
% 0.33/5.21 thf(fact_3005_int__numeral__or__not__num__neg, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ N))))))).
% 0.33/5.21 thf(fact_3006_int__numeral__not__or__num__neg, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ N) @ M))))))).
% 0.33/5.21 thf(fact_3007_numeral__or__not__num__eq, axiom, (! [M:num,N:num]: ((((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ N)) = ((uminus_uminus @ int) @ (((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N)))))))).
% 0.33/5.21 thf(fact_3008_or__minus__numerals_I1_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ one2) @ (bitM @ N)))))))).
% 0.33/5.21 thf(fact_3009_or__minus__numerals_I5_J, axiom, (! [N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) @ (one_one @ int)) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ one2) @ (bitM @ N)))))))).
% 0.33/5.21 thf(fact_3010_horner__sum__of__bool__2__less, axiom, (! [Bs:(list @ $o)]: ((((ord_less @ int) @ (((((groups4207007520872428315er_sum @ $o) @ int) @ (zero_neq_one_of_bool @ int)) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Bs)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((size_size @ (list @ $o)) @ Bs)))))).
% 0.33/5.21 thf(fact_3011_or__minus__numerals_I7_J, axiom, (! [N:num,M:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) @ ((numeral_numeral @ int) @ M)) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ (bitM @ N)))))))).
% 0.33/5.21 thf(fact_3012_or__minus__numerals_I3_J, axiom, (! [M:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ ((bit_or_not_num_neg @ M) @ (bitM @ N)))))))).
% 0.33/5.21 thf(fact_3013_dbl__dec__simps_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: ((((neg_numeral_dbl_dec @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ A) @ (bitM @ K))))))))).
% 0.33/5.21 thf(fact_3014_pred__numeral__simps_I2_J, axiom, (! [K:num]: (((pred_numeral @ (bit0 @ K)) = ((numeral_numeral @ nat) @ (bitM @ K)))))).
% 0.33/5.21 thf(fact_3015_sub__num__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: (((((neg_numeral_sub @ A) @ (bit0 @ K)) @ one2) = ((numeral_numeral @ A) @ (bitM @ K))))))))).
% 0.33/5.21 thf(fact_3016_sub__num__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [L:num]: (((((neg_numeral_sub @ A) @ one2) @ (bit0 @ L)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bitM @ L)))))))))).
% 0.33/5.21 thf(fact_3017_semiring__norm_I26_J, axiom, ((bitM @ one2) = one2)).
% 0.33/5.21 thf(fact_3018_semiring__norm_I27_J, axiom, (! [N:num]: (((bitM @ (bit0 @ N)) = (bit1 @ (bitM @ N)))))).
% 0.33/5.21 thf(fact_3019_semiring__norm_I28_J, axiom, (! [N:num]: (((bitM @ (bit1 @ N)) = (bit1 @ (bit0 @ N)))))).
% 0.33/5.21 thf(fact_3020_inc__BitM__eq, axiom, (! [N:num]: (((inc @ (bitM @ N)) = (bit0 @ N))))).
% 0.33/5.21 thf(fact_3021_BitM__inc__eq, axiom, (! [N:num]: (((bitM @ (inc @ N)) = (bit1 @ N))))).
% 0.33/5.21 thf(fact_3022_eval__nat__numeral_I2_J, axiom, (! [N:num]: ((((numeral_numeral @ nat) @ (bit0 @ N)) = (suc @ ((numeral_numeral @ nat) @ (bitM @ N))))))).
% 0.33/5.21 thf(fact_3023_one__plus__BitM, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bitM @ N)) = (bit0 @ N))))).
% 0.33/5.21 thf(fact_3024_BitM__plus__one, axiom, (! [N:num]: (((((plus_plus @ num) @ (bitM @ N)) @ one2) = (bit0 @ N))))).
% 0.33/5.21 thf(fact_3025_numeral__BitM, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bitM @ N)) = (((minus_minus @ A) @ ((numeral_numeral @ A) @ (bit0 @ N))) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3026_not__numeral__BitM__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: ((((bit_ri4277139882892585799ns_not @ A) @ ((numeral_numeral @ A) @ (bitM @ N))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ N)))))))))).
% 0.33/5.21 thf(fact_3027_sub__BitM__One__eq, axiom, (! [N:num]: (((((neg_numeral_sub @ int) @ (bitM @ N)) @ one2) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((neg_numeral_sub @ int) @ N) @ one2)))))).
% 0.33/5.21 thf(fact_3028_bit__horner__sum__bit__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Bs:(list @ $o),N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((((groups4207007520872428315er_sum @ $o) @ A) @ (zero_neq_one_of_bool @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Bs)) @ N) = ((((ord_less @ nat) @ N) @ ((size_size @ (list @ $o)) @ Bs)) & (((nth @ $o) @ Bs) @ N))))))))).
% 0.33/5.21 thf(fact_3029_bit_Oabstract__boolean__algebra__axioms, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_3030_num__of__nat_Osimps_I2_J, axiom, (! [N:nat]: ((((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((num_of_nat @ (suc @ N)) = (inc @ (num_of_nat @ N)))) & ((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) => ((num_of_nat @ (suc @ N)) = one2)))))).
% 0.33/5.21 thf(fact_3031_eq__numeral__iff__iszero_I7_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = (one_one @ A)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ one2)))))))))).
% 0.33/5.21 thf(fact_3032_eq__numeral__iff__iszero_I8_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y3:num]: ((((one_one @ A) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y3))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ Y3)))))))))).
% 0.33/5.21 thf(fact_3033_num__of__nat__numeral__eq, axiom, (! [Q4:num]: (((num_of_nat @ ((numeral_numeral @ nat) @ Q4)) = Q4)))).
% 0.33/5.21 thf(fact_3034_iszero__neg__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:num]: ((((ring_1_iszero @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ W2))))))))).
% 0.33/5.21 thf(fact_3035_abstract__boolean__algebra_Odisj__conj__distrib2, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Y3:A,Z3:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ ((Conj @ Y3) @ Z3)) @ X) = ((Conj @ ((Disj @ Y3) @ X)) @ ((Disj @ Z3) @ X))))))).
% 0.33/5.21 thf(fact_3036_abstract__boolean__algebra_Oconj__disj__distrib2, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Y3:A,Z3:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ ((Disj @ Y3) @ Z3)) @ X) = ((Disj @ ((Conj @ Y3) @ X)) @ ((Conj @ Z3) @ X))))))).
% 0.33/5.21 thf(fact_3037_abstract__boolean__algebra_Ocompl__eq__compl__iff, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Compl @ X) = (Compl @ Y3)) = (X = Y3)))))).
% 0.33/5.21 thf(fact_3038_abstract__boolean__algebra_Odisj__conj__distrib, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A,Z3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ X) @ ((Conj @ Y3) @ Z3)) = ((Conj @ ((Disj @ X) @ Y3)) @ ((Disj @ X) @ Z3))))))).
% 0.33/5.21 thf(fact_3039_abstract__boolean__algebra_Odisj__cancel__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ X) @ (Compl @ X)) = One2))))).
% 0.33/5.21 thf(fact_3040_abstract__boolean__algebra_Oconj__disj__distrib, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A,Z3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ X) @ ((Disj @ Y3) @ Z3)) = ((Disj @ ((Conj @ X) @ Y3)) @ ((Conj @ X) @ Z3))))))).
% 0.33/5.21 thf(fact_3041_abstract__boolean__algebra_Oconj__cancel__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ X) @ (Compl @ X)) = Zero))))).
% 0.33/5.21 thf(fact_3042_abstract__boolean__algebra_Ocomplement__unique, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,A3:A,X:A,Y3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((((Conj @ A3) @ X) = Zero) => ((((Disj @ A3) @ X) = One2) => ((((Conj @ A3) @ Y3) = Zero) => ((((Disj @ A3) @ Y3) = One2) => (X = Y3))))))))).
% 0.33/5.21 thf(fact_3043_abstract__boolean__algebra_Odisj__cancel__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ (Compl @ X)) @ X) = One2))))).
% 0.33/5.21 thf(fact_3044_abstract__boolean__algebra_Oconj__cancel__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ (Compl @ X)) @ X) = Zero))))).
% 0.33/5.21 thf(fact_3045_abstract__boolean__algebra_Odisj__zero__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ X) @ Zero) = X))))).
% 0.33/5.21 thf(fact_3046_abstract__boolean__algebra_Oconj__zero__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ X) @ Zero) = Zero))))).
% 0.33/5.21 thf(fact_3047_abstract__boolean__algebra_Odisj__one__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ X) @ One2) = One2))))).
% 0.33/5.21 thf(fact_3048_abstract__boolean__algebra_Ode__Morgan__disj, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((Compl @ ((Disj @ X) @ Y3)) = ((Conj @ (Compl @ X)) @ (Compl @ Y3))))))).
% 0.33/5.21 thf(fact_3049_abstract__boolean__algebra_Ode__Morgan__conj, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((Compl @ ((Conj @ X) @ Y3)) = ((Disj @ (Compl @ X)) @ (Compl @ Y3))))))).
% 0.33/5.21 thf(fact_3050_abstract__boolean__algebra_Oconj__zero__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ Zero) @ X) = Zero))))).
% 0.33/5.21 thf(fact_3051_abstract__boolean__algebra_Oconj__one__right, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Conj @ X) @ One2) = X))))).
% 0.33/5.21 thf(fact_3052_abstract__boolean__algebra_Odisj__one__left, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => (((Disj @ One2) @ X) = One2))))).
% 0.33/5.21 thf(fact_3053_abstract__boolean__algebra_Odouble__compl, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((Compl @ (Compl @ X)) = X))))).
% 0.33/5.21 thf(fact_3054_abstract__boolean__algebra_Ocompl__unique, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,X:A,Y3:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((((Conj @ X) @ Y3) = Zero) => ((((Disj @ X) @ Y3) = One2) => ((Compl @ X) = Y3))))))).
% 0.33/5.21 thf(fact_3055_abstract__boolean__algebra_Ocompl__zero, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((Compl @ Zero) = One2))))).
% 0.33/5.21 thf(fact_3056_abstract__boolean__algebra_Ocompl__one, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((Compl @ One2) = Zero))))).
% 0.33/5.21 thf(fact_3057_iszero__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_iszero @ A) = (^ [Z7:A]: ((Z7 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3058_iszero__0, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_iszero @ A) @ (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3059_not__iszero__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W2:num]: ((~ (((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ W2)))))))))).
% 0.33/5.21 thf(fact_3060_not__iszero__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ (one_one @ A)))))))).
% 0.33/5.21 thf(fact_3061_eq__iff__iszero__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => ((^ [Y:A,Z2:A]: ((Y = Z2))) = (^ [X2:A,Y6:A]: (((ring_1_iszero @ A) @ (((minus_minus @ A) @ X2) @ Y6))))))))).
% 0.33/5.21 thf(fact_3062_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A))]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => ((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2))))).
% 0.33/5.21 thf(fact_3063_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I4) = (((nth @ A) @ Ys2) @ I4))))) => (Xs = Ys2)))))).
% 0.33/5.21 thf(fact_3064_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I5:nat]: (((((ord_less @ nat) @ I5) @ K) => (? [X7:A]: (((P @ I5) @ X7)))))) = (? [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ K) => ((P @ I5) @ (((nth @ A) @ Xs3) @ I5)))))))))))).
% 0.33/5.21 thf(fact_3065_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y:(list @ A),Z2:(list @ A)]: ((Y = Z2))) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys3)) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs3)) => ((((nth @ A) @ Xs3) @ I5) = (((nth @ A) @ Ys3) @ I5)))))))))))).
% 0.33/5.21 thf(fact_3066_eq__numeral__iff__iszero_I10_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y3:num]: ((((zero_zero @ A) = ((numeral_numeral @ A) @ Y3)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3067_eq__numeral__iff__iszero_I9_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((numeral_numeral @ A) @ X) = (zero_zero @ A)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3068_num__of__nat_Osimps_I1_J, axiom, ((num_of_nat @ (zero_zero @ nat)) = one2)).
% 0.33/5.21 thf(fact_3069_not__iszero__Numeral1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ one2)))))))).
% 0.33/5.21 thf(fact_3070_not__iszero__neg__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3071_eq__numeral__iff__iszero_I1_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y3:num]: (((((numeral_numeral @ A) @ X) = ((numeral_numeral @ A) @ Y3)) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_3072_eq__numeral__iff__iszero_I12_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y3:num]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y3))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3073_eq__numeral__iff__iszero_I11_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = (zero_zero @ A)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3074_not__iszero__neg__Numeral1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ one2))))))))).
% 0.33/5.21 thf(fact_3075_numeral__num__of__nat, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((numeral_numeral @ nat) @ (num_of_nat @ N)) = N))))).
% 0.33/5.21 thf(fact_3076_num__of__nat__One, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (one_one @ nat)) => ((num_of_nat @ N) = one2))))).
% 0.33/5.21 thf(fact_3077_eq__numeral__iff__iszero_I3_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y3:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = ((numeral_numeral @ A) @ Y3)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ Y3)))))))))).
% 0.33/5.21 thf(fact_3078_eq__numeral__iff__iszero_I2_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y3:num]: (((((numeral_numeral @ A) @ X) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y3))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ Y3)))))))))).
% 0.33/5.21 thf(fact_3079_eq__numeral__iff__iszero_I4_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y3:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y3))) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ Y3) @ X))))))))).
% 0.33/5.21 thf(fact_3080_numeral__num__of__nat__unfold, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_3081_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))))))).
% 0.33/5.21 thf(fact_3082_num__of__nat__plus__distrib, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((num_of_nat @ (((plus_plus @ nat) @ M) @ N)) = (((plus_plus @ num) @ (num_of_nat @ M)) @ (num_of_nat @ N)))))))).
% 0.33/5.21 thf(fact_3083_eq__numeral__iff__iszero_I5_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((numeral_numeral @ A) @ X) = (one_one @ A)) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ X) @ one2))))))))).
% 0.33/5.21 thf(fact_3084_eq__numeral__iff__iszero_I6_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y3:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ Y3)) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ one2) @ Y3))))))))).
% 0.33/5.21 thf(fact_3085_nth__rotate1, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((rotate1 @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ (suc @ N)) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 0.33/5.21 thf(fact_3086_nth__update__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),B2:(array @ B),J:nat,V:B,H2:(heap_ext @ product_unit),I:nat]: ((((((array_noteq @ A) @ B) @ A3) @ B2) => ((((nth @ A) @ (((array_get @ A) @ (((((array_update @ B) @ B2) @ J) @ V) @ H2)) @ A3)) @ I) = (((nth @ A) @ (((array_get @ A) @ H2) @ A3)) @ I))))))))).
% 0.33/5.21 thf(fact_3087_mask__mod__exp, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat,M: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))) @ M)) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((ord_min @ nat) @ M) @ N))) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3088_set__decode__plus__power__2, axiom, (! [N:nat,Z3:nat]: (((~ ((((member @ nat) @ N) @ (nat_set_decode @ Z3)))) => ((nat_set_decode @ (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ Z3)) = (((insert @ nat) @ N) @ (nat_set_decode @ Z3))))))).
% 0.33/5.21 thf(fact_3089_nth__rotate, axiom, (! [A:$tType,N:nat,Xs:(list @ A),M:nat]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((rotate @ A) @ M) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 0.33/5.21 thf(fact_3090_min_Oidem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A]: (((((ord_min @ A) @ A3) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_3091_min_Oleft__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_min @ A) @ A3) @ (((ord_min @ A) @ A3) @ B2)) = (((ord_min @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3092_min_Oright__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_min @ A) @ (((ord_min @ A) @ A3) @ B2)) @ B2) = (((ord_min @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3093_min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2)) = ((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.33/5.21 thf(fact_3094_min_Oabsorb2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_min @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_3095_min_Oabsorb1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_min @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_3096_min__less__iff__conj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z3:A,X:A,Y3:A]: (((((ord_less @ A) @ Z3) @ (((ord_min @ A) @ X) @ Y3)) = ((((ord_less @ A) @ Z3) @ X) & (((ord_less @ A) @ Z3) @ Y3))))))))).
% 0.33/5.21 thf(fact_3097_min_Oabsorb4, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_min @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_3098_min_Oabsorb3, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_min @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_3099_max__min__same_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X:A]: (((((ord_max @ A) @ Y3) @ (((ord_min @ A) @ X) @ Y3)) = Y3))))))).
% 0.33/5.21 thf(fact_3100_max__min__same_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_max @ A) @ (((ord_min @ A) @ X) @ Y3)) @ Y3) = Y3))))))).
% 0.33/5.21 thf(fact_3101_max__min__same_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_max @ A) @ (((ord_min @ A) @ X) @ Y3)) @ X) = X))))))).
% 0.33/5.21 thf(fact_3102_max__min__same_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_max @ A) @ X) @ (((ord_min @ A) @ X) @ Y3)) = X))))))).
% 0.33/5.21 thf(fact_3103_min__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((ord_min @ nat) @ (suc @ M)) @ (suc @ N)) = (suc @ (((ord_min @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_3104_min__0L, axiom, (! [N:nat]: (((((ord_min @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_3105_min__0R, axiom, (! [N:nat]: (((((ord_min @ nat) @ N) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_3106_min__number__of_I1_J, axiom, (! [A:$tType]: ((((numeral @ A) & (ord @ A)) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ U))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ V)))))))))).
% 0.33/5.21 thf(fact_3107_min__0__1_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ X)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_3108_min__0__1_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ ((numeral_numeral @ A) @ X)) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_3109_min__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3110_min__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3111_min__0__1_I6_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_3112_min__0__1_I5_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_3113_rotate__Suc, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((rotate @ A) @ (suc @ N)) @ Xs) = ((rotate1 @ A) @ (((rotate @ A) @ N) @ Xs)))))).
% 0.33/5.21 thf(fact_3114_get__update__elem__neqIndex, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,J:nat,A3:(array @ A),V:A,H2:(heap_ext @ product_unit)]: (((I != J) => ((((nth @ A) @ (((array_get @ A) @ (((((array_update @ A) @ A3) @ J) @ V) @ H2)) @ A3)) @ I) = (((nth @ A) @ (((array_get @ A) @ H2) @ A3)) @ I))))))))).
% 0.33/5.21 thf(fact_3115_min__number__of_I4_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))))))))).
% 0.33/5.21 thf(fact_3116_min__number__of_I3_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V)) = ((numeral_numeral @ A) @ V)))))))))).
% 0.33/5.21 thf(fact_3117_min__number__of_I2_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((numeral_numeral @ A) @ U))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))))))))))).
% 0.33/5.21 thf(fact_3118_rotate__length01, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => ((((rotate @ A) @ N) @ Xs) = Xs))))).
% 0.33/5.21 thf(fact_3119_rotate__id, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((modulo_modulo @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) = (zero_zero @ nat)) => ((((rotate @ A) @ N) @ Xs) = Xs))))).
% 0.33/5.21 thf(fact_3120_min__Suc__numeral, axiom, (! [N:nat,K:num]: (((((ord_min @ nat) @ (suc @ N)) @ ((numeral_numeral @ nat) @ K)) = (suc @ (((ord_min @ nat) @ N) @ (pred_numeral @ K))))))).
% 0.33/5.21 thf(fact_3121_min__numeral__Suc, axiom, (! [K:num,N:nat]: (((((ord_min @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N)) = (suc @ (((ord_min @ nat) @ (pred_numeral @ K)) @ N)))))).
% 0.33/5.21 thf(fact_3122_rotate1__length01, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((rotate1 @ A) @ Xs) = Xs))))).
% 0.33/5.21 thf(fact_3123_min__max__distrib2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_min @ A) @ A3) @ (((ord_max @ A) @ B2) @ C2)) = (((ord_max @ A) @ (((ord_min @ A) @ A3) @ B2)) @ (((ord_min @ A) @ A3) @ C2))))))))).
% 0.33/5.21 thf(fact_3124_min__max__distrib1, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_min @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A3) = (((ord_max @ A) @ (((ord_min @ A) @ B2) @ A3)) @ (((ord_min @ A) @ C2) @ A3))))))))).
% 0.33/5.21 thf(fact_3125_max__min__distrib2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_max @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2)) = (((ord_min @ A) @ (((ord_max @ A) @ A3) @ B2)) @ (((ord_max @ A) @ A3) @ C2))))))))).
% 0.33/5.21 thf(fact_3126_max__min__distrib1, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_max @ A) @ (((ord_min @ A) @ B2) @ C2)) @ A3) = (((ord_min @ A) @ (((ord_max @ A) @ B2) @ A3)) @ (((ord_max @ A) @ C2) @ A3))))))))).
% 0.33/5.21 thf(fact_3127_min__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_min @ A) = (^ [A5:A,B3:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A5) @ B3)) @ A5) @ B3)))))))).
% 0.33/5.21 thf(fact_3128_min_Omono, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A3) @ B2)) @ (((ord_min @ A) @ C2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3129_min_OorderE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (A3 = (((ord_min @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3130_min_OorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((A3 = (((ord_min @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3131_min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2)) => (~ (((((ord_less_eq @ A) @ A3) @ B2) => (~ ((((ord_less_eq @ A) @ A3) @ C2))))))))))))).
% 0.33/5.21 thf(fact_3132_min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ A3) @ C2) => (((ord_less_eq @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2)))))))))).
% 0.33/5.21 thf(fact_3133_min_Oorder__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: ((A5 = (((ord_min @ A) @ A5) @ B3))))))))).
% 0.33/5.21 thf(fact_3134_min_Ocobounded1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A3) @ B2)) @ A3))))))).
% 0.33/5.21 thf(fact_3135_min_Ocobounded2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A3) @ B2)) @ B2))))))).
% 0.33/5.21 thf(fact_3136_min_Oabsorb__iff1, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: (((((ord_min @ A) @ A5) @ B3) = A5)))))))).
% 0.33/5.21 thf(fact_3137_min_Oabsorb__iff2, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: (((((ord_min @ A) @ A5) @ B3) = B3)))))))).
% 0.33/5.21 thf(fact_3138_min_OcoboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_3139_min_OcoboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_3140_min__le__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ (((ord_min @ A) @ X) @ Y3)) @ Z3) = ((((ord_less_eq @ A) @ X) @ Z3) | (((ord_less_eq @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3141_min__absorb1, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_min @ A) @ X) @ Y3) = X)))))))).
% 0.33/5.21 thf(fact_3142_min__absorb2, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_min @ A) @ X) @ Y3) = Y3)))))))).
% 0.33/5.21 thf(fact_3143_of__int__min, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,Y3:int]: ((((ring_1_of_int @ A) @ (((ord_min @ int) @ X) @ Y3)) = (((ord_min @ A) @ ((ring_1_of_int @ A) @ X)) @ ((ring_1_of_int @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3144_min_Oassoc, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_min @ A) @ (((ord_min @ A) @ A3) @ B2)) @ C2) = (((ord_min @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3145_min_Ocommute, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_min @ A) = (^ [A5:A,B3:A]: ((((ord_min @ A) @ B3) @ A5)))))))).
% 0.33/5.21 thf(fact_3146_min_Oleft__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_min @ A) @ B2) @ (((ord_min @ A) @ A3) @ C2)) = (((ord_min @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3147_of__nat__min, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:nat,Y3:nat]: ((((semiring_1_of_nat @ A) @ (((ord_min @ nat) @ X) @ Y3)) = (((ord_min @ A) @ ((semiring_1_of_nat @ A) @ X)) @ ((semiring_1_of_nat @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3148_min__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A,Z3:A]: (((((minus_minus @ A) @ (((ord_min @ A) @ X) @ Y3)) @ Z3) = (((ord_min @ A) @ (((minus_minus @ A) @ X) @ Z3)) @ (((minus_minus @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3149_min__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y3:A,Z3:A]: (((((plus_plus @ A) @ X) @ (((ord_min @ A) @ Y3) @ Z3)) = (((ord_min @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (((plus_plus @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3150_min__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y3:A,Z3:A]: (((((plus_plus @ A) @ (((ord_min @ A) @ X) @ Y3)) @ Z3) = (((ord_min @ A) @ (((plus_plus @ A) @ X) @ Z3)) @ (((plus_plus @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3151_min__less__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less @ A) @ (((ord_min @ A) @ X) @ Y3)) @ Z3) = ((((ord_less @ A) @ X) @ Z3) | (((ord_less @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3152_min_Ostrict__boundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ (((ord_min @ A) @ B2) @ C2)) => (~ (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ A3) @ C2))))))))))))).
% 0.33/5.21 thf(fact_3153_min_Ostrict__order__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A5:A,B3:A]: (((A5 = (((ord_min @ A) @ A5) @ B3)) & (A5 != B3))))))))).
% 0.33/5.21 thf(fact_3154_min_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ A3) @ C2) => (((ord_less @ A) @ (((ord_min @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_3155_min_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ (((ord_min @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_3156_nat__mult__min__left, axiom, (! [M:nat,N:nat,Q4:nat]: (((((times_times @ nat) @ (((ord_min @ nat) @ M) @ N)) @ Q4) = (((ord_min @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ (((times_times @ nat) @ N) @ Q4)))))).
% 0.33/5.21 thf(fact_3157_nat__mult__min__right, axiom, (! [M:nat,N:nat,Q4:nat]: (((((times_times @ nat) @ M) @ (((ord_min @ nat) @ N) @ Q4)) = (((ord_min @ nat) @ (((times_times @ nat) @ M) @ N)) @ (((times_times @ nat) @ M) @ Q4)))))).
% 0.33/5.21 thf(fact_3158_min__diff, axiom, (! [M:nat,I:nat,N:nat]: (((((ord_min @ nat) @ (((minus_minus @ nat) @ M) @ I)) @ (((minus_minus @ nat) @ N) @ I)) = (((minus_minus @ nat) @ (((ord_min @ nat) @ M) @ N)) @ I))))).
% 0.33/5.21 thf(fact_3159_minus__min__eq__max, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X:A,Y3:A]: ((((uminus_uminus @ A) @ (((ord_min @ A) @ X) @ Y3)) = (((ord_max @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3160_minus__max__eq__min, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X:A,Y3:A]: ((((uminus_uminus @ A) @ (((ord_max @ A) @ X) @ Y3)) = (((ord_min @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3161_atLeast0__atMost__Suc, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert @ nat) @ (suc @ N)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.21 thf(fact_3162_atLeastAtMost__insertL, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((insert @ nat) @ M) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_3163_atLeastAtMostSuc__conv, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N)) = (((insert @ nat) @ (suc @ N)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.21 thf(fact_3164_Icc__eq__insert__lb__nat, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((set_or1337092689740270186AtMost @ nat) @ M) @ N) = (((insert @ nat) @ M) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N))))))).
% 0.33/5.21 thf(fact_3165_length__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_length @ A) = (^ [H4:(heap_ext @ product_unit),A5:(array @ A)]: (((size_size @ (list @ A)) @ (((array_get @ A) @ H4) @ A5))))))))).
% 0.33/5.21 thf(fact_3166_max__mult__distrib__left, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((times_times @ A) @ P5) @ (((ord_max @ A) @ X) @ Y3)) = (((ord_max @ A) @ (((times_times @ A) @ P5) @ X)) @ (((times_times @ A) @ P5) @ Y3)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((times_times @ A) @ P5) @ (((ord_max @ A) @ X) @ Y3)) = (((ord_min @ A) @ (((times_times @ A) @ P5) @ X)) @ (((times_times @ A) @ P5) @ Y3))))))))))).
% 0.33/5.21 thf(fact_3167_min__mult__distrib__left, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((times_times @ A) @ P5) @ (((ord_min @ A) @ X) @ Y3)) = (((ord_min @ A) @ (((times_times @ A) @ P5) @ X)) @ (((times_times @ A) @ P5) @ Y3)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((times_times @ A) @ P5) @ (((ord_min @ A) @ X) @ Y3)) = (((ord_max @ A) @ (((times_times @ A) @ P5) @ X)) @ (((times_times @ A) @ P5) @ Y3))))))))))).
% 0.33/5.21 thf(fact_3168_max__mult__distrib__right, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((times_times @ A) @ (((ord_max @ A) @ X) @ Y3)) @ P5) = (((ord_max @ A) @ (((times_times @ A) @ X) @ P5)) @ (((times_times @ A) @ Y3) @ P5)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((times_times @ A) @ (((ord_max @ A) @ X) @ Y3)) @ P5) = (((ord_min @ A) @ (((times_times @ A) @ X) @ P5)) @ (((times_times @ A) @ Y3) @ P5))))))))))).
% 0.33/5.21 thf(fact_3169_min__mult__distrib__right, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((times_times @ A) @ (((ord_min @ A) @ X) @ Y3)) @ P5) = (((ord_min @ A) @ (((times_times @ A) @ X) @ P5)) @ (((times_times @ A) @ Y3) @ P5)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((times_times @ A) @ (((ord_min @ A) @ X) @ Y3)) @ P5) = (((ord_max @ A) @ (((times_times @ A) @ X) @ P5)) @ (((times_times @ A) @ Y3) @ P5))))))))))).
% 0.33/5.21 thf(fact_3170_min__divide__distrib__right, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((divide_divide @ A) @ (((ord_min @ A) @ X) @ Y3)) @ P5) = (((ord_min @ A) @ (((divide_divide @ A) @ X) @ P5)) @ (((divide_divide @ A) @ Y3) @ P5)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((divide_divide @ A) @ (((ord_min @ A) @ X) @ Y3)) @ P5) = (((ord_max @ A) @ (((divide_divide @ A) @ X) @ P5)) @ (((divide_divide @ A) @ Y3) @ P5))))))))))).
% 0.33/5.21 thf(fact_3171_max__divide__distrib__right, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [P5:A,X:A,Y3:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5) => ((((divide_divide @ A) @ (((ord_max @ A) @ X) @ Y3)) @ P5) = (((ord_max @ A) @ (((divide_divide @ A) @ X) @ P5)) @ (((divide_divide @ A) @ Y3) @ P5)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P5))) => ((((divide_divide @ A) @ (((ord_max @ A) @ X) @ Y3)) @ P5) = (((ord_min @ A) @ (((divide_divide @ A) @ X) @ P5)) @ (((divide_divide @ A) @ Y3) @ P5))))))))))).
% 0.33/5.21 thf(fact_3172_and__int_Oelims, axiom, (! [X:int,Xa3:int,Y3:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X) @ Xa3) = Y3) => ((((((member @ int) @ X) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y3 = ((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))) @ Xa3)))))))) & ((~ (((((member @ int) @ X) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y3 = (((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))) @ Xa3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 0.33/5.21 thf(fact_3173_and__int_Osimps, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K3:int,L3:int]: (((((if @ int) @ ((((member @ int) @ K3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ 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))) @ K3))) & (~ ((((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))) @ K3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))).
% 0.33/5.21 thf(fact_3174_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)))))))).
% 0.33/5.21 thf(fact_3175_get__set__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [R:(array @ A),S:(array @ B),X:(list @ B),H2:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ R) @ S) => ((((array_get @ A) @ ((((array_set @ B) @ S) @ X) @ H2)) @ R) = (((array_get @ A) @ H2) @ R))))))))).
% 0.33/5.21 thf(fact_3176_divmod__step__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [L:num,R:A,Q4:A]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q4) @ R)) = ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q4)) @ (one_one @ A))) @ (((minus_minus @ A) @ R) @ ((numeral_numeral @ A) @ L))))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R))) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q4) @ R)) = ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q4)) @ R)))))))))).
% 0.33/5.21 thf(fact_3177_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X2:D]: ((bot_bot @ C)))))))).
% 0.33/5.21 thf(fact_3178_set__same, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),X:(list @ A),Y3:(list @ A),H2:(heap_ext @ product_unit)]: ((((((array_set @ A) @ R) @ X) @ ((((array_set @ A) @ R) @ Y3) @ H2)) = ((((array_set @ A) @ R) @ X) @ H2)))))))).
% 0.33/5.21 thf(fact_3179_max__bot2, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X:A]: (((((ord_max @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 0.33/5.21 thf(fact_3180_max__bot, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X:A]: (((((ord_max @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 0.33/5.21 thf(fact_3181_min__bot2, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X:A]: (((((ord_min @ A) @ X) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3182_min__bot, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X:A]: (((((ord_min @ A) @ (bot_bot @ A)) @ X) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3183_get__set__eq, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),X:(list @ A),H2:(heap_ext @ product_unit)]: (((((array_get @ A) @ ((((array_set @ A) @ R) @ X) @ H2)) @ R) = X))))))).
% 0.33/5.21 thf(fact_3184_atLeastatMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (~ ((((ord_less_eq @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3185_atLeastatMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less_eq @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3186_atLeastatMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A)))))))))).
% 0.33/5.21 thf(fact_3187_nth__Cons__Suc, axiom, (! [A:$tType,X:A,Xs:(list @ A),N:nat]: (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ (suc @ N)) = (((nth @ A) @ Xs) @ N))))).
% 0.33/5.21 thf(fact_3188_nth__Cons__0, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ (zero_zero @ nat)) = X)))).
% 0.33/5.21 thf(fact_3189_subset__Compl__singleton, axiom, (! [A:$tType,A6:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B2) @ A6))))))).
% 0.33/5.21 thf(fact_3190_nth__Cons__numeral, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_3191_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 0.33/5.21 thf(fact_3192_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((A3 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_3193_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ (bot_bot @ A)))))))))).
% 0.33/5.21 thf(fact_3194_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 0.33/5.21 thf(fact_3195_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 0.33/5.21 thf(fact_3196_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 0.33/5.21 thf(fact_3197_not__psubset__empty, axiom, (! [A:$tType,A6:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A6) @ (bot_bot @ (set @ A)))))))).
% 0.33/5.21 thf(fact_3198_diff__shunt__var, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((((minus_minus @ A) @ X) @ Y3) = (bot_bot @ A)) = (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3199_simp__from__to, axiom, ((set_or1337092689740270186AtMost @ int) = (^ [I5:int,J3:int]: (((((if @ (set @ int)) @ (((ord_less @ int) @ J3) @ I5)) @ (bot_bot @ (set @ int))) @ (((insert @ int) @ I5) @ (((set_or1337092689740270186AtMost @ int) @ (((plus_plus @ int) @ I5) @ (one_one @ int))) @ J3))))))).
% 0.33/5.21 thf(fact_3200_length__Suc__conv, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((size_size @ (list @ A)) @ Xs) = (suc @ N)) = (? [Y6:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ Y6) @ Ys3)) & (((size_size @ (list @ A)) @ Ys3) = N)))))))).
% 0.33/5.21 thf(fact_3201_Suc__length__conv, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((suc @ N) = ((size_size @ (list @ A)) @ Xs)) = (? [Y6:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ Y6) @ Ys3)) & (((size_size @ (list @ A)) @ Ys3) = N)))))))).
% 0.33/5.21 thf(fact_3202_impossible__Cons, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),X:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2)) => (Xs != (((cons @ A) @ X) @ Ys2)))))).
% 0.33/5.21 thf(fact_3203_subset__Compl__self__eq, axiom, (! [A:$tType,A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ A6)) = (A6 = (bot_bot @ (set @ A))))))).
% 0.33/5.21 thf(fact_3204_set__set__swap, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [R:(array @ A),R4:(array @ B),X:(list @ A),X5:(list @ B),H2:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ R) @ R4) => (((((array_set @ A) @ R) @ X) @ ((((array_set @ B) @ R4) @ X5) @ H2)) = ((((array_set @ B) @ R4) @ X5) @ ((((array_set @ A) @ R) @ X) @ H2)))))))))).
% 0.33/5.21 thf(fact_3205_Suc__le__length__iff, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ (suc @ N)) @ ((size_size @ (list @ A)) @ Xs)) = (? [X2:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ X2) @ Ys3)) & (((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ Ys3))))))))).
% 0.33/5.21 thf(fact_3206_Compl__insert, axiom, (! [A:$tType,X:A,A6:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X) @ A6)) = (((minus_minus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))).
% 0.33/5.21 thf(fact_3207_list_Osize_I4_J, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_3208_nth__Cons_H, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: ((((N = (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = X)) & ((N != (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))))).
% 0.33/5.21 thf(fact_3209_atLeastAtMostPlus1__int__conv, axiom, (! [M:int,N:int]: (((((ord_less_eq @ int) @ M) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) => ((((set_or1337092689740270186AtMost @ int) @ M) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) = (((insert @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) @ (((set_or1337092689740270186AtMost @ int) @ M) @ N))))))).
% 0.33/5.21 thf(fact_3210_psubset__insert__iff, axiom, (! [A:$tType,A6:(set @ A),X:A,B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ B7)) = (((((member @ A) @ X) @ B7) => (((ord_less @ (set @ A)) @ A6) @ B7)) & ((~ ((((member @ A) @ X) @ B7))) => (((((member @ A) @ X) @ A6) => (((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B7)) & ((~ ((((member @ A) @ X) @ A6))) => (((ord_less_eq @ (set @ A)) @ A6) @ B7))))))))).
% 0.33/5.21 thf(fact_3211_nth__non__equal__first__eq, axiom, (! [A:$tType,X:A,Y3:A,Xs:(list @ A),N:nat]: (((X != Y3) => (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = Y3) = (((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) = Y3) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.21 thf(fact_3212_horner__sum__simps_I2_J, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (! [F:(B > A),A3:A,X:B,Xs:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ (((cons @ B) @ X) @ Xs)) = (((plus_plus @ A) @ (F @ X)) @ (((times_times @ A) @ A3) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Xs)))))))))).
% 0.33/5.21 thf(fact_3213_divmod__algorithm__code_I8_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N:num]: ((((((ord_less @ num) @ M) @ N) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit1 @ M))))) & ((~ ((((ord_less @ num) @ M) @ N))) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit1 @ N)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N)) @ (((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit0 @ (bit1 @ N))))))))))))).
% 0.33/5.21 thf(fact_3214_divmod__algorithm__code_I7_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N:num]: ((((((ord_less_eq @ num) @ M) @ N) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit0 @ M))))) & ((~ ((((ord_less_eq @ num) @ M) @ N))) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit1 @ N)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N)) @ (((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit0 @ (bit1 @ N))))))))))))).
% 0.33/5.21 thf(fact_3215_divides__aux__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [Q4:A,R:A]: ((((unique5940410009612947441es_aux @ A) @ ((((product_Pair @ A) @ A) @ Q4) @ R)) = (R = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3216_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))))))))).
% 0.33/5.21 thf(fact_3217_bit__0__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (((bit_se5641148757651400278ts_bit @ A) @ (zero_zero @ A)) = (bot_bot @ (nat > $o))))))).
% 0.33/5.21 thf(fact_3218_set__decode__zero, axiom, ((nat_set_decode @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))).
% 0.33/5.21 thf(fact_3219_length__product, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B)]: ((((size_size @ (list @ ((product_prod @ A) @ B))) @ ((((product @ A) @ B) @ Xs) @ Ys2)) = (((times_times @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2)))))).
% 0.33/5.21 thf(fact_3220_divmod__algorithm__code_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num]: (((((unique8689654367752047608divmod @ A) @ M) @ one2) = ((((product_Pair @ A) @ A) @ ((numeral_numeral @ A) @ M)) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3221_divmod__algorithm__code_I3_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit0 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))).
% 0.33/5.21 thf(fact_3222_divmod__algorithm__code_I4_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))).
% 0.33/5.21 thf(fact_3223_divmod__BitM__2__eq, axiom, (! [M:num]: (((((unique8689654367752047608divmod @ int) @ (bitM @ M)) @ (bit0 @ one2)) = ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_3224_bot__nat__def, axiom, ((bot_bot @ nat) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_3225_divmod__divmod__step, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique8689654367752047608divmod @ A) = (^ [M2:num,N3:num]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less @ num) @ M2) @ N3)) @ ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ M2))) @ (((unique1321980374590559556d_step @ A) @ N3) @ (((unique8689654367752047608divmod @ A) @ M2) @ (bit0 @ N3))))))))))).
% 0.33/5.21 thf(fact_3226_minus__one__div__numeral, axiom, (! [N:num]: (((((divide_divide @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N)) = ((uminus_uminus @ int) @ (adjust_div @ (((unique8689654367752047608divmod @ int) @ one2) @ N))))))).
% 0.33/5.21 thf(fact_3227_one__div__minus__numeral, axiom, (! [N:num]: (((((divide_divide @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((uminus_uminus @ int) @ (adjust_div @ (((unique8689654367752047608divmod @ int) @ one2) @ N))))))).
% 0.33/5.21 thf(fact_3228_neg__eucl__rel__int__mult__2, axiom, (! [B2:int,A3:int,Q4: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) @ Q4) @ 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) @ Q4) @ (((minus_minus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R)) @ (one_one @ int))))))))).
% 0.33/5.21 thf(fact_3229_Divides_Oadjust__div__eq, axiom, (! [Q4:int,R:int]: (((adjust_div @ ((((product_Pair @ int) @ int) @ Q4) @ R)) = (((plus_plus @ int) @ Q4) @ ((zero_neq_one_of_bool @ int) @ (R != (zero_zero @ int)))))))).
% 0.33/5.21 thf(fact_3230_minus__numeral__div__numeral, axiom, (! [M:num,N:num]: (((((divide_divide @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = ((uminus_uminus @ int) @ (adjust_div @ (((unique8689654367752047608divmod @ int) @ M) @ N))))))).
% 0.33/5.21 thf(fact_3231_numeral__div__minus__numeral, axiom, (! [M:num,N:num]: (((((divide_divide @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((uminus_uminus @ int) @ (adjust_div @ (((unique8689654367752047608divmod @ int) @ M) @ N))))))).
% 0.33/5.21 thf(fact_3232_eucl__rel__int__by0, axiom, (! [K:int]: ((((eucl_rel_int @ K) @ (zero_zero @ int)) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K))))).
% 0.33/5.21 thf(fact_3233_eucl__rel__int__dividesI, axiom, (! [L:int,K:int,Q4:int]: (((L != (zero_zero @ int)) => ((K = (((times_times @ int) @ Q4) @ L)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ (zero_zero @ int)))))))).
% 0.33/5.21 thf(fact_3234_zminus1__lemma, axiom, (! [A3:int,B2:int,Q4:int,R:int]: (((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ 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) @ Q4)) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ Q4)) @ (one_one @ int)))) @ ((((if @ int) @ (R = (zero_zero @ int))) @ (zero_zero @ int)) @ (((minus_minus @ int) @ B2) @ R))))))))).
% 0.33/5.21 thf(fact_3235_eucl__rel__int__iff, axiom, (! [K:int,L:int,Q4:int,R:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R)) = ((K = (((plus_plus @ int) @ (((times_times @ int) @ L) @ Q4)) @ 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)))) => (Q4 = (zero_zero @ int))))))))))).
% 0.33/5.21 thf(fact_3236_eucl__rel__int__remainderI, axiom, (! [R:int,L:int,K:int,Q4: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) @ Q4) @ L)) @ R)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R)))))))).
% 0.33/5.21 thf(fact_3237_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))) => ((! [Q5:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q5) @ (zero_zero @ int))) => ((A22 != (zero_zero @ int)) => (A1 != (((times_times @ int) @ Q5) @ A22)))))) => (~ ((! [R3:int,Q5:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q5) @ R3)) => ((((sgn_sgn @ int) @ R3) = ((sgn_sgn @ int) @ A22)) => ((((ord_less @ int) @ ((abs_abs @ int) @ R3)) @ ((abs_abs @ int) @ A22)) => (A1 != (((plus_plus @ int) @ (((times_times @ int) @ Q5) @ A22)) @ R3))))))))))))))).
% 0.33/5.21 thf(fact_3238_eucl__rel__int_Osimps, axiom, (eucl_rel_int = (^ [A12:int,A23:int,A33:((product_prod @ int) @ int)]: (((? [K3:int]: (((A12 = K3) & ((A23 = (zero_zero @ int)) & (A33 = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K3)))))) | ((? [L3:int,K3:int,Q3:int]: (((A12 = K3) & ((A23 = L3) & ((A33 = ((((product_Pair @ int) @ int) @ Q3) @ (zero_zero @ int))) & ((L3 != (zero_zero @ int)) & (K3 = (((times_times @ int) @ Q3) @ L3)))))))) | (? [R5:int,L3:int,K3:int,Q3:int]: (((A12 = K3) & ((A23 = L3) & ((A33 = ((((product_Pair @ int) @ int) @ Q3) @ R5)) & ((((sgn_sgn @ int) @ R5) = ((sgn_sgn @ int) @ L3)) & ((((ord_less @ int) @ ((abs_abs @ int) @ R5)) @ ((abs_abs @ int) @ L3)) & (K3 = (((plus_plus @ int) @ (((times_times @ int) @ Q3) @ L3)) @ R5))))))))))))))).
% 0.33/5.21 thf(fact_3239_pos__eucl__rel__int__mult__2, axiom, (! [B2:int,A3:int,Q4:int,R:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ B2) => ((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ 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) @ Q4) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R))))))))).
% 0.33/5.21 thf(fact_3240_and__int_Opelims, axiom, (! [X:int,Xa3:int,Y3:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa3)) => (~ ((((((((member @ int) @ X) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y3 = ((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))) @ Xa3)))))))) & ((~ (((((member @ int) @ X) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y3 = (((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))) @ Xa3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3241_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) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ 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) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ 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))))))))))))).
% 0.33/5.21 thf(fact_3242_length__Cons, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) = (suc @ ((size_size @ (list @ A)) @ Xs)))))).
% 0.33/5.21 thf(fact_3243_normalize__negative, axiom, (! [Q4:int,P5:int]: (((((ord_less @ int) @ Q4) @ (zero_zero @ int)) => ((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q4)) = (normalize @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ P5)) @ ((uminus_uminus @ int) @ Q4)))))))).
% 0.33/5.21 thf(fact_3244_upto__aux__rec, axiom, (upto_aux = (^ [I5:int,J3:int,Js:(list @ int)]: (((((if @ (list @ int)) @ (((ord_less @ int) @ J3) @ I5)) @ Js) @ (((upto_aux @ I5) @ (((minus_minus @ int) @ J3) @ (one_one @ int))) @ (((cons @ int) @ J3) @ Js))))))).
% 0.33/5.21 thf(fact_3245_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)))))).
% 0.33/5.21 thf(fact_3246_normalize__denom__pos, axiom, (! [R:((product_prod @ int) @ int),P5:int,Q4:int]: ((((normalize @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q4)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q4))))).
% 0.33/5.21 thf(fact_3247_normalize__crossproduct, axiom, (! [Q4:int,S:int,P5:int,R:int]: (((Q4 != (zero_zero @ int)) => ((S != (zero_zero @ int)) => (((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q4)) = (normalize @ ((((product_Pair @ int) @ int) @ R) @ S))) => ((((times_times @ int) @ P5) @ S) = (((times_times @ int) @ R) @ Q4)))))))).
% 0.33/5.21 thf(fact_3248_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)) => ((! [K2:int,L2:int]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ K2) @ L2)) => (((~ (((((member @ int) @ K2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => ((P @ (((divide_divide @ int) @ K2) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L2) @ ((numeral_numeral @ int) @ (bit0 @ one2))))) => ((P @ K2) @ L2))))) => ((P @ A0) @ A1)))))).
% 0.33/5.21 thf(fact_3249_upto_Opinduct, axiom, (! [A0:int,A1:int,P:(int > (int > $o))]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ A0) @ A1)) => ((! [I4:int,J2:int]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ I4) @ J2)) => (((((ord_less_eq @ int) @ I4) @ J2) => ((P @ (((plus_plus @ int) @ I4) @ (one_one @ int))) @ J2)) => ((P @ I4) @ J2))))) => ((P @ A0) @ A1)))))).
% 0.33/5.21 thf(fact_3250_bit__cut__integer__def, axiom, (code_bit_cut_integer = (^ [K3:code_integer]: (((((product_Pair @ code_integer) @ $o) @ (((divide_divide @ code_integer) @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ (~ ((((dvd_dvd @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ K3)))))))).
% 0.33/5.21 thf(fact_3251_bezw__0, axiom, (! [X:nat]: ((((bezw @ X) @ (zero_zero @ nat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))))).
% 0.33/5.21 thf(fact_3252_nth__zip, axiom, (! [A:$tType,B:$tType,I:nat,Xs:(list @ A),Ys2:(list @ B)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys2)) => ((((nth @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ I) = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ B) @ Ys2) @ I)))))))).
% 0.33/5.21 thf(fact_3253_Gcd__0__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((gcd_Gcd @ A) @ A6) = (zero_zero @ A)) = (((ord_less_eq @ (set @ A)) @ A6) @ (((insert @ A) @ (zero_zero @ A)) @ (bot_bot @ (set @ A))))))))))).
% 0.33/5.21 thf(fact_3254_Gcd__empty, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Gcd @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3255_divmod__integer_H__def, axiom, ((unique8689654367752047608divmod @ code_integer) = (^ [M2:num,N3:num]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((numeral_numeral @ code_integer) @ M2)) @ ((numeral_numeral @ code_integer) @ N3))) @ (((modulo_modulo @ code_integer) @ ((numeral_numeral @ code_integer) @ M2)) @ ((numeral_numeral @ code_integer) @ N3))))))).
% 0.33/5.21 thf(fact_3256_Gcd__nat__eq__one, axiom, (! [N6:(set @ nat)]: (((((member @ nat) @ (one_one @ nat)) @ N6) => (((gcd_Gcd @ nat) @ N6) = (one_one @ nat)))))).
% 0.33/5.21 thf(fact_3257_Gcd__1, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((member @ A) @ (one_one @ A)) @ A6) => (((gcd_Gcd @ A) @ A6) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3258_Gcd__int__greater__eq__0, axiom, (! [K5:(set @ int)]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((gcd_Gcd @ int) @ K5))))).
% 0.33/5.21 thf(fact_3259_Gcd__eq__1__I, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A3:A,A6:(set @ A)]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((member @ A) @ A3) @ A6) => (((gcd_Gcd @ A) @ A6) = (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_3260_Code__Numeral_Opositive__def, axiom, (code_positive = (numeral_numeral @ code_integer))).
% 0.33/5.21 thf(fact_3261_integer__of__num__triv_I2_J, axiom, ((code_integer_of_num @ (bit0 @ one2)) = ((numeral_numeral @ code_integer) @ (bit0 @ one2)))).
% 0.33/5.21 thf(fact_3262_card__atLeastAtMost__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or1337092689740270186AtMost @ int) @ L) @ U)) = (nat2 @ (((plus_plus @ int) @ (((minus_minus @ int) @ U) @ L)) @ (one_one @ int))))))).
% 0.33/5.21 thf(fact_3263_possible__bit__def, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((bit_se6407376104438227557le_bit @ A) = (^ [Tyrep:(itself @ A),N3:nat]: (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N3) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3264_abstract__boolean__algebra__sym__diff_Ointro, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A))]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((((((boolea5476839437570043046axioms @ A) @ Conj) @ Disj) @ Compl) @ Xor) => (((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor)))))).
% 0.33/5.21 thf(fact_3265_abstract__boolean__algebra__sym__diff__axioms_Ointro, axiom, (! [A:$tType,Xor:(A > (A > A)),Disj:(A > (A > A)),Conj:(A > (A > A)),Compl:(A > A)]: (((! [X3:A,Y5:A]: ((((Xor @ X3) @ Y5) = ((Disj @ ((Conj @ X3) @ (Compl @ Y5))) @ ((Conj @ (Compl @ X3)) @ Y5))))) => (((((boolea5476839437570043046axioms @ A) @ Conj) @ Disj) @ Compl) @ Xor))))).
% 0.33/5.21 thf(fact_3266_abstract__boolean__algebra__sym__diff__axioms__def, axiom, (! [A:$tType]: (((boolea5476839437570043046axioms @ A) = (^ [Conj2:(A > (A > A)),Disj2:(A > (A > A)),Compl2:(A > A),Xor2:(A > (A > A))]: ((! [X2:A,Y6:A]: ((((Xor2 @ X2) @ Y6) = ((Disj2 @ ((Conj2 @ X2) @ (Compl2 @ Y6))) @ ((Conj2 @ (Compl2 @ X2)) @ Y6))))))))))).
% 0.33/5.21 thf(fact_3267_possible__bit__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [Ty:(itself @ A)]: ((((bit_se6407376104438227557le_bit @ A) @ Ty) @ (zero_zero @ nat)))))))).
% 0.33/5.21 thf(fact_3268_possible__bit__less__imp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [Tyrep2:(itself @ A),I:nat,J:nat]: (((((bit_se6407376104438227557le_bit @ A) @ Tyrep2) @ I) => ((((ord_less_eq @ nat) @ J) @ I) => (((bit_se6407376104438227557le_bit @ A) @ Tyrep2) @ J))))))))).
% 0.33/5.21 thf(fact_3269_integer__of__num__triv_I1_J, axiom, ((code_integer_of_num @ one2) = (one_one @ code_integer))).
% 0.33/5.21 thf(fact_3270_integer__of__num__def, axiom, (code_integer_of_num = (numeral_numeral @ code_integer))).
% 0.33/5.21 thf(fact_3271_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A,Xor:(A > (A > A))]: (((((((((boolea3799213064322606851m_diff @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) @ Xor) => (((((boolea5476839437570043046axioms @ A) @ Conj) @ Disj) @ Compl) @ Xor))))).
% 0.33/5.21 thf(fact_3272_abstract__boolean__algebra__sym__diff__def, axiom, (! [A:$tType]: (((boolea3799213064322606851m_diff @ A) = (^ [Conj2:(A > (A > A)),Disj2:(A > (A > A)),Compl2:(A > A),Zero2:A,One:A,Xor2:(A > (A > A))]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj2) @ Disj2) @ Compl2) @ Zero2) @ One) & (((((boolea5476839437570043046axioms @ A) @ Conj2) @ Disj2) @ Compl2) @ Xor2)))))))).
% 0.33/5.21 thf(fact_3273_card__Diff__insert, axiom, (! [A:$tType,A3:A,A6:(set @ A),B7:(set @ A)]: (((((member @ A) @ A3) @ A6) => ((~ ((((member @ A) @ A3) @ B7))) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ A3) @ B7))) = (((minus_minus @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ B7))) @ (one_one @ nat)))))))).
% 0.33/5.21 thf(fact_3274_card_Oempty, axiom, (! [A:$tType]: ((((finite_card @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_3275_card__insert__le__m1, axiom, (! [A:$tType,N:nat,Y3:(set @ A),X:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ Y3)) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((insert @ A) @ X) @ Y3))) @ N)))))).
% 0.33/5.21 thf(fact_3276_card__Diff__singleton__if, axiom, (! [A:$tType,X:A,A6:(set @ A)]: ((((((member @ A) @ X) @ A6) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A6)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X) @ A6))) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) = ((finite_card @ A) @ A6))))))).
% 0.33/5.21 thf(fact_3277_card__Diff__singleton, axiom, (! [A:$tType,X:A,A6:(set @ A)]: (((((member @ A) @ X) @ A6) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A6)) @ (one_one @ nat))))))).
% 0.33/5.21 thf(fact_3278_card__atLeastAtMost, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or1337092689740270186AtMost @ nat) @ L) @ U)) = (((minus_minus @ nat) @ (suc @ U)) @ L))))).
% 0.33/5.21 thf(fact_3279_sgn__integer__code, axiom, ((sgn_sgn @ code_integer) = (^ [K3:code_integer]: (((((if @ code_integer) @ (K3 = (zero_zero @ code_integer))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ (one_one @ code_integer))) @ (one_one @ code_integer))))))).
% 0.33/5.21 thf(fact_3280_card__insert__le, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ (((insert @ A) @ X) @ A6)))))).
% 0.33/5.21 thf(fact_3281_card__1__singletonE, axiom, (! [A:$tType,A6:(set @ A)]: (((((finite_card @ A) @ A6) = (one_one @ nat)) => (~ ((! [X3:A]: ((A6 != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 0.33/5.21 thf(fact_3282_card__1__singleton__iff, axiom, (! [A:$tType,A6:(set @ A)]: (((((finite_card @ A) @ A6) = (suc @ (zero_zero @ nat))) = (? [X2:A]: ((A6 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))).
% 0.33/5.21 thf(fact_3283_card__eq__SucD, axiom, (! [A:$tType,A6:(set @ A),K:nat]: (((((finite_card @ A) @ A6) = (suc @ K)) => (? [B5:A,B9:(set @ A)]: (((A6 = (((insert @ A) @ B5) @ B9)) & ((~ ((((member @ A) @ B5) @ B9))) & ((((finite_card @ A) @ B9) = K) & ((K = (zero_zero @ nat)) => (B9 = (bot_bot @ (set @ A))))))))))))).
% 0.33/5.21 thf(fact_3284_card__Suc__eq, axiom, (! [A:$tType,A6:(set @ A),K:nat]: (((((finite_card @ A) @ A6) = (suc @ K)) = (? [B3:A,B6:(set @ A)]: (((A6 = (((insert @ A) @ B3) @ B6)) & ((~ ((((member @ A) @ B3) @ B6))) & ((((finite_card @ A) @ B6) = K) & ((K = (zero_zero @ nat)) => (B6 = (bot_bot @ (set @ A))))))))))))).
% 0.33/5.21 thf(fact_3285_card__Diff1__le, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A6))))).
% 0.33/5.21 thf(fact_3286_drop__bit__exp__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: (((((bit_se4197421643247451524op_bit @ A) @ M) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ ((((ord_less_eq @ nat) @ M) @ N) & (((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ M)))))))))).
% 0.33/5.21 thf(fact_3287_bit__minus__2__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.21 thf(fact_3288_bit__double__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) & ((N != (zero_zero @ nat)) & (((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))))))))).
% 0.33/5.21 thf(fact_3289_bit__mask__sub__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (one_one @ A))) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((ord_less @ nat) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_3290_bit__minus__exp__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((ord_less_eq @ nat) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_3291_bit__minus__1__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = (((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))))))).
% 0.33/5.21 thf(fact_3292_plus__integer__code_I1_J, axiom, (! [K:code_integer]: (((((plus_plus @ code_integer) @ K) @ (zero_zero @ code_integer)) = K)))).
% 0.33/5.21 thf(fact_3293_plus__integer__code_I2_J, axiom, (! [L:code_integer]: (((((plus_plus @ code_integer) @ (zero_zero @ code_integer)) @ L) = L)))).
% 0.33/5.21 thf(fact_3294_less__integer__code_I1_J, axiom, (~ ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))))).
% 0.33/5.21 thf(fact_3295_abs__integer__code, axiom, ((abs_abs @ code_integer) = (^ [K3:code_integer]: (((((if @ code_integer) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ K3)) @ K3))))).
% 0.33/5.21 thf(fact_3296_uminus__integer__code_I1_J, axiom, (((uminus_uminus @ code_integer) @ (zero_zero @ code_integer)) = (zero_zero @ code_integer))).
% 0.33/5.21 thf(fact_3297_minus__integer__code_I1_J, axiom, (! [K:code_integer]: (((((minus_minus @ code_integer) @ K) @ (zero_zero @ code_integer)) = K)))).
% 0.33/5.21 thf(fact_3298_minus__integer__code_I2_J, axiom, (! [L:code_integer]: (((((minus_minus @ code_integer) @ (zero_zero @ code_integer)) @ L) = ((uminus_uminus @ code_integer) @ L))))).
% 0.33/5.21 thf(fact_3299_bit__mask__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((bit_se2239418461657761734s_mask @ A) @ M)) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((ord_less @ nat) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_3300_bit__of__nat__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((bit_se5641148757651400278ts_bit @ nat) @ M) @ N))))))))).
% 0.33/5.21 thf(fact_3301_bit__minus__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (~ ((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ N))))))))))).
% 0.33/5.21 thf(fact_3302_bit__push__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ M) @ A3)) @ N) = ((((ord_less_eq @ nat) @ M) @ N) & ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((bit_se5641148757651400278ts_bit @ A) @ A3) @ (((minus_minus @ nat) @ N) @ M))))))))))).
% 0.33/5.21 thf(fact_3303_fold__possible__bit, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) = (zero_zero @ A)) = (~ ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))))))))).
% 0.33/5.21 thf(fact_3304_bit__2__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_3305_CHAR__eq0__iff, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((semiri4206861660011772517g_char @ A) @ (type2 @ A)) = (zero_zero @ nat)) = (! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => (((semiring_1_of_nat @ A) @ N3) != (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_3306_CHAR__eq__posI, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [C2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ C2) => ((((semiring_1_of_nat @ A) @ C2) = (zero_zero @ A)) => ((! [X3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ X3) => ((((ord_less @ nat) @ X3) @ C2) => (((semiring_1_of_nat @ A) @ X3) != (zero_zero @ A)))))) => (((semiri4206861660011772517g_char @ A) @ (type2 @ A)) = C2)))))))))).
% 0.33/5.21 thf(fact_3307_CHAR__pos__iff, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((semiri4206861660011772517g_char @ A) @ (type2 @ A))) = (? [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((semiring_1_of_nat @ A) @ N3) = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_3308_of__nat__CHAR, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ ((semiri4206861660011772517g_char @ A) @ (type2 @ A))) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3309_CHAR__eqI, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [C2:nat]: (((((semiring_1_of_nat @ A) @ C2) = (zero_zero @ A)) => ((! [X3:nat]: (((((semiring_1_of_nat @ A) @ X3) = (zero_zero @ A)) => (((dvd_dvd @ nat) @ C2) @ X3)))) => (((semiri4206861660011772517g_char @ A) @ (type2 @ A)) = C2))))))))).
% 0.33/5.21 thf(fact_3310_CHAR__eq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiri4206861660011772517g_char @ A) @ (type2 @ A)) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3311_of__nat__eq__0__iff__char__dvd, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((semiring_1_of_nat @ A) @ N) = (zero_zero @ A)) = (((dvd_dvd @ nat) @ ((semiri4206861660011772517g_char @ A) @ (type2 @ A))) @ N)))))))).
% 0.33/5.21 thf(fact_3312_divmod__integer__def, axiom, (code_divmod_integer = (^ [K3:code_integer,L3:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ K3) @ L3)) @ (((modulo_modulo @ code_integer) @ K3) @ L3)))))).
% 0.33/5.21 thf(fact_3313_divmod__abs__def, axiom, (code_divmod_abs = (^ [K3:code_integer,L3:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((abs_abs @ code_integer) @ K3)) @ ((abs_abs @ code_integer) @ L3))) @ (((modulo_modulo @ code_integer) @ ((abs_abs @ code_integer) @ K3)) @ ((abs_abs @ code_integer) @ L3))))))).
% 0.33/5.21 thf(fact_3314_card__greaterThanLessThan__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or5935395276787703475ssThan @ int) @ L) @ U)) = (nat2 @ (((minus_minus @ int) @ U) @ (((plus_plus @ int) @ L) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_3315_nth__enumerate__eq, axiom, (! [A:$tType,M:nat,Xs:(list @ A),N:nat]: (((((ord_less @ nat) @ M) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs)) @ M) = ((((product_Pair @ nat) @ A) @ (((plus_plus @ nat) @ N) @ M)) @ (((nth @ A) @ Xs) @ M))))))).
% 0.33/5.21 thf(fact_3316_count__list_Osimps_I2_J, axiom, (! [A:$tType,X:A,Y3:A,Xs:(list @ A)]: ((((X = Y3) => ((((count_list @ A) @ (((cons @ A) @ X) @ Xs)) @ Y3) = (((plus_plus @ nat) @ (((count_list @ A) @ Xs) @ Y3)) @ (one_one @ nat)))) & ((X != Y3) => ((((count_list @ A) @ (((cons @ A) @ X) @ Xs)) @ Y3) = (((count_list @ A) @ Xs) @ Y3))))))).
% 0.33/5.21 thf(fact_3317_greaterThanLessThan__iff, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3318_greaterThanLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or5935395276787703475ssThan @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))).
% 0.33/5.21 thf(fact_3319_greaterThanLessThan__empty__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: ((((((set_or5935395276787703475ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 0.33/5.21 thf(fact_3320_greaterThanLessThan__empty__iff2, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 0.33/5.21 thf(fact_3321_enumerate__simps_I2_J, axiom, (! [B:$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)))))).
% 0.33/5.21 thf(fact_3322_greaterThanLessThan__subseteq__greaterThanLessThan, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or5935395276787703475ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3323_divmod__abs__code_I6_J, axiom, (! [J:code_integer]: ((((code_divmod_abs @ (zero_zero @ code_integer)) @ J) = ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer)))))).
% 0.33/5.21 thf(fact_3324_count__le__length, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((ord_less_eq @ nat) @ (((count_list @ A) @ Xs) @ X)) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.21 thf(fact_3325_greaterThanLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3326_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)))))).
% 0.33/5.21 thf(fact_3327_sqr_Osimps_I3_J, axiom, (! [N:num]: (((sqr @ (bit1 @ N)) = (bit1 @ (bit0 @ (((plus_plus @ num) @ (sqr @ N)) @ N))))))).
% 0.33/5.21 thf(fact_3328_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)))))).
% 0.33/5.21 thf(fact_3329_prod__decode__aux_Oelims, axiom, (! [X:nat,Xa3:nat,Y3:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X) @ Xa3) = Y3) => (((((ord_less_eq @ nat) @ Xa3) @ X) => (Y3 = ((((product_Pair @ nat) @ nat) @ Xa3) @ (((minus_minus @ nat) @ X) @ Xa3)))) & ((~ ((((ord_less_eq @ nat) @ Xa3) @ X))) => (Y3 = ((nat_prod_decode_aux @ (suc @ X)) @ (((minus_minus @ nat) @ Xa3) @ (suc @ X)))))))))).
% 0.33/5.21 thf(fact_3330_prod__decode__aux_Osimps, axiom, (nat_prod_decode_aux = (^ [K3:nat,M2:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ M2) @ K3)) @ ((((product_Pair @ nat) @ nat) @ M2) @ (((minus_minus @ nat) @ K3) @ M2))) @ ((nat_prod_decode_aux @ (suc @ K3)) @ (((minus_minus @ nat) @ M2) @ (suc @ K3)))))))).
% 0.33/5.21 thf(fact_3331_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)))))).
% 0.33/5.21 thf(fact_3332_nat__of__integer__of__nat, axiom, (! [N:nat]: (((code_nat_of_integer @ ((semiring_1_of_nat @ code_integer) @ N)) = N)))).
% 0.33/5.21 thf(fact_3333_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))))).
% 0.33/5.21 thf(fact_3334_card__greaterThanLessThan, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ L) @ U)) = (((minus_minus @ nat) @ U) @ (suc @ L)))))).
% 0.33/5.21 thf(fact_3335_rat__one__code, axiom, ((quotient_of @ (one_one @ rat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int)))).
% 0.33/5.21 thf(fact_3336_rat__zero__code, axiom, ((quotient_of @ (zero_zero @ rat)) = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))).
% 0.33/5.21 thf(fact_3337_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)))).
% 0.33/5.21 thf(fact_3338_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)))))).
% 0.33/5.21 thf(fact_3339_prod__decode__aux_Ocases, axiom, (! [X:((product_prod @ nat) @ nat)]: ((~ ((! [K2:nat,M6:nat]: ((X != ((((product_Pair @ nat) @ nat) @ K2) @ M6))))))))).
% 0.33/5.21 thf(fact_3340_sgn__rat__def, axiom, ((sgn_sgn @ rat) = (^ [A5:rat]: (((((if @ rat) @ (A5 = (zero_zero @ rat))) @ (zero_zero @ rat)) @ ((((if @ rat) @ (((ord_less @ rat) @ (zero_zero @ rat)) @ A5)) @ (one_one @ rat)) @ ((uminus_uminus @ rat) @ (one_one @ rat)))))))).
% 0.33/5.21 thf(fact_3341_diff__rat__def, axiom, ((minus_minus @ rat) = (^ [Q3:rat,R5:rat]: ((((plus_plus @ rat) @ Q3) @ ((uminus_uminus @ rat) @ R5)))))).
% 0.33/5.21 thf(fact_3342_abs__rat__def, axiom, ((abs_abs @ rat) = (^ [A5:rat]: (((((if @ rat) @ (((ord_less @ rat) @ A5) @ (zero_zero @ rat))) @ ((uminus_uminus @ rat) @ A5)) @ A5))))).
% 0.33/5.21 thf(fact_3343_sqr_Osimps_I2_J, axiom, (! [N:num]: (((sqr @ (bit0 @ N)) = (bit0 @ (bit0 @ (sqr @ N))))))).
% 0.33/5.21 thf(fact_3344_sqr_Osimps_I1_J, axiom, ((sqr @ one2) = one2)).
% 0.33/5.21 thf(fact_3345_sqr__conv__mult, axiom, (sqr = (^ [X2:num]: ((((times_times @ num) @ X2) @ X2))))).
% 0.33/5.21 thf(fact_3346_quotient__of__denom__pos, axiom, (! [R:rat,P5:int,Q4:int]: ((((quotient_of @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q4)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q4))))).
% 0.33/5.21 thf(fact_3347_numeral__sqr, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [K:num]: ((((numeral_numeral @ A) @ (sqr @ K)) = (((times_times @ A) @ ((numeral_numeral @ A) @ K)) @ ((numeral_numeral @ A) @ K))))))))).
% 0.33/5.21 thf(fact_3348_quotient__of__int, axiom, (! [A3:int]: (((quotient_of @ (of_int @ A3)) = ((((product_Pair @ int) @ int) @ A3) @ (one_one @ int)))))).
% 0.33/5.21 thf(fact_3349_pow_Osimps_I3_J, axiom, (! [X:num,Y3:num]: ((((pow @ X) @ (bit1 @ Y3)) = (((times_times @ num) @ (sqr @ ((pow @ X) @ Y3))) @ X))))).
% 0.33/5.21 thf(fact_3350_Frct__code__post_I4_J, axiom, (! [K:num]: (((frct @ ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ (one_one @ int))) = ((numeral_numeral @ rat) @ K))))).
% 0.33/5.21 thf(fact_3351_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)))))).
% 0.33/5.21 thf(fact_3352_prod__encode__prod__decode__aux, axiom, (! [K:nat,M:nat]: (((nat_prod_encode @ ((nat_prod_decode_aux @ K) @ M)) = (((plus_plus @ nat) @ (nat_triangle @ K)) @ M))))).
% 0.33/5.21 thf(fact_3353_prod__encode__eq, axiom, (! [X:((product_prod @ nat) @ nat),Y3:((product_prod @ nat) @ nat)]: ((((nat_prod_encode @ X) = (nat_prod_encode @ Y3)) = (X = Y3))))).
% 0.33/5.21 thf(fact_3354_less__eq__rat__def, axiom, ((ord_less_eq @ rat) = (^ [X2:rat,Y6:rat]: (((((ord_less @ rat) @ X2) @ Y6) | (X2 = Y6)))))).
% 0.33/5.21 thf(fact_3355_Frct__code__post_I9_J, axiom, (! [Q4:((product_prod @ int) @ int)]: ((((uminus_uminus @ rat) @ ((uminus_uminus @ rat) @ (frct @ Q4))) = (frct @ Q4))))).
% 0.33/5.21 thf(fact_3356_pow_Osimps_I1_J, axiom, (! [X:num]: ((((pow @ X) @ one2) = X)))).
% 0.33/5.21 thf(fact_3357_le__prod__encode__2, axiom, (! [B2:nat,A3:nat]: ((((ord_less_eq @ nat) @ B2) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A3) @ B2)))))).
% 0.33/5.21 thf(fact_3358_le__prod__encode__1, axiom, (! [A3:nat,B2:nat]: ((((ord_less_eq @ nat) @ A3) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A3) @ B2)))))).
% 0.33/5.21 thf(fact_3359_Frct__code__post_I2_J, axiom, (! [A3:int]: (((frct @ ((((product_Pair @ int) @ int) @ A3) @ (zero_zero @ int))) = (zero_zero @ rat))))).
% 0.33/5.21 thf(fact_3360_Frct__code__post_I1_J, axiom, (! [A3:int]: (((frct @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ A3)) = (zero_zero @ rat))))).
% 0.33/5.21 thf(fact_3361_Frct__code__post_I3_J, axiom, ((frct @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) = (one_one @ rat))).
% 0.33/5.21 thf(fact_3362_pow_Osimps_I2_J, axiom, (! [X:num,Y3:num]: ((((pow @ X) @ (bit0 @ Y3)) = (sqr @ ((pow @ X) @ Y3)))))).
% 0.33/5.21 thf(fact_3363_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))))))).
% 0.33/5.21 thf(fact_3364_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))))))).
% 0.33/5.21 thf(fact_3365_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))))))).
% 0.33/5.21 thf(fact_3366_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)))))).
% 0.33/5.21 thf(fact_3367_one__div__numeral, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((product_fst @ A) @ A) @ (((unique8689654367752047608divmod @ A) @ one2) @ N))))))))).
% 0.33/5.21 thf(fact_3368_in__measures_I2_J, axiom, (! [A:$tType,X:A,Y3:A,F:(A > nat),Fs:(list @ (A > nat))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs))) = ((((ord_less @ nat) @ (F @ X)) @ (F @ Y3)) | (((F @ X) = (F @ Y3)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measures @ A) @ Fs)))))))).
% 0.33/5.21 thf(fact_3369_Cons__lenlex__iff, axiom, (! [A:$tType,M:A,Ms:(list @ A),N:A,Ns:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ M) @ Ms)) @ (((cons @ A) @ N) @ Ns))) @ ((lenlex @ A) @ R)) = ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ms)) @ ((size_size @ (list @ A)) @ Ns)) | (((((size_size @ (list @ A)) @ Ms) = ((size_size @ (list @ A)) @ Ns)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ M) @ N)) @ R)) | ((M = N) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ms) @ Ns)) @ ((lenlex @ A) @ R))))))))).
% 0.33/5.21 thf(fact_3370_fst__divmod__integer, axiom, (! [K:code_integer,L:code_integer]: (((((product_fst @ code_integer) @ code_integer) @ ((code_divmod_integer @ K) @ L)) = (((divide_divide @ code_integer) @ K) @ L))))).
% 0.33/5.21 thf(fact_3371_fst__divmod__abs, axiom, (! [K:code_integer,L:code_integer]: (((((product_fst @ code_integer) @ code_integer) @ ((code_divmod_abs @ K) @ L)) = (((divide_divide @ code_integer) @ ((abs_abs @ code_integer) @ K)) @ ((abs_abs @ code_integer) @ L)))))).
% 0.33/5.21 thf(fact_3372_Suc__0__div__numeral, axiom, (! [K:num]: (((((divide_divide @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ K)) = (((product_fst @ nat) @ nat) @ (((unique8689654367752047608divmod @ nat) @ one2) @ K)))))).
% 0.33/5.21 thf(fact_3373_lenlex__length, axiom, (! [A:$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)))))).
% 0.33/5.21 thf(fact_3374_measures__less, axiom, (! [A:$tType,F:(A > nat),X:A,Y3:A,Fs:(list @ (A > nat))]: (((((ord_less @ nat) @ (F @ X)) @ (F @ Y3)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs))))))).
% 0.33/5.21 thf(fact_3375_measures__lesseq, axiom, (! [A:$tType,F:(A > nat),X:A,Y3:A,Fs:(list @ (A > nat))]: (((((ord_less_eq @ nat) @ (F @ X)) @ (F @ Y3)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measures @ A) @ Fs)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs)))))))).
% 0.33/5.21 thf(fact_3376_sorted__list__of__set__greaterThanLessThan, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ (suc @ I)) @ J) => (((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ I) @ J)) = (((cons @ nat) @ (suc @ I)) @ ((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ (suc @ I)) @ J)))))))).
% 0.33/5.21 thf(fact_3377_bezw_Osimps, axiom, (bezw = (^ [X2:nat,Y6:nat]: (((((if @ ((product_prod @ int) @ int)) @ (Y6 = (zero_zero @ nat))) @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y6) @ (((modulo_modulo @ nat) @ X2) @ Y6)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y6) @ (((modulo_modulo @ nat) @ X2) @ Y6)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y6) @ (((modulo_modulo @ nat) @ X2) @ Y6)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X2) @ Y6)))))))))).
% 0.33/5.21 thf(fact_3378_bezw_Oelims, axiom, (! [X:nat,Xa3:nat,Y3:((product_prod @ int) @ int)]: (((((bezw @ X) @ Xa3) = Y3) => (((Xa3 = (zero_zero @ nat)) => (Y3 = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa3 != (zero_zero @ nat)) => (Y3 = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3379_bezw__non__0, axiom, (! [Y3:nat,X:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Y3) => (((bezw @ X) @ Y3) = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X) @ Y3)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X) @ Y3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X) @ Y3)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Y3)))))))))).
% 0.33/5.21 thf(fact_3380_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))))))).
% 0.33/5.21 thf(fact_3381_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)) & (! [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ N3)) @ (((nth @ B) @ Ys2) @ N3))) @ R))))))))).
% 0.33/5.21 thf(fact_3382_greaterThanAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))).
% 0.33/5.21 thf(fact_3383_greaterThanAtMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))).
% 0.33/5.21 thf(fact_3384_greaterThanAtMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ K) @ L)))))))))).
% 0.33/5.21 thf(fact_3385_greaterThanAtMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((bot_bot @ (set @ A)) = (((set_or3652927894154168847AtMost @ A) @ K) @ L)) = (~ ((((ord_less @ A) @ K) @ L)))))))))).
% 0.33/5.21 thf(fact_3386_one__mod__numeral, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((product_snd @ A) @ A) @ (((unique8689654367752047608divmod @ A) @ one2) @ N))))))))).
% 0.33/5.21 thf(fact_3387_Ioc__inj, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((set_or3652927894154168847AtMost @ A) @ A3) @ B2) = (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = (((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ D2) @ C2)) | ((A3 = C2) & (B2 = D2)))))))))).
% 0.33/5.21 thf(fact_3388_atLeastSucAtMost__greaterThanAtMost, axiom, (! [L:nat,U:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ L)) @ U) = (((set_or3652927894154168847AtMost @ nat) @ L) @ U))))).
% 0.33/5.21 thf(fact_3389_Ioc__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3390_divides__aux__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique5940410009612947441es_aux @ A) = (^ [Qr:((product_prod @ A) @ A)]: (((((product_snd @ A) @ A) @ Qr) = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3391_quotient__of__denom__pos_H, axiom, (! [R:rat]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ (quotient_of @ R)))))).
% 0.33/5.21 thf(fact_3392_sorted__list__of__set__greaterThanAtMost, axiom, (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ (suc @ I)) @ J) => (((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ I) @ J)) = (((cons @ nat) @ (suc @ I)) @ ((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ (suc @ I)) @ J)))))))).
% 0.33/5.21 thf(fact_3393_greaterThanAtMost__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3394_greaterThanLessThan__subseteq__greaterThanAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3395_minus__one__mod__numeral, axiom, (! [N:num]: (((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N)) = ((adjust_mod @ ((numeral_numeral @ int) @ N)) @ (((product_snd @ int) @ int) @ (((unique8689654367752047608divmod @ int) @ one2) @ N))))))).
% 0.33/5.21 thf(fact_3396_one__mod__minus__numeral, axiom, (! [N:num]: (((((modulo_modulo @ int) @ (one_one @ int)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((uminus_uminus @ int) @ ((adjust_mod @ ((numeral_numeral @ int) @ N)) @ (((product_snd @ int) @ int) @ (((unique8689654367752047608divmod @ int) @ one2) @ N)))))))).
% 0.33/5.21 thf(fact_3397_numeral__mod__minus__numeral, axiom, (! [M:num,N:num]: (((((modulo_modulo @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((uminus_uminus @ int) @ ((adjust_mod @ ((numeral_numeral @ int) @ N)) @ (((product_snd @ int) @ int) @ (((unique8689654367752047608divmod @ int) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_3398_minus__numeral__mod__numeral, axiom, (! [M:num,N:num]: (((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = ((adjust_mod @ ((numeral_numeral @ int) @ N)) @ (((product_snd @ int) @ int) @ (((unique8689654367752047608divmod @ int) @ M) @ N))))))).
% 0.33/5.21 thf(fact_3399_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))))))))).
% 0.33/5.21 thf(fact_3400_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))))).
% 0.33/5.21 thf(fact_3401_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)))))).
% 0.33/5.21 thf(fact_3402_Suc__0__mod__numeral, axiom, (! [K:num]: (((((modulo_modulo @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ K)) = (((product_snd @ nat) @ nat) @ (((unique8689654367752047608divmod @ nat) @ one2) @ K)))))).
% 0.33/5.21 thf(fact_3403_atLeastPlusOneAtMost__greaterThanAtMost__int, axiom, (! [L:int,U:int]: (((((set_or1337092689740270186AtMost @ int) @ (((plus_plus @ int) @ L) @ (one_one @ int))) @ U) = (((set_or3652927894154168847AtMost @ int) @ L) @ U))))).
% 0.33/5.21 thf(fact_3404_Divides_Oadjust__mod__def, axiom, (adjust_mod = (^ [L3:int,R5:int]: (((((if @ int) @ (R5 = (zero_zero @ int))) @ (zero_zero @ int)) @ (((minus_minus @ int) @ L3) @ R5)))))).
% 0.33/5.21 thf(fact_3405_bezw_Opelims, axiom, (! [X:nat,Xa3:nat,Y3:((product_prod @ int) @ int)]: (((((bezw @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)) => (~ (((((Xa3 = (zero_zero @ nat)) => (Y3 = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa3 != (zero_zero @ nat)) => (Y3 = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Xa3)))))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3406_in__set__enumerate__eq, axiom, (! [A:$tType,P5:((product_prod @ nat) @ A),N:nat,Xs:(list @ A)]: (((((member @ ((product_prod @ nat) @ A)) @ P5) @ ((set2 @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs))) = ((((ord_less_eq @ nat) @ N) @ (((product_fst @ nat) @ A) @ P5)) & ((((ord_less @ nat) @ (((product_fst @ nat) @ A) @ P5)) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N)) & ((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ (((product_fst @ nat) @ A) @ P5)) @ N)) = (((product_snd @ nat) @ A) @ P5)))))))).
% 0.33/5.21 thf(fact_3407_list__encode_Osimps_I2_J, axiom, (! [X:nat,Xs:(list @ nat)]: (((nat_list_encode @ (((cons @ nat) @ X) @ Xs)) = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X) @ (nat_list_encode @ Xs)))))))).
% 0.33/5.21 thf(fact_3408_bezw__aux, axiom, (! [X:nat,Y3:nat]: ((((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ X) @ Y3)) = (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ ((bezw @ X) @ Y3))) @ ((semiring_1_of_nat @ int) @ X))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ X) @ Y3))) @ ((semiring_1_of_nat @ int) @ Y3))))))).
% 0.33/5.21 thf(fact_3409_in__set__zip, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B),Xs:(list @ A),Ys2:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ P5) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) = (? [N3:nat]: ((((((nth @ A) @ Xs) @ N3) = (((product_fst @ A) @ B) @ P5)) & (((((nth @ B) @ Ys2) @ N3) = (((product_snd @ A) @ B) @ P5)) & ((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs)) & (((ord_less @ nat) @ N3) @ ((size_size @ (list @ B)) @ Ys2))))))))))).
% 0.33/5.21 thf(fact_3410_gcd__eq__0__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_gcd @ A) @ A3) @ B2) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) & (B2 = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_3411_gcd_Obottom__left__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ (one_one @ A)) @ A3) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_3412_gcd_Obottom__right__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ A3) @ (one_one @ A)) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_3413_gcd__neg2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((gcd_gcd @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((gcd_gcd @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3414_gcd__neg1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((gcd_gcd @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((gcd_gcd @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3415_gcd__0__left__nat, axiom, (! [X:nat]: (((((gcd_gcd @ nat) @ (zero_zero @ nat)) @ X) = X)))).
% 0.33/5.21 thf(fact_3416_gcd__0__nat, axiom, (! [X:nat]: (((((gcd_gcd @ nat) @ X) @ (zero_zero @ nat)) = X)))).
% 0.33/5.21 thf(fact_3417_gcd__nat_Oright__neutral, axiom, (! [A3:nat]: (((((gcd_gcd @ nat) @ A3) @ (zero_zero @ nat)) = A3)))).
% 0.33/5.21 thf(fact_3418_gcd__nat_Oneutr__eq__iff, axiom, (! [A3:nat,B2:nat]: ((((zero_zero @ nat) = (((gcd_gcd @ nat) @ A3) @ B2)) = ((A3 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3419_gcd__nat_Oleft__neutral, axiom, (! [A3:nat]: (((((gcd_gcd @ nat) @ (zero_zero @ nat)) @ A3) = A3)))).
% 0.33/5.21 thf(fact_3420_gcd__nat_Oeq__neutr__iff, axiom, (! [A3:nat,B2:nat]: ((((((gcd_gcd @ nat) @ A3) @ B2) = (zero_zero @ nat)) = ((A3 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3421_gcd__1__nat, axiom, (! [M:nat]: (((((gcd_gcd @ nat) @ M) @ (one_one @ nat)) = (one_one @ nat))))).
% 0.33/5.21 thf(fact_3422_gcd__neg__numeral__2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,N:num]: (((((gcd_gcd @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((gcd_gcd @ A) @ A3) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.21 thf(fact_3423_gcd__neg__numeral__1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [N:num,A3:A]: (((((gcd_gcd @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) @ A3) = (((gcd_gcd @ A) @ ((numeral_numeral @ A) @ N)) @ A3)))))))).
% 0.33/5.21 thf(fact_3424_is__unit__gcd__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (one_one @ A)) = ((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3425_gcd__Suc__0, axiom, (! [M:nat]: (((((gcd_gcd @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3426_gcd__pos__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((gcd_gcd @ nat) @ M) @ N)) = ((M != (zero_zero @ nat)) | (N != (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3427_count__notin, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) => ((((count_list @ A) @ Xs) @ X) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3428_list__encode__eq, axiom, (! [X:(list @ nat),Y3:(list @ nat)]: ((((nat_list_encode @ X) = (nat_list_encode @ Y3)) = (X = Y3))))).
% 0.33/5.21 thf(fact_3429_gcd__mult__distrib__nat, axiom, (! [K:nat,M:nat,N:nat]: (((((times_times @ nat) @ K) @ (((gcd_gcd @ nat) @ M) @ N)) = (((gcd_gcd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)))))).
% 0.33/5.21 thf(fact_3430_gcd__add__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M:A,K:A,N:A]: (((((gcd_gcd @ A) @ M) @ (((plus_plus @ A) @ (((times_times @ A) @ K) @ M)) @ N)) = (((gcd_gcd @ A) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_3431_gcd__dvd__prod, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,K:A]: ((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (((times_times @ A) @ K) @ B2)))))))).
% 0.33/5.21 thf(fact_3432_gcd__le1__nat, axiom, (! [A3:nat,B2:nat]: (((A3 != (zero_zero @ nat)) => (((ord_less_eq @ nat) @ (((gcd_gcd @ nat) @ A3) @ B2)) @ A3))))).
% 0.33/5.21 thf(fact_3433_gcd__le2__nat, axiom, (! [B2:nat,A3:nat]: (((B2 != (zero_zero @ nat)) => (((ord_less_eq @ nat) @ (((gcd_gcd @ nat) @ A3) @ B2)) @ B2))))).
% 0.33/5.21 thf(fact_3434_gcd__diff2__nat, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((gcd_gcd @ nat) @ (((minus_minus @ nat) @ N) @ M)) @ N) = (((gcd_gcd @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_3435_gcd__diff1__nat, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((gcd_gcd @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ N) = (((gcd_gcd @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_3436_gcd__non__0__nat, axiom, (! [Y3:nat,X:nat]: (((Y3 != (zero_zero @ nat)) => ((((gcd_gcd @ nat) @ X) @ Y3) = (((gcd_gcd @ nat) @ Y3) @ (((modulo_modulo @ nat) @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_3437_gcd__nat_Osimps, axiom, ((gcd_gcd @ nat) = (^ [X2:nat,Y6:nat]: (((((if @ nat) @ (Y6 = (zero_zero @ nat))) @ X2) @ (((gcd_gcd @ nat) @ Y6) @ (((modulo_modulo @ nat) @ X2) @ Y6))))))).
% 0.33/5.21 thf(fact_3438_gcd__nat_Oelims, axiom, (! [X:nat,Xa3:nat,Y3:nat]: ((((((gcd_gcd @ nat) @ X) @ Xa3) = Y3) => (((Xa3 = (zero_zero @ nat)) => (Y3 = X)) & ((Xa3 != (zero_zero @ nat)) => (Y3 = (((gcd_gcd @ nat) @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3))))))))).
% 0.33/5.21 thf(fact_3439_gcd__mult__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3440_gcd__mult__unit2, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3441_gcd__div__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3442_gcd__div__unit2, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3443_bezout__nat, axiom, (! [A3:nat,B2:nat]: (((A3 != (zero_zero @ nat)) => (? [X3:nat,Y5:nat]: (((((times_times @ nat) @ A3) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y5)) @ (((gcd_gcd @ nat) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3444_bezout__gcd__nat_H, axiom, (! [B2:nat,A3:nat]: ((? [X3:nat,Y5:nat]: ((((((ord_less_eq @ nat) @ (((times_times @ nat) @ B2) @ Y5)) @ (((times_times @ nat) @ A3) @ X3)) & ((((minus_minus @ nat) @ (((times_times @ nat) @ A3) @ X3)) @ (((times_times @ nat) @ B2) @ Y5)) = (((gcd_gcd @ nat) @ A3) @ B2))) | ((((ord_less_eq @ nat) @ (((times_times @ nat) @ A3) @ Y5)) @ (((times_times @ nat) @ B2) @ X3)) & ((((minus_minus @ nat) @ (((times_times @ nat) @ B2) @ X3)) @ (((times_times @ nat) @ A3) @ Y5)) = (((gcd_gcd @ nat) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3445_length__pos__if__in__set, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)))))).
% 0.33/5.21 thf(fact_3446_card__length, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ ((set2 @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.21 thf(fact_3447_all__set__conv__all__nth, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (P @ X2)))) = (! [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I5))))))))).
% 0.33/5.21 thf(fact_3448_all__nth__imp__all__set, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),X:A]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I4))))) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (P @ X)))))).
% 0.33/5.21 thf(fact_3449_in__set__conv__nth, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) = (? [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ I5) = X)))))))).
% 0.33/5.21 thf(fact_3450_list__ball__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A),P:(A > $o)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (P @ X3)))) => (P @ (((nth @ A) @ Xs) @ N))))))).
% 0.33/5.21 thf(fact_3451_nth__mem, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ N)) @ ((set2 @ A) @ Xs)))))).
% 0.33/5.21 thf(fact_3452_nth__equal__first__eq, axiom, (! [A:$tType,X:A,Xs:(list @ A),N:nat]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) => ((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = X) = (N = (zero_zero @ nat)))))))).
% 0.33/5.21 thf(fact_3453_prod__decode__aux_Opelims, axiom, (! [X:nat,Xa3:nat,Y3:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)) => (~ (((((((ord_less_eq @ nat) @ Xa3) @ X) => (Y3 = ((((product_Pair @ nat) @ nat) @ Xa3) @ (((minus_minus @ nat) @ X) @ Xa3)))) & ((~ ((((ord_less_eq @ nat) @ Xa3) @ X))) => (Y3 = ((nat_prod_decode_aux @ (suc @ X)) @ (((minus_minus @ nat) @ Xa3) @ (suc @ X)))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3454_gcd__nat_Opelims, axiom, (! [X:nat,Xa3:nat,Y3:nat]: ((((((gcd_gcd @ nat) @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)) => (~ (((((Xa3 = (zero_zero @ nat)) => (Y3 = X)) & ((Xa3 != (zero_zero @ nat)) => (Y3 = (((gcd_gcd @ nat) @ Xa3) @ (((modulo_modulo @ nat) @ X) @ Xa3))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3455_list__encode_Oelims, axiom, (! [X:(list @ nat),Y3:nat]: ((((nat_list_encode @ X) = Y3) => (((X = (nil @ nat)) => (Y3 != (zero_zero @ nat))) => (~ ((! [X3:nat,Xs2:(list @ nat)]: (((X = (((cons @ nat) @ X3) @ Xs2)) => (Y3 != (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X3) @ (nat_list_encode @ Xs2))))))))))))))).
% 0.33/5.21 thf(fact_3456_zero__rat__def, axiom, ((zero_zero @ rat) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))))).
% 0.33/5.21 thf(fact_3457_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)))))).
% 0.33/5.21 thf(fact_3458_gcd__1__int, axiom, (! [M:int]: (((((gcd_gcd @ int) @ M) @ (one_one @ int)) = (one_one @ int))))).
% 0.33/5.21 thf(fact_3459_gcd__neg2__int, axiom, (! [X:int,Y3:int]: (((((gcd_gcd @ int) @ X) @ ((uminus_uminus @ int) @ Y3)) = (((gcd_gcd @ int) @ X) @ Y3))))).
% 0.33/5.21 thf(fact_3460_gcd__neg1__int, axiom, (! [X:int,Y3:int]: (((((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X)) @ Y3) = (((gcd_gcd @ int) @ X) @ Y3))))).
% 0.33/5.21 thf(fact_3461_gcd__pos__int, axiom, (! [M:int,N:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ (((gcd_gcd @ int) @ M) @ N)) = ((M != (zero_zero @ int)) | (N != (zero_zero @ int))))))).
% 0.33/5.21 thf(fact_3462_gcd__neg__numeral__1__int, axiom, (! [N:num,X:int]: (((((gcd_gcd @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) @ X) = (((gcd_gcd @ int) @ ((numeral_numeral @ int) @ N)) @ X))))).
% 0.33/5.21 thf(fact_3463_gcd__neg__numeral__2__int, axiom, (! [X:int,N:num]: (((((gcd_gcd @ int) @ X) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((gcd_gcd @ int) @ X) @ ((numeral_numeral @ int) @ N)))))).
% 0.33/5.21 thf(fact_3464_gcd__0__left__int, axiom, (! [X:int]: (((((gcd_gcd @ int) @ (zero_zero @ int)) @ X) = ((abs_abs @ int) @ X))))).
% 0.33/5.21 thf(fact_3465_gcd__0__int, axiom, (! [X:int]: (((((gcd_gcd @ int) @ X) @ (zero_zero @ int)) = ((abs_abs @ int) @ X))))).
% 0.33/5.21 thf(fact_3466_gcd__int__int__eq, axiom, (! [M:nat,N:nat]: (((((gcd_gcd @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ M) @ N)))))).
% 0.33/5.21 thf(fact_3467_length__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = (zero_zero @ nat)) = (Xs = (nil @ A)))))).
% 0.33/5.21 thf(fact_3468_horner__sum__simps_I1_J, axiom, (! [B:$tType,A:$tType]: (((comm_semiring_0 @ A) => (! [F:(B > A),A3:A]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ (nil @ B)) = (zero_zero @ A)))))))).
% 0.33/5.21 thf(fact_3469_length__greater__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)) = (Xs != (nil @ A)))))).
% 0.33/5.21 thf(fact_3470_gcd__nat__abs__right__eq, axiom, (! [N:nat,K:int]: (((((gcd_gcd @ nat) @ N) @ (nat2 @ ((abs_abs @ int) @ K))) = (nat2 @ (((gcd_gcd @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 0.33/5.21 thf(fact_3471_gcd__nat__abs__left__eq, axiom, (! [K:int,N:nat]: (((((gcd_gcd @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N) = (nat2 @ (((gcd_gcd @ int) @ K) @ ((semiring_1_of_nat @ int) @ N))))))).
% 0.33/5.21 thf(fact_3472_pair__less__iff1, axiom, (! [X:nat,Y3:nat,Z3:nat]: (((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X) @ Y3)) @ ((((product_Pair @ nat) @ nat) @ X) @ Z3))) @ fun_pair_less) = (((ord_less @ nat) @ Y3) @ Z3))))).
% 0.33/5.21 thf(fact_3473_list__encode_Ocases, axiom, (! [X:(list @ nat)]: (((X != (nil @ nat)) => (~ ((! [X3:nat,Xs2:(list @ nat)]: ((X != (((cons @ nat) @ X3) @ Xs2)))))))))).
% 0.33/5.21 thf(fact_3474_gcd__ge__0__int, axiom, (! [X:int,Y3:int]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((gcd_gcd @ int) @ X) @ Y3))))).
% 0.33/5.21 thf(fact_3475_bezout__int, axiom, (! [X:int,Y3:int]: ((? [U2:int,V4:int]: (((((plus_plus @ int) @ (((times_times @ int) @ U2) @ X)) @ (((times_times @ int) @ V4) @ Y3)) = (((gcd_gcd @ int) @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_3476_gcd__mult__distrib__int, axiom, (! [K:int,M:int,N:int]: (((((times_times @ int) @ ((abs_abs @ int) @ K)) @ (((gcd_gcd @ int) @ M) @ N)) = (((gcd_gcd @ int) @ (((times_times @ int) @ K) @ M)) @ (((times_times @ int) @ K) @ N)))))).
% 0.33/5.21 thf(fact_3477_list_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (list @ A)) @ (nil @ A)) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_3478_list__encode_Osimps_I1_J, axiom, ((nat_list_encode @ (nil @ nat)) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_3479_count__list_Osimps_I1_J, axiom, (! [A:$tType,Y3:A]: (((((count_list @ A) @ (nil @ A)) @ Y3) = (zero_zero @ nat))))).
% 0.33/5.21 thf(fact_3480_gcd__le1__int, axiom, (! [A3:int,B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ A3) => (((ord_less_eq @ int) @ (((gcd_gcd @ int) @ A3) @ B2)) @ A3))))).
% 0.33/5.21 thf(fact_3481_gcd__le2__int, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((gcd_gcd @ int) @ A3) @ B2)) @ B2))))).
% 0.33/5.21 thf(fact_3482_gcd__cases__int, axiom, (! [X:int,Y3:int,P:(int > $o)]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (P @ (((gcd_gcd @ int) @ X) @ Y3)))) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ Y3) @ (zero_zero @ int)) => (P @ (((gcd_gcd @ int) @ X) @ ((uminus_uminus @ int) @ Y3))))) => (((((ord_less_eq @ int) @ X) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (P @ (((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X)) @ Y3)))) => (((((ord_less_eq @ int) @ X) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ Y3) @ (zero_zero @ int)) => (P @ (((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X)) @ ((uminus_uminus @ int) @ Y3))))) => (P @ (((gcd_gcd @ int) @ X) @ Y3))))))))).
% 0.33/5.21 thf(fact_3483_gcd__unique__int, axiom, (! [D2:int,A3:int,B2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ D2) & ((((dvd_dvd @ int) @ D2) @ A3) & ((((dvd_dvd @ int) @ D2) @ B2) & (! [E3:int]: ((((((dvd_dvd @ int) @ E3) @ A3) & (((dvd_dvd @ int) @ E3) @ B2)) => (((dvd_dvd @ int) @ E3) @ D2))))))) = (D2 = (((gcd_gcd @ int) @ A3) @ B2)))))).
% 0.33/5.21 thf(fact_3484_gcd__non__0__int, axiom, (! [Y3:int,X:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Y3) => ((((gcd_gcd @ int) @ X) @ Y3) = (((gcd_gcd @ int) @ Y3) @ (((modulo_modulo @ int) @ X) @ Y3))))))).
% 0.33/5.21 thf(fact_3485_gcd__code__int, axiom, ((gcd_gcd @ int) = (^ [K3:int,L3:int]: (((abs_abs @ int) @ ((((if @ int) @ (L3 = (zero_zero @ int))) @ K3) @ (((gcd_gcd @ int) @ L3) @ (((modulo_modulo @ int) @ ((abs_abs @ int) @ K3)) @ ((abs_abs @ int) @ L3))))))))).
% 0.33/5.21 thf(fact_3486_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))))).
% 0.33/5.21 thf(fact_3487_gcd__int__def, axiom, ((gcd_gcd @ int) = (^ [X2:int,Y6:int]: (((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ (nat2 @ ((abs_abs @ int) @ X2))) @ (nat2 @ ((abs_abs @ int) @ Y6)))))))).
% 0.33/5.21 thf(fact_3488_one__rat__def, axiom, ((one_one @ rat) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))))).
% 0.33/5.21 thf(fact_3489_plus__rat_Oabs__eq, axiom, (! [Xa3:((product_prod @ int) @ int),X:((product_prod @ int) @ int)]: ((((ratrel @ Xa3) @ Xa3) => (((ratrel @ X) @ X) => ((((plus_plus @ rat) @ (abs_Rat @ Xa3)) @ (abs_Rat @ X)) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Xa3)) @ (((product_snd @ int) @ int) @ X))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X)) @ (((product_snd @ int) @ int) @ Xa3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ Xa3)) @ (((product_snd @ int) @ int) @ X)))))))))).
% 0.33/5.21 thf(fact_3490_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)))))).
% 0.33/5.21 thf(fact_3491_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))))).
% 0.33/5.21 thf(fact_3492_smax__insertI, axiom, (! [Y3:((product_prod @ nat) @ nat),Y7:(set @ ((product_prod @ nat) @ nat)),X:((product_prod @ nat) @ nat),X8:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ Y3) @ Y7) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y3)) @ 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))) @ X8) @ Y7)) @ 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))) @ (((insert @ ((product_prod @ nat) @ nat)) @ X) @ X8)) @ Y7)) @ fun_max_strict))))))).
% 0.33/5.21 thf(fact_3493_smin__insertI, axiom, (! [X:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y3:((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) @ Y3)) @ 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) @ (((insert @ ((product_prod @ nat) @ nat)) @ Y3) @ YS))) @ fun_min_strict))))))).
% 0.33/5.21 thf(fact_3494_ratrel__iff, axiom, (ratrel = (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: ((((((product_snd @ int) @ int) @ X2) != (zero_zero @ int)) & (((((product_snd @ int) @ int) @ Y6) != (zero_zero @ int)) & ((((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)) = (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2))))))))).
% 0.33/5.21 thf(fact_3495_smin__emptyI, axiom, (! [X8:(set @ ((product_prod @ nat) @ nat))]: (((X8 != (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))) @ X8) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_strict))))).
% 0.33/5.21 thf(fact_3496_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)))).
% 0.33/5.21 thf(fact_3497_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)))).
% 0.33/5.21 thf(fact_3498_times__rat_Oabs__eq, axiom, (! [Xa3:((product_prod @ int) @ int),X:((product_prod @ int) @ int)]: ((((ratrel @ Xa3) @ Xa3) => (((ratrel @ X) @ X) => ((((times_times @ rat) @ (abs_Rat @ Xa3)) @ (abs_Rat @ X)) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Xa3)) @ (((product_fst @ int) @ int) @ X))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ Xa3)) @ (((product_snd @ int) @ int) @ X)))))))))).
% 0.33/5.21 thf(fact_3499_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)))))))).
% 0.33/5.21 thf(fact_3500_wmin__insertI, axiom, (! [X:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y3:((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) @ Y3)) @ 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) @ (((insert @ ((product_prod @ nat) @ nat)) @ Y3) @ YS))) @ fun_min_weak))))))).
% 0.33/5.21 thf(fact_3501_wmax__insertI, axiom, (! [Y3:((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)) @ Y3) @ YS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y3)) @ 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))) @ (((insert @ ((product_prod @ nat) @ nat)) @ X) @ XS)) @ YS)) @ fun_max_weak))))))).
% 0.33/5.21 thf(fact_3502_positive_Oabs__eq, axiom, (! [X:((product_prod @ int) @ int)]: ((((ratrel @ X) @ X) => ((positive @ (abs_Rat @ X)) = (((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X)) @ (((product_snd @ int) @ int) @ X)))))))).
% 0.33/5.21 thf(fact_3503_upto_Opelims, axiom, (! [X:int,Xa3:int,Y3:(list @ int)]: (((((upto @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa3)) => (~ (((((((ord_less_eq @ int) @ X) @ Xa3) => (Y3 = (((cons @ int) @ X) @ ((upto @ (((plus_plus @ int) @ X) @ (one_one @ int))) @ Xa3)))) & ((~ ((((ord_less_eq @ int) @ X) @ Xa3))) => (Y3 = (nil @ int)))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa3)))))))))))).
% 0.33/5.21 thf(fact_3504_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)))))))).
% 0.33/5.21 thf(fact_3505_upto__Nil, axiom, (! [I:int,J:int]: (((((upto @ I) @ J) = (nil @ int)) = (((ord_less @ int) @ J) @ I))))).
% 0.33/5.21 thf(fact_3506_upto__Nil2, axiom, (! [I:int,J:int]: ((((nil @ int) = ((upto @ I) @ J)) = (((ord_less @ int) @ J) @ I))))).
% 0.33/5.21 thf(fact_3507_upto__empty, axiom, (! [J:int,I:int]: (((((ord_less @ int) @ J) @ I) => (((upto @ I) @ J) = (nil @ int)))))).
% 0.33/5.21 thf(fact_3508_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))))))).
% 0.33/5.21 thf(fact_3509_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))))))).
% 0.33/5.21 thf(fact_3510_upto__rec__numeral_I1_J, axiom, (! [M:num,N:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N)) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N)) = (((cons @ int) @ ((numeral_numeral @ int) @ M)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N)))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N)) = (nil @ int))))))).
% 0.33/5.21 thf(fact_3511_upto__rec__numeral_I4_J, axiom, (! [M:num,N:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N)))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (nil @ int))))))).
% 0.33/5.21 thf(fact_3512_upto__rec__numeral_I3_J, axiom, (! [M:num,N:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = (nil @ int))))))).
% 0.33/5.21 thf(fact_3513_upto__rec__numeral_I2_J, axiom, (! [M:num,N:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((cons @ int) @ ((numeral_numeral @ int) @ M)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N)))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (nil @ int))))))).
% 0.33/5.21 thf(fact_3514_less__by__empty, axiom, (! [A:$tType,A6:(set @ ((product_prod @ A) @ A)),B7:(set @ ((product_prod @ A) @ A))]: (((A6 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A6) @ B7))))).
% 0.33/5.21 thf(fact_3515_wmin__emptyI, axiom, (! [X8:(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))) @ X8) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_weak)))).
% 0.33/5.21 thf(fact_3516_positive__minus, axiom, (! [X:rat]: (((~ ((positive @ X))) => ((X != (zero_zero @ rat)) => (positive @ ((uminus_uminus @ rat) @ X))))))).
% 0.33/5.21 thf(fact_3517_greaterThanAtMost__upto, axiom, ((set_or3652927894154168847AtMost @ int) = (^ [I5:int,J3:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I5) @ (one_one @ int))) @ J3)))))).
% 0.33/5.21 thf(fact_3518_less__rat__def, axiom, ((ord_less @ rat) = (^ [X2:rat,Y6:rat]: ((positive @ (((minus_minus @ rat) @ Y6) @ X2)))))).
% 0.33/5.21 thf(fact_3519_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))))))).
% 0.33/5.21 thf(fact_3520_upto_Oelims, axiom, (! [X:int,Xa3:int,Y3:(list @ int)]: (((((upto @ X) @ Xa3) = Y3) => (((((ord_less_eq @ int) @ X) @ Xa3) => (Y3 = (((cons @ int) @ X) @ ((upto @ (((plus_plus @ int) @ X) @ (one_one @ int))) @ Xa3)))) & ((~ ((((ord_less_eq @ int) @ X) @ Xa3))) => (Y3 = (nil @ int)))))))).
% 0.33/5.21 thf(fact_3521_upto_Osimps, axiom, (upto = (^ [I5:int,J3:int]: (((((if @ (list @ int)) @ (((ord_less_eq @ int) @ I5) @ J3)) @ (((cons @ int) @ I5) @ ((upto @ (((plus_plus @ int) @ I5) @ (one_one @ int))) @ J3))) @ (nil @ int)))))).
% 0.33/5.21 thf(fact_3522_greaterThanLessThan__upto, axiom, ((set_or5935395276787703475ssThan @ int) = (^ [I5:int,J3:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I5) @ (one_one @ int))) @ (((minus_minus @ int) @ J3) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_3523_positive_Orep__eq, axiom, (positive = (^ [X2:rat]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ (rep_Rat @ X2))) @ (((product_snd @ int) @ int) @ (rep_Rat @ X2)))))))).
% 0.33/5.21 thf(fact_3524_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))))))))).
% 0.33/5.21 thf(fact_3525_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))).
% 0.33/5.21 thf(fact_3526_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))).
% 0.33/5.21 thf(fact_3527_upto__rec2, axiom, (! [I:int,J:int]: (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ (((cons @ int) @ J) @ (nil @ int)))))))).
% 0.33/5.21 thf(fact_3528_inverse__eq__iff__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((((inverse_inverse @ A) @ A3) = ((inverse_inverse @ A) @ B2)) = (A3 = B2)))))))).
% 0.33/5.21 thf(fact_3529_inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: ((((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A3)) = A3))))))).
% 0.33/5.21 thf(fact_3530_inverse__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3531_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((((inverse_inverse @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3532_inverse__mult__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((inverse_inverse @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2))))))))).
% 0.33/5.21 thf(fact_3533_inverse__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.21 thf(fact_3534_inverse__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [X:A]: (((((inverse_inverse @ A) @ X) = (one_one @ A)) = (X = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3535_inverse__minus__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: ((((inverse_inverse @ A) @ ((uminus_uminus @ A) @ A3)) = ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_3536_inverse__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((inverse_inverse @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ B2) @ A3)))))))).
% 0.33/5.21 thf(fact_3537_abs__inverse, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((inverse_inverse @ A) @ A3)) = ((inverse_inverse @ A) @ ((abs_abs @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_3538_sgn__inverse, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A3:A]: ((((sgn_sgn @ A) @ ((inverse_inverse @ A) @ A3)) = ((inverse_inverse @ A) @ ((sgn_sgn @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_3539_inverse__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: ((((inverse_inverse @ A) @ ((sgn_sgn @ A) @ A3)) = ((sgn_sgn @ A) @ A3)))))))).
% 0.33/5.21 thf(fact_3540_inverse__nonnegative__iff__nonnegative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A3)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_3541_inverse__nonpositive__iff__nonpositive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3542_inverse__less__iff__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.21 thf(fact_3543_inverse__less__iff__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))))).
% 0.33/5.21 thf(fact_3544_inverse__negative__iff__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3545_inverse__positive__iff__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 0.33/5.21 thf(fact_3546_inverse__le__iff__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.21 thf(fact_3547_inverse__le__iff__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 0.33/5.21 thf(fact_3548_left__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ A3) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3549_right__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ ((inverse_inverse @ A) @ A3)) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3550_inverse__eq__divide__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num]: ((((inverse_inverse @ A) @ ((numeral_numeral @ A) @ W2)) = (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ W2))))))))).
% 0.33/5.21 thf(fact_3551_inverse__eq__divide__neg__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num]: ((((inverse_inverse @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((divide_divide @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))).
% 0.33/5.21 thf(fact_3552_mult__commute__imp__mult__inverse__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Y3:A,X:A]: ((((((times_times @ A) @ Y3) @ X) = (((times_times @ A) @ X) @ Y3)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ Y3)) @ X) = (((times_times @ A) @ X) @ ((inverse_inverse @ A) @ Y3)))))))))).
% 0.33/5.21 thf(fact_3553_inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((((inverse_inverse @ A) @ A3) = ((inverse_inverse @ A) @ B2)) => (A3 = B2)))))))).
% 0.33/5.21 thf(fact_3554_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: (((field @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3555_inverse__zero__imp__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((((inverse_inverse @ A) @ A3) = (zero_zero @ A)) => (A3 = (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3556_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((((inverse_inverse @ A) @ A3) = ((inverse_inverse @ A) @ B2)) => ((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (A3 = B2)))))))))).
% 0.33/5.21 thf(fact_3557_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A3)) = A3)))))))).
% 0.33/5.21 thf(fact_3558_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A3) != (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3559_inverse__less__imp__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ B2) @ A3))))))))).
% 0.33/5.21 thf(fact_3560_less__imp__inverse__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3561_inverse__less__imp__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A3))))))))).
% 0.33/5.21 thf(fact_3562_less__imp__inverse__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3563_inverse__negative__imp__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ (zero_zero @ A)) => ((A3 != (zero_zero @ A)) => (((ord_less @ A) @ A3) @ (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_3564_inverse__positive__imp__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A3)) => ((A3 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A3))))))))).
% 0.33/5.21 thf(fact_3565_negative__imp__inverse__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_3566_positive__imp__inverse__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A3))))))))).
% 0.33/5.21 thf(fact_3567_nonzero__inverse__mult__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A3))))))))))).
% 0.33/5.21 thf(fact_3568_nonzero__inverse__minus__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((uminus_uminus @ A) @ A3)) = ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3569_inverse__unique, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ B2) = (one_one @ A)) => (((inverse_inverse @ A) @ A3) = B2)))))))).
% 0.33/5.21 thf(fact_3570_inverse__numeral__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ ((numeral_numeral @ A) @ one2)) = ((numeral_numeral @ A) @ one2)))))).
% 0.33/5.21 thf(fact_3571_divide__inverse__commute, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A5:A,B3:A]: ((((times_times @ A) @ ((inverse_inverse @ A) @ B3)) @ A5)))))))).
% 0.33/5.21 thf(fact_3572_divide__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => ((divide_divide @ A) = (^ [A5:A,B3:A]: ((((times_times @ A) @ A5) @ ((inverse_inverse @ A) @ B3))))))))).
% 0.33/5.21 thf(fact_3573_field__class_Ofield__divide__inverse, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A5:A,B3:A]: ((((times_times @ A) @ A5) @ ((inverse_inverse @ A) @ B3))))))))).
% 0.33/5.21 thf(fact_3574_inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => ((inverse_inverse @ A) = ((divide_divide @ A) @ (one_one @ A))))))).
% 0.33/5.21 thf(fact_3575_power__mult__power__inverse__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:nat,N:nat]: (((((times_times @ A) @ (((power_power @ A) @ X) @ M)) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X)) @ N)) = (((times_times @ A) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X)) @ N)) @ (((power_power @ A) @ X) @ M))))))))).
% 0.33/5.21 thf(fact_3576_power__mult__inverse__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:nat]: (((((times_times @ A) @ (((power_power @ A) @ X) @ M)) @ ((inverse_inverse @ A) @ X)) = (((times_times @ A) @ ((inverse_inverse @ A) @ X)) @ (((power_power @ A) @ X) @ M))))))))).
% 0.33/5.21 thf(fact_3577_mult__inverse__of__nat__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Xa3:nat,X:A]: (((((times_times @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ Xa3))) @ X) = (((times_times @ A) @ X) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ Xa3)))))))))).
% 0.33/5.21 thf(fact_3578_nonzero__abs__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((abs_abs @ A) @ ((inverse_inverse @ A) @ A3)) = ((inverse_inverse @ A) @ ((abs_abs @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3579_mult__inverse__of__int__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Xa3:int,X:A]: (((((times_times @ A) @ ((inverse_inverse @ A) @ ((ring_1_of_int @ A) @ Xa3))) @ X) = (((times_times @ A) @ X) @ ((inverse_inverse @ A) @ ((ring_1_of_int @ A) @ Xa3)))))))))).
% 0.33/5.21 thf(fact_3580_inverse__le__imp__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 0.33/5.21 thf(fact_3581_le__imp__inverse__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3582_inverse__le__imp__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 0.33/5.21 thf(fact_3583_le__imp__inverse__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3584_inverse__le__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ X)) @ (one_one @ A)) = ((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X))))))))).
% 0.33/5.21 thf(fact_3585_one__less__inverse, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_3586_one__less__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X) & (((ord_less @ A) @ X) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_3587_inverse__add, axiom, (! [A:$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))))))))))).
% 0.33/5.21 thf(fact_3588_division__ring__inverse__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ (((plus_plus @ A) @ A3) @ B2))) @ ((inverse_inverse @ A) @ B2))))))))))).
% 0.33/5.21 thf(fact_3589_field__class_Ofield__inverse, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ A3) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_3590_division__ring__inverse__diff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ (((minus_minus @ A) @ B2) @ A3))) @ ((inverse_inverse @ A) @ B2))))))))))).
% 0.33/5.21 thf(fact_3591_nonzero__inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A3) = (((divide_divide @ A) @ (one_one @ A)) @ A3))))))))).
% 0.33/5.21 thf(fact_3592_inverse__le__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ B2) @ A3)) & ((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3593_inverse__less__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A3) @ B2)) => (((ord_less @ A) @ B2) @ A3)) & ((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (zero_zero @ A)) => (((ord_less @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3594_one__le__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 0.33/5.21 thf(fact_3595_inverse__less__1__iff, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3596_one__le__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X) & (((ord_less_eq @ A) @ X) @ (one_one @ A)))))))))).
% 0.33/5.21 thf(fact_3597_reals__Archimedean, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (? [N4:nat]: ((((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N4)))) @ X)))))))))).
% 0.33/5.21 thf(fact_3598_length__append__singleton, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((size_size @ (list @ A)) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) = (suc @ ((size_size @ (list @ A)) @ Xs)))))).
% 0.33/5.21 thf(fact_3599_length__Suc__conv__rev, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((size_size @ (list @ A)) @ Xs) = (suc @ N)) = (? [Y6:A,Ys3:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ Y6) @ (nil @ A)))) & (((size_size @ (list @ A)) @ Ys3) = N)))))))).
% 0.33/5.21 thf(fact_3600_nth__append, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys2:(list @ A)]: ((((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys2)) @ N) = (((nth @ A) @ Xs) @ N))) & ((~ ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)))) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys2)) @ N) = (((nth @ A) @ Ys2) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 0.33/5.21 thf(fact_3601_upto__split2, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ J)) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K)))))))).
% 0.33/5.21 thf(fact_3602_upto__split1, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ ((upto @ J) @ K)))))))).
% 0.33/5.21 thf(fact_3603_ex__inverse__of__nat__less, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (? [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) & (((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N4))) @ X))))))))))).
% 0.33/5.21 thf(fact_3604_power__diff__conv__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:nat,N:nat]: (((X != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ M) @ N) => ((((power_power @ A) @ X) @ (((minus_minus @ nat) @ N) @ M)) = (((times_times @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X)) @ M))))))))))).
% 0.33/5.21 thf(fact_3605_horner__sum__append, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [F:(B > A),A3:A,Xs:(list @ B),Ys2:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ (((append @ B) @ Xs) @ Ys2)) = (((plus_plus @ A) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Xs)) @ (((times_times @ A) @ (((power_power @ A) @ A3) @ ((size_size @ (list @ B)) @ Xs))) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Ys2)))))))))).
% 0.33/5.21 thf(fact_3606_upto__split3, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ (((cons @ int) @ J) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K))))))))).
% 0.33/5.21 thf(fact_3607_list__encode_Opelims, axiom, (! [X:(list @ nat),Y3:nat]: ((((nat_list_encode @ X) = Y3) => ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ X) => (((X = (nil @ nat)) => ((Y3 = (zero_zero @ nat)) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (nil @ nat)))))) => (~ ((! [X3:nat,Xs2:(list @ nat)]: (((X = (((cons @ nat) @ X3) @ Xs2)) => ((Y3 = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X3) @ (nat_list_encode @ Xs2))))) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (((cons @ nat) @ X3) @ Xs2)))))))))))))))).
% 0.33/5.21 thf(fact_3608_rp__inv__image__rp, axiom, (! [A:$tType,B:$tType,P:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),F:(B > A)]: ((((fun_reduction_pair @ A) @ P) => ((fun_reduction_pair @ B) @ ((((fun_rp_inv_image @ A) @ B) @ P) @ F)))))).
% 0.33/5.21 thf(fact_3609_take__Suc__conv__app__nth, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((take @ A) @ (suc @ I)) @ Xs) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (nil @ A)))))))).
% 0.33/5.21 thf(fact_3610_min__strict__def, axiom, (fun_min_strict = ((min_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_less))).
% 0.33/5.21 thf(fact_3611_upt__rec__numeral, axiom, (! [M:num,N:num]: ((((((ord_less @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N)) => (((upt @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N)) = (((cons @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((upt @ (suc @ ((numeral_numeral @ nat) @ M))) @ ((numeral_numeral @ nat) @ N))))) & ((~ ((((ord_less @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N)))) => (((upt @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N)) = (nil @ nat))))))).
% 0.33/5.21 thf(fact_3612_take__Suc__Cons, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: (((((take @ A) @ (suc @ N)) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ (((take @ A) @ N) @ Xs)))))).
% 0.33/5.21 thf(fact_3613_take__eq__Nil2, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((nil @ A) = (((take @ A) @ N) @ Xs)) = ((N = (zero_zero @ nat)) | (Xs = (nil @ A))))))).
% 0.33/5.21 thf(fact_3614_take__eq__Nil, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((take @ A) @ N) @ Xs) = (nil @ A)) = ((N = (zero_zero @ nat)) | (Xs = (nil @ A))))))).
% 0.33/5.21 thf(fact_3615_take0, axiom, (! [A:$tType]: ((((take @ A) @ (zero_zero @ nat)) = (^ [Xs3:(list @ A)]: ((nil @ A))))))).
% 0.33/5.21 thf(fact_3616_take__all, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((take @ A) @ N) @ Xs) = Xs))))).
% 0.33/5.21 thf(fact_3617_take__all__iff, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((take @ A) @ N) @ Xs) = Xs) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N))))).
% 0.33/5.21 thf(fact_3618_take__upt, axiom, (! [I:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ M)) @ N) => ((((take @ nat) @ M) @ ((upt @ I) @ N)) = ((upt @ I) @ (((plus_plus @ nat) @ I) @ M))))))).
% 0.33/5.21 thf(fact_3619_nth__take, axiom, (! [A:$tType,I:nat,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ N) => ((((nth @ A) @ (((take @ A) @ N) @ Xs)) @ I) = (((nth @ A) @ Xs) @ I)))))).
% 0.33/5.21 thf(fact_3620_upt__conv__Nil, axiom, (! [J:nat,I:nat]: (((((ord_less_eq @ nat) @ J) @ I) => (((upt @ I) @ J) = (nil @ nat)))))).
% 0.33/5.21 thf(fact_3621_upt__eq__Nil__conv, axiom, (! [I:nat,J:nat]: (((((upt @ I) @ J) = (nil @ nat)) = ((J = (zero_zero @ nat)) | (((ord_less_eq @ nat) @ J) @ I)))))).
% 0.33/5.21 thf(fact_3622_nth__upt, axiom, (! [I:nat,K:nat,J:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J) => ((((nth @ nat) @ ((upt @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ K)))))).
% 0.33/5.21 thf(fact_3623_take__Cons__numeral, axiom, (! [A:$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)))))).
% 0.33/5.21 thf(fact_3624_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)))))))).
% 0.33/5.21 thf(fact_3625_take__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((take @ A) @ (zero_zero @ nat)) @ Xs) = (nil @ A))))).
% 0.33/5.21 thf(fact_3626_atLeastAtMost__upt, axiom, ((set_or1337092689740270186AtMost @ nat) = (^ [N3:nat,M2:nat]: (((set2 @ nat) @ ((upt @ N3) @ (suc @ M2))))))).
% 0.33/5.21 thf(fact_3627_upt__Suc, axiom, (! [I:nat,J:nat]: ((((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (suc @ J)) = (((append @ nat) @ ((upt @ I) @ J)) @ (((cons @ nat) @ J) @ (nil @ nat))))) & ((~ ((((ord_less_eq @ nat) @ I) @ J))) => (((upt @ I) @ (suc @ J)) = (nil @ nat))))))).
% 0.33/5.21 thf(fact_3628_upt__Suc__append, axiom, (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (suc @ J)) = (((append @ nat) @ ((upt @ I) @ J)) @ (((cons @ nat) @ J) @ (nil @ nat)))))))).
% 0.33/5.21 thf(fact_3629_upt__conv__Cons__Cons, axiom, (! [M:nat,N:nat,Ns:(list @ nat),Q4:nat]: ((((((cons @ nat) @ M) @ (((cons @ nat) @ N) @ Ns)) = ((upt @ M) @ Q4)) = ((((cons @ nat) @ N) @ Ns) = ((upt @ (suc @ M)) @ Q4)))))).
% 0.33/5.21 thf(fact_3630_upt__0, axiom, (! [I:nat]: ((((upt @ I) @ (zero_zero @ nat)) = (nil @ nat))))).
% 0.33/5.21 thf(fact_3631_greaterThanAtMost__upt, axiom, ((set_or3652927894154168847AtMost @ nat) = (^ [N3:nat,M2:nat]: (((set2 @ nat) @ ((upt @ (suc @ N3)) @ (suc @ M2))))))).
% 0.33/5.21 thf(fact_3632_greaterThanLessThan__upt, axiom, ((set_or5935395276787703475ssThan @ nat) = (^ [N3:nat,M2:nat]: (((set2 @ nat) @ ((upt @ (suc @ N3)) @ M2)))))).
% 0.33/5.21 thf(fact_3633_set__take__subset__set__take, axiom, (! [A:$tType,M:nat,N:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ M) @ Xs))) @ ((set2 @ A) @ (((take @ A) @ N) @ Xs))))))).
% 0.33/5.21 thf(fact_3634_upt__conv__Cons, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((upt @ I) @ J) = (((cons @ nat) @ I) @ ((upt @ (suc @ I)) @ J))))))).
% 0.33/5.21 thf(fact_3635_nth__take__lemma, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Ys2)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => ((((nth @ A) @ Xs) @ I4) = (((nth @ A) @ Ys2) @ I4))))) => ((((take @ A) @ K) @ Xs) = (((take @ A) @ K) @ Ys2)))))))).
% 0.33/5.21 thf(fact_3636_upt__eq__Cons__conv, axiom, (! [I:nat,J:nat,X:nat,Xs:(list @ nat)]: (((((upt @ I) @ J) = (((cons @ nat) @ X) @ Xs)) = ((((ord_less @ nat) @ I) @ J) & ((I = X) & (((upt @ (((plus_plus @ nat) @ I) @ (one_one @ nat))) @ J) = Xs))))))).
% 0.33/5.21 thf(fact_3637_upt__rec, axiom, (upt = (^ [I5:nat,J3:nat]: (((((if @ (list @ nat)) @ (((ord_less @ nat) @ I5) @ J3)) @ (((cons @ nat) @ I5) @ ((upt @ (suc @ I5)) @ J3))) @ (nil @ nat)))))).
% 0.33/5.21 thf(fact_3638_take__Cons_H, axiom, (! [A:$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)))))))).
% 0.33/5.21 thf(fact_3639_horner__sum__bit__eq__take__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,N:nat]: (((((((groups4207007520872428315er_sum @ $o) @ A) @ (zero_neq_one_of_bool @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((map @ nat) @ $o) @ ((bit_se5641148757651400278ts_bit @ A) @ A3)) @ ((upt @ (zero_zero @ nat)) @ N))) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)))))))).
% 0.33/5.21 thf(fact_3640_lexord__take__index__conv, axiom, (! [A:$tType,X:(list @ A),Y3:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y3)) @ ((lexord @ A) @ R)) = (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ X)) @ ((size_size @ (list @ A)) @ Y3)) & ((((take @ A) @ ((size_size @ (list @ A)) @ X)) @ Y3) = X)) | (? [I5:nat]: (((((ord_less @ nat) @ I5) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ X)) @ ((size_size @ (list @ A)) @ Y3))) & (((((take @ A) @ I5) @ X) = (((take @ A) @ I5) @ Y3)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ X) @ I5)) @ (((nth @ A) @ Y3) @ I5))) @ R)))))))))).
% 0.33/5.21 thf(fact_3641_lex__take__index, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lex @ A) @ R)) => (~ ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Ys2)) => (((((take @ A) @ I4) @ Xs) = (((take @ A) @ I4) @ Ys2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ I4)) @ (((nth @ A) @ Ys2) @ I4))) @ R)))))))))))))).
% 0.33/5.21 thf(fact_3642_min__weak__def, axiom, (fun_min_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((min_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))).
% 0.33/5.21 thf(fact_3643_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_3644_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 0.33/5.21 thf(fact_3645_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B2)) = (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3646_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3647_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ B2) = (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3648_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.33/5.21 thf(fact_3649_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ Z3) = ((((ord_less_eq @ A) @ X) @ Z3) & (((ord_less_eq @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3650_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A3) = ((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 0.33/5.21 thf(fact_3651_sup__bot__left, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 0.33/5.21 thf(fact_3652_sup__bot__right, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 0.33/5.21 thf(fact_3653_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X:A,Y3:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y3)) = ((X = (bot_bot @ A)) & (Y3 = (bot_bot @ A)))))))))).
% 0.33/5.21 thf(fact_3654_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X:A,Y3:A]: ((((((sup_sup @ A) @ X) @ Y3) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y3 = (bot_bot @ A)))))))))).
% 0.33/5.21 thf(fact_3655_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A3:A,B2:A]: ((((((sup_sup @ A) @ A3) @ B2) = (bot_bot @ A)) = ((A3 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 0.33/5.21 thf(fact_3656_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_3657_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A3:A,B2:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A3) @ B2)) = ((A3 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))).
% 0.33/5.21 thf(fact_3658_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ (bot_bot @ A)) = A3))))))).
% 0.33/5.21 thf(fact_3659_Compl__Diff__eq, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ B7)) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ B7))))).
% 0.33/5.21 thf(fact_3660_nth__map, axiom, (! [B:$tType,A:$tType,N:nat,Xs:(list @ A),F:(A > B)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ B) @ ((((map @ A) @ B) @ F) @ Xs)) @ N) = (F @ (((nth @ A) @ Xs) @ N))))))).
% 0.33/5.21 thf(fact_3661_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3662_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3)) = (((sup_sup @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3663_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ Z3) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3664_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X2:A,Y6:A]: ((((sup_sup @ A) @ Y6) @ X2)))))))).
% 0.33/5.21 thf(fact_3665_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ C2) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3666_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ Z3) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3667_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A5:A,B3:A]: ((((sup_sup @ A) @ B3) @ A5)))))))).
% 0.33/5.21 thf(fact_3668_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X2:A,Y6:A]: ((((sup_sup @ A) @ Y6) @ X2)))))))).
% 0.33/5.21 thf(fact_3669_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A,C2:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A3) @ C2)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3670_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3)) = (((sup_sup @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3671_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.33/5.21 thf(fact_3672_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:A,K:A,A3:A,B2:A]: (((A6 = (((sup_sup @ A) @ K) @ A3)) => ((((sup_sup @ A) @ A6) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3673_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B7:A,K:A,B2:A,A3:A]: (((B7 = (((sup_sup @ A) @ K) @ B2)) => ((((sup_sup @ A) @ A3) @ B7) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3674_less__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,A3:A,B2:A]: (((((ord_less @ A) @ X) @ A3) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3675_less__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,B2:A,A3:A]: (((((ord_less @ A) @ X) @ B2) => (((ord_less @ A) @ X) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3676_sup_Oabsorb3, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((sup_sup @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_3677_sup_Oabsorb4, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((sup_sup @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_3678_sup_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A3) => (~ (((((ord_less @ A) @ B2) @ A3) => (~ ((((ord_less @ A) @ C2) @ A3))))))))))))).
% 0.33/5.21 thf(fact_3679_sup_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less @ A) = (^ [B3:A,A5:A]: (((A5 = (((sup_sup @ A) @ A5) @ B3)) & (A5 != B3))))))))).
% 0.33/5.21 thf(fact_3680_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ C2) @ A3) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3681_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3682_sup__max, axiom, (! [A:$tType]: ((((semilattice_sup @ A) & (linorder @ A)) => ((sup_sup @ A) = (ord_max @ A)))))).
% 0.33/5.21 thf(fact_3683_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3684_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3685_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: (((((sup_sup @ A) @ A5) @ B3) = B3)))))))).
% 0.33/5.21 thf(fact_3686_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: (((((sup_sup @ A) @ A5) @ B3) = A5)))))))).
% 0.33/5.21 thf(fact_3687_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A]: ((((ord_less_eq @ A) @ B2) @ (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3688_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3689_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: ((A5 = (((sup_sup @ A) @ A5) @ B3))))))))).
% 0.33/5.21 thf(fact_3690_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A3))))))))).
% 0.33/5.21 thf(fact_3691_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A3) => (~ (((((ord_less_eq @ A) @ B2) @ A3) => (~ ((((ord_less_eq @ A) @ C2) @ A3))))))))))))).
% 0.33/5.21 thf(fact_3692_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((sup_sup @ A) @ X) @ Y3) = Y3)))))))).
% 0.33/5.21 thf(fact_3693_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((sup_sup @ A) @ X) @ Y3) = X)))))))).
% 0.33/5.21 thf(fact_3694_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((sup_sup @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_3695_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((sup_sup @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_3696_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F:(A > (A > A)),X:A,Y3:A]: (((! [X3:A,Y5:A]: ((((ord_less_eq @ A) @ X3) @ ((F @ X3) @ Y5)))) => ((! [X3:A,Y5:A]: ((((ord_less_eq @ A) @ Y5) @ ((F @ X3) @ Y5)))) => ((! [X3:A,Y5:A,Z4:A]: (((((ord_less_eq @ A) @ Y5) @ X3) => ((((ord_less_eq @ A) @ Z4) @ X3) => (((ord_less_eq @ A) @ ((F @ Y5) @ Z4)) @ X3))))) => ((((sup_sup @ A) @ X) @ Y3) = ((F @ X) @ Y3))))))))))).
% 0.33/5.21 thf(fact_3697_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((A3 = (((sup_sup @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 0.33/5.21 thf(fact_3698_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => (A3 = (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3699_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y6:A]: (((((sup_sup @ A) @ X2) @ Y6) = Y6)))))))).
% 0.33/5.21 thf(fact_3700_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y3:A,X:A,Z3:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_less_eq @ A) @ Z3) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y3) @ Z3)) @ X))))))))).
% 0.33/5.21 thf(fact_3701_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ (((sup_sup @ A) @ C2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3702_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A3:A,D2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A3) => ((((ord_less_eq @ A) @ D2) @ B2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D2)) @ (((sup_sup @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3703_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,B2:A,A3:A]: (((((ord_less_eq @ A) @ X) @ B2) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3704_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_3705_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y3:A,X:A]: ((((ord_less_eq @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3706_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3707_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,X:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ X) => ((((ord_less_eq @ A) @ B2) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ X))))))))).
% 0.33/5.21 thf(fact_3708_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ X) => (~ (((((ord_less_eq @ A) @ A3) @ X) => (~ ((((ord_less_eq @ A) @ B2) @ X))))))))))))).
% 0.33/5.21 thf(fact_3709_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3710_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y3:A,X:A]: ((((ord_less_eq @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3711_boolean__algebra_Odisj__zero__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 0.33/5.21 thf(fact_3712_ivl__disj__un__two__touch_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3713_ivl__disj__un__two_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3714_card__Un__le, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A6) @ B7))) @ (((plus_plus @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7)))))).
% 0.33/5.21 thf(fact_3715_ivl__disj__un__two_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3716_ivl__disj__un__two__touch_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3717_ivl__disj__un__two_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3718_nth__map__upt, axiom, (! [A:$tType,I:nat,N:nat,M:nat,F:(nat > A)]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ N) @ M)) => ((((nth @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ M) @ N))) @ I) = (F @ (((plus_plus @ nat) @ M) @ I))))))).
% 0.33/5.21 thf(fact_3719_lexord__sufI, axiom, (! [A:$tType,U:(list @ A),W2:(list @ A),R:(set @ ((product_prod @ A) @ A)),V:(list @ A),Z3:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ W2)) @ ((lexord @ A) @ R)) => ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ W2)) @ ((size_size @ (list @ A)) @ U)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ U) @ V)) @ (((append @ A) @ W2) @ Z3))) @ ((lexord @ A) @ R))))))).
% 0.33/5.21 thf(fact_3720_ivl__disj__un__singleton_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))).
% 0.33/5.21 thf(fact_3721_ivl__disj__un__two_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3722_ivl__disj__un__singleton_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ (((insert @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or3652927894154168847AtMost @ A) @ L) @ U))))))))).
% 0.33/5.21 thf(fact_3723_map__upt__eqI, axiom, (! [A:$tType,Xs:(list @ A),N:nat,M:nat,F:(nat > A)]: (((((size_size @ (list @ A)) @ Xs) = (((minus_minus @ nat) @ N) @ M)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I4) = (F @ (((plus_plus @ nat) @ M) @ I4)))))) => (((((map @ nat) @ A) @ F) @ ((upt @ M) @ N)) = Xs)))))).
% 0.33/5.21 thf(fact_3724_max__weak__def, axiom, (fun_max_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((max_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))).
% 0.33/5.21 thf(fact_3725_sorted__list__of__set__atMost__Suc, axiom, (! [K:nat]: ((((linord4507533701916653071of_set @ nat) @ ((set_ord_atMost @ nat) @ (suc @ K))) = (((append @ nat) @ ((linord4507533701916653071of_set @ nat) @ ((set_ord_atMost @ nat) @ K))) @ (((cons @ nat) @ (suc @ K)) @ (nil @ nat))))))).
% 0.33/5.21 thf(fact_3726_sorted__list__of__set__lessThan__Suc, axiom, (! [K:nat]: ((((linord4507533701916653071of_set @ nat) @ ((set_ord_lessThan @ nat) @ (suc @ K))) = (((append @ nat) @ ((linord4507533701916653071of_set @ nat) @ ((set_ord_lessThan @ nat) @ K))) @ (((cons @ nat) @ K) @ (nil @ nat))))))).
% 0.33/5.21 thf(fact_3727_id__take__nth__drop, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (Xs = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (((drop @ A) @ (suc @ I)) @ Xs)))))))).
% 0.33/5.21 thf(fact_3728_atMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I) @ K)))))))).
% 0.33/5.21 thf(fact_3729_lessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))).
% 0.33/5.21 thf(fact_3730_drop0, axiom, (! [A:$tType]: ((((drop @ A) @ (zero_zero @ nat)) = (^ [X2:(list @ A)]: (X2)))))).
% 0.33/5.21 thf(fact_3731_atMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X)) @ ((set_ord_atMost @ A) @ Y3)) = (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3732_lessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X)) @ ((set_ord_lessThan @ A) @ Y3)) = (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3733_drop__Suc__Cons, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: (((((drop @ A) @ (suc @ N)) @ (((cons @ A) @ X) @ Xs)) = (((drop @ A) @ N) @ Xs))))).
% 0.33/5.21 thf(fact_3734_lessThan__0, axiom, (((set_ord_lessThan @ nat) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))).
% 0.33/5.21 thf(fact_3735_card__atMost, axiom, (! [U:nat]: ((((finite_card @ nat) @ ((set_ord_atMost @ nat) @ U)) = (suc @ U))))).
% 0.33/5.21 thf(fact_3736_Icc__subset__Iic__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H2:A,H:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H2)) @ ((set_ord_atMost @ A) @ H)) = ((~ ((((ord_less_eq @ A) @ L) @ H2))) | (((ord_less_eq @ A) @ H2) @ H))))))))).
% 0.33/5.21 thf(fact_3737_drop__eq__Nil2, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((nil @ A) = (((drop @ A) @ N) @ Xs)) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N))))).
% 0.33/5.21 thf(fact_3738_drop__eq__Nil, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((drop @ A) @ N) @ Xs) = (nil @ A)) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N))))).
% 0.33/5.21 thf(fact_3739_drop__all, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((drop @ A) @ N) @ Xs) = (nil @ A)))))).
% 0.33/5.21 thf(fact_3740_atMost__0, axiom, (((set_ord_atMost @ nat) @ (zero_zero @ nat)) = (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))).
% 0.33/5.21 thf(fact_3741_drop__Cons__numeral, axiom, (! [A:$tType,V:num,X:A,Xs:(list @ A)]: (((((drop @ A) @ ((numeral_numeral @ nat) @ V)) @ (((cons @ A) @ X) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat))) @ Xs))))).
% 0.33/5.21 thf(fact_3742_nth__drop, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_3743_sup__nat__def, axiom, ((sup_sup @ nat) = (ord_max @ nat))).
% 0.33/5.21 thf(fact_3744_Iic__subset__Iio__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ A3)) @ ((set_ord_lessThan @ A) @ B2)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3745_drop__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((drop @ A) @ (zero_zero @ nat)) @ Xs) = Xs)))).
% 0.33/5.21 thf(fact_3746_lessThan__Suc__atMost, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = ((set_ord_atMost @ nat) @ K))))).
% 0.33/5.21 thf(fact_3747_ivl__disj__un__one_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))).
% 0.33/5.21 thf(fact_3748_ivl__disj__un__one_I1_J, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3749_sup__int__def, axiom, ((sup_sup @ int) = (ord_max @ int))).
% 0.33/5.21 thf(fact_3750_atMost__atLeast0, axiom, ((set_ord_atMost @ nat) = ((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)))).
% 0.33/5.21 thf(fact_3751_map__Suc__upt, axiom, (! [M:nat,N:nat]: ((((((map @ nat) @ nat) @ suc) @ ((upt @ M) @ N)) = ((upt @ (suc @ M)) @ (suc @ N)))))).
% 0.33/5.21 thf(fact_3752_lessThan__Suc, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = (((insert @ nat) @ K) @ ((set_ord_lessThan @ nat) @ K)))))).
% 0.33/5.21 thf(fact_3753_lessThan__empty__iff, axiom, (! [N:nat]: (((((set_ord_lessThan @ nat) @ N) = (bot_bot @ (set @ nat))) = (N = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3754_atMost__Suc, axiom, (! [K:nat]: ((((set_ord_atMost @ nat) @ (suc @ K)) = (((insert @ nat) @ (suc @ K)) @ ((set_ord_atMost @ nat) @ K)))))).
% 0.33/5.21 thf(fact_3755_lessThan__strict__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [M:A,N:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M)) @ ((set_ord_lessThan @ A) @ N)) = (((ord_less @ A) @ M) @ N)))))))).
% 0.33/5.21 thf(fact_3756_set__drop__subset__set__drop, axiom, (! [A:$tType,N:nat,M:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ N) @ M) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((drop @ A) @ M) @ Xs))) @ ((set2 @ A) @ (((drop @ A) @ N) @ Xs))))))).
% 0.33/5.21 thf(fact_3757_atLeast__upt, axiom, ((set_ord_lessThan @ nat) = (^ [N3:nat]: (((set2 @ nat) @ ((upt @ (zero_zero @ nat)) @ N3)))))).
% 0.33/5.21 thf(fact_3758_ivl__disj__un__one_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))).
% 0.33/5.21 thf(fact_3759_drop__Cons_H, axiom, (! [A:$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))))))).
% 0.33/5.21 thf(fact_3760_append__eq__append__conv__if, axiom, (! [A:$tType,Xs_1:(list @ A),Xs_2:(list @ A),Ys_1:(list @ A),Ys_2:(list @ A)]: ((((((append @ A) @ Xs_1) @ Xs_2) = (((append @ A) @ Ys_1) @ Ys_2)) = (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs_1)) @ ((size_size @ (list @ A)) @ Ys_1)) => ((Xs_1 = (((take @ A) @ ((size_size @ (list @ A)) @ Xs_1)) @ Ys_1)) & (Xs_2 = (((append @ A) @ (((drop @ A) @ ((size_size @ (list @ A)) @ Xs_1)) @ Ys_1)) @ Ys_2)))) & ((~ ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs_1)) @ ((size_size @ (list @ A)) @ Ys_1)))) => (((((take @ A) @ ((size_size @ (list @ A)) @ Ys_1)) @ Xs_1) = Ys_1) & ((((append @ A) @ (((drop @ A) @ ((size_size @ (list @ A)) @ Ys_1)) @ Xs_1)) @ Xs_2) = Ys_2)))))))).
% 0.33/5.21 thf(fact_3761_atMost__upto, axiom, ((set_ord_atMost @ nat) = (^ [N3:nat]: (((set2 @ nat) @ ((upt @ (zero_zero @ nat)) @ (suc @ N3))))))).
% 0.33/5.21 thf(fact_3762_max__strict__def, axiom, (fun_max_strict = ((max_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_less))).
% 0.33/5.21 thf(fact_3763_Cons__nth__drop__Suc, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (((drop @ A) @ (suc @ I)) @ Xs)) = (((drop @ A) @ I) @ Xs)))))).
% 0.33/5.21 thf(fact_3764_atLeast1__atMost__eq__remove0, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((minus_minus @ (set @ nat)) @ ((set_ord_atMost @ nat) @ N)) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))).
% 0.33/5.21 thf(fact_3765_take__hd__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((append @ A) @ (((take @ A) @ N) @ Xs)) @ (((cons @ A) @ ((hd @ A) @ (((drop @ A) @ N) @ Xs))) @ (nil @ A))) = (((take @ A) @ (suc @ N)) @ Xs)))))).
% 0.33/5.21 thf(fact_3766_upd__conv__take__nth__drop, axiom, (! [A:$tType,I:nat,Xs:(list @ A),A3:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ Xs) @ I) @ A3) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ A3) @ (((drop @ A) @ (suc @ I)) @ Xs)))))))).
% 0.33/5.21 thf(fact_3767_effect__makeE, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H2) @ H) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H2))) => ((H = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H2))) => (((((array_get @ A) @ H) @ R) = ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) => ((((array_present @ A) @ H) @ R) => ((~ ((((array_present @ A) @ H2) @ R))) => ((((plus_plus @ nat) @ N) @ (one_one @ nat)) != N2))))))))))))))).
% 0.33/5.21 thf(fact_3768_n__lists__Nil, axiom, (! [A:$tType,N:nat]: ((((N = (zero_zero @ nat)) => ((((n_lists @ A) @ N) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))) & ((N != (zero_zero @ nat)) => ((((n_lists @ A) @ N) @ (nil @ A)) = (nil @ (list @ A)))))))).
% 0.33/5.21 thf(fact_3769_hd__upt, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((hd @ nat) @ ((upt @ I) @ J)) = I))))).
% 0.33/5.21 thf(fact_3770_list__update__beyond, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I) => (((((list_update @ A) @ Xs) @ I) @ X) = Xs))))).
% 0.33/5.21 thf(fact_3771_take__update__cancel, axiom, (! [A:$tType,N:nat,M:nat,Xs:(list @ A),Y3:A]: (((((ord_less_eq @ nat) @ N) @ M) => ((((take @ A) @ N) @ ((((list_update @ A) @ Xs) @ M) @ Y3)) = (((take @ A) @ N) @ Xs)))))).
% 0.33/5.21 thf(fact_3772_drop__update__cancel, axiom, (! [A:$tType,N:nat,M:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ N) @ M) => ((((drop @ A) @ M) @ ((((list_update @ A) @ Xs) @ N) @ X)) = (((drop @ A) @ M) @ Xs)))))).
% 0.33/5.21 thf(fact_3773_get__update__eq, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),I:nat,V:A,H2:(heap_ext @ product_unit)]: (((((array_get @ A) @ (((((array_update @ A) @ A3) @ I) @ V) @ H2)) @ A3) = ((((list_update @ A) @ (((array_get @ A) @ H2) @ A3)) @ I) @ V)))))))).
% 0.33/5.21 thf(fact_3774_nth__list__update__eq, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X)) @ I) = X))))).
% 0.33/5.21 thf(fact_3775_hd__take, axiom, (! [A:$tType,J:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ J) => (((hd @ A) @ (((take @ A) @ J) @ Xs)) = ((hd @ A) @ Xs)))))).
% 0.33/5.21 thf(fact_3776_set__swap, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((set2 @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ J))) @ J) @ (((nth @ A) @ Xs) @ I))) = ((set2 @ A) @ Xs))))))).
% 0.33/5.21 thf(fact_3777_get__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit),Ys2:(list @ A)]: (((((array_get @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ys2) @ H2))) = Xs))))))).
% 0.33/5.21 thf(fact_3778_set, axiom, (! [A:$tType]: (((heap @ A) => (! [Ls:(list @ A),H2:(heap_ext @ product_unit),New_ls:(list @ A)]: ((((((array_set @ A) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ls) @ H2))) @ New_ls) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ls) @ H2))) = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ New_ls) @ H2))))))))).
% 0.33/5.21 thf(fact_3779_not__present__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [H2:(heap_ext @ product_unit),Xs:(list @ A)]: ((~ ((((array_present @ A) @ H2) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))))))))))).
% 0.33/5.21 thf(fact_3780_present__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit)]: ((((array_present @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))))))))).
% 0.33/5.21 thf(fact_3781_list__update__code_I3_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),I:nat,Y3:A]: ((((((list_update @ A) @ (((cons @ A) @ X) @ Xs)) @ (suc @ I)) @ Y3) = (((cons @ A) @ X) @ ((((list_update @ A) @ Xs) @ I) @ Y3)))))).
% 0.33/5.21 thf(fact_3782_list__update__code_I2_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y3:A]: ((((((list_update @ A) @ (((cons @ A) @ X) @ Xs)) @ (zero_zero @ nat)) @ Y3) = (((cons @ A) @ Y3) @ Xs))))).
% 0.33/5.21 thf(fact_3783_set__update__memI, axiom, (! [A:$tType,N:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ N) @ X))))))).
% 0.33/5.21 thf(fact_3784_list__update__append1, axiom, (! [A:$tType,I:nat,Xs:(list @ A),Ys2:(list @ A),X:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys2)) @ I) @ X) = (((append @ A) @ ((((list_update @ A) @ Xs) @ I) @ X)) @ Ys2)))))).
% 0.33/5.21 thf(fact_3785_length__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit),Ys2:(list @ A)]: (((((array_length @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ys2) @ H2))) = ((size_size @ (list @ A)) @ Xs)))))))).
% 0.33/5.21 thf(fact_3786_list__update__same__conv, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((((list_update @ A) @ Xs) @ I) @ X) = Xs) = ((((nth @ A) @ Xs) @ I) = X)))))).
% 0.33/5.21 thf(fact_3787_nth__list__update, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat,X:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((I = J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X)) @ J) = X)) & ((I != J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X)) @ J) = (((nth @ A) @ Xs) @ J)))))))).
% 0.33/5.21 thf(fact_3788_hd__conv__nth, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((hd @ A) @ Xs) = (((nth @ A) @ Xs) @ (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3789_drop__update__swap, axiom, (! [A:$tType,M:nat,N:nat,Xs:(list @ A),X:A]: (((((ord_less_eq @ nat) @ M) @ N) => ((((drop @ A) @ M) @ ((((list_update @ A) @ Xs) @ N) @ X)) = ((((list_update @ A) @ (((drop @ A) @ M) @ Xs)) @ (((minus_minus @ nat) @ N) @ M)) @ X)))))).
% 0.33/5.21 thf(fact_3790_present__alloc__noteq, axiom, (! [A:$tType,B:$tType]: ((((heap @ B) & (heap @ A)) => (! [H2:(heap_ext @ product_unit),A3:(array @ A),Xs:(list @ B)]: (((((array_present @ A) @ H2) @ A3) => ((((array_noteq @ A) @ B) @ A3) @ (((product_fst @ (array @ B)) @ (heap_ext @ product_unit)) @ (((array_alloc @ B) @ Xs) @ H2)))))))))).
% 0.33/5.21 thf(fact_3791_update__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_update @ A) = (^ [A5:(array @ A),I5:nat,X2:A,H4:(heap_ext @ product_unit)]: (((((array_set @ A) @ A5) @ ((((list_update @ A) @ (((array_get @ A) @ H4) @ A5)) @ I5) @ X2)) @ H4)))))))).
% 0.33/5.21 thf(fact_3792_list__update__append, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys2:(list @ A),X:A]: ((((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys2)) @ N) @ X) = (((append @ A) @ ((((list_update @ A) @ Xs) @ N) @ X)) @ Ys2))) & ((~ ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)))) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys2)) @ N) @ X) = (((append @ A) @ Xs) @ ((((list_update @ A) @ Ys2) @ (((minus_minus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))) @ X)))))))).
% 0.33/5.21 thf(fact_3793_n__lists_Osimps_I1_J, axiom, (! [A:$tType,Xs:(list @ A)]: (((((n_lists @ A) @ (zero_zero @ nat)) @ Xs) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 0.33/5.21 thf(fact_3794_effect__makeI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H:(heap_ext @ product_unit),F:(nat > A),N:nat,H2:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H) = (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H2)) => ((((((heap_Time_effect @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H2) @ H) @ A3) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))))))).
% 0.33/5.21 thf(fact_3795_hd__drop__conv__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((hd @ A) @ (((drop @ A) @ N) @ Xs)) = (((nth @ A) @ Xs) @ N)))))).
% 0.33/5.21 thf(fact_3796_effect__of__listE, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H2) @ H) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))) => ((H = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H2))) => (((((array_get @ A) @ H) @ R) = Xs) => ((((array_present @ A) @ H) @ R) => ((~ ((((array_present @ A) @ H2) @ R))) => (N2 != (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))))))))))))).
% 0.33/5.21 thf(fact_3797_effect__of__listI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H:(heap_ext @ product_unit),Xs:(list @ A),H2:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H) = (((array_alloc @ A) @ Xs) @ H2)) => ((((((heap_Time_effect @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H2) @ H) @ A3) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs)))))))))).
% 0.33/5.21 thf(fact_3798_set__update__distinct, axiom, (! [A:$tType,Xs:(list @ A),N:nat,X:A]: ((((distinct @ A) @ Xs) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((set2 @ A) @ ((((list_update @ A) @ Xs) @ N) @ X)) = (((insert @ A) @ X) @ (((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ (((nth @ A) @ Xs) @ N)) @ (bot_bot @ (set @ A))))))))))).
% 0.33/5.21 thf(fact_3799_effect__newE, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H2) @ H) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H2))) => ((H = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H2))) => (((((array_get @ A) @ H) @ R) = (((replicate @ A) @ N) @ X)) => ((((array_present @ A) @ H) @ R) => ((~ ((((array_present @ A) @ H2) @ R))) => ((((plus_plus @ nat) @ N) @ (one_one @ nat)) != N2))))))))))))))).
% 0.33/5.21 thf(fact_3800_replicate__eq__replicate, axiom, (! [A:$tType,M:nat,X:A,N:nat,Y3:A]: ((((((replicate @ A) @ M) @ X) = (((replicate @ A) @ N) @ Y3)) = ((M = N) & ((M != (zero_zero @ nat)) => (X = Y3))))))).
% 0.33/5.21 thf(fact_3801_replicate__empty, axiom, (! [A:$tType,N:nat,X:A]: ((((((replicate @ A) @ N) @ X) = (nil @ A)) = (N = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3802_empty__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((nil @ A) = (((replicate @ A) @ N) @ X)) = (N = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3803_Ball__set__replicate, axiom, (! [A:$tType,N:nat,A3:A,P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ (((replicate @ A) @ N) @ A3))) => (P @ X2)))) = ((P @ A3) | (N = (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3804_Bex__set__replicate, axiom, (! [A:$tType,N:nat,A3:A,P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ (((replicate @ A) @ N) @ A3))) & (P @ X2)))) = ((P @ A3) & (N != (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3805_in__set__replicate, axiom, (! [A:$tType,X:A,N:nat,Y3:A]: (((((member @ A) @ X) @ ((set2 @ A) @ (((replicate @ A) @ N) @ Y3))) = ((X = Y3) & (N != (zero_zero @ nat))))))).
% 0.33/5.21 thf(fact_3806_nth__replicate, axiom, (! [A:$tType,I:nat,N:nat,X:A]: (((((ord_less @ nat) @ I) @ N) => ((((nth @ A) @ (((replicate @ A) @ N) @ X)) @ I) = X))))).
% 0.33/5.21 thf(fact_3807_hd__replicate, axiom, (! [A:$tType,N:nat,X:A]: (((N != (zero_zero @ nat)) => (((hd @ A) @ (((replicate @ A) @ N) @ X)) = X))))).
% 0.33/5.21 thf(fact_3808_set__replicate, axiom, (! [A:$tType,N:nat,X:A]: (((N != (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N) @ X)) = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))).
% 0.33/5.21 thf(fact_3809_distinct__swap, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((distinct @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ J))) @ J) @ (((nth @ A) @ Xs) @ I))) = ((distinct @ A) @ Xs))))))).
% 0.33/5.21 thf(fact_3810_replicate__Suc, axiom, (! [A:$tType,N:nat,X:A]: (((((replicate @ A) @ (suc @ N)) @ X) = (((cons @ A) @ X) @ (((replicate @ A) @ N) @ X)))))).
% 0.33/5.21 thf(fact_3811_replicate__0, axiom, (! [A:$tType,X:A]: (((((replicate @ A) @ (zero_zero @ nat)) @ X) = (nil @ A))))).
% 0.33/5.21 thf(fact_3812_distinct__conv__nth, axiom, (! [A:$tType]: (((distinct @ A) = (^ [Xs3:(list @ A)]: ((! [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs3)) => (! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => ((I5 != J3) => ((((nth @ A) @ Xs3) @ I5) != (((nth @ A) @ Xs3) @ J3))))))))))))))).
% 0.33/5.21 thf(fact_3813_nth__eq__iff__index__eq, axiom, (! [A:$tType,Xs:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I) = (((nth @ A) @ Xs) @ J)) = (I = J)))))))).
% 0.33/5.21 thf(fact_3814_effect__newI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H:(heap_ext @ product_unit),N:nat,X:A,H2:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H) = (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H2)) => ((((((heap_Time_effect @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H2) @ H) @ A3) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))))))).
% 0.33/5.21 thf(fact_3815_distinct__Ex1, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((distinct @ A) @ Xs) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (? [X3:nat]: (((((ord_less @ nat) @ X3) @ ((size_size @ (list @ A)) @ Xs)) & (((((nth @ A) @ Xs) @ X3) = X) & (! [Y4:nat]: ((((((ord_less @ nat) @ Y4) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ Y4) = X)) => (Y4 = X3))))))))))))).
% 0.33/5.21 thf(fact_3816_set__replicate__Suc, axiom, (! [A:$tType,N:nat,X:A]: ((((set2 @ A) @ (((replicate @ A) @ (suc @ N)) @ X)) = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 0.33/5.21 thf(fact_3817_set__replicate__conv__if, axiom, (! [A:$tType,N:nat,X:A]: ((((N = (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N) @ X)) = (bot_bot @ (set @ A)))) & ((N != (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N) @ X)) = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 0.33/5.21 thf(fact_3818_Cons__replicate__eq, axiom, (! [A:$tType,X:A,Xs:(list @ A),N:nat,Y3:A]: ((((((cons @ A) @ X) @ Xs) = (((replicate @ A) @ N) @ Y3)) = ((X = Y3) & ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (Xs = (((replicate @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ X)))))))).
% 0.33/5.21 thf(fact_3819_atLeast1__lessThan__eq__remove0, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((minus_minus @ (set @ nat)) @ ((set_ord_lessThan @ nat) @ N)) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))).
% 0.33/5.21 thf(fact_3820_listrel1__iff__update, axiom, (! [A:$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)) = (? [Y6:A,N3:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ N3)) @ Y6)) @ R) & ((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs)) & (Ys2 = ((((list_update @ A) @ Xs) @ N3) @ Y6)))))))))).
% 0.33/5.21 thf(fact_3821_Gcd__remove0__nat, axiom, (! [M7:(set @ nat)]: ((((finite_finite2 @ nat) @ M7) => (((gcd_Gcd @ nat) @ M7) = ((gcd_Gcd @ nat) @ (((minus_minus @ (set @ nat)) @ M7) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))))).
% 0.33/5.21 thf(fact_3822_remdups__adj__singleton__iff, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs)) = (suc @ (zero_zero @ nat))) = ((Xs != (nil @ A)) & (Xs = (((replicate @ A) @ ((size_size @ (list @ A)) @ Xs)) @ ((hd @ A) @ Xs)))))))).
% 0.33/5.21 thf(fact_3823_atLeastLessThan__iff, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3824_infinite__Icc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3825_atLeastLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A)))))))))).
% 0.33/5.21 thf(fact_3826_atLeastLessThan__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3827_atLeastLessThan__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) = (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3828_ivl__subset, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,J:A,M:A,N:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ N)) = ((((ord_less_eq @ A) @ J) @ I) | ((((ord_less_eq @ A) @ M) @ I) & (((ord_less_eq @ A) @ J) @ N)))))))))).
% 0.33/5.21 thf(fact_3829_infinite__Ico__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3830_card_Oinfinite, axiom, (! [A:$tType,A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((finite_card @ A) @ A6) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3831_ivl__diff, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,N:A,M:A]: (((((ord_less_eq @ A) @ I) @ N) => ((((minus_minus @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ N)) = (((set_or7035219750837199246ssThan @ A) @ N) @ M))))))))).
% 0.33/5.21 thf(fact_3832_infinite__Ioc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3833_infinite__Ioo__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3834_card__0__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((((finite_card @ A) @ A6) = (zero_zero @ nat)) = (A6 = (bot_bot @ (set @ A)))))))).
% 0.33/5.21 thf(fact_3835_card__insert__disjoint, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => (((finite_card @ A) @ (((insert @ A) @ X) @ A6)) = (suc @ ((finite_card @ A) @ A6)))))))).
% 0.33/5.21 thf(fact_3836_atLeastLessThan__singleton, axiom, (! [M:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ M)) = (((insert @ nat) @ M) @ (bot_bot @ (set @ nat))))))).
% 0.33/5.21 thf(fact_3837_finite__psubset__induct, axiom, (! [A:$tType,A6:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A6) => ((! [A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [B10:(set @ A)]: (((((ord_less @ (set @ A)) @ B10) @ A9) => (P @ B10)))) => (P @ A9))))) => (P @ A6)))))).
% 0.33/5.21 thf(fact_3838_atLeastLessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((A3 = C2) & (B2 = D2))))))))))).
% 0.33/5.21 thf(fact_3839_atLeastLessThan__inj_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (A3 = C2)))))))))).
% 0.33/5.21 thf(fact_3840_atLeastLessThan__inj_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (B2 = D2)))))))))).
% 0.33/5.21 thf(fact_3841_infinite__Ico, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_3842_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N7:(set @ nat)]: ((? [M2:nat]: ((! [X2:nat]: (((((member @ nat) @ X2) @ N7) => (((ord_less @ nat) @ X2) @ M2)))))))))).
% 0.33/5.21 thf(fact_3843_bounded__nat__set__is__finite, axiom, (! [N6:(set @ nat),N:nat]: (((! [X3:nat]: (((((member @ nat) @ X3) @ N6) => (((ord_less @ nat) @ X3) @ N)))) => ((finite_finite2 @ nat) @ N6))))).
% 0.33/5.21 thf(fact_3844_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N7:(set @ nat)]: ((? [M2:nat]: ((! [X2:nat]: (((((member @ nat) @ X2) @ N7) => (((ord_less_eq @ nat) @ X2) @ M2)))))))))).
% 0.33/5.21 thf(fact_3845_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (? [X3:A]: (((((member @ A) @ X3) @ A6) & ((((ord_less_eq @ A) @ X3) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => ((((ord_less_eq @ A) @ Xa) @ X3) => (X3 = Xa))))))))))))))))).
% 0.33/5.21 thf(fact_3846_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (? [X3:A]: (((((member @ A) @ X3) @ A6) & ((((ord_less_eq @ A) @ A3) @ X3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => ((((ord_less_eq @ A) @ X3) @ Xa) => (X3 = Xa))))))))))))))))).
% 0.33/5.21 thf(fact_3847_subset__eq__atLeast0__lessThan__finite, axiom, (! [N6:(set @ nat),N:nat]: (((((ord_less_eq @ (set @ nat)) @ N6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) => ((finite_finite2 @ nat) @ N6))))).
% 0.33/5.21 thf(fact_3848_finite__set__decode, axiom, (! [N:nat]: (((finite_finite2 @ nat) @ (nat_set_decode @ N))))).
% 0.33/5.21 thf(fact_3849_atLeastLessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3850_ivl__disj__un__two_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3851_ex__nat__less__eq, axiom, (! [N:nat,P:(nat > $o)]: (((? [M2:nat]: (((((ord_less @ nat) @ M2) @ N) & (P @ M2)))) = (? [X2:nat]: (((((member @ nat) @ X2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) & (P @ X2)))))))).
% 0.33/5.21 thf(fact_3852_all__nat__less__eq, axiom, (! [N:nat,P:(nat > $o)]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N) => (P @ M2)))) = (! [X2:nat]: (((((member @ nat) @ X2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) => (P @ X2)))))))).
% 0.33/5.21 thf(fact_3853_atLeastLessThan__add__Un, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((set_or7035219750837199246ssThan @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)) = (((sup_sup @ (set @ nat)) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ nat) @ J) @ (((plus_plus @ nat) @ J) @ K)))))))).
% 0.33/5.21 thf(fact_3854_atLeastLessThanSuc__atLeastAtMost, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ L) @ (suc @ U)) = (((set_or1337092689740270186AtMost @ nat) @ L) @ U))))).
% 0.33/5.21 thf(fact_3855_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A6) & (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => ((((ord_less_eq @ A) @ Xa) @ X3) => (X3 = Xa)))))))))))))))).
% 0.33/5.21 thf(fact_3856_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A6) & (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => ((((ord_less_eq @ A) @ X3) @ Xa) => (X3 = Xa)))))))))))))))).
% 0.33/5.21 thf(fact_3857_infinite__growing, axiom, (! [A:$tType]: (((linorder @ A) => (! [X8:(set @ A)]: (((X8 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (? [Xa:A]: (((((member @ A) @ Xa) @ X8) & (((ord_less @ A) @ X3) @ Xa))))))) => (~ (((finite_finite2 @ A) @ X8))))))))))).
% 0.33/5.21 thf(fact_3858_ex__min__if__finite, axiom, (! [A:$tType]: (((order @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ S3) & (~ ((? [Xa:A]: (((((member @ A) @ Xa) @ S3) & (((ord_less @ A) @ Xa) @ X3))))))))))))))))).
% 0.33/5.21 thf(fact_3859_infinite__Icc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_3860_lessThan__atLeast0, axiom, ((set_ord_lessThan @ nat) = ((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)))).
% 0.33/5.21 thf(fact_3861_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B7:(set @ A),A6:(set @ B),R:(B > (A > $o))]: ((((finite_finite2 @ A) @ B7) => ((! [A8:B]: (((((member @ B) @ A8) @ A6) => (? [B11:A]: (((((member @ A) @ B11) @ B7) & ((R @ A8) @ B11))))))) => ((! [A13:B,A24:B,B5:A]: (((((member @ B) @ A13) @ A6) => ((((member @ B) @ A24) @ A6) => ((((member @ A) @ B5) @ B7) => (((R @ A13) @ B5) => (((R @ A24) @ B5) => (A13 = A24)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A6)) @ ((finite_card @ A) @ B7)))))))).
% 0.33/5.21 thf(fact_3862_remdups__adj__length, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.21 thf(fact_3863_infinite__Ioc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_3864_atLeastLessThan0, axiom, (! [M:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))))).
% 0.33/5.21 thf(fact_3865_infinite__Ioo, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2))))))))))).
% 0.33/5.21 thf(fact_3866_atLeastSucLessThan__greaterThanLessThan, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ L)) @ U) = (((set_or5935395276787703475ssThan @ nat) @ L) @ U))))).
% 0.33/5.21 thf(fact_3867_ivl__disj__un__two_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3868_ivl__disj__un__one_I2_J, axiom, (! [A:$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))))))))).
% 0.33/5.21 thf(fact_3869_finite__ranking__induct, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [S3:(set @ B),P:((set @ B) > $o),F:(B > A)]: ((((finite_finite2 @ B) @ S3) => ((P @ (bot_bot @ (set @ B))) => ((! [X3:B,S4:(set @ B)]: ((((finite_finite2 @ B) @ S4) => ((! [Y4:B]: (((((member @ B) @ Y4) @ S4) => (((ord_less_eq @ A) @ (F @ Y4)) @ (F @ X3))))) => ((P @ S4) => (P @ (((insert @ B) @ X3) @ S4))))))) => (P @ S3)))))))))).
% 0.33/5.21 thf(fact_3870_finite__linorder__max__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A6) => ((P @ (bot_bot @ (set @ A))) => ((! [B5:A,A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [X4:A]: (((((member @ A) @ X4) @ A9) => (((ord_less @ A) @ X4) @ B5)))) => ((P @ A9) => (P @ (((insert @ A) @ B5) @ A9))))))) => (P @ A6)))))))))).
% 0.33/5.21 thf(fact_3871_finite__linorder__min__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A6) => ((P @ (bot_bot @ (set @ A))) => ((! [B5:A,A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [X4:A]: (((((member @ A) @ X4) @ A9) => (((ord_less @ A) @ B5) @ X4)))) => ((P @ A9) => (P @ (((insert @ A) @ B5) @ A9))))))) => (P @ A6)))))))))).
% 0.33/5.21 thf(fact_3872_card__eq__0__iff, axiom, (! [A:$tType,A6:(set @ A)]: (((((finite_card @ A) @ A6) = (zero_zero @ nat)) = ((A6 = (bot_bot @ (set @ A))) | (~ (((finite_finite2 @ A) @ A6)))))))).
% 0.33/5.21 thf(fact_3873_atLeast0__lessThan__Suc, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert @ nat) @ N) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.21 thf(fact_3874_card__Suc__eq__finite, axiom, (! [A:$tType,A6:(set @ A),K:nat]: (((((finite_card @ A) @ A6) = (suc @ K)) = (? [B3:A,B6:(set @ A)]: (((A6 = (((insert @ A) @ B3) @ B6)) & ((~ ((((member @ A) @ B3) @ B6))) & ((((finite_card @ A) @ B6) = K) & ((finite_finite2 @ A) @ B6)))))))))).
% 0.33/5.21 thf(fact_3875_card__insert__if, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((((member @ A) @ X) @ A6) => (((finite_card @ A) @ (((insert @ A) @ X) @ A6)) = ((finite_card @ A) @ A6))) & ((~ ((((member @ A) @ X) @ A6))) => (((finite_card @ A) @ (((insert @ A) @ X) @ A6)) = (suc @ ((finite_card @ A) @ A6))))))))).
% 0.33/5.21 thf(fact_3876_card__ge__0__finite, axiom, (! [A:$tType,A6:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A6)) => ((finite_finite2 @ A) @ A6))))).
% 0.33/5.21 thf(fact_3877_obtain__subset__with__card__n, axiom, (! [A:$tType,N:nat,S3:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ S3)) => (~ ((! [T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T4) @ S3) => ((((finite_card @ A) @ T4) = N) => (~ (((finite_finite2 @ A) @ T4))))))))))))).
% 0.33/5.21 thf(fact_3878_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F4:(set @ A),C6:nat]: (((! [G3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G3) @ F4) => (((finite_finite2 @ A) @ G3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G3)) @ C6))))) => (((finite_finite2 @ A) @ F4) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F4)) @ C6)))))).
% 0.33/5.21 thf(fact_3879_card__seteq, axiom, (! [A:$tType,B7:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B7)) @ ((finite_card @ A) @ A6)) => (A6 = B7))))))).
% 0.33/5.21 thf(fact_3880_card__mono, axiom, (! [A:$tType,B7:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7))))))).
% 0.33/5.21 thf(fact_3881_card__less__sym__Diff, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ A) @ B7) => ((((ord_less @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7)) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ B7))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ B7) @ A6))))))))).
% 0.33/5.21 thf(fact_3882_card__le__sym__Diff, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7)) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ B7))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ B7) @ A6))))))))).
% 0.33/5.21 thf(fact_3883_psubset__card__mono, axiom, (! [A:$tType,B7:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less @ (set @ A)) @ A6) @ B7) => (((ord_less @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7))))))).
% 0.33/5.21 thf(fact_3884_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))))).
% 0.33/5.21 thf(fact_3885_atLeastAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3886_atLeastLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3887_ivl__disj__un__two__touch_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3888_greaterThanAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3889_greaterThanLessThan__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D2)))))))))).
% 0.33/5.21 thf(fact_3890_ivl__disj__un__two_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3891_card__gt__0__iff, axiom, (! [A:$tType,A6:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A6)) = ((A6 != (bot_bot @ (set @ A))) & ((finite_finite2 @ A) @ A6)))))).
% 0.33/5.21 thf(fact_3892_card__le__Suc0__iff__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ (suc @ (zero_zero @ nat))) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (! [Y6:A]: (((((member @ A) @ Y6) @ A6) => (X2 = Y6)))))))))))).
% 0.33/5.21 thf(fact_3893_card__le__Suc__iff, axiom, (! [A:$tType,N:nat,A6:(set @ A)]: (((((ord_less_eq @ nat) @ (suc @ N)) @ ((finite_card @ A) @ A6)) = (? [A5:A,B6:(set @ A)]: (((A6 = (((insert @ A) @ A5) @ B6)) & ((~ ((((member @ A) @ A5) @ B6))) & ((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ B6)) & ((finite_finite2 @ A) @ B6)))))))))).
% 0.33/5.21 thf(fact_3894_ivl__disj__un__two__touch_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3895_finite__induct__select, axiom, (! [A:$tType,S3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ S3) => ((P @ (bot_bot @ (set @ A))) => ((! [T4:(set @ A)]: (((((ord_less @ (set @ A)) @ T4) @ S3) => ((P @ T4) => (? [X4:A]: (((((member @ A) @ X4) @ (((minus_minus @ (set @ A)) @ S3) @ T4)) & (P @ (((insert @ A) @ X4) @ T4))))))))) => (P @ S3))))))).
% 0.33/5.21 thf(fact_3896_card__psubset, axiom, (! [A:$tType,B7:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((((ord_less @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7)) => (((ord_less @ (set @ A)) @ A6) @ B7))))))).
% 0.33/5.21 thf(fact_3897_diff__card__le__card__Diff, axiom, (! [A:$tType,B7:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B7) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ A) @ B7))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ B7))))))).
% 0.33/5.21 thf(fact_3898_remdups__adj__adjacent, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ (suc @ I)) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) => ((((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ I) != (((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ (suc @ I))))))).
% 0.33/5.21 thf(fact_3899_remdups__adj__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((N = (zero_zero @ nat)) => (((remdups_adj @ A) @ (((replicate @ A) @ N) @ X)) = (nil @ A))) & ((N != (zero_zero @ nat)) => (((remdups_adj @ A) @ (((replicate @ A) @ N) @ X)) = (((cons @ A) @ X) @ (nil @ A)))))))).
% 0.33/5.21 thf(fact_3900_atLeastLessThanSuc, axiom, (! [M:nat,N:nat]: ((((((ord_less_eq @ nat) @ M) @ N) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N)) = (((insert @ nat) @ N) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)))) & ((~ ((((ord_less_eq @ nat) @ M) @ N))) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N)) = (bot_bot @ (set @ nat)))))))).
% 0.33/5.21 thf(fact_3901_subset__eq__atLeast0__lessThan__card, axiom, (! [N6:(set @ nat),N:nat]: (((((ord_less_eq @ (set @ nat)) @ N6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ N6)) @ N))))).
% 0.33/5.21 thf(fact_3902_ivl__disj__un__singleton_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ (((insert @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))).
% 0.33/5.21 thf(fact_3903_ivl__disj__un__two_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 0.33/5.21 thf(fact_3904_ivl__disj__un__singleton_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U))))))))).
% 0.33/5.21 thf(fact_3905_card_Oremove, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((finite_card @ A) @ A6) = (suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 0.33/5.21 thf(fact_3906_card_Oinsert__remove, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((finite_card @ A) @ (((insert @ A) @ X) @ A6)) = (suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 0.33/5.21 thf(fact_3907_card__Suc__Diff1, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) = ((finite_card @ A) @ A6))))))).
% 0.33/5.21 thf(fact_3908_card__Diff1__less, axiom, (! [A:$tType,A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A6))))))).
% 0.33/5.21 thf(fact_3909_card__Diff2__less, axiom, (! [A:$tType,A6:(set @ A),X:A,Y3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((member @ A) @ Y3) @ A6) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ (((insert @ A) @ Y3) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A6)))))))).
% 0.33/5.21 thf(fact_3910_card__Diff1__less__iff, axiom, (! [A:$tType,A6:(set @ A),X:A]: (((((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A6)) = (((finite_finite2 @ A) @ A6) & (((member @ A) @ X) @ A6)))))).
% 0.33/5.21 thf(fact_3911_remdups__adj__length__ge1, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))))))).
% 0.33/5.21 thf(fact_3912_atLeastLessThan__nat__numeral, axiom, (! [M:nat,K:num]: ((((((ord_less_eq @ nat) @ M) @ (pred_numeral @ K)) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ ((numeral_numeral @ nat) @ K)) = (((insert @ nat) @ (pred_numeral @ K)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (pred_numeral @ K))))) & ((~ ((((ord_less_eq @ nat) @ M) @ (pred_numeral @ K)))) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ ((numeral_numeral @ nat) @ K)) = (bot_bot @ (set @ nat)))))))).
% 0.33/5.21 thf(fact_3913_set__encode__insert, axiom, (! [A6:(set @ nat),N:nat]: ((((finite_finite2 @ nat) @ A6) => ((~ ((((member @ nat) @ N) @ A6))) => ((nat_set_encode @ (((insert @ nat) @ N) @ A6)) = (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (nat_set_encode @ A6)))))))).
% 0.33/5.21 thf(fact_3914_even__set__encode__iff, axiom, (! [A6:(set @ nat)]: ((((finite_finite2 @ nat) @ A6) => ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (nat_set_encode @ A6)) = (~ ((((member @ nat) @ (zero_zero @ nat)) @ A6)))))))).
% 0.33/5.21 thf(fact_3915_arg__min__if__finite_I2_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S3:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (~ ((? [X4:A]: (((((member @ A) @ X4) @ S3) & (((ord_less @ B) @ (F @ X4)) @ (F @ ((((lattic7623131987881927897min_on @ A) @ B) @ F) @ S3)))))))))))))))).
% 0.33/5.21 thf(fact_3916_arg__min__least, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S3:(set @ A),Y3:A,F:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => ((((member @ A) @ Y3) @ S3) => (((ord_less_eq @ B) @ (F @ ((((lattic7623131987881927897min_on @ A) @ B) @ F) @ S3))) @ (F @ Y3))))))))))).
% 0.33/5.21 thf(fact_3917_set__decode__inverse, axiom, (! [N:nat]: (((nat_set_encode @ (nat_set_decode @ N)) = N)))).
% 0.33/5.21 thf(fact_3918_set__encode__empty, axiom, ((nat_set_encode @ (bot_bot @ (set @ nat))) = (zero_zero @ nat))).
% 0.33/5.21 thf(fact_3919_set__encode__inverse, axiom, (! [A6:(set @ nat)]: ((((finite_finite2 @ nat) @ A6) => ((nat_set_decode @ (nat_set_encode @ A6)) = A6))))).
% 0.33/5.21 thf(fact_3920_finite__atLeastZeroLessThan__int, axiom, (! [U:int]: (((finite_finite2 @ int) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U))))).
% 0.33/5.21 thf(fact_3921_finite__maxlen, axiom, (! [A:$tType,M7:(set @ (list @ A))]: ((((finite_finite2 @ (list @ A)) @ M7) => (? [N4:nat]: ((! [X4:(list @ A)]: (((((member @ (list @ A)) @ X4) @ M7) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ X4)) @ N4)))))))))).
% 0.33/5.21 thf(fact_3922_set__encode__eq, axiom, (! [A6:(set @ nat),B7:(set @ nat)]: ((((finite_finite2 @ nat) @ A6) => (((finite_finite2 @ nat) @ B7) => (((nat_set_encode @ A6) = (nat_set_encode @ B7)) = (A6 = B7))))))).
% 0.33/5.21 thf(fact_3923_atLeastLessThanPlusOne__atLeastAtMost__int, axiom, (! [L:int,U:int]: (((((set_or7035219750837199246ssThan @ int) @ L) @ (((plus_plus @ int) @ U) @ (one_one @ int))) = (((set_or1337092689740270186AtMost @ int) @ L) @ U))))).
% 0.33/5.21 thf(fact_3924_card__atLeastZeroLessThan__int, axiom, (! [U:int]: ((((finite_card @ int) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U)) = (nat2 @ U))))).
% 0.33/5.21 thf(fact_3925_atLeastPlusOneLessThan__greaterThanLessThan__int, axiom, (! [L:int,U:int]: (((((set_or7035219750837199246ssThan @ int) @ (((plus_plus @ int) @ L) @ (one_one @ int))) @ U) = (((set_or5935395276787703475ssThan @ int) @ L) @ U))))).
% 0.33/5.21 thf(fact_3926_set__encode__inf, axiom, (! [A6:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ A6))) => ((nat_set_encode @ A6) = (zero_zero @ nat)))))).
% 0.33/5.21 thf(fact_3927_atLeastLessThan__upto, axiom, ((set_or7035219750837199246ssThan @ int) = (^ [I5:int,J3:int]: (((set2 @ int) @ ((upto @ I5) @ (((minus_minus @ int) @ J3) @ (one_one @ int)))))))).
% 0.33/5.21 thf(fact_3928_wmax__emptyI, axiom, (! [X8:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ X8) => (((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)))) @ X8)) @ fun_max_weak))))).
% 0.33/5.21 thf(fact_3929_smax__emptyI, axiom, (! [Y7:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ Y7) => ((Y7 != (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)))) @ Y7)) @ fun_max_strict)))))).
% 0.33/5.21 thf(fact_3930_arg__min__if__finite_I1_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S3:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((member @ A) @ ((((lattic7623131987881927897min_on @ A) @ B) @ F) @ S3)) @ S3))))))))).
% 0.33/5.21 thf(fact_3931_nth__image, axiom, (! [A:$tType,L:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ L) @ ((size_size @ (list @ A)) @ Xs)) => (((((image @ nat) @ A) @ ((nth @ A) @ Xs)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ L)) = ((set2 @ A) @ (((take @ A) @ L) @ Xs))))))).
% 0.33/5.21 thf(fact_3932_fact__split, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((semiring_char_0_fact @ A) @ N) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (((minus_minus @ nat) @ N) @ K)) @ N)))) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N) @ K))))))))))).
% 0.33/5.21 thf(fact_3933_sum__list__update, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [K:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((groups8242544230860333062m_list @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ X)) @ (((nth @ A) @ Xs) @ K)))))))))).
% 0.33/5.21 thf(fact_3934_set__take__disj__set__drop__if__distinct, axiom, (! [A:$tType,Vs:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ Vs) => ((((ord_less_eq @ nat) @ I) @ J) => ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ I) @ Vs))) @ ((set2 @ A) @ (((drop @ A) @ J) @ Vs))) = (bot_bot @ (set @ A)))))))).
% 0.33/5.21 thf(fact_3935_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A]: (((((inf_inf @ A) @ A3) @ A3) = A3))))))).
% 0.33/5.21 thf(fact_3936_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 0.33/5.21 thf(fact_3937_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ A3) @ B2)) = (((inf_inf @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3938_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3939_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ B2) = (((inf_inf @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_3940_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Y3) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3941_inf__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.33/5.21 thf(fact_3942_le__inf__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = ((((ord_less_eq @ A) @ X) @ Y3) & (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3943_inf_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2)) = ((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 0.33/5.21 thf(fact_3944_boolean__algebra_Oconj__zero__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3945_boolean__algebra_Oconj__zero__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3946_inf__bot__right, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3947_inf__bot__left, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3948_inf__sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = X))))))).
% 0.33/5.21 thf(fact_3949_sup__inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = X))))))).
% 0.33/5.21 thf(fact_3950_image__Suc__atLeastAtMost, axiom, (! [I:nat,J:nat]: ((((((image @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ J)) = (((set_or1337092689740270186AtMost @ nat) @ (suc @ I)) @ (suc @ J)))))).
% 0.33/5.21 thf(fact_3951_image__Suc__atLeastLessThan, axiom, (! [I:nat,J:nat]: ((((((image @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) = (((set_or7035219750837199246ssThan @ nat) @ (suc @ I)) @ (suc @ J)))))).
% 0.33/5.21 thf(fact_3952_Ints__prod, axiom, (! [A:$tType,B:$tType]: ((((comm_monoid_mult @ B) & (ring_1 @ B)) => (! [A6:(set @ A),F:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ B) @ (F @ X3)) @ (ring_1_Ints @ B))))) => (((member @ B) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ A6)) @ (ring_1_Ints @ B))))))))).
% 0.33/5.21 thf(fact_3953_image__add__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ (zero_zero @ A))) @ S3) = S3))))))).
% 0.33/5.21 thf(fact_3954_inf__compl__bot__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ (((inf_inf @ A) @ X) @ Y3)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3955_inf__compl__bot__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3956_inf__compl__bot__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ ((uminus_uminus @ A) @ X))) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3957_boolean__algebra_Oconj__cancel__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ X) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3958_boolean__algebra_Oconj__cancel__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ ((uminus_uminus @ A) @ X)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_3959_boolean__algebra_Ode__Morgan__conj, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((uminus_uminus @ A) @ (((inf_inf @ A) @ X) @ Y3)) = (((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3960_boolean__algebra_Ode__Morgan__disj, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((uminus_uminus @ A) @ (((sup_sup @ A) @ X) @ Y3)) = (((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_3961_image__uminus__atLeastAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (((set_or1337092689740270186AtMost @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3962_sum__list_ONil, axiom, (! [A:$tType]: (((monoid_add @ A) => (((groups8242544230860333062m_list @ A) @ (nil @ A)) = (zero_zero @ A)))))).
% 0.33/5.21 thf(fact_3963_sum__list__eq__0__iff, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [Ns:(list @ A)]: (((((groups8242544230860333062m_list @ A) @ Ns) = (zero_zero @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Ns)) => (X2 = (zero_zero @ A)))))))))))).
% 0.33/5.21 thf(fact_3964_image__uminus__greaterThanLessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y3)) = (((set_or5935395276787703475ssThan @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3965_Compl__disjoint, axiom, (! [A:$tType,A6:(set @ A)]: (((((inf_inf @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ A6)) = (bot_bot @ (set @ A)))))).
% 0.33/5.21 thf(fact_3966_Compl__disjoint2, axiom, (! [A:$tType,A6:(set @ A)]: (((((inf_inf @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ A6) = (bot_bot @ (set @ A)))))).
% 0.33/5.21 thf(fact_3967_Diff__Compl, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((minus_minus @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ B7)) = (((inf_inf @ (set @ A)) @ A6) @ B7))))).
% 0.33/5.21 thf(fact_3968_image__uminus__greaterThanAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y3)) = (((set_or7035219750837199246ssThan @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3969_image__uminus__atLeastLessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y3:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y3)) = (((set_or3652927894154168847AtMost @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.21 thf(fact_3970_prod_OatMost__Suc, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_3971_prod_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))))).
% 0.33/5.21 thf(fact_3972_image__mult__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (((((image @ A) @ A) @ ((times_times @ A) @ D2)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ D2) @ A3)) @ (((times_times @ A) @ D2) @ B2)))))))))).
% 0.33/5.21 thf(fact_3973_prod_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N)) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N)))))))))))).
% 0.33/5.21 thf(fact_3974_prod_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))))).
% 0.33/5.21 thf(fact_3975_distrib__imp1, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((! [X3:A,Y5:A,Z4:A]: (((((inf_inf @ A) @ X3) @ (((sup_sup @ A) @ Y5) @ Z4)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X3) @ Y5)) @ (((inf_inf @ A) @ X3) @ Z4))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z3)))))))))).
% 0.33/5.21 thf(fact_3976_distrib__imp2, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((! [X3:A,Y5:A,Z4:A]: (((((sup_sup @ A) @ X3) @ (((inf_inf @ A) @ Y5) @ Z4)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X3) @ Y5)) @ (((sup_sup @ A) @ X3) @ Z4))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z3)))))))))).
% 0.33/5.21 thf(fact_3977_inf__sup__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3978_inf__sup__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y3:A,Z3:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ Z3)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ X)) @ (((inf_inf @ A) @ Z3) @ X))))))))).
% 0.33/5.21 thf(fact_3979_sup__inf__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3980_sup__inf__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y3:A,Z3:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ Z3)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ X)) @ (((sup_sup @ A) @ Z3) @ X))))))))).
% 0.33/5.21 thf(fact_3981_boolean__algebra_Oconj__disj__distrib, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3982_boolean__algebra_Odisj__conj__distrib, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3983_boolean__algebra_Oconj__disj__distrib2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,Z3:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ Z3)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ X)) @ (((inf_inf @ A) @ Z3) @ X))))))))).
% 0.33/5.21 thf(fact_3984_boolean__algebra_Odisj__conj__distrib2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y3:A,Z3:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ Z3)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ X)) @ (((sup_sup @ A) @ Z3) @ X))))))))).
% 0.33/5.21 thf(fact_3985_inf__min, axiom, (! [A:$tType]: ((((semilattice_inf @ A) & (linorder @ A)) => ((inf_inf @ A) = (ord_min @ A)))))).
% 0.33/5.21 thf(fact_3986_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.33/5.21 thf(fact_3987_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = (((inf_inf @ A) @ Y3) @ (((inf_inf @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3988_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Z3) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3989_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X2:A,Y6:A]: ((((inf_inf @ A) @ Y6) @ X2)))))))).
% 0.33/5.21 thf(fact_3990_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C2) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3991_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Z3) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_3992_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A5:A,B3:A]: ((((inf_inf @ A) @ B3) @ A5)))))))).
% 0.33/5.21 thf(fact_3993_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X2:A,Y6:A]: ((((inf_inf @ A) @ Y6) @ X2)))))))).
% 0.33/5.21 thf(fact_3994_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A3:A,C2:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A3) @ C2)) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2))))))))).
% 0.33/5.21 thf(fact_3995_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)) = (((inf_inf @ A) @ Y3) @ (((inf_inf @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_3996_inf__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.33/5.21 thf(fact_3997_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:A,K:A,A3:A,B2:A]: (((A6 = (((inf_inf @ A) @ K) @ A3)) => ((((inf_inf @ A) @ A6) @ B2) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3998_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B7:A,K:A,B2:A,A3:A]: (((B7 = (((inf_inf @ A) @ K) @ B2)) => ((((inf_inf @ A) @ A3) @ B7) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_3999_less__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,X:A,B2:A]: (((((ord_less @ A) @ A3) @ X) => (((ord_less @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ X)))))))).
% 0.33/5.21 thf(fact_4000_less__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X:A,A3:A]: (((((ord_less @ A) @ B2) @ X) => (((ord_less @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ X)))))))).
% 0.33/5.21 thf(fact_4001_inf_Oabsorb3, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((inf_inf @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_4002_inf_Oabsorb4, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((inf_inf @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_4003_inf_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2)) => (~ (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ A3) @ C2))))))))))))).
% 0.33/5.21 thf(fact_4004_inf_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less @ A) = (^ [A5:A,B3:A]: (((A5 = (((inf_inf @ A) @ A5) @ B3)) & (A5 != B3))))))))).
% 0.33/5.21 thf(fact_4005_inf_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ A3) @ C2) => (((ord_less @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_4006_inf_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_4007_inf_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_4008_inf_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C2)))))))).
% 0.33/5.21 thf(fact_4009_inf_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [B3:A,A5:A]: (((((inf_inf @ A) @ A5) @ B3) = B3)))))))).
% 0.33/5.21 thf(fact_4010_inf_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: (((((inf_inf @ A) @ A5) @ B3) = A5)))))))).
% 0.33/5.21 thf(fact_4011_inf_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ B2))))))).
% 0.33/5.21 thf(fact_4012_inf_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ A3))))))).
% 0.33/5.21 thf(fact_4013_inf_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A5:A,B3:A]: ((A5 = (((inf_inf @ A) @ A5) @ B3))))))))).
% 0.33/5.21 thf(fact_4014_inf__greatest, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ X) @ Z3) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3)))))))))).
% 0.33/5.21 thf(fact_4015_inf_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ A3) @ C2) => (((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2)))))))))).
% 0.33/5.21 thf(fact_4016_inf_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C2)) => (~ (((((ord_less_eq @ A) @ A3) @ B2) => (~ ((((ord_less_eq @ A) @ A3) @ C2))))))))))))).
% 0.33/5.21 thf(fact_4017_inf__absorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((inf_inf @ A) @ X) @ Y3) = Y3)))))))).
% 0.33/5.21 thf(fact_4018_inf__absorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((inf_inf @ A) @ X) @ Y3) = X)))))))).
% 0.33/5.21 thf(fact_4019_inf_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((inf_inf @ A) @ A3) @ B2) = B2)))))))).
% 0.33/5.21 thf(fact_4020_inf_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((inf_inf @ A) @ A3) @ B2) = A3)))))))).
% 0.33/5.21 thf(fact_4021_le__iff__inf, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y6:A]: (((((inf_inf @ A) @ X2) @ Y6) = X2)))))))).
% 0.33/5.21 thf(fact_4022_inf__unique, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [F:(A > (A > A)),X:A,Y3:A]: (((! [X3:A,Y5:A]: ((((ord_less_eq @ A) @ ((F @ X3) @ Y5)) @ X3))) => ((! [X3:A,Y5:A]: ((((ord_less_eq @ A) @ ((F @ X3) @ Y5)) @ Y5))) => ((! [X3:A,Y5:A,Z4:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => ((((ord_less_eq @ A) @ X3) @ Z4) => (((ord_less_eq @ A) @ X3) @ ((F @ Y5) @ Z4)))))) => ((((inf_inf @ A) @ X) @ Y3) = ((F @ X) @ Y3))))))))))).
% 0.33/5.21 thf(fact_4023_inf_OorderI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((A3 = (((inf_inf @ A) @ A3) @ B2)) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 0.33/5.21 thf(fact_4024_inf_OorderE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (A3 = (((inf_inf @ A) @ A3) @ B2))))))))).
% 0.33/5.21 thf(fact_4025_le__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ X)))))))).
% 0.33/5.21 thf(fact_4026_le__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,X:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ X) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ X)))))))).
% 0.33/5.21 thf(fact_4027_inf__mono, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ (((inf_inf @ A) @ C2) @ D2)))))))))).
% 0.33/5.21 thf(fact_4028_le__infI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ X) @ A3) => ((((ord_less_eq @ A) @ X) @ B2) => (((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A3) @ B2)))))))))).
% 0.33/5.21 thf(fact_4029_le__infE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ X) @ (((inf_inf @ A) @ A3) @ B2)) => (~ (((((ord_less_eq @ A) @ X) @ A3) => (~ ((((ord_less_eq @ A) @ X) @ B2))))))))))))).
% 0.33/5.21 thf(fact_4030_inf__le2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Y3))))))).
% 0.33/5.21 thf(fact_4031_inf__le1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ X))))))).
% 0.33/5.21 thf(fact_4032_inf__sup__ord_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ X))))))).
% 0.33/5.21 thf(fact_4033_inf__sup__ord_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Y3))))))).
% 0.33/5.21 thf(fact_4034_distrib__sup__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z3))) @ (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 0.33/5.21 thf(fact_4035_distrib__inf__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z3:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z3))) @ (((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_4036_diff__eq, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => ((minus_minus @ A) = (^ [X2:A,Y6:A]: ((((inf_inf @ A) @ X2) @ ((uminus_uminus @ A) @ Y6))))))))).
% 0.33/5.21 thf(fact_4037_inf__cancel__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,A3:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ A3)) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ B2)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_4038_inf__cancel__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,A3:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X)) @ A3)) @ (((inf_inf @ A) @ X) @ B2)) = (bot_bot @ A)))))))).
% 0.33/5.21 thf(fact_4039_zero__notin__Suc__image, axiom, (! [A6:(set @ nat)]: ((~ ((((member @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ A6))))))).
% 0.33/5.21 thf(fact_4040_image__int__atLeastLessThan, axiom, (! [A3:nat,B2:nat]: ((((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2)) = (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 0.33/5.21 thf(fact_4041_member__le__sum__list, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))).
% 0.33/5.21 thf(fact_4042_Compl__Un, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = (((inf_inf @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ ((uminus_uminus @ (set @ A)) @ B7)))))).
% 0.33/5.21 thf(fact_4043_Compl__Int, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B7)) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ ((uminus_uminus @ (set @ A)) @ B7)))))).
% 0.33/5.21 thf(fact_4044_Diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((((inf_inf @ (set @ A)) @ A7) @ ((uminus_uminus @ (set @ A)) @ B6)))))))).
% 0.33/5.21 thf(fact_4045_prod_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_mult @ A)) => (! [A3:B,C2:B,B2:B,D2:B,G:(B > A),H2:(B > A)]: (((A3 = C2) => ((B2 = D2) => ((! [X3:B]: (((((ord_less_eq @ B) @ C2) @ X3) => ((((ord_less @ B) @ X3) @ D2) => ((G @ X3) = (H2 @ X3)))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D2)))))))))))).
% 0.33/5.21 thf(fact_4046_prod_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,P5:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P5))))))))))).
% 0.33/5.21 thf(fact_4047_inf__shunt, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((((inf_inf @ A) @ X) @ Y3) = (bot_bot @ A)) = (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y3))))))))).
% 0.33/5.21 thf(fact_4048_shunt1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Z3) = (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ Y3)) @ Z3))))))))).
% 0.33/5.21 thf(fact_4049_shunt2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ (((inf_inf @ A) @ X) @ ((uminus_uminus @ A) @ Y3))) @ Z3) = (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z3))))))))).
% 0.33/5.21 thf(fact_4050_sup__neg__inf, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [P5:A,Q4:A,R:A]: (((((ord_less_eq @ A) @ P5) @ (((sup_sup @ A) @ Q4) @ R)) = (((ord_less_eq @ A) @ (((inf_inf @ A) @ P5) @ ((uminus_uminus @ A) @ Q4))) @ R)))))))).
% 0.33/5.21 thf(fact_4051_card__image__le, axiom, (! [B:$tType,A:$tType,A6:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ A6) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ ((((image @ A) @ B) @ F) @ A6))) @ ((finite_card @ A) @ A6)))))).
% 0.33/5.21 thf(fact_4052_Groups__List_Osum__list__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ X3)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))).
% 0.33/5.21 thf(fact_4053_sum__list__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ X3)))) => ((((groups8242544230860333062m_list @ A) @ Xs) = (zero_zero @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (X2 = (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_4054_sum__list__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X3) @ (zero_zero @ A))))) => (((ord_less_eq @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ (zero_zero @ A))))))))).
% 0.33/5.21 thf(fact_4055_Ioc__disjoint, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = (bot_bot @ (set @ A))) = ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ D2) @ C2) | ((((ord_less_eq @ A) @ B2) @ C2) | (((ord_less_eq @ A) @ D2) @ A3))))))))))).
% 0.33/5.21 thf(fact_4056_disjoint__eq__subset__Compl, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A6) @ B7) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ B7)))))).
% 0.33/5.21 thf(fact_4057_image__int__atLeastAtMost, axiom, (! [A3:nat,B2:nat]: ((((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 0.33/5.21 thf(fact_4058_sum__list__abs, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [Xs:(list @ A)]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((groups8242544230860333062m_list @ A) @ Xs))) @ ((groups8242544230860333062m_list @ A) @ ((((map @ A) @ A) @ (abs_abs @ A)) @ Xs))))))))).
% 0.33/5.21 thf(fact_4059_sum__list__replicate, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat,C2:A]: ((((groups8242544230860333062m_list @ A) @ (((replicate @ A) @ N) @ C2)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ C2)))))))).
% 0.33/5.21 thf(fact_4060_prod_OatLeast0__atMost__Suc, axiom, (! [A:$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)))))))))).
% 0.33/5.21 thf(fact_4061_prod_OatLeast0__lessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ N))))))))).
% 0.33/5.21 thf(fact_4062_prod_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less @ nat) @ M) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ N))))))))))).
% 0.33/5.21 thf(fact_4063_prod_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N))))))))))).
% 0.33/5.21 thf(fact_4064_prod_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (((times_times @ A) @ (G @ (suc @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.21 thf(fact_4065_prod_OatLeastLessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:nat,B2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ A3) @ B2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ (suc @ B2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2))) @ (G @ B2)))))))))).
% 0.33/5.21 thf(fact_4066_prod_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((times_times @ A) @ (G @ N)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))))).
% 0.33/5.21 thf(fact_4067_surj__card__le, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B7:(set @ B),F:(A > B)]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ B)) @ B7) @ ((((image @ A) @ B) @ F) @ A6)) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ B7)) @ ((finite_card @ A) @ A6))))))).
% 0.33/5.21 thf(fact_4068_prod__Suc__fact, axiom, (! [N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) = ((semiring_char_0_fact @ nat) @ N))))).
% 0.33/5.21 thf(fact_4069_prod__Suc__Suc__fact, axiom, (! [N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((semiring_char_0_fact @ nat) @ N))))).
% 0.33/5.21 thf(fact_4070_image__Suc__atMost, axiom, (! [N:nat]: ((((((image @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ (suc @ N)))))).
% 0.33/5.21 thf(fact_4071_Iio__Int__singleton, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,K:A]: ((((((ord_less @ A) @ X) @ K) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) & ((~ ((((ord_less @ A) @ X) @ K))) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))))))))))).
% 0.33/5.21 thf(fact_4072_image__Suc__lessThan, axiom, (! [N:nat]: ((((((image @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N))))).
% 0.33/5.21 thf(fact_4073_atLeast0__atMost__Suc__eq__insert__0, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.21 thf(fact_4074_atLeast0__lessThan__Suc__eq__insert__0, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.21 thf(fact_4075_prod_Ohead, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.21 thf(fact_4076_atMost__Suc__eq__insert__0, axiom, (! [N:nat]: ((((set_ord_atMost @ nat) @ (suc @ N)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N))))))).
% 0.33/5.21 thf(fact_4077_lessThan__Suc__eq__insert__0, axiom, (! [N:nat]: ((((set_ord_lessThan @ nat) @ (suc @ N)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N))))))).
% 0.33/5.21 thf(fact_4078_sum__list__strict__mono, axiom, (! [B:$tType,A:$tType]: ((((monoid_add @ B) & (strict9044650504122735259up_add @ B)) => (! [Xs:(list @ A),F:(A > B),G:(A > B)]: (((Xs != (nil @ A)) => ((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less @ B) @ (F @ X3)) @ (G @ X3))))) => (((ord_less @ B) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ F) @ Xs))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ G) @ Xs))))))))))).
% 0.33/5.21 thf(fact_4079_elem__le__sum__list, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [K:nat,Ns:(list @ A)]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Ns)) => (((ord_less_eq @ A) @ (((nth @ A) @ Ns) @ K)) @ ((groups8242544230860333062m_list @ A) @ Ns))))))))).
% 0.33/5.21 thf(fact_4080_prod_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))))).
% 0.33/5.21 thf(fact_4081_prod_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A),P5:nat]: (((((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ N) @ P5))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))))).
% 0.33/5.21 thf(fact_4082_fact__prod__Suc, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N3:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))))))))).
% 0.33/5.21 thf(fact_4083_image__atLeastZeroLessThan__int, axiom, (! [U:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ U) => ((((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U) = ((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ ((set_ord_lessThan @ nat) @ (nat2 @ U)))))))).
% 0.33/5.21 thf(fact_4084_image__mult__atLeastAtMost__if, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,X:A,Y3:A]: ((((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ X)) @ (((times_times @ A) @ C2) @ Y3)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less_eq @ A) @ X) @ Y3) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ Y3)) @ (((times_times @ A) @ C2) @ X)))) & ((~ ((((ord_less_eq @ A) @ X) @ Y3))) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (bot_bot @ (set @ A))))))))))))).
% 0.33/5.21 thf(fact_4085_prod__pos__nat__iff, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((((groups7121269368397514597t_prod @ A) @ nat) @ F) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (F @ X2)))))))))).
% 0.33/5.21 thf(fact_4086_prod_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((~ ((((member @ B) @ X) @ A6))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X) @ A6)) = (((times_times @ A) @ (G @ X)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6))))))))))).
% 0.33/5.21 thf(fact_4087_prod__eq__1__iff, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => ((((((groups7121269368397514597t_prod @ A) @ nat) @ F) @ A6) = (one_one @ nat)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => ((F @ X2) = (one_one @ nat)))))))))).
% 0.33/5.21 thf(fact_4088_prod_Oinfinite, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A)]: (((~ (((finite_finite2 @ B) @ A6))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_4089_prod__zero__iff, axiom, (! [A:$tType,B:$tType]: (((semidom @ A) => (! [A6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6) = (zero_zero @ A)) = (? [X2:B]: (((((member @ B) @ X2) @ A6) & ((F @ X2) = (zero_zero @ A))))))))))))).
% 0.33/5.21 thf(fact_4090_prod_Oempty, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))).
% 0.33/5.21 thf(fact_4091_inf__nat__def, axiom, ((inf_inf @ nat) = (ord_min @ nat))).
% 0.33/5.21 thf(fact_4092_inf__int__def, axiom, ((inf_inf @ int) = (ord_min @ int))).
% 0.33/5.21 thf(fact_4093_prod_Oneutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => ((G @ X3) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = (one_one @ A))))))))).
% 0.33/5.21 thf(fact_4094_prod_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A),A6:(set @ B)]: (((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) != (one_one @ A)) => (~ ((! [A8:B]: (((((member @ B) @ A8) @ A6) => ((G @ A8) = (one_one @ A)))))))))))))).
% 0.33/5.21 thf(fact_4095_prod__nonneg, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X3))))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.21 thf(fact_4096_prod__mono, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A6:(set @ B),F:(B > A),G:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4)) & (((ord_less_eq @ A) @ (F @ I4)) @ (G @ I4)))))) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6))))))))).
% 0.33/5.21 thf(fact_4097_prod__pos, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less @ A) @ (zero_zero @ A)) @ (F @ X3))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.21 thf(fact_4098_prod__ge__1, axiom, (! [A:$tType,B:$tType]: (((linord181362715937106298miring @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (one_one @ A)) @ (F @ X3))))) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.21 thf(fact_4099_prod__zero, axiom, (! [B:$tType,A:$tType]: (((comm_semiring_1 @ A) => (! [A6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((? [X4:B]: (((((member @ B) @ X4) @ A6) & ((F @ X4) = (zero_zero @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6) = (zero_zero @ A)))))))))).
% 0.33/5.21 thf(fact_4100_prod__le__1, axiom, (! [B:$tType,A:$tType]: (((linord181362715937106298miring @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X3)) & (((ord_less_eq @ A) @ (F @ X3)) @ (one_one @ A)))))) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ (one_one @ A))))))))).
% 0.33/5.21 thf(fact_4101_prod_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [R2:(A > (A > $o)),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((R2 @ (one_one @ A)) @ (one_one @ A)) => ((! [X1:A,Y1:A,X23:A,Y22:A]: (((((R2 @ X1) @ X23) & ((R2 @ Y1) @ Y22)) => ((R2 @ (((times_times @ A) @ X1) @ Y1)) @ (((times_times @ A) @ X23) @ Y22))))) => (((finite_finite2 @ B) @ S3) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((R2 @ (H2 @ X3)) @ (G @ X3))))) => ((R2 @ ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3)))))))))))).
% 0.33/5.21 thf(fact_4102_prod_Oinsert__if, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((member @ B) @ X) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X) @ A6)) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6))) & ((~ ((((member @ B) @ X) @ A6))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X) @ A6)) = (((times_times @ A) @ (G @ X)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6)))))))))))).
% 0.33/5.21 thf(fact_4103_prod_Oreindex__bij__witness__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S5:(set @ B),T5:(set @ C),S3:(set @ B),I:(C > B),J:(B > C),T6:(set @ C),G:(B > A),H2:(C > A)]: ((((finite_finite2 @ B) @ S5) => (((finite_finite2 @ C) @ T5) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) => ((I @ (J @ A8)) = A8)))) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) => (((member @ C) @ (J @ A8)) @ (((minus_minus @ (set @ C)) @ T6) @ T5))))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((J @ (I @ B5)) = B5)))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => (((member @ B) @ (I @ B5)) @ (((minus_minus @ (set @ B)) @ S3) @ S5))))) => ((! [A8:B]: (((((member @ B) @ A8) @ S5) => ((G @ A8) = (one_one @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T5) => ((H2 @ B5) = (one_one @ A))))) => ((! [A8:B]: (((((member @ B) @ A8) @ S3) => ((H2 @ (J @ A8)) = (G @ A8))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ C) @ A) @ H2) @ T6))))))))))))))))).
% 0.33/5.21 thf(fact_4104_less__1__prod2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),I:A,F:(A > B)]: ((((finite_finite2 @ A) @ I6) => ((((member @ A) @ I) @ I6) => ((((ord_less @ B) @ (one_one @ B)) @ (F @ I)) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F @ I4))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ I6)))))))))))).
% 0.33/5.21 thf(fact_4105_less__1__prod, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ I6) => ((I6 != (bot_bot @ (set @ A))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less @ B) @ (one_one @ B)) @ (F @ I4))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ I6))))))))))).
% 0.33/5.21 thf(fact_4106_prod_Osubset__diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B7:(set @ B),A6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B7) @ A6) => (((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ B7))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B7))))))))))).
% 0.33/5.21 thf(fact_4107_prod_Omono__neutral__cong__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3)))))))))))).
% 0.33/5.21 thf(fact_4108_prod_Omono__neutral__cong__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ X3) = (one_one @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ T6)))))))))))).
% 0.33/5.21 thf(fact_4109_prod_Omono__neutral__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3))))))))))).
% 0.33/5.21 thf(fact_4110_prod_Omono__neutral__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6))))))))))).
% 0.33/5.21 thf(fact_4111_prod_Osame__carrierI, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C6:(set @ B),A6:(set @ B),B7:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C6) => ((((ord_less_eq @ (set @ B)) @ A6) @ C6) => ((((ord_less_eq @ (set @ B)) @ B7) @ C6) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ C6) @ A6)) => ((G @ A8) = (one_one @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C6) @ B7)) => ((H2 @ B5) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ C6)) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ B7)))))))))))))).
% 0.33/5.21 thf(fact_4112_prod_Osame__carrier, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C6:(set @ B),A6:(set @ B),B7:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C6) => ((((ord_less_eq @ (set @ B)) @ A6) @ C6) => ((((ord_less_eq @ (set @ B)) @ B7) @ C6) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ C6) @ A6)) => ((G @ A8) = (one_one @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C6) @ B7)) => ((H2 @ B5) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ B7)) = (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ C6)))))))))))))).
% 0.33/5.21 thf(fact_4113_prod_Ounion__inter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),B7:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A6) @ B7))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ B7))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B7))))))))))).
% 0.33/5.21 thf(fact_4114_prod_OInt__Diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A),B7:(set @ B)]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ B7))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ B7))))))))))).
% 0.33/5.21 thf(fact_4115_prod_Omono__neutral__cong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => (((finite_finite2 @ B) @ S3) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ I4) = (one_one @ A))))) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ S3) @ T6)) => ((G @ I4) = (one_one @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ (((inf_inf @ (set @ B)) @ S3) @ T6)) => ((G @ X3) = (H2 @ X3))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ T6))))))))))))).
% 0.33/5.21 thf(fact_4116_prod__mono__strict, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A6:(set @ B),F:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((! [I4:B]: (((((member @ B) @ I4) @ A6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4)) & (((ord_less @ A) @ (F @ I4)) @ (G @ I4)))))) => ((A6 != (bot_bot @ (set @ B))) => (((ord_less @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6))))))))))).
% 0.33/5.21 thf(fact_4117_prod_Oinsert__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A),X:B]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X) @ A6)) = (((times_times @ A) @ (G @ X)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ (((insert @ B) @ X) @ (bot_bot @ (set @ B)))))))))))))).
% 0.33/5.21 thf(fact_4118_prod_Oremove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((((member @ B) @ X) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6) = (((times_times @ A) @ (G @ X)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ (((insert @ B) @ X) @ (bot_bot @ (set @ B))))))))))))))).
% 0.33/5.21 thf(fact_4119_prod_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),B7:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ (((inf_inf @ (set @ B)) @ A6) @ B7)) => ((G @ X3) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A6) @ B7)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B7)))))))))))).
% 0.33/5.21 thf(fact_4120_prod_Ounion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),B7:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => (((((inf_inf @ (set @ B)) @ A6) @ B7) = (bot_bot @ (set @ B))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A6) @ B7)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B7)))))))))))).
% 0.33/5.21 thf(fact_4121_prod_Ounion__diff2, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),B7:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A6) @ B7)) = (((times_times @ A) @ (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ B7))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ B7) @ A6)))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ B7)))))))))))).
% 0.33/5.21 thf(fact_4122_prod__mono2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [B7:(set @ A),A6:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((! [B5:A]: (((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B7) @ A6)) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F @ B5))))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F @ A8))))) => (((ord_less_eq @ B) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ A6)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ B7)))))))))))).
% 0.33/5.21 thf(fact_4123_prod__le__power, axiom, (! [B:$tType,A:$tType]: (((linordered_semidom @ A) => (! [A6:(set @ B),F:(B > A),N:A,K:nat]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4)) & (((ord_less_eq @ A) @ (F @ I4)) @ N))))) => ((((ord_less_eq @ nat) @ ((finite_card @ B) @ A6)) @ K) => ((((ord_less_eq @ A) @ (one_one @ A)) @ N) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ (((power_power @ A) @ N) @ K))))))))))).
% 0.33/5.21 thf(fact_4124_prod__diff1, axiom, (! [A:$tType,B:$tType]: (((semidom_divide @ A) => (! [A6:(set @ B),F:(B > A),A3:B]: ((((finite_finite2 @ B) @ A6) => (((F @ A3) != (zero_zero @ A)) => (((((member @ B) @ A3) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ F) @ (((minus_minus @ (set @ B)) @ A6) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B))))) = (((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ (F @ A3)))) & ((~ ((((member @ B) @ A3) @ A6))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F) @ (((minus_minus @ (set @ B)) @ A6) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)))))))))))).
% 0.33/5.21 thf(fact_4125_prod__Un, axiom, (! [A:$tType,B:$tType]: (((field @ A) => (! [A6:(set @ B),B7:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ (((inf_inf @ (set @ B)) @ A6) @ B7)) => ((F @ X3) != (zero_zero @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A6) @ B7)) = (((divide_divide @ A) @ (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ B7))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A6) @ B7))))))))))))).
% 0.33/5.21 thf(fact_4126_translation__Compl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,T2:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A3)) @ ((uminus_uminus @ (set @ A)) @ T2)) = ((uminus_uminus @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A3)) @ T2))))))))).
% 0.33/5.21 thf(fact_4127_remdups__adj__altdef, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((remdups_adj @ A) @ Xs) = Ys2) = (? [F2:(nat > nat)]: (((((order_mono @ nat) @ nat) @ F2) & ((((((image @ 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))) & ((! [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I5) = (((nth @ A) @ Ys2) @ (F2 @ I5)))))) & (! [I5:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I5) @ (one_one @ nat))) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I5) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ I5) @ (one_one @ nat)))) = ((F2 @ I5) = (F2 @ (((plus_plus @ nat) @ I5) @ (one_one @ nat))))))))))))))))).
% 0.33/5.22 thf(fact_4128_pochhammer__times__pochhammer__half, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z3:A,N:nat]: (((((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z3) @ (suc @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: ((((plus_plus @ A) @ Z3) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ K3)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))))))).
% 0.33/5.22 thf(fact_4129_butlast__take, axiom, (! [A:$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)))))).
% 0.33/5.22 thf(fact_4130_prod_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ (^ [Uu:B]: ((one_one @ A)))) @ A6) = (one_one @ A)))))))).
% 0.33/5.22 thf(fact_4131_of__int__prod, axiom, (! [A:$tType,B:$tType]: (((comm_ring_1 @ A) => (! [F:(B > int),A6:(set @ B)]: ((((ring_1_of_int @ A) @ ((((groups7121269368397514597t_prod @ B) @ int) @ F) @ A6)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: (((ring_1_of_int @ A) @ (F @ X2))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4132_finite__Collect__less__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less @ nat) @ N3) @ K)))))))).
% 0.33/5.22 thf(fact_4133_finite__Collect__le__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ K)))))))).
% 0.33/5.22 thf(fact_4134_card__Collect__less__nat, axiom, (! [N:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N))))) = N)))).
% 0.33/5.22 thf(fact_4135_finite__interval__int1, axiom, (! [A3:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I5:int]: (((((ord_less_eq @ int) @ A3) @ I5) & (((ord_less_eq @ int) @ I5) @ B2))))))))).
% 0.33/5.22 thf(fact_4136_finite__interval__int4, axiom, (! [A3:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I5:int]: (((((ord_less @ int) @ A3) @ I5) & (((ord_less @ int) @ I5) @ B2))))))))).
% 0.33/5.22 thf(fact_4137_prod_Odelta, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (one_one @ A))))))))))).
% 0.33/5.22 thf(fact_4138_prod_Odelta_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A3 = K3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A3 = K3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (one_one @ A))))))))))).
% 0.33/5.22 thf(fact_4139_sum__list__0, axiom, (! [B:$tType,A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X2:B]: ((zero_zero @ A)))) @ Xs)) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_4140_card__Collect__le__nat, axiom, (! [N:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less_eq @ nat) @ I5) @ N))))) = (suc @ N))))).
% 0.33/5.22 thf(fact_4141_of__nat__prod, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [F:(B > nat),A6:(set @ B)]: ((((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ B) @ nat) @ F) @ A6)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: (((semiring_1_of_nat @ A) @ (F @ X2))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4142_finite__interval__int2, axiom, (! [A3:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I5:int]: (((((ord_less_eq @ int) @ A3) @ I5) & (((ord_less @ int) @ I5) @ B2))))))))).
% 0.33/5.22 thf(fact_4143_finite__interval__int3, axiom, (! [A3:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I5:int]: (((((ord_less @ int) @ A3) @ I5) & (((ord_less_eq @ int) @ I5) @ B2))))))))).
% 0.33/5.22 thf(fact_4144_length__butlast, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((butlast @ A) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))).
% 0.33/5.22 thf(fact_4145_image__divide__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D2:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (((((image @ A) @ A) @ (^ [C4:A]: ((((divide_divide @ A) @ C4) @ D2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((divide_divide @ A) @ A3) @ D2)) @ (((divide_divide @ A) @ B2) @ D2)))))))))).
% 0.33/5.22 thf(fact_4146_Gcd__int__eq, axiom, (! [N6:(set @ nat)]: ((((gcd_Gcd @ int) @ ((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ N6)) = ((semiring_1_of_nat @ int) @ ((gcd_Gcd @ nat) @ N6)))))).
% 0.33/5.22 thf(fact_4147_prod_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),X:(B > A),Y3:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((X @ I5) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((Y3 @ I5) != (one_one @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((((times_times @ A) @ (X @ I5)) @ (Y3 @ I5)) != (one_one @ A))))))))))))))).
% 0.33/5.22 thf(fact_4148_prod_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4149_nat__seg__image__imp__finite, axiom, (! [A:$tType,A6:(set @ A),F:(nat > A),N:nat]: (((A6 = ((((image @ nat) @ A) @ F) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N)))))) => ((finite_finite2 @ A) @ A6))))).
% 0.33/5.22 thf(fact_4150_finite__conv__nat__seg__image, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A7:(set @ A)]: ((? [N3:nat,F2:(nat > A)]: ((A7 = ((((image @ nat) @ A) @ F2) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N3)))))))))))))).
% 0.33/5.22 thf(fact_4151_sum_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),X:(B > A),Y3:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((X @ I5) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((Y3 @ I5) != (zero_zero @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I5:B]: (((((member @ B) @ I5) @ I6) & ((((plus_plus @ A) @ (X @ I5)) @ (Y3 @ I5)) != (zero_zero @ A))))))))))))))).
% 0.33/5.22 thf(fact_4152_prod_Ointer__filter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ A6) & (P @ X2)))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (G @ X2)) @ (one_one @ A))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4153_prod_Odistrib, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A),H2:(B > A),A6:(set @ B)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: ((((times_times @ A) @ (G @ X2)) @ (H2 @ X2))))) @ A6) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ A6))))))))).
% 0.33/5.22 thf(fact_4154_prod_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4155_mod__prod__eq, axiom, (! [B:$tType,A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [F:(B > A),A3:A,A6:(set @ B)]: (((((modulo_modulo @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [I5:B]: ((((modulo_modulo @ A) @ (F @ I5)) @ A3)))) @ A6)) @ A3) = (((modulo_modulo @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A6)) @ A3)))))))).
% 0.33/5.22 thf(fact_4156_int__prod, axiom, (! [B:$tType,F:(B > nat),A6:(set @ B)]: ((((semiring_1_of_nat @ int) @ ((((groups7121269368397514597t_prod @ B) @ nat) @ F) @ A6)) = ((((groups7121269368397514597t_prod @ B) @ int) @ (^ [X2:B]: (((semiring_1_of_nat @ int) @ (F @ X2))))) @ A6))))).
% 0.33/5.22 thf(fact_4157_prod__int__eq, axiom, (! [I:nat,J:nat]: ((((((groups7121269368397514597t_prod @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ J)) = ((((groups7121269368397514597t_prod @ int) @ int) @ (^ [X2:int]: (X2))) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ I)) @ ((semiring_1_of_nat @ int) @ J))))))).
% 0.33/5.22 thf(fact_4158_translation__subtract__Compl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,T2:(set @ A)]: ((((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ X2) @ A3)))) @ ((uminus_uminus @ (set @ A)) @ T2)) = ((uminus_uminus @ (set @ A)) @ ((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ X2) @ A3)))) @ T2))))))))).
% 0.33/5.22 thf(fact_4159_Collect__imp__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: (((P @ X2) => (Q @ X2))))) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P))) @ ((collect @ A) @ Q)))))).
% 0.33/5.22 thf(fact_4160_nat__less__as__int, axiom, ((ord_less @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 0.33/5.22 thf(fact_4161_len__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_len @ A) = (^ [A5:(array @ A)]: (((heap_Time_tap @ nat) @ (^ [H4:(heap_ext @ product_unit)]: ((((array_length @ A) @ H4) @ A5))))))))))).
% 0.33/5.22 thf(fact_4162_lambda__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((^ [X2:A]: (X2)) = ((times_times @ A) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_4163_lambda__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => ((^ [H4:A]: ((zero_zero @ A))) = ((times_times @ A) @ (zero_zero @ A))))))).
% 0.33/5.22 thf(fact_4164_Compl__eq, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A7:(set @ A)]: (((collect @ A) @ (^ [X2:A]: ((~ ((((member @ A) @ X2) @ A7)))))))))))).
% 0.33/5.22 thf(fact_4165_Collect__neg__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: ((~ ((P @ X2)))))) = ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P)))))).
% 0.33/5.22 thf(fact_4166_uminus__set__def, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A7:(set @ A)]: (((collect @ A) @ ((uminus_uminus @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A7))))))))))).
% 0.33/5.22 thf(fact_4167_min__of__mono, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),M:A,N:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_min @ B) @ (F @ M)) @ (F @ N)) = (F @ (((ord_min @ A) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_4168_mono__Suc, axiom, (((order_mono @ nat) @ nat) @ suc)).
% 0.33/5.22 thf(fact_4169_strict__subset__divisors__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A]: (((((ord_less @ (set @ A)) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ A3))))) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ B2))))) = ((((dvd_dvd @ A) @ A3) @ B2) & (~ ((((dvd_dvd @ A) @ B2) @ A3))))))))))).
% 0.33/5.22 thf(fact_4170_mono__add, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A]: ((((order_mono @ A) @ A) @ ((plus_plus @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_4171_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A7:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A7)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B6)))))))))).
% 0.33/5.22 thf(fact_4172_mono__strict__invE, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y3)) => (((ord_less @ A) @ X) @ Y3))))))))).
% 0.33/5.22 thf(fact_4173_max__of__mono, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),M:A,N:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_max @ B) @ (F @ M)) @ (F @ N)) = (F @ (((ord_max @ A) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_4174_numeral__code_I2_J, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit0 @ N)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.22 thf(fact_4175_mono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (F2 @ Y6)))))))))))).
% 0.33/5.22 thf(fact_4176_monoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B)]: (((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y5))))) => (((order_mono @ A) @ B) @ F)))))))).
% 0.33/5.22 thf(fact_4177_monoE, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y3)))))))))).
% 0.33/5.22 thf(fact_4178_monoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y3)))))))))).
% 0.33/5.22 thf(fact_4179_min__def__raw, axiom, (! [A:$tType]: (((ord @ A) => ((ord_min @ A) = (^ [A5:A,B3:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A5) @ B3)) @ A5) @ B3)))))))).
% 0.33/5.22 thf(fact_4180_subset__divisors__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ A3))))) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ B2))))) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_4181_max__def__raw, axiom, (! [A:$tType]: (((ord @ A) => ((ord_max @ A) = (^ [A5:A,B3:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A5) @ B3)) @ B3) @ A5)))))))).
% 0.33/5.22 thf(fact_4182_nat__leq__as__int, axiom, ((ord_less_eq @ nat) = (^ [A5:nat,B3:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3)))))).
% 0.33/5.22 thf(fact_4183_effect__LetI, axiom, (! [B:$tType,A:$tType,X:A,T2:A,F:(A > (heap_Time_Heap @ B)),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:B,N:nat]: (((X = T2) => (((((((heap_Time_effect @ B) @ (F @ X)) @ H2) @ H) @ R) @ N) => ((((((heap_Time_effect @ B) @ (F @ T2)) @ H2) @ H) @ R) @ N)))))).
% 0.33/5.22 thf(fact_4184_lessThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_lessThan @ A) = (^ [U3:A]: (((collect @ A) @ (^ [X2:A]: ((((ord_less @ A) @ X2) @ U3))))))))))).
% 0.33/5.22 thf(fact_4185_atMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atMost @ A) = (^ [U3:A]: (((collect @ A) @ (^ [X2:A]: ((((ord_less_eq @ A) @ X2) @ U3))))))))))).
% 0.33/5.22 thf(fact_4186_sum__list__const__mult, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [C2:A,F:(B > A),Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X2:B]: ((((times_times @ A) @ C2) @ (F @ X2))))) @ Xs)) = (((times_times @ A) @ C2) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4187_sum__list__mult__const, axiom, (! [B:$tType,A:$tType]: (((semiring_0 @ A) => (! [F:(B > A),C2:A,Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X2:B]: ((((times_times @ A) @ (F @ X2)) @ C2)))) @ Xs)) = (((times_times @ A) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs))) @ C2)))))))).
% 0.33/5.22 thf(fact_4188_finite__less__ub, axiom, (! [F:(nat > nat),U:nat]: (((! [N4:nat]: ((((ord_less_eq @ nat) @ N4) @ (F @ N4)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ (F @ N3)) @ U))))))))).
% 0.33/5.22 thf(fact_4189_finite__int__segment, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A3:A,B2:A]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ (ring_1_Ints @ A)) & ((((ord_less_eq @ A) @ A3) @ X2) & (((ord_less_eq @ A) @ X2) @ B2))))))))))))).
% 0.33/5.22 thf(fact_4190_finite__M__bounded__by__nat, axiom, (! [P:(nat > $o),I:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((P @ K3) & (((ord_less @ nat) @ K3) @ I))))))))).
% 0.33/5.22 thf(fact_4191_finite__divisors__int, axiom, (! [I:int]: (((I != (zero_zero @ int)) => ((finite_finite2 @ int) @ ((collect @ int) @ (^ [D5:int]: ((((dvd_dvd @ int) @ D5) @ I))))))))).
% 0.33/5.22 thf(fact_4192_integer__of__num_I2_J, axiom, (! [N:num]: (((code_integer_of_num @ (bit0 @ N)) = (((plus_plus @ code_integer) @ (code_integer_of_num @ N)) @ (code_integer_of_num @ N)))))).
% 0.33/5.22 thf(fact_4193_numeral__code_I3_J, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_4194_power__numeral__even, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [Z3:A,W2:num]: (((((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ (bit0 @ W2))) = (((times_times @ A) @ (((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ W2))) @ (((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ W2)))))))))).
% 0.33/5.22 thf(fact_4195_mono__ge2__power__minus__self, axiom, (! [K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((order_mono @ nat) @ nat) @ (^ [M2:nat]: ((((minus_minus @ nat) @ (((power_power @ nat) @ K) @ M2)) @ M2)))))))).
% 0.33/5.22 thf(fact_4196_prod_Ointer__restrict, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A),B7:(set @ B)]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ B7)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (((member @ B) @ X2) @ B7)) @ (G @ X2)) @ (one_one @ A))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4197_prod_Osetdiff__irrelevant, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ ((collect @ B) @ (^ [X2:B]: (((G @ X2) = (one_one @ A))))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A6))))))))).
% 0.33/5.22 thf(fact_4198_power__numeral__odd, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [Z3:A,W2:num]: (((((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ (bit1 @ W2))) = (((times_times @ A) @ (((times_times @ A) @ Z3) @ (((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ W2)))) @ (((power_power @ A) @ Z3) @ ((numeral_numeral @ nat) @ W2)))))))))).
% 0.33/5.22 thf(fact_4199_card__length__sum__list__rec, axiom, (! [M:nat,N6:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M) => (((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M) & (((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) @ M) @ (one_one @ nat))) & (((groups8242544230860333062m_list @ nat) @ L3) = N6))))))) @ ((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M) & ((((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ L3)) @ (one_one @ nat)) = N6)))))))))))).
% 0.33/5.22 thf(fact_4200_sum__list__mono, axiom, (! [B:$tType,A:$tType]: ((((monoid_add @ B) & (ordere6658533253407199908up_add @ B)) => (! [Xs:(list @ A),F:(A > B),G:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ B) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ F) @ Xs))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ G) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4201_finite__abs__int__segment, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A3:A]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [K3:A]: (((((member @ A) @ K3) @ (ring_1_Ints @ A)) & (((ord_less_eq @ A) @ ((abs_abs @ A) @ K3)) @ A3)))))))))))).
% 0.33/5.22 thf(fact_4202_nat__plus__as__int, axiom, ((plus_plus @ nat) = (^ [A5:nat,B3:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3))))))).
% 0.33/5.22 thf(fact_4203_nat__times__as__int, axiom, ((times_times @ nat) = (^ [A5:nat,B3:nat]: ((nat2 @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3))))))).
% 0.33/5.22 thf(fact_4204_card__length__sum__list, axiom, (! [M:nat,N6:nat]: ((((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M) & (((groups8242544230860333062m_list @ nat) @ L3) = N6)))))) = ((binomial @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N6) @ M)) @ (one_one @ nat))) @ N6))))).
% 0.33/5.22 thf(fact_4205_nat__minus__as__int, axiom, ((minus_minus @ nat) = (^ [A5:nat,B3:nat]: ((nat2 @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3))))))).
% 0.33/5.22 thf(fact_4206_prod_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I5)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4207_finite__divisors__nat, axiom, (! [M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ M))))))))).
% 0.33/5.22 thf(fact_4208_card__less, axiom, (! [M7:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M7) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I))))))) != (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_4209_card__less__Suc, axiom, (! [M7:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M7) => ((suc @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ (suc @ K3)) @ M7) & (((ord_less @ nat) @ K3) @ I))))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I)))))))))))).
% 0.33/5.22 thf(fact_4210_card__less__Suc2, axiom, (! [M7:(set @ nat),I:nat]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ M7))) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ (suc @ K3)) @ M7) & (((ord_less @ nat) @ K3) @ I)))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I)))))))))))).
% 0.33/5.22 thf(fact_4211_sum__list__Suc, axiom, (! [A:$tType,F:(A > nat),Xs:(list @ A)]: ((((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ (^ [X2:A]: ((suc @ (F @ X2))))) @ Xs)) = (((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)))))).
% 0.33/5.22 thf(fact_4212_prod__int__plus__eq, axiom, (! [I:nat,J:nat]: ((((((groups7121269368397514597t_prod @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ (((plus_plus @ nat) @ I) @ J))) = ((((groups7121269368397514597t_prod @ int) @ int) @ (^ [X2:int]: (X2))) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ I)) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ I) @ J)))))))).
% 0.33/5.22 thf(fact_4213_nat__div__as__int, axiom, ((divide_divide @ nat) = (^ [A5:nat,B3:nat]: ((nat2 @ (((divide_divide @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3))))))).
% 0.33/5.22 thf(fact_4214_nat__mod__as__int, axiom, ((modulo_modulo @ nat) = (^ [A5:nat,B3:nat]: ((nat2 @ (((modulo_modulo @ int) @ ((semiring_1_of_nat @ int) @ A5)) @ ((semiring_1_of_nat @ int) @ B3))))))).
% 0.33/5.22 thf(fact_4215_mono__invE, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y3)) => (((ord_less_eq @ A) @ X) @ Y3))))))))).
% 0.33/5.22 thf(fact_4216_mono__iff__le__Suc, axiom, (! [A:$tType]: (((order @ A) => (((order_mono @ nat) @ A) = (^ [F2:(nat > A)]: ((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ N3)) @ (F2 @ (suc @ N3)))))))))))).
% 0.33/5.22 thf(fact_4217_image__add__int__atLeastLessThan, axiom, (! [L:int,U:int]: ((((((image @ int) @ int) @ (^ [X2:int]: ((((plus_plus @ int) @ X2) @ L)))) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ (((minus_minus @ int) @ U) @ L))) = (((set_or7035219750837199246ssThan @ int) @ L) @ U))))).
% 0.33/5.22 thf(fact_4218_mono__inf, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ A) & (semilattice_inf @ B)) => (! [F:(A > B),A6:A,B7:A]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ (F @ (((inf_inf @ A) @ A6) @ B7))) @ (((inf_inf @ B) @ (F @ A6)) @ (F @ B7)))))))))).
% 0.33/5.22 thf(fact_4219_map__replicate__trivial, axiom, (! [A:$tType,X:A,I:nat]: ((((((map @ nat) @ A) @ (^ [I5:nat]: (X))) @ ((upt @ (zero_zero @ nat)) @ I)) = (((replicate @ A) @ I) @ X))))).
% 0.33/5.22 thf(fact_4220_mono__sup, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ A) & (semilattice_sup @ B)) => (! [F:(A > B),A6:A,B7:A]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ (((sup_sup @ B) @ (F @ A6)) @ (F @ B7))) @ (F @ (((sup_sup @ A) @ A6) @ B7)))))))))).
% 0.33/5.22 thf(fact_4221_semiring__char__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiri4206861660011772517g_char @ A) = (^ [Uu2:(itself @ A)]: (((gcd_Gcd @ nat) @ ((collect @ nat) @ (^ [N3:nat]: ((((semiring_1_of_nat @ A) @ N3) = (zero_zero @ A))))))))))))).
% 0.33/5.22 thf(fact_4222_map__add__upt, axiom, (! [N:nat,M:nat]: ((((((map @ nat) @ nat) @ (^ [I5:nat]: ((((plus_plus @ nat) @ I5) @ N)))) @ ((upt @ (zero_zero @ nat)) @ M)) = ((upt @ N) @ (((plus_plus @ nat) @ M) @ N)))))).
% 0.33/5.22 thf(fact_4223_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)))))).
% 0.33/5.22 thf(fact_4224_prod_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N))) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.22 thf(fact_4225_prod_OIf__cases, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),P:(B > $o),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (H2 @ X2)) @ (G @ X2))))) @ A6) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P)))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))))).
% 0.33/5.22 thf(fact_4226_sum__list__triv, axiom, (! [C:$tType,B:$tType]: (((semiring_1 @ B) => (! [R:B,Xs:(list @ C)]: ((((groups8242544230860333062m_list @ B) @ ((((map @ C) @ B) @ (^ [X2:C]: (R))) @ Xs)) = (((times_times @ B) @ ((semiring_1_of_nat @ B) @ ((size_size @ (list @ C)) @ Xs))) @ R)))))))).
% 0.33/5.22 thf(fact_4227_prod_OatMost__Suc__shift, axiom, (! [A:$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) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_atMost @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4228_prod_OlessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4229_prod_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat,M:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ (suc @ I5)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M))))))))).
% 0.33/5.22 thf(fact_4230_prod_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A3 @ I5)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.22 thf(fact_4231_prod_OatLeast1__atMost__eq, axiom, (! [A:$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) @ (^ [K3:nat]: ((G @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4232_prod_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [M2:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))))).
% 0.33/5.22 thf(fact_4233_map__upt__Suc, axiom, (! [A:$tType,F:(nat > A),N:nat]: ((((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ (suc @ N))) = (((cons @ A) @ (F @ (zero_zero @ nat))) @ ((((map @ nat) @ A) @ (^ [I5:nat]: ((F @ (suc @ I5))))) @ ((upt @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.22 thf(fact_4234_prod_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A3 @ I5)) @ ((set_ord_lessThan @ nat) @ I5))))) @ ((set_ord_atMost @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4235_prod__atLeastAtMost__code, axiom, (! [A:$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) @ (^ [A5:nat]: (((times_times @ A) @ (F @ A5))))) @ A3) @ B2) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_4236_map__nth, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((map @ nat) @ A) @ ((nth @ A) @ Xs)) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))) = Xs)))).
% 0.33/5.22 thf(fact_4237_ratrel__def, axiom, (ratrel = (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: ((((((product_snd @ int) @ int) @ X2) != (zero_zero @ int)) & (((((product_snd @ int) @ int) @ Y6) != (zero_zero @ int)) & ((((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)) = (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2))))))))).
% 0.33/5.22 thf(fact_4238_fact__prod, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N3:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N3)))))))))).
% 0.33/5.22 thf(fact_4239_map__decr__upt, axiom, (! [M:nat,N:nat]: ((((((map @ nat) @ nat) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat)))))) @ ((upt @ (suc @ M)) @ (suc @ N))) = ((upt @ M) @ N))))).
% 0.33/5.22 thf(fact_4240_prod_Odelta__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A3:B,B2:(B > A),C2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = (((times_times @ A) @ (B2 @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B))))))))))))))).
% 0.33/5.22 thf(fact_4241_diff__nat__eq__if, axiom, (! [Z6:int,Z3:int]: ((((((ord_less @ int) @ Z6) @ (zero_zero @ int)) => ((((minus_minus @ nat) @ (nat2 @ Z3)) @ (nat2 @ Z6)) = (nat2 @ Z3))) & ((~ ((((ord_less @ int) @ Z6) @ (zero_zero @ int)))) => ((((minus_minus @ nat) @ (nat2 @ Z3)) @ (nat2 @ Z6)) = ((((if @ nat) @ (((ord_less @ int) @ (((minus_minus @ int) @ Z3) @ Z6)) @ (zero_zero @ int))) @ (zero_zero @ nat)) @ (nat2 @ (((minus_minus @ int) @ Z3) @ Z6))))))))).
% 0.33/5.22 thf(fact_4242_prod_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat,M:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M))))))))).
% 0.33/5.22 thf(fact_4243_prod_OatMost__shift, axiom, (! [A:$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) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4244_pochhammer__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A5:A,N3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3))))))))).
% 0.33/5.22 thf(fact_4245_mono__mult, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((order_mono @ A) @ A) @ ((times_times @ A) @ A3))))))))).
% 0.33/5.22 thf(fact_4246_mono__times__nat, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((order_mono @ nat) @ nat) @ ((times_times @ nat) @ N)))))).
% 0.33/5.22 thf(fact_4247_finite__lists__length__le, axiom, (! [A:$tType,A6:(set @ A),N:nat]: ((((finite_finite2 @ A) @ A6) => ((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A6) & (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ N)))))))))).
% 0.33/5.22 thf(fact_4248_mono__image__least, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [F:(A > B),M:A,N:A,M5:B,N2:B]: (((((order_mono @ A) @ B) @ F) => ((((((image @ A) @ B) @ F) @ (((set_or7035219750837199246ssThan @ A) @ M) @ N)) = (((set_or7035219750837199246ssThan @ B) @ M5) @ N2)) => ((((ord_less @ A) @ M) @ N) => ((F @ M) = M5)))))))))).
% 0.33/5.22 thf(fact_4249_fact__prod__rev, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N3:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ ((minus_minus @ nat) @ N3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))))))))).
% 0.33/5.22 thf(fact_4250_fact__eq__fact__times, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((semiring_char_0_fact @ nat) @ M) = (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ N)) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M)))))))).
% 0.33/5.22 thf(fact_4251_image__mult__atLeastAtMost__if_H, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,C2:A]: ((((((ord_less_eq @ A) @ X) @ Y3) => (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image @ A) @ A) @ (^ [X2:A]: ((((times_times @ A) @ X2) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ X) @ C2)) @ (((times_times @ A) @ Y3) @ C2)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((times_times @ A) @ X2) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ Y3) @ C2)) @ (((times_times @ A) @ X) @ C2)))))) & ((~ ((((ord_less_eq @ A) @ X) @ Y3))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((times_times @ A) @ X2) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = (bot_bot @ (set @ A))))))))))).
% 0.33/5.22 thf(fact_4252_image__affinity__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X2)) @ 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)) @ M) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ A3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ A3)) @ C2))))))))))))).
% 0.33/5.22 thf(fact_4253_image__affinity__atLeastAtMost__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ A3)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ A3)) @ C2))))))))))))).
% 0.33/5.22 thf(fact_4254_image__affinity__atLeastAtMost__div, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ 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)) @ M) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ M)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ M)) @ C2))))))))))))).
% 0.33/5.22 thf(fact_4255_image__affinity__atLeastAtMost__div__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ M)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X2:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X2) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ M)) @ C2))))))))))))).
% 0.33/5.22 thf(fact_4256_prod__gen__delta, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A3:B,B2:(B > A),C2:A]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ C2)))) @ S3) = (((times_times @ A) @ (B2 @ A3)) @ (((power_power @ A) @ C2) @ (((minus_minus @ nat) @ ((finite_card @ B) @ S3)) @ (one_one @ nat)))))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ C2)))) @ S3) = (((power_power @ A) @ C2) @ ((finite_card @ B) @ S3)))))))))))).
% 0.33/5.22 thf(fact_4257_pochhammer__Suc__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.22 thf(fact_4258_pochhammer__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A5:A,N3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N3) @ I5)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N3))))))))).
% 0.33/5.22 thf(fact_4259_set__decode__def, axiom, (nat_set_decode = (^ [X2:nat]: (((collect @ nat) @ (^ [N3:nat]: ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ X2) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))))))))).
% 0.33/5.22 thf(fact_4260_image__minus__const__atLeastLessThan__nat, axiom, (! [C2:nat,Y3:nat,X:nat]: ((((((ord_less @ nat) @ C2) @ Y3) => (((((image @ nat) @ nat) @ (^ [I5:nat]: ((((minus_minus @ nat) @ I5) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y3)) = (((set_or7035219750837199246ssThan @ nat) @ (((minus_minus @ nat) @ X) @ C2)) @ (((minus_minus @ nat) @ Y3) @ C2)))) & ((~ ((((ord_less @ nat) @ C2) @ Y3))) => (((((ord_less @ nat) @ X) @ Y3) => (((((image @ nat) @ nat) @ (^ [I5:nat]: ((((minus_minus @ nat) @ I5) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y3)) = (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))) & ((~ ((((ord_less @ nat) @ X) @ Y3))) => (((((image @ nat) @ nat) @ (^ [I5:nat]: ((((minus_minus @ nat) @ I5) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y3)) = (bot_bot @ (set @ nat)))))))))).
% 0.33/5.22 thf(fact_4261_fact__div__fact, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ M))))))).
% 0.33/5.22 thf(fact_4262_prod_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4263_prod_Oin__pairs__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4264_gbinomial__altdef__of__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((divide_divide @ A) @ (((minus_minus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ I5))) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ K3) @ I5)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K3))))))))).
% 0.33/5.22 thf(fact_4265_signed__take__bit__code, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N3:nat,A5:A]: (((((if @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N3)) @ A5)) @ N3)) @ (((plus_plus @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N3)) @ A5)) @ (((bit_se4730199178511100633sh_bit @ A) @ (suc @ N3)) @ ((uminus_uminus @ A) @ (one_one @ A))))) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N3)) @ A5))))))))).
% 0.33/5.22 thf(fact_4266_pochhammer__Suc__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N) @ I5)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.22 thf(fact_4267_binomial__altdef__of__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => (((semiring_1_of_nat @ A) @ ((binomial @ N) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N) @ I5))) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ K) @ I5)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)))))))))).
% 0.33/5.22 thf(fact_4268_gbinomial__mult__fact, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ (((gbinomial @ A) @ A3) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K))))))))).
% 0.33/5.22 thf(fact_4269_gbinomial__mult__fact_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K)) @ ((semiring_char_0_fact @ A) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K))))))))).
% 0.33/5.22 thf(fact_4270_gbinomial__prod__rev, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: ((((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K3))) @ ((semiring_char_0_fact @ A) @ K3))))))))).
% 0.33/5.22 thf(fact_4271_nth__butlast, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ ((butlast @ A) @ Xs))) => ((((nth @ A) @ ((butlast @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ N)))))).
% 0.33/5.22 thf(fact_4272_take__butlast, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((take @ A) @ N) @ ((butlast @ A) @ Xs)) = (((take @ A) @ N) @ Xs)))))).
% 0.33/5.22 thf(fact_4273_prod_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [P5:nat,K:nat,G:(nat > A),H2:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ ((((if @ A) @ (J3 = K)) @ (one_one @ A)) @ (H2 @ (((minus_minus @ nat) @ J3) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ (H2 @ J3))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))))).
% 0.33/5.22 thf(fact_4274_gbinomial__Suc, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ A3) @ (suc @ K)) = (((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ K))) @ ((semiring_char_0_fact @ A) @ (suc @ K)))))))))).
% 0.33/5.22 thf(fact_4275_pochhammer__code, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A5:A,N3:nat]: (((((if @ A) @ (N3 = (zero_zero @ nat))) @ (one_one @ A)) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [O:nat]: (((times_times @ A) @ (((plus_plus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ O)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) @ (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_4276_normalize__def, axiom, (normalize = (^ [P6:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ P6))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6)))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ ((((if @ ((product_prod @ int) @ int)) @ ((((product_snd @ int) @ int) @ P6) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))))))))).
% 0.33/5.22 thf(fact_4277_butlast__conv__take, axiom, (! [A:$tType]: (((butlast @ A) = (^ [Xs3:(list @ A)]: ((((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ (one_one @ nat))) @ Xs3))))))).
% 0.33/5.22 thf(fact_4278_butlast__list__update, axiom, (! [A:$tType,K:nat,Xs:(list @ A),X:A]: ((((K = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((butlast @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((butlast @ A) @ Xs))) & ((K != (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((butlast @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((((list_update @ A) @ ((butlast @ A) @ Xs)) @ K) @ X))))))).
% 0.33/5.22 thf(fact_4279_gbinomial__code, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A5:A,K3:nat]: (((((if @ A) @ (K3 = (zero_zero @ nat))) @ (one_one @ A)) @ (((divide_divide @ A) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [L3:nat]: (((times_times @ A) @ (((minus_minus @ A) @ A5) @ ((semiring_1_of_nat @ A) @ L3)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ K3) @ (one_one @ nat))) @ (one_one @ A))) @ ((semiring_char_0_fact @ A) @ K3)))))))))).
% 0.33/5.22 thf(fact_4280_card__lists__distinct__length__eq_H, axiom, (! [A:$tType,K:nat,A6:(set @ A)]: (((((ord_less @ nat) @ K) @ ((finite_card @ A) @ A6)) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (((distinct @ A) @ Xs3) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A6))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A6)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_4281_card__lists__distinct__length__eq, axiom, (! [A:$tType,A6:(set @ A),K:nat]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ nat) @ K) @ ((finite_card @ A) @ A6)) => (((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)) @ A6))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A6)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_4282_bit__cut__integer__code, axiom, (code_bit_cut_integer = (^ [K3:code_integer]: (((((if @ ((product_prod @ code_integer) @ $o)) @ (K3 = (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)) @ (^ [R5:code_integer,S6:code_integer]: (((((product_Pair @ code_integer) @ $o) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ R5) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ S6))) @ (S6 = (one_one @ code_integer)))))) @ ((code_divmod_abs @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 0.33/5.22 thf(fact_4283_integer__of__int__code, axiom, (code_integer_of_int = (^ [K3:int]: (((((if @ code_integer) @ (((ord_less @ int) @ K3) @ (zero_zero @ int))) @ ((uminus_uminus @ code_integer) @ (code_integer_of_int @ ((uminus_uminus @ int) @ K3)))) @ ((((if @ code_integer) @ (K3 = (zero_zero @ int))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ ((((modulo_modulo @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (zero_zero @ int))) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (one_one @ code_integer))))))))).
% 0.33/5.22 thf(fact_4284_swap__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_swap @ A) = (^ [I5:nat,X2:A,A5:(array @ A)]: ((((heap_Time_guard @ A) @ (^ [H4:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I5) @ (((array_length @ A) @ H4) @ A5))))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H4) @ A5)) @ I5)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A5) @ I5) @ X2) @ H4)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))).
% 0.33/5.22 thf(fact_4285_choose__even__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((((if @ A) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I5))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ N))) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.22 thf(fact_4286_integer__of__int__eq__of__int, axiom, (code_integer_of_int = (ring_1_of_int @ code_integer))).
% 0.33/5.22 thf(fact_4287_Ints__sum, axiom, (! [A:$tType,B:$tType]: (((ring_1 @ B) => (! [A6:(set @ A),F:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ B) @ (F @ X3)) @ (ring_1_Ints @ B))))) => (((member @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ A6)) @ (ring_1_Ints @ B))))))))).
% 0.33/5.22 thf(fact_4288_sum_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [Uu:B]: ((zero_zero @ A)))) @ A6) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_4289_of__nat__sum, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [F:(B > nat),A6:(set @ B)]: ((((semiring_1_of_nat @ A) @ ((((groups7311177749621191930dd_sum @ B) @ nat) @ F) @ A6)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((semiring_1_of_nat @ A) @ (F @ X2))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4290_of__int__sum, axiom, (! [A:$tType,B:$tType]: (((ring_1 @ A) => (! [F:(B > int),A6:(set @ B)]: ((((ring_1_of_int @ A) @ ((((groups7311177749621191930dd_sum @ B) @ int) @ F) @ A6)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((ring_1_of_int @ A) @ (F @ X2))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4291_sum_Oempty, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (bot_bot @ (set @ B))) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_4292_sum__eq__0__iff, axiom, (! [A:$tType,B:$tType]: (((canoni5634975068530333245id_add @ A) => (! [F4:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ F4) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ F4) = (zero_zero @ A)) = (! [X2:B]: (((((member @ B) @ X2) @ F4) => ((F @ X2) = (zero_zero @ A))))))))))))).
% 0.33/5.22 thf(fact_4293_sum_Oinfinite, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),G:(B > A)]: (((~ (((finite_finite2 @ B) @ A6))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4294_sum_Odelta, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (zero_zero @ A))))))))))).
% 0.33/5.22 thf(fact_4295_sum_Odelta_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A3) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A3 = K3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A3 = K3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (zero_zero @ A))))))))))).
% 0.33/5.22 thf(fact_4296_sum__abs, axiom, (! [B:$tType,A:$tType]: (((ordere166539214618696060dd_abs @ B) => (! [F:(A > B),A6:(set @ A)]: ((((ord_less_eq @ B) @ ((abs_abs @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ A6))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I5:A]: (((abs_abs @ B) @ (F @ I5))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4297_sum_OatMost__Suc, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4298_sum_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))))).
% 0.33/5.22 thf(fact_4299_sum__abs__ge__zero, axiom, (! [B:$tType,A:$tType]: (((ordere166539214618696060dd_abs @ B) => (! [F:(A > B),A6:(set @ A)]: ((((ord_less_eq @ B) @ (zero_zero @ B)) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I5:A]: (((abs_abs @ B) @ (F @ I5))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4300_sum__constant, axiom, (! [B:$tType,A:$tType]: (((semiring_1 @ A) => (! [Y3:A,A6:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (Y3))) @ A6) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A6))) @ Y3)))))))).
% 0.33/5.22 thf(fact_4301_sum__list__upt, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((groups8242544230860333062m_list @ nat) @ ((upt @ M) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))).
% 0.33/5.22 thf(fact_4302_sum__of__bool__mult__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A6:(set @ B),P:(B > $o),F:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: ((((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (P @ X2))) @ (F @ X2))))) @ A6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P))))))))))).
% 0.33/5.22 thf(fact_4303_sum__mult__of__bool__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A6:(set @ B),F:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: ((((times_times @ A) @ (F @ X2)) @ ((zero_neq_one_of_bool @ A) @ (P @ X2)))))) @ A6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P))))))))))).
% 0.33/5.22 thf(fact_4304_sum_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N)) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N)))))))))))).
% 0.33/5.22 thf(fact_4305_sum_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))))).
% 0.33/5.22 thf(fact_4306_sum__of__bool__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A6:(set @ B),P:(B > $o)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((zero_neq_one_of_bool @ A) @ (P @ X2))))) @ A6) = ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P))))))))))))).
% 0.33/5.22 thf(fact_4307_sum__zero__power, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A6:(set @ nat),C2:(nat > A)]: ((((((finite_finite2 @ nat) @ A6) & (((member @ nat) @ (zero_zero @ nat)) @ A6)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (C2 @ I5)) @ (((power_power @ A) @ (zero_zero @ A)) @ I5))))) @ A6) = (C2 @ (zero_zero @ nat)))) & ((~ ((((finite_finite2 @ nat) @ A6) & (((member @ nat) @ (zero_zero @ nat)) @ A6)))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (C2 @ I5)) @ (((power_power @ A) @ (zero_zero @ A)) @ I5))))) @ A6) = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_4308_sum__zero__power_H, axiom, (! [A:$tType]: (((field @ A) => (! [A6:(set @ nat),C2:(nat > A),D2:(nat > A)]: ((((((finite_finite2 @ nat) @ A6) & (((member @ nat) @ (zero_zero @ nat)) @ A6)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (C2 @ I5)) @ (((power_power @ A) @ (zero_zero @ A)) @ I5))) @ (D2 @ I5))))) @ A6) = (((divide_divide @ A) @ (C2 @ (zero_zero @ nat))) @ (D2 @ (zero_zero @ nat))))) & ((~ ((((finite_finite2 @ nat) @ A6) & (((member @ nat) @ (zero_zero @ nat)) @ A6)))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (C2 @ I5)) @ (((power_power @ A) @ (zero_zero @ A)) @ I5))) @ (D2 @ I5))))) @ A6) = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_4309_divmod__algorithm__code_I5_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N:num]: (((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit0 @ N)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q3:A,R5:A]: (((((product_Pair @ A) @ A) @ Q3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R5))))) @ (((unique8689654367752047608divmod @ A) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4310_divmod__algorithm__code_I6_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N:num]: (((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit0 @ N)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q3:A,R5:A]: (((((product_Pair @ A) @ A) @ Q3) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R5)) @ (one_one @ A)))))) @ (((unique8689654367752047608divmod @ A) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4311_sum__mono, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [K5:(set @ B),F:(B > A),G:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ K5) => (((ord_less_eq @ A) @ (F @ I4)) @ (G @ I4))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ K5)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ K5))))))))).
% 0.33/5.22 thf(fact_4312_sum__negf, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [F:(B > A),A6:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((uminus_uminus @ A) @ (F @ X2))))) @ A6) = ((uminus_uminus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_4313_sum__distrib__left, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [R:A,F:(B > A),A6:(set @ B)]: (((((times_times @ A) @ R) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [N3:B]: ((((times_times @ A) @ R) @ (F @ N3))))) @ A6)))))))).
% 0.33/5.22 thf(fact_4314_sum__distrib__right, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [F:(B > A),A6:(set @ B),R:A]: (((((times_times @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ R) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [N3:B]: ((((times_times @ A) @ (F @ N3)) @ R)))) @ A6)))))))).
% 0.33/5.22 thf(fact_4315_sum__product, axiom, (! [B:$tType,C:$tType,A:$tType]: (((semiring_0 @ B) => (! [F:(A > B),A6:(set @ A),G:(C > B),B7:(set @ C)]: (((((times_times @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ B7)) = ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I5:A]: (((((groups7311177749621191930dd_sum @ C) @ B) @ (^ [J3:C]: ((((times_times @ B) @ (F @ I5)) @ (G @ J3))))) @ B7)))) @ A6)))))))).
% 0.33/5.22 thf(fact_4316_mod__sum__eq, axiom, (! [B:$tType,A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [F:(B > A),A3:A,A6:(set @ B)]: (((((modulo_modulo @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [I5:B]: ((((modulo_modulo @ A) @ (F @ I5)) @ A3)))) @ A6)) @ A3) = (((modulo_modulo @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ A3)))))))).
% 0.33/5.22 thf(fact_4317_unset__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se2638667681897837118et_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se2638667681897837118et_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4318_bit__integer_Oabs__eq, axiom, (! [X:int]: ((((bit_se5641148757651400278ts_bit @ code_integer) @ (code_integer_of_int @ X)) = ((bit_se5641148757651400278ts_bit @ int) @ X))))).
% 0.33/5.22 thf(fact_4319_or__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((bit_se1065995026697491101ons_or @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se1065995026697491101ons_or @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4320_and__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((bit_se5824344872417868541ns_and @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5824344872417868541ns_and @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4321_euclidean__size__integer_Oabs__eq, axiom, (! [X:int]: ((((euclid6346220572633701492n_size @ code_integer) @ (code_integer_of_int @ X)) = ((euclid6346220572633701492n_size @ int) @ X))))).
% 0.33/5.22 thf(fact_4322_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)))))).
% 0.33/5.22 thf(fact_4323_take__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se2584673776208193580ke_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se2584673776208193580ke_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4324_xor__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((bit_se5824344971392196577ns_xor @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5824344971392196577ns_xor @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4325_times__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((times_times @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((times_times @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4326_push__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se4730199178511100633sh_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se4730199178511100633sh_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4327_drop__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se4197421643247451524op_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se4197421643247451524op_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4328_mask__integer_Oabs__eq, axiom, ((bit_se2239418461657761734s_mask @ code_integer) = (^ [X2:nat]: ((code_integer_of_int @ ((bit_se2239418461657761734s_mask @ int) @ X2)))))).
% 0.33/5.22 thf(fact_4329_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q4:((product_prod @ A) @ B),F:(A > (B > C)),G:(A > (B > C)),P5:((product_prod @ A) @ B)]: (((! [X3:A,Y5:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y5) = Q4) => (((F @ X3) @ Y5) = ((G @ X3) @ Y5))))) => ((P5 = Q4) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P5) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q4))))))).
% 0.33/5.22 thf(fact_4330_set__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se5668285175392031749et_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5668285175392031749et_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4331_flip__bit__integer_Oabs__eq, axiom, (! [Xa3:nat,X:int]: (((((bit_se8732182000553998342ip_bit @ code_integer) @ Xa3) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se8732182000553998342ip_bit @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4332_division__segment__integer_Oabs__eq, axiom, (! [X:int]: ((((euclid7384307370059645450egment @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((euclid7384307370059645450egment @ int) @ X)))))).
% 0.33/5.22 thf(fact_4333_sum_Oneutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),G:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => ((G @ X3) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4334_sum_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),A6:(set @ B)]: (((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6) != (zero_zero @ A)) => (~ ((! [A8:B]: (((((member @ B) @ A8) @ A6) => ((G @ A8) = (zero_zero @ A)))))))))))))).
% 0.33/5.22 thf(fact_4335_sum__nonpos, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ (zero_zero @ A))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4336_sum__nonneg, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A6:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X3))))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_4337_sum__mono__inv, axiom, (! [A:$tType,I7:$tType]: (((ordere8940638589300402666id_add @ A) => (! [F:(I7 > A),I6:(set @ I7),G:(I7 > A),I:I7]: (((((((groups7311177749621191930dd_sum @ I7) @ A) @ F) @ I6) = ((((groups7311177749621191930dd_sum @ I7) @ A) @ G) @ I6)) => ((! [I4:I7]: (((((member @ I7) @ I4) @ I6) => (((ord_less_eq @ A) @ (F @ I4)) @ (G @ I4))))) => ((((member @ I7) @ I) @ I6) => (((finite_finite2 @ I7) @ I6) => ((F @ I) = (G @ I)))))))))))).
% 0.33/5.22 thf(fact_4338_sum__cong__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ nat),F:(nat > A),G:(nat > A)]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ A6))) => ((! [X3:nat]: (((((member @ nat) @ (suc @ X3)) @ A6) => ((F @ (suc @ X3)) = (G @ (suc @ X3)))))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ A6) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ A6)))))))))).
% 0.33/5.22 thf(fact_4339_zero__integer__def, axiom, ((zero_zero @ code_integer) = (code_integer_of_int @ (zero_zero @ int)))).
% 0.33/5.22 thf(fact_4340_less__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((ord_less @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (((ord_less @ int) @ Xa3) @ X))))).
% 0.33/5.22 thf(fact_4341_plus__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((plus_plus @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((plus_plus @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4342_uminus__integer_Oabs__eq, axiom, (! [X:int]: ((((uminus_uminus @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((uminus_uminus @ int) @ X)))))).
% 0.33/5.22 thf(fact_4343_sum_Ointer__filter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),G:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ A6) & (P @ X2)))))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (G @ X2)) @ (zero_zero @ A))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4344_one__integer__def, axiom, ((one_one @ code_integer) = (code_integer_of_int @ (one_one @ int)))).
% 0.33/5.22 thf(fact_4345_less__eq__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((ord_less_eq @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (((ord_less_eq @ int) @ Xa3) @ X))))).
% 0.33/5.22 thf(fact_4346_minus__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((minus_minus @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((minus_minus @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4347_abs__integer_Oabs__eq, axiom, (! [X:int]: ((((abs_abs @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((abs_abs @ int) @ X)))))).
% 0.33/5.22 thf(fact_4348_divide__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((divide_divide @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((divide_divide @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4349_nat__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_nat_of_integer @ (code_integer_of_int @ X)) = (nat2 @ X))))).
% 0.33/5.22 thf(fact_4350_sum_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4351_sum_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4352_modulo__integer_Oabs__eq, axiom, (! [Xa3:int,X:int]: (((((modulo_modulo @ code_integer) @ (code_integer_of_int @ Xa3)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((modulo_modulo @ int) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4353_sgn__integer_Oabs__eq, axiom, (! [X:int]: ((((sgn_sgn @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((sgn_sgn @ int) @ X)))))).
% 0.33/5.22 thf(fact_4354_effect__guardE, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H2) @ H) @ R) @ N) => (~ (((H = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2))) => (~ ((P @ H2)))))))))))).
% 0.33/5.22 thf(fact_4355_effect__guardI, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),N:nat,R:A]: (((P @ H2) => ((H = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2))) => ((((((heap_Time_effect @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H2) @ H) @ R) @ N)))))))).
% 0.33/5.22 thf(fact_4356_sum__le__included, axiom, (! [B:$tType,A:$tType,C:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),T2:(set @ C),G:(C > A),I:(C > B),F:(B > A)]: ((((finite_finite2 @ B) @ S) => (((finite_finite2 @ C) @ T2) => ((! [X3:C]: (((((member @ C) @ X3) @ T2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (G @ X3))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S) => (? [Xa:C]: (((((member @ C) @ Xa) @ T2) & (((I @ Xa) = X3) & (((ord_less_eq @ A) @ (F @ X3)) @ (G @ Xa))))))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ S)) @ ((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ T2)))))))))))).
% 0.33/5.22 thf(fact_4357_sum__nonneg__eq__0__iff, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X3))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6) = (zero_zero @ A)) = (! [X2:B]: (((((member @ B) @ X2) @ A6) => ((F @ X2) = (zero_zero @ A)))))))))))))).
% 0.33/5.22 thf(fact_4358_sum__strict__mono__ex1, axiom, (! [A:$tType,I7:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A6:(set @ I7),F:(I7 > A),G:(I7 > A)]: ((((finite_finite2 @ I7) @ A6) => ((! [X3:I7]: (((((member @ I7) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3))))) => ((? [X4:I7]: (((((member @ I7) @ X4) @ A6) & (((ord_less @ A) @ (F @ X4)) @ (G @ X4))))) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ I7) @ A) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ I7) @ A) @ G) @ A6))))))))))).
% 0.33/5.22 thf(fact_4359_sum_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [R2:(A > (A > $o)),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((R2 @ (zero_zero @ A)) @ (zero_zero @ A)) => ((! [X1:A,Y1:A,X23:A,Y22:A]: (((((R2 @ X1) @ X23) & ((R2 @ Y1) @ Y22)) => ((R2 @ (((plus_plus @ A) @ X1) @ Y1)) @ (((plus_plus @ A) @ X23) @ Y22))))) => (((finite_finite2 @ B) @ S3) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((R2 @ (H2 @ X3)) @ (G @ X3))))) => ((R2 @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3)))))))))))).
% 0.33/5.22 thf(fact_4360_sum__strict__mono, axiom, (! [A:$tType,B:$tType]: (((strict7427464778891057005id_add @ A) => (! [A6:(set @ B),F:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => ((A6 != (bot_bot @ (set @ B))) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less @ A) @ (F @ X3)) @ (G @ X3))))) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6))))))))))).
% 0.33/5.22 thf(fact_4361_sum_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_add @ A)) => (! [A3:B,C2:B,B2:B,D2:B,G:(B > A),H2:(B > A)]: (((A3 = C2) => ((B2 = D2) => ((! [X3:B]: (((((ord_less_eq @ B) @ C2) @ X3) => ((((ord_less @ B) @ X3) @ D2) => ((G @ X3) = (H2 @ X3)))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D2)))))))))))).
% 0.33/5.22 thf(fact_4362_sum_Oreindex__bij__witness__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S5:(set @ B),T5:(set @ C),S3:(set @ B),I:(C > B),J:(B > C),T6:(set @ C),G:(B > A),H2:(C > A)]: ((((finite_finite2 @ B) @ S5) => (((finite_finite2 @ C) @ T5) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) => ((I @ (J @ A8)) = A8)))) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) => (((member @ C) @ (J @ A8)) @ (((minus_minus @ (set @ C)) @ T6) @ T5))))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((J @ (I @ B5)) = B5)))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => (((member @ B) @ (I @ B5)) @ (((minus_minus @ (set @ B)) @ S3) @ S5))))) => ((! [A8:B]: (((((member @ B) @ A8) @ S5) => ((G @ A8) = (zero_zero @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T5) => ((H2 @ B5) = (zero_zero @ A))))) => ((! [A8:B]: (((((member @ B) @ A8) @ S3) => ((H2 @ (J @ A8)) = (G @ A8))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ H2) @ T6))))))))))))))))).
% 0.33/5.22 thf(fact_4363_sum_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,P5:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P5))))))))))).
% 0.33/5.22 thf(fact_4364_sum__diff__nat__ivl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N:nat,P5:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P5))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))))))))))).
% 0.33/5.22 thf(fact_4365_sum__nonneg__0, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),F:(B > A),I:B]: ((((finite_finite2 @ B) @ S) => ((! [I4:B]: (((((member @ B) @ I4) @ S) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ S) = (zero_zero @ A)) => ((((member @ B) @ I) @ S) => ((F @ I) = (zero_zero @ A)))))))))))).
% 0.33/5.22 thf(fact_4366_sum__nonneg__leq__bound, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),F:(B > A),B7:A,I:B]: ((((finite_finite2 @ B) @ S) => ((! [I4:B]: (((((member @ B) @ I4) @ S) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ S) = B7) => ((((member @ B) @ I) @ S) => (((ord_less_eq @ A) @ (F @ I)) @ B7))))))))))).
% 0.33/5.22 thf(fact_4367_sum_Ointer__restrict, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),G:(B > A),B7:(set @ B)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ B7)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (((member @ B) @ X2) @ B7)) @ (G @ X2)) @ (zero_zero @ A))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4368_sum_Osetdiff__irrelevant, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A6) @ ((collect @ B) @ (^ [X2:B]: (((G @ X2) = (zero_zero @ A))))))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6))))))))).
% 0.33/5.22 thf(fact_4369_sum__power__add, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,M:nat,I6:(set @ nat)]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((power_power @ A) @ X) @ (((plus_plus @ nat) @ M) @ I5))))) @ I6) = (((times_times @ A) @ (((power_power @ A) @ X) @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ I6))))))))).
% 0.33/5.22 thf(fact_4370_sum_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I5)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4371_sum__choose__upper, axiom, (! [M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ K3) @ M)))) @ ((set_ord_atMost @ nat) @ N)) = ((binomial @ (suc @ N)) @ (suc @ M)))))).
% 0.33/5.22 thf(fact_4372_sum__pos2, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [I6:(set @ B),I:B,F:(B > A)]: ((((finite_finite2 @ B) @ I6) => ((((member @ B) @ I) @ I6) => ((((ord_less @ A) @ (zero_zero @ A)) @ (F @ I)) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I4))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ I6)))))))))))).
% 0.33/5.22 thf(fact_4373_sum__pos, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [I6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ I6) => ((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less @ A) @ (zero_zero @ A)) @ (F @ I4))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ I6))))))))))).
% 0.33/5.22 thf(fact_4374_sum_Omono__neutral__cong__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3)))))))))))).
% 0.33/5.22 thf(fact_4375_sum_Omono__neutral__cong__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T6:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ X3) = (zero_zero @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ T6)))))))))))).
% 0.33/5.22 thf(fact_4376_sum_Omono__neutral__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3))))))))))).
% 0.33/5.22 thf(fact_4377_sum_Omono__neutral__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T6:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T6))))))))))).
% 0.33/5.22 thf(fact_4378_sum_Osame__carrierI, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [C6:(set @ B),A6:(set @ B),B7:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C6) => ((((ord_less_eq @ (set @ B)) @ A6) @ C6) => ((((ord_less_eq @ (set @ B)) @ B7) @ C6) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ C6) @ A6)) => ((G @ A8) = (zero_zero @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C6) @ B7)) => ((H2 @ B5) = (zero_zero @ A))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ C6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ C6)) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ B7)))))))))))))).
% 0.33/5.22 thf(fact_4379_sum_Osame__carrier, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [C6:(set @ B),A6:(set @ B),B7:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C6) => ((((ord_less_eq @ (set @ B)) @ A6) @ C6) => ((((ord_less_eq @ (set @ B)) @ B7) @ C6) => ((! [A8:B]: (((((member @ B) @ A8) @ (((minus_minus @ (set @ B)) @ C6) @ A6)) => ((G @ A8) = (zero_zero @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C6) @ B7)) => ((H2 @ B5) = (zero_zero @ A))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ B7)) = (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ C6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ C6)))))))))))))).
% 0.33/5.22 thf(fact_4380_sum__bounded__below, axiom, (! [A:$tType,B:$tType]: ((((ordere6911136660526730532id_add @ A) & (semiring_1 @ A)) => (! [A6:(set @ B),K5:A,F:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ K5) @ (F @ I4))))) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A6))) @ K5)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_4381_sum__bounded__above, axiom, (! [B:$tType,A:$tType]: ((((ordere6911136660526730532id_add @ A) & (semiring_1 @ A)) => (! [A6:(set @ B),F:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ (F @ I4)) @ K5)))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A6))) @ K5))))))))).
% 0.33/5.22 thf(fact_4382_sum_Omono__neutral__cong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T6:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => (((finite_finite2 @ B) @ S3) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ I4) = (zero_zero @ A))))) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ S3) @ T6)) => ((G @ I4) = (zero_zero @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ (((inf_inf @ (set @ B)) @ S3) @ T6)) => ((G @ X3) = (H2 @ X3))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ T6))))))))))))).
% 0.33/5.22 thf(fact_4383_sum__shift__lb__Suc0__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),K:nat]: ((((F @ (zero_zero @ nat)) = (zero_zero @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ K)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ K)))))))))).
% 0.33/5.22 thf(fact_4384_sum__shift__lb__Suc0__0__upt, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4385_sum_OatLeast0__atMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ (suc @ N)))))))))).
% 0.33/5.22 thf(fact_4386_sum_OatLeast0__lessThan__Suc, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_4387_sum_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ N))))))))))).
% 0.33/5.22 thf(fact_4388_sum_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N))))))))))).
% 0.33/5.22 thf(fact_4389_sum_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (suc @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.22 thf(fact_4390_sum_OatLeastLessThan__Suc, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4391_sum_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((plus_plus @ A) @ (G @ N)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))))).
% 0.33/5.22 thf(fact_4392_sum_OIf__cases, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),P:(B > $o),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (H2 @ X2)) @ (G @ X2))))) @ A6) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P)))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))))).
% 0.33/5.22 thf(fact_4393_sum_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N))) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.22 thf(fact_4394_sum__Suc__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ (F @ (suc @ I5))) @ (F @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((minus_minus @ A) @ (F @ (suc @ N))) @ (F @ M)))))))))).
% 0.33/5.22 thf(fact_4395_sum__Suc__diff_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ (F @ (suc @ I5))) @ (F @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = (((minus_minus @ A) @ (F @ N)) @ (F @ M)))))))))).
% 0.33/5.22 thf(fact_4396_sum_OatMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_atMost @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4397_sum__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),I:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((minus_minus @ A) @ (F @ I5)) @ (F @ (suc @ I5)))))) @ ((set_ord_atMost @ nat) @ I)) = (((minus_minus @ A) @ (F @ (zero_zero @ nat))) @ (F @ (suc @ I)))))))))).
% 0.33/5.22 thf(fact_4398_sum_Ohead, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M) @ N))))))))))).
% 0.33/5.22 thf(fact_4399_sum_OlessThan__Suc__shift, axiom, (! [A:$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) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4400_sum__lessThan__telescope_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N3:nat]: ((((minus_minus @ A) @ (F @ N3)) @ (F @ (suc @ N3)))))) @ ((set_ord_lessThan @ nat) @ M)) = (((minus_minus @ A) @ (F @ (zero_zero @ nat))) @ (F @ M))))))))).
% 0.33/5.22 thf(fact_4401_sum__lessThan__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N3:nat]: ((((minus_minus @ A) @ (F @ (suc @ N3))) @ (F @ N3))))) @ ((set_ord_lessThan @ nat) @ M)) = (((minus_minus @ A) @ (F @ M)) @ (F @ (zero_zero @ nat)))))))))).
% 0.33/5.22 thf(fact_4402_sum_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ (suc @ I5)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M))))))))).
% 0.33/5.22 thf(fact_4403_sum_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A3 @ I5)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 0.33/5.22 thf(fact_4404_sum_OatLeast1__atMost__eq, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((G @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4405_sum__bounds__lt__plus1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),Mm:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((F @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ Mm)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ Mm))))))))).
% 0.33/5.22 thf(fact_4406_sum_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [M2:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))))).
% 0.33/5.22 thf(fact_4407_sum__choose__lower, axiom, (! [R:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ (((plus_plus @ nat) @ R) @ K3)) @ K3)))) @ ((set_ord_atMost @ nat) @ N)) = ((binomial @ (suc @ (((plus_plus @ nat) @ R) @ N))) @ N))))).
% 0.33/5.22 thf(fact_4408_sum_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A3 @ I5)) @ ((set_ord_lessThan @ nat) @ I5))))) @ ((set_ord_atMost @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4409_choose__rising__sum_I1_J, axiom, (! [N:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J3:nat]: (((binomial @ (((plus_plus @ nat) @ N) @ J3)) @ N)))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N) @ M)) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))).
% 0.33/5.22 thf(fact_4410_choose__rising__sum_I2_J, axiom, (! [N:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J3:nat]: (((binomial @ (((plus_plus @ nat) @ N) @ J3)) @ N)))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N) @ M)) @ (one_one @ nat))) @ M))))).
% 0.33/5.22 thf(fact_4411_sum__atLeastAtMost__code, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),A3:nat,B2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2)) = (((((set_fo6178422350223883121st_nat @ A) @ (^ [A5:nat]: (((plus_plus @ A) @ (F @ A5))))) @ A3) @ B2) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4412_card__sum__le__nat__sum, axiom, (! [S3:(set @ nat)]: ((((ord_less_eq @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ nat) @ S3)))) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ S3))))).
% 0.33/5.22 thf(fact_4413_sum__mono2, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [B7:(set @ B),A6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ B7) => ((((ord_less_eq @ (set @ B)) @ A6) @ B7) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ B7) @ A6)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ B5))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ B7))))))))))).
% 0.33/5.22 thf(fact_4414_sum_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),B7:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ B) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ (((inf_inf @ (set @ B)) @ A6) @ B7)) => ((G @ X3) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A6) @ B7)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A6)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B7)))))))))))).
% 0.33/5.22 thf(fact_4415_sum_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))))).
% 0.33/5.22 thf(fact_4416_sum_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A),P5:nat]: (((((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ N) @ P5))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))))).
% 0.33/5.22 thf(fact_4417_sum__div__partition, axiom, (! [B:$tType,A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A6:(set @ B),F:(B > A),B2:A]: ((((finite_finite2 @ B) @ A6) => ((((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ B2) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [A5:B]: ((((divide_divide @ A) @ (F @ A5)) @ B2)))) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ (^ [A5:B]: ((((dvd_dvd @ A) @ B2) @ (F @ A5)))))))) @ (((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ (^ [A5:B]: ((~ ((((dvd_dvd @ A) @ B2) @ (F @ A5)))))))))) @ B2)))))))))).
% 0.33/5.22 thf(fact_4418_one__diff__power__eq, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4419_power__diff__1__eq, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ N)) @ (one_one @ A)) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ (one_one @ A))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4420_geometric__sum, axiom, (! [A:$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))))))))))).
% 0.33/5.22 thf(fact_4421_set__encode__def, axiom, (nat_set_encode = (((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))).
% 0.33/5.22 thf(fact_4422_sum_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M))))))))).
% 0.33/5.22 thf(fact_4423_sum_OatMost__shift, axiom, (! [A:$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) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4424_gbinomial__parallel__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K3))) @ K3)))) @ ((set_ord_atMost @ nat) @ N)) = (((gbinomial @ A) @ (((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ N))) @ (one_one @ A))) @ N)))))))).
% 0.33/5.22 thf(fact_4425_sum__list__sum__nth, axiom, (! [B:$tType]: (((comm_monoid_add @ B) => ((groups8242544230860333062m_list @ B) = (^ [Xs3:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ B) @ ((nth @ B) @ Xs3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs3)))))))))).
% 0.33/5.22 thf(fact_4426_sum__choose__diagonal, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ (((minus_minus @ nat) @ N) @ K3)) @ (((minus_minus @ nat) @ M) @ K3))))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (suc @ N)) @ M)))))).
% 0.33/5.22 thf(fact_4427_vandermonde, axiom, (! [M:nat,N:nat,R:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((times_times @ nat) @ ((binomial @ M) @ K3)) @ ((binomial @ N) @ (((minus_minus @ nat) @ R) @ K3)))))) @ ((set_ord_atMost @ nat) @ R)) = ((binomial @ (((plus_plus @ nat) @ M) @ N)) @ R))))).
% 0.33/5.22 thf(fact_4428_sum__strict__mono2, axiom, (! [B:$tType,A:$tType]: (((ordere8940638589300402666id_add @ B) => (! [B7:(set @ A),A6:(set @ A),B2:A,F:(A > B)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((((member @ A) @ B2) @ (((minus_minus @ (set @ A)) @ B7) @ A6)) => ((((ord_less @ B) @ (zero_zero @ B)) @ (F @ B2)) => ((! [X3:A]: (((((member @ A) @ X3) @ B7) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F @ X3))))) => (((ord_less @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ B7))))))))))))).
% 0.33/5.22 thf(fact_4429_member__le__sum, axiom, (! [B:$tType,C:$tType]: ((((ordere6911136660526730532id_add @ B) & (semiring_1 @ B)) => (! [I:C,A6:(set @ C),F:(C > B)]: (((((member @ C) @ I) @ A6) => ((! [X3:C]: (((((member @ C) @ X3) @ (((minus_minus @ (set @ C)) @ A6) @ (((insert @ C) @ I) @ (bot_bot @ (set @ C))))) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F @ X3))))) => (((finite_finite2 @ C) @ A6) => (((ord_less_eq @ B) @ (F @ I)) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ F) @ A6))))))))))).
% 0.33/5.22 thf(fact_4430_sum__bounded__above__divide, axiom, (! [B:$tType,A:$tType]: (((linordered_field @ A) => (! [A6:(set @ B),F:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ (F @ I4)) @ (((divide_divide @ A) @ K5) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A6))))))) => (((finite_finite2 @ B) @ A6) => ((A6 != (bot_bot @ (set @ B))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ K5)))))))))).
% 0.33/5.22 thf(fact_4431_sum__bounded__above__strict, axiom, (! [B:$tType,A:$tType]: ((((ordere8940638589300402666id_add @ A) & (semiring_1 @ A)) => (! [A6:(set @ B),F:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less @ A) @ (F @ I4)) @ K5)))) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ B) @ A6)) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A6)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A6))) @ K5)))))))))).
% 0.33/5.22 thf(fact_4432_convex__sum__bound__le, axiom, (! [A:$tType,B:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),X:(A > B),A3:(A > B),B2:B,Delta:B]: (((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (X @ I4))))) => ((((((groups7311177749621191930dd_sum @ A) @ B) @ X) @ I6) = (one_one @ B)) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ (A3 @ I4)) @ B2))) @ Delta)))) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I5:A]: ((((times_times @ B) @ (A3 @ I5)) @ (X @ I5))))) @ I6)) @ B2))) @ Delta)))))))))).
% 0.33/5.22 thf(fact_4433_sum__fun__comp, axiom, (! [A:$tType,C:$tType,B:$tType]: (((semiring_1 @ C) => (! [S3:(set @ A),R2:(set @ B),G:(A > B),F:(B > C)]: ((((finite_finite2 @ A) @ S3) => (((finite_finite2 @ B) @ R2) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ G) @ S3)) @ R2) => (((((groups7311177749621191930dd_sum @ A) @ C) @ (^ [X2:A]: ((F @ (G @ X2))))) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ C) @ (^ [Y6:B]: ((((times_times @ C) @ ((semiring_1_of_nat @ C) @ ((finite_card @ A) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ S3) & ((G @ X2) = Y6)))))))) @ (F @ Y6))))) @ R2))))))))))).
% 0.33/5.22 thf(fact_4434_sum__gp__basic, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4435_sum__natinterval__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N:nat,F:(nat > A)]: ((((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F @ K3)) @ (F @ (((plus_plus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((minus_minus @ A) @ (F @ M)) @ (F @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))) & ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F @ K3)) @ (F @ (((plus_plus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_4436_sum__gp__strict, axiom, (! [A:$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))))))))))).
% 0.33/5.22 thf(fact_4437_diff__power__eq__sum, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat,Y3:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ (suc @ N))) @ (((power_power @ A) @ Y3) @ (suc @ N))) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ Y3)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: ((((times_times @ A) @ (((power_power @ A) @ X) @ P6)) @ (((power_power @ A) @ Y3) @ (((minus_minus @ nat) @ N) @ P6)))))) @ ((set_ord_lessThan @ nat) @ (suc @ N))))))))))).
% 0.33/5.22 thf(fact_4438_power__diff__sumr2, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat,Y3:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ Y3) @ N)) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ Y3)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (((power_power @ A) @ Y3) @ (((minus_minus @ nat) @ N) @ (suc @ I5)))) @ (((power_power @ A) @ X) @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4439_sum__telescope_H_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F @ K3)) @ (F @ (((minus_minus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N)) = (((minus_minus @ A) @ (F @ N)) @ (F @ M)))))))))).
% 0.33/5.22 thf(fact_4440_sum__power__shift, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M:nat,N:nat,X:A]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((times_times @ A) @ (((power_power @ A) @ X) @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ N) @ M)))))))))))).
% 0.33/5.22 thf(fact_4441_take__bit__sum, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => ((bit_se2584673776208193580ke_bit @ A) = (^ [N3:nat,A5:A]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((bit_se4730199178511100633sh_bit @ A) @ K3) @ ((zero_neq_one_of_bool @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ A5) @ K3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3))))))))).
% 0.33/5.22 thf(fact_4442_binomial, axiom, (! [A3:nat,B2:nat,N:nat]: (((((power_power @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((times_times @ nat) @ (((times_times @ nat) @ ((semiring_1_of_nat @ nat) @ ((binomial @ N) @ K3))) @ (((power_power @ nat) @ A3) @ K3))) @ (((power_power @ nat) @ B2) @ (((minus_minus @ nat) @ N) @ K3)))))) @ ((set_ord_atMost @ nat) @ N)))))).
% 0.33/5.22 thf(fact_4443_mask__eq__sum__exp, axiom, (! [A:$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) @ (^ [Q3:nat]: ((((ord_less @ nat) @ Q3) @ N)))))))))))).
% 0.33/5.22 thf(fact_4444_sum__gp__multiplied, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M:nat,N:nat,X:A]: (((((ord_less_eq @ nat) @ M) @ N) => ((((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) = (((minus_minus @ A) @ (((power_power @ A) @ X) @ M)) @ (((power_power @ A) @ X) @ (suc @ N))))))))))).
% 0.33/5.22 thf(fact_4445_sum_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4446_one__diff__power__eq_H, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ N)) = (((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((power_power @ A) @ X) @ (((minus_minus @ nat) @ N) @ (suc @ I5)))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 0.33/5.22 thf(fact_4447_sum_Oin__pairs__0, axiom, (! [A:$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) @ (^ [I5:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4448_gbinomial__sum__lower__neg, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K3)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K3))))) @ ((set_ord_atMost @ nat) @ M)) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ M))))))))).
% 0.33/5.22 thf(fact_4449_binomial__ring, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,B2:A,N:nat]: (((((power_power @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K3))) @ (((power_power @ A) @ A3) @ K3))) @ (((power_power @ A) @ B2) @ (((minus_minus @ nat) @ N) @ K3)))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4450_pochhammer__binomial__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [A3:A,B2:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K3))) @ (((comm_s3205402744901411588hammer @ A) @ A3) @ K3))) @ (((comm_s3205402744901411588hammer @ A) @ B2) @ (((minus_minus @ nat) @ N) @ K3)))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4451_horner__sum__eq__sum, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F2:(B > A),A5:A,Xs3:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N3:nat]: ((((times_times @ A) @ (F2 @ (((nth @ B) @ Xs3) @ N3))) @ (((power_power @ A) @ A5) @ N3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs3)))))))))).
% 0.33/5.22 thf(fact_4452_choose__square__sum, axiom, (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((power_power @ nat) @ ((binomial @ N) @ K3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((set_ord_atMost @ nat) @ N)) = ((binomial @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ N))))).
% 0.33/5.22 thf(fact_4453_sum_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [P5:nat,K:nat,G:(nat > A),H2:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ ((((if @ A) @ (J3 = K)) @ (zero_zero @ A)) @ (H2 @ (((minus_minus @ nat) @ J3) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ (H2 @ J3))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))))).
% 0.33/5.22 thf(fact_4454_mask__eq__sum__exp__nat, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (suc @ (zero_zero @ nat))) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ ((collect @ nat) @ (^ [Q3:nat]: ((((ord_less @ nat) @ Q3) @ N))))))))).
% 0.33/5.22 thf(fact_4455_gauss__sum__nat, axiom, (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ nat) @ (((times_times @ nat) @ N) @ (suc @ N))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.22 thf(fact_4456_gbinomial__sum__up__index, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: ((((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ J3)) @ K)))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ nat) @ K) @ (one_one @ nat)))))))))).
% 0.33/5.22 thf(fact_4457_gbinomial__partial__sum__poly, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat,A3:A,X:A,Y3:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ A3)) @ K3)) @ (((power_power @ A) @ X) @ K3))) @ (((power_power @ A) @ Y3) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ ((uminus_uminus @ A) @ A3)) @ K3)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ K3))) @ (((power_power @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M))))))))).
% 0.33/5.22 thf(fact_4458_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)))))).
% 0.33/5.22 thf(fact_4459_Sum__Ico__nat, axiom, (! [M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = (((divide_divide @ nat) @ (((minus_minus @ nat) @ (((times_times @ nat) @ N) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))) @ (((times_times @ nat) @ M) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.22 thf(fact_4460_card__lists__length__le, axiom, (! [A:$tType,A6:(set @ A),N:nat]: ((((finite_finite2 @ A) @ A6) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A6) & (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ N)))))) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((finite_card @ A) @ A6))) @ ((set_ord_atMost @ nat) @ N))))))).
% 0.33/5.22 thf(fact_4461_double__gauss__sum, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4462_double__arith__series, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,D2:A,N:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ D2))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) = (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ D2)))))))))).
% 0.33/5.22 thf(fact_4463_sum__gp0, axiom, (! [A:$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))))))))))).
% 0.33/5.22 thf(fact_4464_choose__alternating__linear__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((N != (one_one @ nat)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I5)) @ ((semiring_1_of_nat @ A) @ I5))) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I5)))))) @ ((set_ord_atMost @ nat) @ N)) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4465_gbinomial__partial__sum__poly__xpos, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat,A3:A,X:A,Y3:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ A3)) @ K3)) @ (((power_power @ A) @ X) @ K3))) @ (((power_power @ A) @ Y3) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ K3)) @ A3)) @ (one_one @ A))) @ K3)) @ (((power_power @ A) @ X) @ K3))) @ (((power_power @ A) @ (((plus_plus @ A) @ X) @ Y3)) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M))))))))).
% 0.33/5.22 thf(fact_4466_gbinomial__sum__nat__pow2, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((divide_divide @ A) @ (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M) @ K3))) @ K3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ K3))))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)))))))).
% 0.33/5.22 thf(fact_4467_arith__series__nat, axiom, (! [A3:nat,D2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I5:nat]: ((((plus_plus @ nat) @ A3) @ (((times_times @ nat) @ I5) @ D2))))) @ (((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) @ D2)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.22 thf(fact_4468_Sum__Icc__nat, axiom, (! [M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X2:nat]: (X2))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((divide_divide @ nat) @ (((minus_minus @ nat) @ (((times_times @ nat) @ N) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))) @ (((times_times @ nat) @ M) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.22 thf(fact_4469_binomial__r__part__sum, axiom, (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (binomial @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (one_one @ nat)))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)))))).
% 0.33/5.22 thf(fact_4470_choose__linear__sum, axiom, (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I5:nat]: ((((times_times @ nat) @ I5) @ ((binomial @ N) @ I5))))) @ ((set_ord_atMost @ nat) @ N)) = (((times_times @ nat) @ N) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 0.33/5.22 thf(fact_4471_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)))))))))).
% 0.33/5.22 thf(fact_4472_arith__series, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,D2:A,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ D2))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ D2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 0.33/5.22 thf(fact_4473_gauss__sum, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4474_sum__gp__offset, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X:A,M:nat,N:nat]: ((((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ M) @ N))) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ M) @ N))) = (((divide_divide @ A) @ (((times_times @ A) @ (((power_power @ A) @ X) @ M)) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ (suc @ N))))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))).
% 0.33/5.22 thf(fact_4475_choose__alternating__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: ((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I5)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I5)))))) @ ((set_ord_atMost @ nat) @ N)) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4476_divmod__step__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique1321980374590559556d_step @ A) = (^ [L3:num]: (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q3:A,R5:A]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L3)) @ R5)) @ ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q3)) @ (one_one @ A))) @ (((minus_minus @ A) @ R5) @ ((numeral_numeral @ A) @ L3)))) @ ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q3)) @ R5)))))))))))).
% 0.33/5.22 thf(fact_4477_gauss__sum__from__Suc__0, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4478_gchoose__row__sum__weighted, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [R:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ R) @ K3)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ R) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ A) @ K3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ M)) = (((times_times @ A) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ (suc @ M))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((gbinomial @ A) @ R) @ (suc @ M)))))))))).
% 0.33/5.22 thf(fact_4479_sum__gp, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [N:nat,M:nat,X:A]: ((((((ord_less @ nat) @ N) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ M)))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ X) @ M)) @ (((power_power @ A) @ X) @ (suc @ N)))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))))).
% 0.33/5.22 thf(fact_4480_gbinomial__r__part__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ ((gbinomial @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ M))) @ (one_one @ A)))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))))))))).
% 0.33/5.22 thf(fact_4481_gbinomial__partial__row__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K3)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ A) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((gbinomial @ A) @ A3) @ (((plus_plus @ nat) @ M) @ (one_one @ nat))))))))))).
% 0.33/5.22 thf(fact_4482_choose__odd__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((((if @ A) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5)))) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I5))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ N))) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))))))))).
% 0.33/5.22 thf(fact_4483_map__entry__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_map_entry @ A) = (^ [I5:nat,F2:(A > A),A5:(array @ A)]: ((((heap_Time_guard @ (array @ A)) @ (^ [H4:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I5) @ (((array_length @ A) @ H4) @ A5))))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A5) @ I5) @ (F2 @ (((nth @ A) @ (((array_get @ A) @ H4) @ A5)) @ I5))) @ H4)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))).
% 0.33/5.22 thf(fact_4484_Array__Time_Onth__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_nth @ A) = (^ [A5:(array @ A),I5:nat]: ((((heap_Time_guard @ A) @ (^ [H4:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I5) @ (((array_length @ A) @ H4) @ A5))))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H4) @ A5)) @ I5)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (one_one @ nat))))))))))))).
% 0.33/5.22 thf(fact_4485_upd__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_upd @ A) = (^ [I5:nat,X2:A,A5:(array @ A)]: ((((heap_Time_guard @ (array @ A)) @ (^ [H4:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I5) @ (((array_length @ A) @ H4) @ A5))))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A5) @ I5) @ X2) @ H4)) @ (one_one @ nat))))))))))))).
% 0.33/5.22 thf(fact_4486_nat__of__integer__code, axiom, (code_nat_of_integer = (^ [K3:code_integer]: (((((if @ nat) @ (((ord_less_eq @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ (zero_zero @ nat)) @ (((((product_case_prod @ code_integer) @ code_integer) @ nat) @ (^ [L3:code_integer,J3:code_integer]: (((((if @ nat) @ (J3 = (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 @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 0.33/5.22 thf(fact_4487_int__sum, axiom, (! [B:$tType,F:(B > nat),A6:(set @ B)]: ((((semiring_1_of_nat @ int) @ ((((groups7311177749621191930dd_sum @ B) @ nat) @ F) @ A6)) = ((((groups7311177749621191930dd_sum @ B) @ int) @ (^ [X2:B]: (((semiring_1_of_nat @ int) @ (F @ X2))))) @ A6))))).
% 0.33/5.22 thf(fact_4488_sum_Otriangle__reindex__eq, axiom, (! [A:$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) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4489_sum_Otriangle__reindex, axiom, (! [A:$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) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4490_sum__subtractf__nat, axiom, (! [A:$tType,A6:(set @ A),G:(A > nat),F:(A > nat)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ nat) @ (G @ X3)) @ (F @ X3))))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X2:A]: ((((minus_minus @ nat) @ (F @ X2)) @ (G @ X2))))) @ A6) = (((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ G) @ A6))))))).
% 0.33/5.22 thf(fact_4491_card__eq__sum, axiom, (! [A:$tType]: (((finite_card @ A) = (((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X2:A]: ((one_one @ nat)))))))).
% 0.33/5.22 thf(fact_4492_prod_Otriangle__reindex__eq, axiom, (! [A:$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) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4493_sum__eq__Suc0__iff, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => ((((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6) = (suc @ (zero_zero @ nat))) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((F @ X2) = (suc @ (zero_zero @ nat))) & (! [Y6:A]: (((((member @ A) @ Y6) @ A6) => ((X2 != Y6) => ((F @ Y6) = (zero_zero @ nat))))))))))))))).
% 0.33/5.22 thf(fact_4494_sum__SucD, axiom, (! [A:$tType,F:(A > nat),A6:(set @ A),N:nat]: (((((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6) = (suc @ N)) => (? [X3:A]: (((((member @ A) @ X3) @ A6) & (((ord_less @ nat) @ (zero_zero @ nat)) @ (F @ X3))))))))).
% 0.33/5.22 thf(fact_4495_sum__eq__1__iff, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => ((((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6) = (one_one @ nat)) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((F @ X2) = (one_one @ nat)) & (! [Y6:A]: (((((member @ A) @ Y6) @ A6) => ((X2 != Y6) => ((F @ Y6) = (zero_zero @ nat))))))))))))))).
% 0.33/5.22 thf(fact_4496_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) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_4497_sum__Suc, axiom, (! [A:$tType,F:(A > nat),A6:(set @ A)]: ((((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X2:A]: ((suc @ (F @ X2))))) @ A6) = (((plus_plus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6)) @ ((finite_card @ A) @ A6)))))).
% 0.33/5.22 thf(fact_4498_sum__multicount, axiom, (! [A:$tType,B:$tType,S3:(set @ A),T6:(set @ B),R2:(A > (B > $o)),K:nat]: ((((finite_finite2 @ A) @ S3) => (((finite_finite2 @ B) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ T6) => (((finite_card @ A) @ ((collect @ A) @ (^ [I5:A]: (((((member @ A) @ I5) @ S3) & ((R2 @ I5) @ X3)))))) = K)))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I5:A]: (((finite_card @ B) @ ((collect @ B) @ (^ [J3:B]: (((((member @ B) @ J3) @ T6) & ((R2 @ I5) @ J3))))))))) @ S3) = (((times_times @ nat) @ K) @ ((finite_card @ B) @ T6))))))))).
% 0.33/5.22 thf(fact_4499_sum__diff__distrib, axiom, (! [A:$tType]: (((ord @ A) => (! [Q:(A > nat),P:(A > nat),N:A]: (((! [X3:A]: ((((ord_less_eq @ nat) @ (Q @ X3)) @ (P @ X3)))) => ((((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) @ (^ [X2:A]: ((((minus_minus @ nat) @ (P @ X2)) @ (Q @ X2))))) @ ((set_ord_lessThan @ A) @ N)))))))))).
% 0.33/5.22 thf(fact_4500_prod__encode__def, axiom, (nat_prod_encode = ((((product_case_prod @ nat) @ nat) @ nat) @ (^ [M2:nat,N3:nat]: ((((plus_plus @ nat) @ (nat_triangle @ (((plus_plus @ nat) @ M2) @ N3))) @ M2)))))).
% 0.33/5.22 thf(fact_4501_rat__less__code, axiom, ((ord_less @ rat) = (^ [P6:rat,Q3:rat]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [B3:int,D5:int]: ((((ord_less @ int) @ (((times_times @ int) @ A5) @ D5)) @ (((times_times @ int) @ C4) @ B3))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P6)))))).
% 0.33/5.22 thf(fact_4502_rat__less__eq__code, axiom, ((ord_less_eq @ rat) = (^ [P6:rat,Q3:rat]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [B3:int,D5:int]: ((((ord_less_eq @ int) @ (((times_times @ int) @ A5) @ D5)) @ (((times_times @ int) @ C4) @ B3))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P6)))))).
% 0.33/5.22 thf(fact_4503_rat__uminus__code, axiom, (! [P5:rat]: (((quotient_of @ ((uminus_uminus @ rat) @ P5)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int]: ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ A5))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4504_lenlex__conv, axiom, (! [A:$tType]: (((lenlex @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ ((size_size @ (list @ A)) @ Ys3)) | ((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys3)) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys3)) @ ((lex @ A) @ R5)))))))))))))).
% 0.33/5.22 thf(fact_4505_Divides_Oadjust__div__def, axiom, (adjust_div = ((((product_case_prod @ int) @ int) @ int) @ (^ [Q3:int,R5:int]: ((((plus_plus @ int) @ Q3) @ ((zero_neq_one_of_bool @ int) @ (R5 != (zero_zero @ int))))))))).
% 0.33/5.22 thf(fact_4506_sum__list__map__eq__sum__count, axiom, (! [A:$tType,F:(A > nat),Xs:(list @ A)]: ((((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ Xs)) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X2:A]: ((((times_times @ nat) @ (((count_list @ A) @ Xs) @ X2)) @ (F @ X2))))) @ ((set2 @ A) @ Xs)))))).
% 0.33/5.22 thf(fact_4507_rat__divide__code, axiom, (! [P5:rat,Q4:rat]: (((quotient_of @ (((divide_divide @ rat) @ P5) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B3:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A5) @ D5)) @ (((times_times @ int) @ C4) @ B3)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4508_rat__times__code, axiom, (! [P5:rat,Q4:rat]: (((quotient_of @ (((times_times @ rat) @ P5) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B3:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A5) @ B3)) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4509_rat__plus__code, axiom, (! [P5:rat,Q4:rat]: (((quotient_of @ (((plus_plus @ rat) @ P5) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B3:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ A5) @ D5)) @ (((times_times @ int) @ B3) @ C4))) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4510_rat__minus__code, axiom, (! [P5:rat,Q4:rat]: (((quotient_of @ (((minus_minus @ rat) @ P5) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B3:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ (((times_times @ int) @ A5) @ D5)) @ (((times_times @ int) @ B3) @ C4))) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4511_sum__list__map__eq__sum__count2, axiom, (! [A:$tType,Xs:(list @ A),X8:(set @ A),F:(A > nat)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ X8) => (((finite_finite2 @ A) @ X8) => (((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ Xs)) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X2:A]: ((((times_times @ nat) @ (((count_list @ A) @ Xs) @ X2)) @ (F @ X2))))) @ X8))))))).
% 0.33/5.22 thf(fact_4512_divmod__step__nat__def, axiom, ((unique1321980374590559556d_step @ nat) = (^ [L3:num]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q3:nat,R5:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ L3)) @ R5)) @ ((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q3)) @ (one_one @ nat))) @ (((minus_minus @ nat) @ R5) @ ((numeral_numeral @ nat) @ L3)))) @ ((((product_Pair @ nat) @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q3)) @ R5))))))))).
% 0.33/5.22 thf(fact_4513_rat__inverse__code, axiom, (! [P5:rat]: (((quotient_of @ ((inverse_inverse @ rat) @ P5)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A5:int,B3:int]: (((((if @ ((product_prod @ int) @ int)) @ (A5 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ A5)) @ B3)) @ ((abs_abs @ int) @ A5)))))) @ (quotient_of @ P5)))))).
% 0.33/5.22 thf(fact_4514_divmod__step__int__def, axiom, ((unique1321980374590559556d_step @ int) = (^ [L3:num]: (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [Q3:int,R5:int]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ L3)) @ R5)) @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q3)) @ (one_one @ int))) @ (((minus_minus @ int) @ R5) @ ((numeral_numeral @ int) @ L3)))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q3)) @ R5))))))))).
% 0.33/5.22 thf(fact_4515_Sum__Icc__int, axiom, (! [M:int,N:int]: (((((ord_less_eq @ int) @ M) @ N) => (((((groups7311177749621191930dd_sum @ int) @ int) @ (^ [X2:int]: (X2))) @ (((set_or1337092689740270186AtMost @ int) @ M) @ N)) = (((divide_divide @ int) @ (((minus_minus @ int) @ (((times_times @ int) @ N) @ (((plus_plus @ int) @ N) @ (one_one @ int)))) @ (((times_times @ int) @ M) @ (((minus_minus @ int) @ M) @ (one_one @ int))))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))).
% 0.33/5.22 thf(fact_4516_divmod__step__integer__def, axiom, ((unique1321980374590559556d_step @ code_integer) = (^ [L3:num]: (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [Q3:code_integer,R5:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less_eq @ code_integer) @ ((numeral_numeral @ code_integer) @ L3)) @ R5)) @ ((((product_Pair @ code_integer) @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q3)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ R5) @ ((numeral_numeral @ code_integer) @ L3)))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q3)) @ R5))))))))).
% 0.33/5.22 thf(fact_4517_num__of__integer__code, axiom, (code_num_of_integer = (^ [K3:code_integer]: (((((if @ num) @ (((ord_less_eq @ code_integer) @ K3) @ (one_one @ code_integer))) @ one2) @ (((((product_case_prod @ code_integer) @ code_integer) @ num) @ (^ [L3:code_integer,J3:code_integer]: (((((if @ num) @ (J3 = (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 @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 0.33/5.22 thf(fact_4518_Array__Time_Omake__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_make @ A) = (^ [N3:nat,F2:(nat > A)]: (((heap_Time_heap @ (array @ A)) @ (^ [H4:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F2) @ ((upt @ (zero_zero @ nat)) @ N3))) @ H4)))))))))))).
% 0.33/5.22 thf(fact_4519_new__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_new @ A) = (^ [N3:nat,X2:A]: (((heap_Time_heap @ (array @ A)) @ (^ [H4:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ (((replicate @ A) @ N3) @ X2)) @ H4)))))))))))).
% 0.33/5.22 thf(fact_4520_of__list__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_of_list @ A) = (^ [Xs3:(list @ A)]: (((heap_Time_heap @ (array @ A)) @ (^ [H4:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs3))))))) @ (((array_alloc @ A) @ Xs3) @ H4)))))))))))).
% 0.33/5.22 thf(fact_4521_num__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_num_of_integer @ (code_integer_of_int @ X)) = (num_of_nat @ (nat2 @ X)))))).
% 0.33/5.22 thf(fact_4522_return__def, axiom, (! [A:$tType]: (((heap_Time_return @ A) = (^ [X2:A]: (((heap_Time_heap @ A) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (one_one @ nat)))))))))))).
% 0.33/5.22 thf(fact_4523_ureturn__def, axiom, (! [A:$tType]: (((heap_Time_ureturn @ A) = (^ [X2:A]: (((heap_Time_heap @ A) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (zero_zero @ nat)))))))))))).
% 0.33/5.22 thf(fact_4524_effect__heapE, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_heap @ A) @ F)) @ H2) @ H) @ R) @ N) => (~ (((H = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => (R != (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2))))))))))).
% 0.33/5.22 thf(fact_4525_effect__heapI, axiom, (! [A:$tType,N:nat,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H2:(heap_ext @ product_unit),H:(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 @ H2)))) => ((H = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2))) => ((((((heap_Time_effect @ A) @ ((heap_Time_heap @ A) @ F)) @ H2) @ H) @ R) @ N))))))).
% 0.33/5.22 thf(fact_4526_divmod__nat__if, axiom, (divmod_nat = (^ [M2:nat,N3:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ ((N3 = (zero_zero @ nat)) | (((ord_less @ nat) @ M2) @ N3))) @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ M2)) @ (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q3:nat]: ((((product_Pair @ nat) @ nat) @ (suc @ Q3))))) @ ((divmod_nat @ (((minus_minus @ nat) @ M2) @ N3)) @ N3))))))).
% 0.33/5.22 thf(fact_4527_int__ge__less__than__def, axiom, (int_ge_less_than = (^ [D5:int]: (((collect @ ((product_prod @ int) @ int)) @ ((((product_case_prod @ int) @ int) @ $o) @ (^ [Z8:int,Z7:int]: (((((ord_less_eq @ int) @ D5) @ Z8) & (((ord_less @ int) @ Z8) @ Z7)))))))))).
% 0.33/5.22 thf(fact_4528_int__ge__less__than2__def, axiom, (int_ge_less_than2 = (^ [D5:int]: (((collect @ ((product_prod @ int) @ int)) @ ((((product_case_prod @ int) @ int) @ $o) @ (^ [Z8:int,Z7:int]: (((((ord_less_eq @ int) @ D5) @ Z7) & (((ord_less @ int) @ Z8) @ Z7)))))))))).
% 0.33/5.22 thf(fact_4529_divmod__integer__code, axiom, (code_divmod_integer = (^ [K3:code_integer,L3:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K3 = (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)) @ K3)) @ ((code_divmod_abs @ K3) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S6:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S6 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ L3) @ S6)))))) @ ((code_divmod_abs @ K3) @ L3)))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (L3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ (((((product_apsnd @ code_integer) @ code_integer) @ code_integer) @ (uminus_uminus @ code_integer)) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((code_divmod_abs @ K3) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S6:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S6 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ L3)) @ S6)))))) @ ((code_divmod_abs @ K3) @ L3))))))))))).
% 0.33/5.22 thf(fact_4530_freeze__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_freeze @ A) = (^ [A5:(array @ A)]: (((heap_Time_heap @ (list @ A)) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((array_get @ A) @ H4) @ A5)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (((plus_plus @ nat) @ (one_one @ nat)) @ (((array_length @ A) @ H4) @ A5)))))))))))))).
% 0.33/5.22 thf(fact_4531_int__of__integer__code, axiom, (code_int_of_integer = (^ [K3:code_integer]: (((((if @ int) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ int) @ (code_int_of_integer @ ((uminus_uminus @ code_integer) @ K3)))) @ ((((if @ int) @ (K3 = (zero_zero @ code_integer))) @ (zero_zero @ int)) @ (((((product_case_prod @ code_integer) @ code_integer) @ int) @ (^ [L3:code_integer,J3:code_integer]: (((((if @ int) @ (J3 = (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 @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))))))))).
% 0.33/5.22 thf(fact_4532_of__nat__code, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N3:nat]: (((((semiri8178284476397505188at_aux @ A) @ (^ [I5:A]: ((((plus_plus @ A) @ I5) @ (one_one @ A))))) @ N3) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4533_inverse__rat__def, axiom, ((inverse_inverse @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat) @ (^ [X2:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X2) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ X2)))))))).
% 0.33/5.22 thf(fact_4534_int__of__integer__inverse, axiom, (! [X:code_integer]: (((code_integer_of_int @ (code_int_of_integer @ X)) = X)))).
% 0.33/5.22 thf(fact_4535_int__of__integer__integer__of__int, axiom, (! [K:int]: (((code_int_of_integer @ (code_integer_of_int @ K)) = K)))).
% 0.33/5.22 thf(fact_4536_integer__of__int__int__of__integer, axiom, (! [K:code_integer]: (((code_integer_of_int @ (code_int_of_integer @ K)) = K)))).
% 0.33/5.22 thf(fact_4537_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))))).
% 0.33/5.22 thf(fact_4538_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)))))).
% 0.33/5.22 thf(fact_4539_of__int__integer__of, axiom, (! [K:code_integer]: ((((ring_1_of_int @ code_integer) @ (code_int_of_integer @ K)) = K)))).
% 0.33/5.22 thf(fact_4540_int__of__integer__of__int, axiom, (! [K:int]: (((code_int_of_integer @ ((ring_1_of_int @ code_integer) @ K)) = K)))).
% 0.33/5.22 thf(fact_4541_zero__integer_Orep__eq, axiom, ((code_int_of_integer @ (zero_zero @ code_integer)) = (zero_zero @ int))).
% 0.33/5.22 thf(fact_4542_int__of__integer__numeral, axiom, (! [K:num]: (((code_int_of_integer @ ((numeral_numeral @ code_integer) @ K)) = ((numeral_numeral @ int) @ K))))).
% 0.33/5.22 thf(fact_4543_plus__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((plus_plus @ code_integer) @ X) @ Xa3)) = (((plus_plus @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4544_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)))))).
% 0.33/5.22 thf(fact_4545_times__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((times_times @ code_integer) @ X) @ Xa3)) = (((times_times @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4546_one__integer_Orep__eq, axiom, ((code_int_of_integer @ (one_one @ code_integer)) = (one_one @ int))).
% 0.33/5.22 thf(fact_4547_minus__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((minus_minus @ code_integer) @ X) @ Xa3)) = (((minus_minus @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4548_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)))))).
% 0.33/5.22 thf(fact_4549_divide__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((divide_divide @ code_integer) @ X) @ Xa3)) = (((divide_divide @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4550_modulo__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((modulo_modulo @ code_integer) @ X) @ Xa3)) = (((modulo_modulo @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4551_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)))))).
% 0.33/5.22 thf(fact_4552_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))))).
% 0.33/5.22 thf(fact_4553_division__segment__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((euclid7384307370059645450egment @ code_integer) @ X)) = ((euclid7384307370059645450egment @ int) @ (code_int_of_integer @ X)))))).
% 0.33/5.22 thf(fact_4554_euclidean__size__integer_Orep__eq, axiom, ((euclid6346220572633701492n_size @ code_integer) = (^ [X2:code_integer]: (((euclid6346220572633701492n_size @ int) @ (code_int_of_integer @ X2)))))).
% 0.33/5.22 thf(fact_4555_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)))))).
% 0.33/5.22 thf(fact_4556_integer__eq__iff, axiom, ((^ [Y:code_integer,Z2:code_integer]: ((Y = Z2))) = (^ [K3:code_integer,L3:code_integer]: (((code_int_of_integer @ K3) = (code_int_of_integer @ L3)))))).
% 0.33/5.22 thf(fact_4557_integer__eqI, axiom, (! [K:code_integer,L:code_integer]: ((((code_int_of_integer @ K) = (code_int_of_integer @ L)) => (K = L))))).
% 0.33/5.22 thf(fact_4558_int__of__integer__inject, axiom, (! [X:code_integer,Y3:code_integer]: ((((code_int_of_integer @ X) = (code_int_of_integer @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_4559_less__integer_Orep__eq, axiom, ((ord_less @ code_integer) = (^ [X2:code_integer,Xa4:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ X2)) @ (code_int_of_integer @ Xa4)))))).
% 0.33/5.22 thf(fact_4560_integer__less__iff, axiom, ((ord_less @ code_integer) = (^ [K3:code_integer,L3:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ K3)) @ (code_int_of_integer @ L3)))))).
% 0.33/5.22 thf(fact_4561_integer__less__eq__iff, axiom, ((ord_less_eq @ code_integer) = (^ [K3:code_integer,L3:code_integer]: ((((ord_less_eq @ int) @ (code_int_of_integer @ K3)) @ (code_int_of_integer @ L3)))))).
% 0.33/5.22 thf(fact_4562_less__eq__integer_Orep__eq, axiom, ((ord_less_eq @ code_integer) = (^ [X2:code_integer,Xa4:code_integer]: ((((ord_less_eq @ int) @ (code_int_of_integer @ X2)) @ (code_int_of_integer @ Xa4)))))).
% 0.33/5.22 thf(fact_4563_nat__of__integer_Orep__eq, axiom, (code_nat_of_integer = (^ [X2:code_integer]: ((nat2 @ (code_int_of_integer @ X2)))))).
% 0.33/5.22 thf(fact_4564_take__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se2584673776208193580ke_bit @ code_integer) @ X) @ Xa3)) = (((bit_se2584673776208193580ke_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4565_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)))))).
% 0.33/5.22 thf(fact_4566_and__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se5824344872417868541ns_and @ code_integer) @ X) @ Xa3)) = (((bit_se5824344872417868541ns_and @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4567_bit__integer_Orep__eq, axiom, ((bit_se5641148757651400278ts_bit @ code_integer) = (^ [X2:code_integer]: (((bit_se5641148757651400278ts_bit @ int) @ (code_int_of_integer @ X2)))))).
% 0.33/5.22 thf(fact_4568_or__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se1065995026697491101ons_or @ code_integer) @ X) @ Xa3)) = (((bit_se1065995026697491101ons_or @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4569_xor__integer_Orep__eq, axiom, (! [X:code_integer,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se5824344971392196577ns_xor @ code_integer) @ X) @ Xa3)) = (((bit_se5824344971392196577ns_xor @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4570_push__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se4730199178511100633sh_bit @ code_integer) @ X) @ Xa3)) = (((bit_se4730199178511100633sh_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4571_drop__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se4197421643247451524op_bit @ code_integer) @ X) @ Xa3)) = (((bit_se4197421643247451524op_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4572_mask__integer_Orep__eq, axiom, (! [X:nat]: (((code_int_of_integer @ ((bit_se2239418461657761734s_mask @ code_integer) @ X)) = ((bit_se2239418461657761734s_mask @ int) @ X))))).
% 0.33/5.22 thf(fact_4573_unset__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se2638667681897837118et_bit @ code_integer) @ X) @ Xa3)) = (((bit_se2638667681897837118et_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4574_set__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se5668285175392031749et_bit @ code_integer) @ X) @ Xa3)) = (((bit_se5668285175392031749et_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4575_flip__bit__integer_Orep__eq, axiom, (! [X:nat,Xa3:code_integer]: (((code_int_of_integer @ (((bit_se8732182000553998342ip_bit @ code_integer) @ X) @ Xa3)) = (((bit_se8732182000553998342ip_bit @ int) @ X) @ (code_int_of_integer @ Xa3)))))).
% 0.33/5.22 thf(fact_4576_of__nat__aux_Osimps_I2_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),N:nat,I:A]: ((((((semiri8178284476397505188at_aux @ A) @ Inc) @ (suc @ N)) @ I) = ((((semiri8178284476397505188at_aux @ A) @ Inc) @ N) @ (Inc @ I))))))))).
% 0.33/5.22 thf(fact_4577_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),I:A]: ((((((semiri8178284476397505188at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))).
% 0.33/5.22 thf(fact_4578_num__of__integer_Orep__eq, axiom, (code_num_of_integer = (^ [X2:code_integer]: ((num_of_nat @ (nat2 @ (code_int_of_integer @ X2))))))).
% 0.33/5.22 thf(fact_4579_uminus__rat__def, axiom, ((uminus_uminus @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat) @ (^ [X2:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X2))) @ (((product_snd @ int) @ int) @ X2))))))).
% 0.33/5.22 thf(fact_4580_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))))).
% 0.33/5.22 thf(fact_4581_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))))).
% 0.33/5.22 thf(fact_4582_transpose__rectangle, axiom, (! [A:$tType,Xs:(list @ (list @ A)),N:nat]: ((((Xs = (nil @ (list @ A))) => (N = (zero_zero @ nat))) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ I4)) = N)))) => (((transpose @ A) @ Xs) = ((((map @ nat) @ (list @ A)) @ (^ [I5:nat]: (((((map @ nat) @ A) @ (^ [J3:nat]: ((((nth @ A) @ (((nth @ (list @ A)) @ Xs) @ J3)) @ I5)))) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ (list @ A))) @ Xs)))))) @ ((upt @ (zero_zero @ nat)) @ N)))))))).
% 0.33/5.22 thf(fact_4583_sorted__wrt__less__sum__mono__lowerbound, axiom, (! [B:$tType]: (((ordere6911136660526730532id_add @ B) => (! [F:(nat > B),Ns:(list @ nat)]: (((! [X3:nat,Y5:nat]: (((((ord_less_eq @ nat) @ X3) @ Y5) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y5))))) => ((((sorted_wrt @ nat) @ (ord_less @ nat)) @ Ns) => (((ord_less_eq @ B) @ ((((groups7311177749621191930dd_sum @ nat) @ B) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ nat)) @ Ns)))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ nat) @ B) @ F) @ Ns))))))))))).
% 0.33/5.22 thf(fact_4584_strict__sorted__imp__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))).
% 0.33/5.22 thf(fact_4585_sorted2, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Zs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Zs))) = ((((ord_less_eq @ A) @ X) @ Y3) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ Y3) @ Zs)))))))))).
% 0.33/5.22 thf(fact_4586_sorted0, axiom, (! [A:$tType]: (((linorder @ A) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (nil @ A)))))).
% 0.33/5.22 thf(fact_4587_strict__sorted__simps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (((sorted_wrt @ A) @ (ord_less @ A)) @ (nil @ A)))))).
% 0.33/5.22 thf(fact_4588_strict__sorted__equal, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ Xs) => ((((sorted_wrt @ A) @ (ord_less @ A)) @ Ys2) => ((((set2 @ A) @ Ys2) = ((set2 @ A) @ Xs)) => (Ys2 = Xs)))))))))).
% 0.33/5.22 thf(fact_4589_sorted__take, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),N:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((take @ A) @ N) @ Xs))))))))).
% 0.33/5.22 thf(fact_4590_sorted__replicate, axiom, (! [A:$tType]: (((linorder @ A) => (! [N:nat,X:A]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((replicate @ A) @ N) @ X)))))))).
% 0.33/5.22 thf(fact_4591_sorted__drop, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),N:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((drop @ A) @ N) @ Xs))))))))).
% 0.33/5.22 thf(fact_4592_sorted__remdups__adj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((remdups_adj @ A) @ Xs))))))))).
% 0.33/5.22 thf(fact_4593_sorted__wrt__upt, axiom, (! [M:nat,N:nat]: ((((sorted_wrt @ nat) @ (ord_less @ nat)) @ ((upt @ M) @ N))))).
% 0.33/5.22 thf(fact_4594_sorted__upt, axiom, (! [M:nat,N:nat]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((upt @ M) @ N))))).
% 0.33/5.22 thf(fact_4595_sorted__list__of__set_Osorted__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((linord4507533701916653071of_set @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_4596_sorted__list__of__set_Ostrict__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((sorted_wrt @ A) @ (ord_less @ A)) @ ((linord4507533701916653071of_set @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_4597_sorted__map, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) = (((sorted_wrt @ B) @ (^ [X2:B,Y6:B]: ((((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y6))))) @ Xs)))))))).
% 0.33/5.22 thf(fact_4598_sorted1, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X) @ (nil @ A))))))))).
% 0.33/5.22 thf(fact_4599_sorted__simps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Ys2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X) @ Ys2)) = ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Ys2)) => (((ord_less_eq @ A) @ X) @ X2)))) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys2))))))))).
% 0.33/5.22 thf(fact_4600_strict__sorted__simps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Ys2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ (((cons @ A) @ X) @ Ys2)) = ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Ys2)) => (((ord_less @ A) @ X) @ X2)))) & (((sorted_wrt @ A) @ (ord_less @ A)) @ Ys2))))))))).
% 0.33/5.22 thf(fact_4601_sorted__append, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((append @ A) @ Xs) @ Ys2)) = ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys2) & (! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (! [Y6:A]: (((((member @ A) @ Y6) @ ((set2 @ A) @ Ys2)) => (((ord_less_eq @ A) @ X2) @ Y6)))))))))))))))).
% 0.33/5.22 thf(fact_4602_strict__sorted__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ L) = ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) & ((distinct @ A) @ L))))))))).
% 0.33/5.22 thf(fact_4603_sorted__distinct__set__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((distinct @ A) @ Xs) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys2) => (((distinct @ A) @ Ys2) => ((((set2 @ A) @ Xs) = ((set2 @ A) @ Ys2)) => (Xs = Ys2)))))))))))).
% 0.33/5.22 thf(fact_4604_sorted__wrt01, axiom, (! [A:$tType,Xs:(list @ A),P:(A > (A > $o))]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((sorted_wrt @ A) @ P) @ Xs))))).
% 0.33/5.22 thf(fact_4605_sorted__wrt__iff__nth__less, axiom, (! [A:$tType]: (((sorted_wrt @ A) = (^ [P4:(A > (A > $o)),Xs3:(list @ A)]: ((! [I5:nat,J3:nat]: (((((ord_less @ nat) @ I5) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs3)) => ((P4 @ (((nth @ A) @ Xs3) @ I5)) @ (((nth @ A) @ Xs3) @ J3)))))))))))).
% 0.33/5.22 thf(fact_4606_sorted__wrt__nth__less, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ P) @ Xs) => ((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => ((P @ (((nth @ A) @ Xs) @ I)) @ (((nth @ A) @ Xs) @ J)))))))).
% 0.33/5.22 thf(fact_4607_sorted__butlast, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((butlast @ A) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4608_sorted__enumerate, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((((map @ ((product_prod @ nat) @ A)) @ nat) @ ((product_fst @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs)))))).
% 0.33/5.22 thf(fact_4609_sorted01, axiom, (! [A:$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)))))))).
% 0.33/5.22 thf(fact_4610_finite__sorted__distinct__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (? [X3:(list @ A)]: (((((set2 @ A) @ X3) = A6) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ X3) & (((distinct @ A) @ X3) & (! [Y4:(list @ A)]: ((((((set2 @ A) @ Y4) = A6) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Y4) & ((distinct @ A) @ Y4))) => (Y4 = X3)))))))))))))))).
% 0.33/5.22 thf(fact_4611_sorted__iff__nth__mono__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I5:nat,J3:nat]: (((((ord_less @ nat) @ I5) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ A) @ Xs) @ J3))))))))))))).
% 0.33/5.22 thf(fact_4612_sorted__list__of__set_Oidem__if__sorted__distinct, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((distinct @ A) @ Xs) => (((linord4507533701916653071of_set @ A) @ ((set2 @ A) @ Xs)) = Xs))))))))).
% 0.33/5.22 thf(fact_4613_sorted__list__of__set_Ofinite__set__strict__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (~ ((! [L2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ L2) => ((((set2 @ A) @ L2) = A6) => (((size_size @ (list @ A)) @ L2) != ((finite_card @ A) @ A6))))))))))))))).
% 0.33/5.22 thf(fact_4614_sorted__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I5:nat]: (((((ord_less @ nat) @ (suc @ I5)) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ A) @ Xs) @ (suc @ I5))))))))))))).
% 0.33/5.22 thf(fact_4615_sorted__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ A) @ Xs) @ J))))))))))).
% 0.33/5.22 thf(fact_4616_sorted__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I5:nat,J3:nat]: (((((ord_less_eq @ nat) @ I5) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ A) @ Xs) @ J3))))))))))))).
% 0.33/5.22 thf(fact_4617_sorted__wrt__less__idx, axiom, (! [Ns:(list @ nat),I:nat]: (((((sorted_wrt @ nat) @ (ord_less @ nat)) @ Ns) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ nat)) @ Ns)) => (((ord_less_eq @ nat) @ I) @ (((nth @ nat) @ Ns) @ I))))))).
% 0.33/5.22 thf(fact_4618_sorted__list__of__set_Osorted__key__list__of__set__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),L:(list @ A)]: ((((finite_finite2 @ A) @ A6) => (((((sorted_wrt @ A) @ (ord_less @ A)) @ L) & ((((set2 @ A) @ L) = A6) & (((size_size @ (list @ A)) @ L) = ((finite_card @ A) @ A6)))) = (((linord4507533701916653071of_set @ A) @ A6) = L))))))))).
% 0.33/5.22 thf(fact_4619_length__transpose__sorted, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => (((Xs = (nil @ (list @ A))) => (((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = (zero_zero @ nat))) & ((Xs != (nil @ (list @ A))) => (((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = ((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ (zero_zero @ nat)))))))))).
% 0.33/5.22 thf(fact_4620_sum__diff1_H__aux, axiom, (! [B:$tType,A:$tType]: (((ab_group_add @ B) => (! [F4:(set @ A),I6:(set @ A),F:(A > B),I:A]: ((((finite_finite2 @ A) @ F4) => ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [I5:A]: (((((member @ A) @ I5) @ I6) & ((F @ I5) != (zero_zero @ B))))))) @ F4) => (((((member @ A) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = (((minus_minus @ B) @ ((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ I6)) @ (F @ I)))) & ((~ ((((member @ A) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = ((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ I6)))))))))))).
% 0.33/5.22 thf(fact_4621_finite__mono__remains__stable__implies__strict__prefix, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A)]: ((((finite_finite2 @ A) @ ((((image @ nat) @ A) @ F) @ (top_top @ (set @ nat)))) => ((((order_mono @ nat) @ A) @ F) => ((! [N4:nat]: ((((F @ N4) = (F @ (suc @ N4))) => ((F @ (suc @ N4)) = (F @ (suc @ (suc @ N4))))))) => (? [N8:nat]: (((! [N5:nat]: (((((ord_less_eq @ nat) @ N5) @ N8) => (! [M3:nat]: (((((ord_less_eq @ nat) @ M3) @ N8) => ((((ord_less @ nat) @ M3) @ N5) => (((ord_less @ A) @ (F @ M3)) @ (F @ N5))))))))) & (! [N5:nat]: (((((ord_less_eq @ nat) @ N8) @ N5) => ((F @ N8) = (F @ N5))))))))))))))))).
% 0.33/5.22 thf(fact_4622_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 0.33/5.22 thf(fact_4623_inf__top__left, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [X:A]: (((((inf_inf @ A) @ (top_top @ A)) @ X) = X))))))).
% 0.33/5.22 thf(fact_4624_inf__top__right, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (top_top @ A)) = X))))))).
% 0.33/5.22 thf(fact_4625_inf__eq__top__iff, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [X:A,Y3:A]: ((((((inf_inf @ A) @ X) @ Y3) = (top_top @ A)) = ((X = (top_top @ A)) & (Y3 = (top_top @ A)))))))))).
% 0.33/5.22 thf(fact_4626_top__eq__inf__iff, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [X:A,Y3:A]: ((((top_top @ A) = (((inf_inf @ A) @ X) @ Y3)) = ((X = (top_top @ A)) & (Y3 = (top_top @ A)))))))))).
% 0.33/5.22 thf(fact_4627_inf__top_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [A3:A,B2:A]: ((((((inf_inf @ A) @ A3) @ B2) = (top_top @ A)) = ((A3 = (top_top @ A)) & (B2 = (top_top @ A)))))))))).
% 0.33/5.22 thf(fact_4628_inf__top_Oleft__neutral, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [A3:A]: (((((inf_inf @ A) @ (top_top @ A)) @ A3) = A3))))))).
% 0.33/5.22 thf(fact_4629_inf__top_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [A3:A,B2:A]: ((((top_top @ A) = (((inf_inf @ A) @ A3) @ B2)) = ((A3 = (top_top @ A)) & (B2 = (top_top @ A)))))))))).
% 0.33/5.22 thf(fact_4630_inf__top_Oright__neutral, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (! [A3:A]: (((((inf_inf @ A) @ A3) @ (top_top @ A)) = A3))))))).
% 0.33/5.22 thf(fact_4631_boolean__algebra_Odisj__one__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4632_boolean__algebra_Odisj__one__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4633_sup__top__right, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4634_sup__top__left, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4635_max__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [X:A]: (((((ord_max @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4636_max__top2, axiom, (! [A:$tType]: (((order_top @ A) => (! [X:A]: (((((ord_max @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4637_min__top2, axiom, (! [A:$tType]: (((order_top @ A) => (! [X:A]: (((((ord_min @ A) @ X) @ (top_top @ A)) = X))))))).
% 0.33/5.22 thf(fact_4638_min__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [X:A]: (((((ord_min @ A) @ (top_top @ A)) @ X) = X))))))).
% 0.33/5.22 thf(fact_4639_boolean__algebra_Ocompl__zero, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (((uminus_uminus @ A) @ (bot_bot @ A)) = (top_top @ A)))))).
% 0.33/5.22 thf(fact_4640_boolean__algebra_Ocompl__one, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (((uminus_uminus @ A) @ (top_top @ A)) = (bot_bot @ A)))))).
% 0.33/5.22 thf(fact_4641_boolean__algebra_Odisj__cancel__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ ((uminus_uminus @ A) @ X)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4642_boolean__algebra_Odisj__cancel__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ X) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4643_sup__compl__top__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4644_sup__compl__top__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ (((sup_sup @ A) @ X) @ Y3)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4645_finite__compl, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ A) @ ((uminus_uminus @ (set @ A)) @ A6)) = ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))).
% 0.33/5.22 thf(fact_4646_Gcd__UNIV, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Gcd @ A) @ (top_top @ (set @ A))) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_4647_sum_Oempty_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [P5:(B > A)]: ((((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ (bot_bot @ (set @ B))) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_4648_sum_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),P5:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((P5 @ X2) != (zero_zero @ A))))))) => (((((member @ B) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ (((insert @ B) @ I) @ I6)) = ((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ I6))) & ((~ ((((member @ B) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ (((insert @ B) @ I) @ I6)) = (((plus_plus @ A) @ (P5 @ I)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ I6)))))))))))).
% 0.33/5.22 thf(fact_4649_boolean__algebra_Oconj__one__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (top_top @ A)) = X))))))).
% 0.33/5.22 thf(fact_4650_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4651_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 0.33/5.22 thf(fact_4652_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 0.33/5.22 thf(fact_4653_top_Oextremum__strict, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A3))))))))).
% 0.33/5.22 thf(fact_4654_top_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((A3 != (top_top @ A)) = (((ord_less @ A) @ A3) @ (top_top @ A))))))))).
% 0.33/5.22 thf(fact_4655_sum_Onon__neutral_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),I6:(set @ B)]: ((((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((G @ X2) != (zero_zero @ A))))))) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I6)))))))).
% 0.33/5.22 thf(fact_4656_finite__fun__UNIVD1, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((((finite_card @ B) @ (top_top @ (set @ B))) != (suc @ (zero_zero @ nat))) => ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))).
% 0.33/5.22 thf(fact_4657_sorted__upto, axiom, (! [M:int,N:int]: ((((sorted_wrt @ int) @ (ord_less_eq @ int)) @ ((upto @ M) @ N))))).
% 0.33/5.22 thf(fact_4658_sorted__wrt__upto, axiom, (! [I:int,J:int]: ((((sorted_wrt @ int) @ (ord_less @ int)) @ ((upto @ I) @ J))))).
% 0.33/5.22 thf(fact_4659_sup__cancel__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,A3:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ A3)) @ (((sup_sup @ A) @ X) @ B2)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4660_sup__cancel__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,A3:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ A3)) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X)) @ B2)) = (top_top @ A)))))))).
% 0.33/5.22 thf(fact_4661_Compl__empty__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ A)))))).
% 0.33/5.22 thf(fact_4662_Compl__UNIV__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 0.33/5.22 thf(fact_4663_Compl__partition2, axiom, (! [A:$tType,A6:(set @ A)]: (((((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ A6) = (top_top @ (set @ A)))))).
% 0.33/5.22 thf(fact_4664_Compl__partition, axiom, (! [A:$tType,A6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ A6)) = (top_top @ (set @ A)))))).
% 0.33/5.22 thf(fact_4665_Compl__eq__Diff__UNIV, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = ((minus_minus @ (set @ A)) @ (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_4666_sup__shunt, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((((sup_sup @ A) @ X) @ Y3) = (top_top @ A)) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)))))))).
% 0.33/5.22 thf(fact_4667_boolean__algebra_Ocomplement__unique, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [A3:A,X:A,Y3:A]: ((((((inf_inf @ A) @ A3) @ X) = (bot_bot @ A)) => (((((sup_sup @ A) @ A3) @ X) = (top_top @ A)) => (((((inf_inf @ A) @ A3) @ Y3) = (bot_bot @ A)) => (((((sup_sup @ A) @ A3) @ Y3) = (top_top @ A)) => (X = Y3))))))))))).
% 0.33/5.22 thf(fact_4668_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F:(B > A),A6:(set @ B)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A6))) @ ((((image @ B) @ A) @ F) @ ((uminus_uminus @ (set @ B)) @ A6))))))).
% 0.33/5.22 thf(fact_4669_sum_Omono__neutral__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T6)))))))))).
% 0.33/5.22 thf(fact_4670_sum_Omono__neutral__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T6) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3)))))))))).
% 0.33/5.22 thf(fact_4671_sum_Omono__neutral__cong__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T6:(set @ B),H2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ I4) = (zero_zero @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3) = ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ T6))))))))))).
% 0.33/5.22 thf(fact_4672_sum_Omono__neutral__cong__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A),H2:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (zero_zero @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T6) = ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ S3))))))))))).
% 0.33/5.22 thf(fact_4673_boolean__algebra__class_Oboolean__algebra_Ocompl__unique, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X:A,Y3:A]: ((((((inf_inf @ A) @ X) @ Y3) = (bot_bot @ A)) => (((((sup_sup @ A) @ X) @ Y3) = (top_top @ A)) => (((uminus_uminus @ A) @ X) = Y3))))))))).
% 0.33/5.22 thf(fact_4674_sum_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((G @ X2) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((H2 @ X2) != (zero_zero @ A))))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ (^ [I5:B]: ((((plus_plus @ A) @ (G @ I5)) @ (H2 @ I5))))) @ I6) = (((plus_plus @ A) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I6)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ I6))))))))))).
% 0.33/5.22 thf(fact_4675_sum_OG__def, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (((groups1027152243600224163dd_sum @ B) @ A) = (^ [P6:(B > A),I8:(set @ B)]: (((((if @ A) @ ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I8) & ((P6 @ X2) != (zero_zero @ A)))))))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ P6) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I8) & ((P6 @ X2) != (zero_zero @ A)))))))) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4676_finite__UNIV__card__ge__0, axiom, (! [A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ (top_top @ (set @ A)))))))).
% 0.33/5.22 thf(fact_4677_sorted__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ ((transpose @ A) @ Xs))))))).
% 0.33/5.22 thf(fact_4678_rev__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((rev @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (suc @ N)))))))).
% 0.33/5.22 thf(fact_4679_rev__update, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Y3:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((rev @ A) @ ((((list_update @ A) @ Xs) @ K) @ Y3)) = ((((list_update @ A) @ ((rev @ A) @ Xs)) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K)) @ (one_one @ nat))) @ Y3)))))).
% 0.33/5.22 thf(fact_4680_boolean__algebra_Oabstract__boolean__algebra__axioms, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => ((((((boolea2506097494486148201lgebra @ A) @ (inf_inf @ A)) @ (sup_sup @ A)) @ (uminus_uminus @ A)) @ (bot_bot @ A)) @ (top_top @ A)))))).
% 0.33/5.22 thf(fact_4681_card__range__greater__zero, axiom, (! [A:$tType,B:$tType,F:(B > A)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))))).
% 0.33/5.22 thf(fact_4682_sorted__rev__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) = (! [I5:nat]: (((((ord_less @ nat) @ (suc @ I5)) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ (suc @ I5))) @ (((nth @ A) @ Xs) @ I5)))))))))))).
% 0.33/5.22 thf(fact_4683_sorted__rev__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) = (! [I5:nat,J3:nat]: (((((ord_less_eq @ nat) @ I5) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J3)) @ (((nth @ A) @ Xs) @ I5))))))))))))).
% 0.33/5.22 thf(fact_4684_sorted__rev__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J)) @ (((nth @ A) @ Xs) @ I))))))))))).
% 0.33/5.22 thf(fact_4685_range__mod, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((((image @ nat) @ nat) @ (^ [M2:nat]: ((((modulo_modulo @ nat) @ M2) @ N)))) @ (top_top @ (set @ nat))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.33/5.22 thf(fact_4686_sum__diff1_H, axiom, (! [B:$tType,A:$tType]: (((ab_group_add @ B) => (! [I6:(set @ A),F:(A > B),I:A]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [I5:A]: (((((member @ A) @ I5) @ I6) & ((F @ I5) != (zero_zero @ B))))))) => (((((member @ A) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = (((minus_minus @ B) @ ((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ I6)) @ (F @ I)))) & ((~ ((((member @ A) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = ((((groups1027152243600224163dd_sum @ A) @ B) @ F) @ I6))))))))))).
% 0.33/5.22 thf(fact_4687_nth__nth__transpose__sorted, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat,J:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs))) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ (list @ A))) @ (((filter @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ Xs))) => ((((nth @ A) @ (((nth @ (list @ A)) @ ((transpose @ A) @ Xs)) @ I)) @ J) = (((nth @ A) @ (((nth @ (list @ A)) @ Xs) @ J)) @ I)))))))).
% 0.33/5.22 thf(fact_4688_transpose__column, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((((map @ (list @ A)) @ A) @ (^ [Ys3:(list @ A)]: ((((nth @ A) @ Ys3) @ I)))) @ (((filter @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ ((transpose @ A) @ Xs))) = (((nth @ (list @ A)) @ Xs) @ I))))))).
% 0.33/5.22 thf(fact_4689_transpose__column__length, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((size_size @ (list @ (list @ A))) @ (((filter @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ ((transpose @ A) @ Xs))) = ((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ I)))))))).
% 0.33/5.22 thf(fact_4690_card__UNIV__unit, axiom, (((finite_card @ product_unit) @ (top_top @ (set @ product_unit))) = (one_one @ nat))).
% 0.33/5.22 thf(fact_4691_infinite__UNIV__int, axiom, (~ (((finite_finite2 @ int) @ (top_top @ (set @ int)))))).
% 0.33/5.22 thf(fact_4692_integer__of__int__cases, axiom, (! [X:code_integer]: ((~ ((! [Y5:int]: (((X = (code_integer_of_int @ Y5)) => (~ ((((member @ int) @ Y5) @ (top_top @ (set @ int))))))))))))).
% 0.33/5.22 thf(fact_4693_integer__of__int__induct, axiom, (! [P:(code_integer > $o),X:code_integer]: (((! [Y5:int]: (((((member @ int) @ Y5) @ (top_top @ (set @ int))) => (P @ (code_integer_of_int @ Y5))))) => (P @ X))))).
% 0.33/5.22 thf(fact_4694_integer__of__int__inject, axiom, (! [X:int,Y3:int]: (((((member @ int) @ X) @ (top_top @ (set @ int))) => ((((member @ int) @ Y3) @ (top_top @ (set @ int))) => (((code_integer_of_int @ X) = (code_integer_of_int @ Y3)) = (X = Y3))))))).
% 0.33/5.22 thf(fact_4695_int__of__integer, axiom, (! [X:code_integer]: ((((member @ int) @ (code_int_of_integer @ X)) @ (top_top @ (set @ int)))))).
% 0.33/5.22 thf(fact_4696_int__of__integer__cases, axiom, (! [Y3:int]: (((((member @ int) @ Y3) @ (top_top @ (set @ int))) => (~ ((! [X3:code_integer]: ((Y3 != (code_int_of_integer @ X3)))))))))).
% 0.33/5.22 thf(fact_4697_int__of__integer__induct, axiom, (! [Y3:int,P:(int > $o)]: (((((member @ int) @ Y3) @ (top_top @ (set @ int))) => ((! [X3:code_integer]: ((P @ (code_int_of_integer @ X3)))) => (P @ Y3)))))).
% 0.33/5.22 thf(fact_4698_length__filter__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((filter @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.22 thf(fact_4699_sorted__same, axiom, (! [A:$tType]: (((linorder @ A) => (! [G:((list @ A) > A),Xs:(list @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((filter @ A) @ (^ [X2:A]: ((X2 = (G @ Xs))))) @ Xs)))))))).
% 0.33/5.22 thf(fact_4700_integer__of__int__inverse, axiom, (! [Y3:int]: (((((member @ int) @ Y3) @ (top_top @ (set @ int))) => ((code_int_of_integer @ (code_integer_of_int @ Y3)) = Y3))))).
% 0.33/5.22 thf(fact_4701_length__filter__less, axiom, (! [A:$tType,X:A,Xs:(list @ A),P:(A > $o)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => ((~ ((P @ X))) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((filter @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))))).
% 0.33/5.22 thf(fact_4702_surj__list__encode, axiom, (((((image @ (list @ nat)) @ nat) @ nat_list_encode) @ (top_top @ (set @ (list @ nat)))) = (top_top @ (set @ nat)))).
% 0.33/5.22 thf(fact_4703_surj__prod__encode, axiom, (((((image @ ((product_prod @ nat) @ nat)) @ nat) @ nat_prod_encode) @ (top_top @ (set @ ((product_prod @ nat) @ nat)))) = (top_top @ (set @ nat)))).
% 0.33/5.22 thf(fact_4704_sorted__filter, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),P:(B > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ (((filter @ B) @ P) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4705_sorted__map__same, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),G:((list @ B) > A),Xs:(list @ B)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ (((filter @ B) @ (^ [X2:B]: (((F @ X2) = (G @ Xs))))) @ Xs))))))))).
% 0.33/5.22 thf(fact_4706_sum__list__map__filter_H, axiom, (! [A:$tType,B:$tType]: (((monoid_add @ A) => (! [F:(B > A),P:(B > $o),Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ (((filter @ B) @ P) @ Xs))) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (F @ X2)) @ (zero_zero @ A))))) @ Xs))))))))).
% 0.33/5.22 thf(fact_4707_sum__list__filter__le__nat, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ (((filter @ A) @ P) @ Xs)))) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ Xs)))))).
% 0.33/5.22 thf(fact_4708_sum__list__map__filter, axiom, (! [A:$tType,B:$tType]: (((monoid_add @ A) => (! [Xs:(list @ B),P:(B > $o),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ ((set2 @ B) @ Xs)) => ((~ ((P @ X3))) => ((F @ X3) = (zero_zero @ A)))))) => (((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ (((filter @ B) @ P) @ Xs))) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4709_Ints__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_Ints @ A) = ((((image @ int) @ A) @ (ring_1_of_int @ A)) @ (top_top @ (set @ int)))))))).
% 0.33/5.22 thf(fact_4710_int__in__range__abs, axiom, (! [N:nat]: ((((member @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((((image @ int) @ int) @ (abs_abs @ int)) @ (top_top @ (set @ int))))))).
% 0.33/5.22 thf(fact_4711_length__filter__conv__card, axiom, (! [A:$tType,P5:(A > $o),Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((filter @ A) @ P5) @ Xs)) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & (P5 @ (((nth @ A) @ Xs) @ I5))))))))))).
% 0.33/5.22 thf(fact_4712_nth__transpose, axiom, (! [A:$tType,I:nat,Xs:(list @ (list @ A))]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs))) => ((((nth @ (list @ A)) @ ((transpose @ A) @ Xs)) @ I) = ((((map @ (list @ A)) @ A) @ (^ [Xs3:(list @ A)]: ((((nth @ A) @ Xs3) @ I)))) @ (((filter @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ Xs))))))).
% 0.33/5.22 thf(fact_4713_transpose__max__length, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((((foldr @ (list @ A)) @ nat) @ (^ [Xs3:(list @ A)]: (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs3))))) @ ((transpose @ A) @ Xs)) @ (zero_zero @ nat)) = ((size_size @ (list @ (list @ A))) @ (((filter @ (list @ A)) @ (^ [X2:(list @ A)]: ((X2 != (nil @ A))))) @ Xs)))))).
% 0.33/5.22 thf(fact_4714_transpose__aux__max, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Xss:(list @ (list @ B))]: (((((ord_max @ nat) @ (suc @ ((size_size @ (list @ A)) @ Xs))) @ (((((foldr @ (list @ B)) @ nat) @ (^ [Xs3:(list @ B)]: (((ord_max @ nat) @ ((size_size @ (list @ B)) @ Xs3))))) @ Xss) @ (zero_zero @ nat))) = (suc @ (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (((((foldr @ (list @ B)) @ nat) @ (^ [X2:(list @ B)]: (((ord_max @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ B)) @ X2)) @ (suc @ (zero_zero @ nat))))))) @ (((filter @ (list @ B)) @ (^ [Ys3:(list @ B)]: ((Ys3 != (nil @ B))))) @ Xss)) @ (zero_zero @ nat)))))))).
% 0.33/5.22 thf(fact_4715_transpose__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => (((transpose @ A) @ ((transpose @ A) @ Xs)) = (((takeWhile @ (list @ A)) @ (^ [X2:(list @ A)]: ((X2 != (nil @ A))))) @ Xs)))))).
% 0.33/5.22 thf(fact_4716_length__takeWhile__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.22 thf(fact_4717_sorted__takeWhile, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((takeWhile @ A) @ P) @ Xs))))))))).
% 0.33/5.22 thf(fact_4718_takeWhile__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) => ((((nth @ A) @ (((takeWhile @ A) @ P) @ Xs)) @ J) = (((nth @ A) @ Xs) @ J)))))).
% 0.33/5.22 thf(fact_4719_nth__length__takeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)) => (~ ((P @ (((nth @ A) @ Xs) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs)))))))))).
% 0.33/5.22 thf(fact_4720_sum__list_Oeq__foldr, axiom, (! [A:$tType]: (((monoid_add @ A) => ((groups8242544230860333062m_list @ A) = (^ [Xs3:(list @ A)]: ((((((foldr @ A) @ A) @ (plus_plus @ A)) @ Xs3) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4721_length__takeWhile__less__P__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ J) => (P @ (((nth @ A) @ Xs) @ I4))))) => ((((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs)))))))).
% 0.33/5.22 thf(fact_4722_takeWhile__eq__take__P__nth, axiom, (! [A:$tType,N:nat,Xs:(list @ A),P:(A > $o)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N) => ((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I4)))))) => (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (~ ((P @ (((nth @ A) @ Xs) @ N))))) => ((((takeWhile @ A) @ P) @ Xs) = (((take @ A) @ N) @ Xs))))))).
% 0.33/5.22 thf(fact_4723_nths__shift__lemma__Suc, axiom, (! [A:$tType,P:(nat > $o),Xs:(list @ A),Is:(list @ nat)]: ((((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((P @ (suc @ (((product_snd @ A) @ nat) @ P6)))))) @ ((((zip @ A) @ nat) @ Xs) @ Is))) = ((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((P @ (((product_snd @ A) @ nat) @ P6))))) @ ((((zip @ A) @ nat) @ Xs) @ ((((map @ nat) @ nat) @ suc) @ Is)))))))).
% 0.33/5.22 thf(fact_4724_horner__sum__foldr, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F2:(B > A),A5:A,Xs3:(list @ B)]: ((((((foldr @ B) @ A) @ (^ [X2:B,B3:A]: ((((plus_plus @ A) @ (F2 @ X2)) @ (((times_times @ A) @ A5) @ B3))))) @ Xs3) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4725_nths__shift__lemma, axiom, (! [A:$tType,A6:(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)) @ A6)))) @ ((((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)) @ A6)))) @ ((((zip @ A) @ nat) @ Xs) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 0.33/5.22 thf(fact_4726_filter__equals__takeWhile__sorted__rev, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),T2:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ ((((map @ B) @ A) @ F) @ Xs))) => ((((filter @ B) @ (^ [X2:B]: ((((ord_less @ A) @ T2) @ (F @ X2))))) @ Xs) = (((takeWhile @ B) @ (^ [X2:B]: ((((ord_less @ A) @ T2) @ (F @ X2))))) @ Xs))))))))).
% 0.33/5.22 thf(fact_4727_length__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = (((((foldr @ (list @ A)) @ nat) @ (^ [Xs3:(list @ A)]: (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs3))))) @ Xs) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_4728_foldr__max__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Y3:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) => (((Xs = (nil @ A)) => ((((((foldr @ A) @ A) @ (ord_max @ A)) @ Xs) @ Y3) = Y3)) & ((Xs != (nil @ A)) => ((((((foldr @ A) @ A) @ (ord_max @ A)) @ Xs) @ Y3) = (((ord_max @ A) @ (((nth @ A) @ Xs) @ (zero_zero @ nat))) @ Y3))))))))))).
% 0.33/5.22 thf(fact_4729_rec__nat__add__eq__if, axiom, (! [A:$tType,A3:A,F:(nat > (A > A)),V:num,N:nat]: ((((((rec_nat @ A) @ A3) @ F) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V)) @ N)) = ((F @ (((plus_plus @ nat) @ (pred_numeral @ V)) @ N)) @ ((((rec_nat @ A) @ A3) @ F) @ (((plus_plus @ nat) @ (pred_numeral @ V)) @ N))))))).
% 0.33/5.22 thf(fact_4730_case__nat__add__eq__if, axiom, (! [A:$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)))))).
% 0.33/5.22 thf(fact_4731_divmod__integer__eq__cases, axiom, (code_divmod_integer = (^ [K3:code_integer,L3:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K3 = (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)) @ K3)) @ (((((((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) @ K3) = ((sgn_sgn @ code_integer) @ L3))) @ ((code_divmod_abs @ K3) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S6:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S6 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((abs_abs @ code_integer) @ L3)) @ S6)))))) @ ((code_divmod_abs @ K3) @ L3)))))))))).
% 0.33/5.22 thf(fact_4732_old_Onat_Osimps_I7_J, axiom, (! [T:$tType,F1:T,F22:(nat > (T > T)),Nat:nat]: ((((((rec_nat @ T) @ F1) @ F22) @ (suc @ Nat)) = ((F22 @ Nat) @ ((((rec_nat @ T) @ F1) @ F22) @ Nat)))))).
% 0.33/5.22 thf(fact_4733_old_Onat_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(nat > (T > T))]: ((((((rec_nat @ T) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))).
% 0.33/5.22 thf(fact_4734_case__nat__numeral, axiom, (! [A:$tType,A3:A,F:(nat > A),V:num]: ((((((case_nat @ A) @ A3) @ F) @ ((numeral_numeral @ nat) @ V)) = (F @ (pred_numeral @ V)))))).
% 0.33/5.22 thf(fact_4735_rec__nat__numeral, axiom, (! [A:$tType,A3:A,F:(nat > (A > A)),V:num]: ((((((rec_nat @ A) @ A3) @ F) @ ((numeral_numeral @ nat) @ V)) = ((F @ (pred_numeral @ V)) @ ((((rec_nat @ A) @ A3) @ F) @ (pred_numeral @ V))))))).
% 0.33/5.22 thf(fact_4736_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E4:$tType,F:(B > A),G:(C > B),X:C,F5:(D > A),G4:(E4 > D),X5:E4]: ((((F @ (G @ X)) = (F5 @ (G4 @ X5))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ E4) @ F5) @ G4) @ X5)))))).
% 0.33/5.22 thf(fact_4737_nat_Ocase__distrib, axiom, (! [B:$tType,A:$tType,H2:(A > B),F1:A,F22:(nat > A),Nat:nat]: (((H2 @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = ((((case_nat @ B) @ (H2 @ F1)) @ (^ [X2:nat]: ((H2 @ (F22 @ X2))))) @ Nat))))).
% 0.33/5.22 thf(fact_4738_old_Onat_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F22:(nat > A),X22:nat]: ((((((case_nat @ A) @ F1) @ F22) @ (suc @ X22)) = (F22 @ X22))))).
% 0.33/5.22 thf(fact_4739_old_Onat_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:(nat > A)]: ((((((case_nat @ A) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))).
% 0.33/5.22 thf(fact_4740_nat_Odisc__eq__case_I2_J, axiom, (! [Nat:nat]: (((Nat != (zero_zero @ nat)) = ((((case_nat @ $o) @ $false) @ (^ [Uu:nat]: ($true))) @ Nat))))).
% 0.33/5.22 thf(fact_4741_nat_Odisc__eq__case_I1_J, axiom, (! [Nat:nat]: (((Nat = (zero_zero @ nat)) = ((((case_nat @ $o) @ $true) @ (^ [Uu:nat]: ($false))) @ Nat))))).
% 0.33/5.22 thf(fact_4742_rec__nat__Suc__imp, axiom, (! [A:$tType,F:(nat > A),F1:A,F22:(nat > (A > A)),N:nat]: (((F = (((rec_nat @ A) @ F1) @ F22)) => ((F @ (suc @ N)) = ((F22 @ N) @ (F @ N))))))).
% 0.33/5.22 thf(fact_4743_rec__nat__0__imp, axiom, (! [A:$tType,F:(nat > A),F1:A,F22:(nat > (A > A))]: (((F = (((rec_nat @ A) @ F1) @ F22)) => ((F @ (zero_zero @ nat)) = F1))))).
% 0.33/5.22 thf(fact_4744_sum__comp__morphism, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((comm_monoid_add @ B) & (comm_monoid_add @ A)) => (! [H2:(B > A),G:(C > B),A6:(set @ C)]: ((((H2 @ (zero_zero @ B)) = (zero_zero @ A)) => ((! [X3:B,Y5:B]: (((H2 @ (((plus_plus @ B) @ X3) @ Y5)) = (((plus_plus @ A) @ (H2 @ X3)) @ (H2 @ Y5))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ (((((comp @ B) @ A) @ C) @ H2) @ G)) @ A6) = (H2 @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ A6))))))))))).
% 0.33/5.22 thf(fact_4745_uminus__sum__list__map, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_4746_less__eq__nat_Osimps_I2_J, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) = ((((case_nat @ $o) @ $false) @ ((ord_less_eq @ nat) @ M)) @ N))))).
% 0.33/5.22 thf(fact_4747_max__Suc1, axiom, (! [N:nat,M:nat]: (((((ord_max @ nat) @ (suc @ N)) @ M) = ((((case_nat @ nat) @ (suc @ N)) @ (^ [M4:nat]: ((suc @ (((ord_max @ nat) @ N) @ M4))))) @ M))))).
% 0.33/5.22 thf(fact_4748_max__Suc2, axiom, (! [M:nat,N:nat]: (((((ord_max @ nat) @ M) @ (suc @ N)) = ((((case_nat @ nat) @ (suc @ N)) @ (^ [M4:nat]: ((suc @ (((ord_max @ nat) @ M4) @ N))))) @ M))))).
% 0.33/5.22 thf(fact_4749_sum_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A6:(set @ B),H2:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A6) => ((! [X3:B,Y5:B]: (((((member @ B) @ X3) @ A6) => ((((member @ B) @ Y5) @ A6) => ((X3 != Y5) => (((H2 @ X3) = (H2 @ Y5)) => ((G @ (H2 @ X3)) = (zero_zero @ A)))))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ ((((image @ B) @ C) @ H2) @ A6)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H2)) @ A6)))))))))).
% 0.33/5.22 thf(fact_4750_prod_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),H2:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A6) => ((! [X3:B,Y5:B]: (((((member @ B) @ X3) @ A6) => ((((member @ B) @ Y5) @ A6) => ((X3 != Y5) => (((H2 @ X3) = (H2 @ Y5)) => ((G @ (H2 @ X3)) = (one_one @ A)))))))) => (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ ((((image @ B) @ C) @ H2) @ A6)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H2)) @ A6)))))))))).
% 0.33/5.22 thf(fact_4751_sum__image__le, axiom, (! [A:$tType,B:$tType,C:$tType]: (((ordere6911136660526730532id_add @ B) => (! [I6:(set @ C),G:(A > B),F:(C > A)]: ((((finite_finite2 @ C) @ I6) => ((! [I4:C]: (((((member @ C) @ I4) @ I6) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (G @ (F @ I4)))))) => (((ord_less_eq @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ G) @ ((((image @ C) @ A) @ F) @ I6))) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ (((((comp @ A) @ B) @ C) @ G) @ F)) @ I6)))))))))).
% 0.33/5.22 thf(fact_4752_diff__Suc, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ M) @ (suc @ N)) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [K3:nat]: (K3))) @ (((minus_minus @ nat) @ M) @ N)))))).
% 0.33/5.22 thf(fact_4753_min__Suc1, axiom, (! [N:nat,M:nat]: (((((ord_min @ nat) @ (suc @ N)) @ M) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [M4:nat]: ((suc @ (((ord_min @ nat) @ N) @ M4))))) @ M))))).
% 0.33/5.22 thf(fact_4754_min__Suc2, axiom, (! [M:nat,N:nat]: (((((ord_min @ nat) @ M) @ (suc @ N)) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [M4:nat]: ((suc @ (((ord_min @ nat) @ M4) @ N))))) @ M))))).
% 0.33/5.22 thf(fact_4755_Nitpick_Ocase__nat__unfold, axiom, (! [A:$tType]: (((case_nat @ A) = (^ [X2:A,F2:(nat > A),N3:nat]: (((((if @ A) @ (N3 = (zero_zero @ nat))) @ X2) @ (F2 @ (((minus_minus @ nat) @ N3) @ (one_one @ nat)))))))))).
% 0.33/5.22 thf(fact_4756_nths__def, axiom, (! [A:$tType]: (((nths @ A) = (^ [Xs3:(list @ A),A7:(set @ 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)) @ A7)))) @ ((((zip @ A) @ nat) @ Xs3) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs3))))))))))).
% 0.33/5.22 thf(fact_4757_length__product__lists, axiom, (! [B:$tType,Xss:(list @ (list @ B))]: ((((size_size @ (list @ (list @ B))) @ ((product_lists @ B) @ Xss)) = (((((foldr @ nat) @ nat) @ (times_times @ nat)) @ ((((map @ (list @ B)) @ nat) @ (size_size @ (list @ B))) @ Xss)) @ (one_one @ nat)))))).
% 0.33/5.22 thf(fact_4758_nat_Osplit__sels_I2_J, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:(nat > A),Nat:nat]: (((P @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = (~ ((((Nat = (zero_zero @ nat)) & (~ ((P @ F1)))) | ((Nat = (suc @ (pred @ Nat))) & (~ ((P @ (F22 @ (pred @ Nat))))))))))))).
% 0.33/5.22 thf(fact_4759_nths__singleton, axiom, (! [A:$tType,A6:(set @ nat),X:A]: ((((((member @ nat) @ (zero_zero @ nat)) @ A6) => ((((nths @ A) @ (((cons @ A) @ X) @ (nil @ A))) @ A6) = (((cons @ A) @ X) @ (nil @ A)))) & ((~ ((((member @ nat) @ (zero_zero @ nat)) @ A6))) => ((((nths @ A) @ (((cons @ A) @ X) @ (nil @ A))) @ A6) = (nil @ A))))))).
% 0.33/5.22 thf(fact_4760_card_Ocomp__fun__commute__on, axiom, ((((((comp @ nat) @ nat) @ nat) @ suc) @ suc) = (((((comp @ nat) @ nat) @ nat) @ suc) @ suc))).
% 0.33/5.22 thf(fact_4761_nths__all, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ nat) @ I4) @ I6)))) => ((((nths @ A) @ Xs) @ I6) = Xs))))).
% 0.33/5.22 thf(fact_4762_sorted__nths, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I6:(set @ nat)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((nths @ A) @ Xs) @ I6))))))))).
% 0.33/5.22 thf(fact_4763_sum_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4764_sum_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4765_prod_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4766_prod_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4767_lift__def, axiom, (! [A:$tType,B:$tType]: ((((heap_Time_lift @ A) @ B) = ((((comp @ B) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_return @ B)))))).
% 0.33/5.22 thf(fact_4768_drop__eq__nths, axiom, (! [A:$tType]: (((drop @ A) = (^ [N3:nat,Xs3:(list @ A)]: ((((nths @ A) @ Xs3) @ ((collect @ nat) @ ((ord_less_eq @ nat) @ N3))))))))).
% 0.33/5.22 thf(fact_4769_euclidean__size__int__def, axiom, ((euclid6346220572633701492n_size @ int) = (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int)))).
% 0.33/5.22 thf(fact_4770_pred__def, axiom, (pred = (((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [X24:nat]: (X24))))).
% 0.33/5.22 thf(fact_4771_length__nths, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((size_size @ (list @ A)) @ (((nths @ A) @ Xs) @ I6)) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & (((member @ nat) @ I5) @ I6)))))))))).
% 0.33/5.22 thf(fact_4772_sum_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 0.33/5.22 thf(fact_4773_sum_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4774_prod_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 0.33/5.22 thf(fact_4775_prod_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_4776_sum_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)))))))))).
% 0.33/5.22 thf(fact_4777_prod_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)))))))))).
% 0.33/5.22 thf(fact_4778_sum_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4779_sum_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4780_prod_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4781_prod_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4782_sum_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(int > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ int) @ A) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4783_filter__eq__nths, axiom, (! [A:$tType]: (((filter @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((((nths @ A) @ Xs3) @ ((collect @ nat) @ (^ [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs3)) & (P4 @ (((nth @ A) @ Xs3) @ I5))))))))))))).
% 0.33/5.22 thf(fact_4784_prod_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(int > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ int) @ A) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4785_sum_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(int > A),M:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ int) @ A) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4786_sum_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M))))))))))).
% 0.33/5.22 thf(fact_4787_prod_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M))))))))))).
% 0.33/5.22 thf(fact_4788_prod_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(int > A),M:nat,N:nat]: ((((((groups7121269368397514597t_prod @ int) @ A) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.22 thf(fact_4789_Gcd__int__def, axiom, ((gcd_Gcd @ int) = (^ [K6:(set @ int)]: (((semiring_1_of_nat @ int) @ ((gcd_Gcd @ nat) @ ((((image @ int) @ nat) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int))) @ K6))))))).
% 0.33/5.22 thf(fact_4790_nths__Cons, axiom, (! [A:$tType,X:A,L:(list @ A),A6:(set @ nat)]: (((((nths @ A) @ (((cons @ A) @ X) @ L)) @ A6) = (((append @ A) @ ((((if @ (list @ A)) @ (((member @ nat) @ (zero_zero @ nat)) @ A6)) @ (((cons @ A) @ X) @ (nil @ A))) @ (nil @ A))) @ (((nths @ A) @ L) @ ((collect @ nat) @ (^ [J3:nat]: ((((member @ nat) @ (suc @ J3)) @ A6)))))))))).
% 0.33/5.22 thf(fact_4791_nat_Osplit__sels_I1_J, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:(nat > A),Nat:nat]: (((P @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = (((Nat = (zero_zero @ nat)) => (P @ F1)) & ((Nat = (suc @ (pred @ Nat))) => (P @ (F22 @ (pred @ Nat))))))))).
% 0.33/5.22 thf(fact_4792_Code__Numeral_Onegative__def, axiom, (code_negative = (((((comp @ code_integer) @ code_integer) @ num) @ (uminus_uminus @ code_integer)) @ (numeral_numeral @ code_integer)))).
% 0.33/5.22 thf(fact_4793_finite__mono__strict__prefix__implies__finite__fixpoint, axiom, (! [A:$tType,F:(nat > (set @ A)),S3:(set @ A)]: (((! [I4:nat]: ((((ord_less_eq @ (set @ A)) @ (F @ I4)) @ S3))) => (((finite_finite2 @ A) @ S3) => ((? [N9:nat]: (((! [N4:nat]: (((((ord_less_eq @ nat) @ N4) @ N9) => (! [M6:nat]: (((((ord_less_eq @ nat) @ M6) @ N9) => ((((ord_less @ nat) @ M6) @ N4) => (((ord_less @ (set @ A)) @ (F @ M6)) @ (F @ N4))))))))) & (! [N4:nat]: (((((ord_less_eq @ nat) @ N9) @ N4) => ((F @ N9) = (F @ N4)))))))) => ((F @ ((finite_card @ A) @ S3)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ F) @ (top_top @ (set @ nat))))))))))).
% 0.33/5.22 thf(fact_4794_range__abs__Nats, axiom, (((((image @ int) @ int) @ (abs_abs @ int)) @ (top_top @ (set @ int))) = (semiring_1_Nats @ int))).
% 0.33/5.22 thf(fact_4795_Sup__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((complete_Sup_Sup @ A) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = Y3)))))))).
% 0.33/5.22 thf(fact_4796_cSup__atLeastAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => (((complete_Sup_Sup @ A) @ (((set_or1337092689740270186AtMost @ A) @ Y3) @ X)) = X)))))))).
% 0.33/5.22 thf(fact_4797_cSup__atLeastLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Sup_Sup @ A) @ (((set_or7035219750837199246ssThan @ A) @ Y3) @ X)) = X)))))))).
% 0.33/5.22 thf(fact_4798_Sup__atLeastLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Sup_Sup @ A) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y3)) = Y3)))))))).
% 0.33/5.22 thf(fact_4799_cSup__greaterThanAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Sup_Sup @ A) @ (((set_or3652927894154168847AtMost @ A) @ Y3) @ X)) = X)))))))).
% 0.33/5.22 thf(fact_4800_Sup__greaterThanAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Sup_Sup @ A) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y3)) = Y3)))))))).
% 0.33/5.22 thf(fact_4801_Sup__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Sup_Sup @ A) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y3)) = Y3)))))))).
% 0.33/5.22 thf(fact_4802_cSup__greaterThanLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Sup_Sup @ A) @ (((set_or5935395276787703475ssThan @ A) @ Y3) @ X)) = X)))))))).
% 0.33/5.22 thf(fact_4803_Nats__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (semiring_1_Nats @ A)))))).
% 0.33/5.22 thf(fact_4804_cSup__eq, axiom, (! [A:$tType]: ((((condit1219197933456340205attice @ A) & (no_bot @ A)) => (! [X8:(set @ A),A3:A]: (((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ X3) @ A3)))) => ((! [Y5:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X8) => (((ord_less_eq @ A) @ X4) @ Y5)))) => (((ord_less_eq @ A) @ A3) @ Y5)))) => (((complete_Sup_Sup @ A) @ X8) = A3))))))))).
% 0.33/5.22 thf(fact_4805_cSup__eq__maximum, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Z3:A,X8:(set @ A)]: (((((member @ A) @ Z3) @ X8) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ X3) @ Z3)))) => (((complete_Sup_Sup @ A) @ X8) = Z3))))))))).
% 0.33/5.22 thf(fact_4806_Nats__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (semiring_1_Nats @ A)))))).
% 0.33/5.22 thf(fact_4807_Nats__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => (((member @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (semiring_1_Nats @ A)))))))))).
% 0.33/5.22 thf(fact_4808_Nats__mult, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => (((member @ A) @ (((times_times @ A) @ A3) @ B2)) @ (semiring_1_Nats @ A)))))))))).
% 0.33/5.22 thf(fact_4809_Nats__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [W2:num]: ((((member @ A) @ ((numeral_numeral @ A) @ W2)) @ (semiring_1_Nats @ A)))))))).
% 0.33/5.22 thf(fact_4810_of__nat__in__Nats, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: ((((member @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (semiring_1_Nats @ A)))))))).
% 0.33/5.22 thf(fact_4811_Nats__induct, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X:A,P:(A > $o)]: (((((member @ A) @ X) @ (semiring_1_Nats @ A)) => ((! [N4:nat]: ((P @ ((semiring_1_of_nat @ A) @ N4)))) => (P @ X))))))))).
% 0.33/5.22 thf(fact_4812_Nats__cases, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X:A]: (((((member @ A) @ X) @ (semiring_1_Nats @ A)) => (~ ((! [N4:nat]: ((X != ((semiring_1_of_nat @ A) @ N4))))))))))))).
% 0.33/5.22 thf(fact_4813_fact__in__Nats, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((member @ A) @ ((semiring_char_0_fact @ A) @ N)) @ (semiring_1_Nats @ A)))))))).
% 0.33/5.22 thf(fact_4814_cSup__eq__non__empty, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A3:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ X3) @ A3)))) => ((! [Y5:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X8) => (((ord_less_eq @ A) @ X4) @ Y5)))) => (((ord_less_eq @ A) @ A3) @ Y5)))) => (((complete_Sup_Sup @ A) @ X8) = A3)))))))))).
% 0.33/5.22 thf(fact_4815_cSup__least, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),Z3:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ X3) @ Z3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ Z3))))))))).
% 0.33/5.22 thf(fact_4816_le__cSup__finite, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),X:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X) @ X8) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ X8)))))))))).
% 0.33/5.22 thf(fact_4817_less__cSupD, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Z3:A]: (((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ Z3) @ ((complete_Sup_Sup @ A) @ X8)) => (? [X3:A]: (((((member @ A) @ X3) @ X8) & (((ord_less @ A) @ Z3) @ X3)))))))))))).
% 0.33/5.22 thf(fact_4818_less__cSupE, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [Y3:A,X8:(set @ A)]: (((((ord_less @ A) @ Y3) @ ((complete_Sup_Sup @ A) @ X8)) => ((X8 != (bot_bot @ (set @ A))) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (~ ((((ord_less @ A) @ Y3) @ X3)))))))))))))))).
% 0.33/5.22 thf(fact_4819_finite__imp__Sup__less, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),X:A,A3:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X) @ X8) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less @ A) @ X3) @ A3)))) => (((ord_less @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ A3)))))))))).
% 0.33/5.22 thf(fact_4820_inf__Sup, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [A3:A,B7:(set @ A)]: (((((inf_inf @ A) @ A3) @ ((complete_Sup_Sup @ A) @ B7)) = ((complete_Sup_Sup @ A) @ ((((image @ A) @ A) @ ((inf_inf @ A) @ A3)) @ B7))))))))).
% 0.33/5.22 thf(fact_4821_Sup__inf__eq__bot__iff, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [B7:(set @ A),A3:A]: ((((((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ B7)) @ A3) = (bot_bot @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ B7) => ((((inf_inf @ A) @ X2) @ A3) = (bot_bot @ A)))))))))))).
% 0.33/5.22 thf(fact_4822_finite__subset__Union, axiom, (! [A:$tType,A6:(set @ A),B12:(set @ (set @ A))]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ ((complete_Sup_Sup @ (set @ A)) @ B12)) => (~ ((! [F6:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ F6) => ((((ord_less_eq @ (set @ (set @ A))) @ F6) @ B12) => (~ ((((ord_less_eq @ (set @ A)) @ A6) @ ((complete_Sup_Sup @ (set @ A)) @ F6))))))))))))))).
% 0.33/5.22 thf(fact_4823_card__Union__le__sum__card, axiom, (! [A:$tType,U4:(set @ (set @ A))]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ U4))) @ ((((groups7311177749621191930dd_sum @ (set @ A)) @ nat) @ (finite_card @ A)) @ U4))))).
% 0.33/5.22 thf(fact_4824_SUP__inf, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [F:(B > A),B7:(set @ B),A3:A]: (((((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ B7))) @ A3) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [B3:B]: ((((inf_inf @ A) @ (F @ B3)) @ A3)))) @ B7))))))))).
% 0.33/5.22 thf(fact_4825_Sup__inf, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [B7:(set @ A),A3:A]: (((((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ B7)) @ A3) = ((complete_Sup_Sup @ A) @ ((((image @ A) @ A) @ (^ [B3:A]: ((((inf_inf @ A) @ B3) @ A3)))) @ B7))))))))).
% 0.33/5.22 thf(fact_4826_inf__SUP, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [A3:A,F:(B > A),B7:(set @ B)]: (((((inf_inf @ A) @ A3) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ B7))) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [B3:B]: ((((inf_inf @ A) @ A3) @ (F @ B3))))) @ B7))))))))).
% 0.33/5.22 thf(fact_4827_SUP__inf__distrib2, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [F:(B > A),A6:(set @ B),G:(C > A),B7:(set @ C)]: (((((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B7))) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [A5:B]: (((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (^ [B3:C]: ((((inf_inf @ A) @ (F @ A5)) @ (G @ B3))))) @ B7))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4828_cSUP__least, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),F:(B > A),M7:A]: (((A6 != (bot_bot @ (set @ B))) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ M7)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ M7))))))))).
% 0.33/5.22 thf(fact_4829_finite__Sup__less__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),A3:A]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ A3) = (! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less @ A) @ X2) @ A3))))))))))))).
% 0.33/5.22 thf(fact_4830_cSup__abs__le, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),A3:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ A3)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ ((complete_Sup_Sup @ A) @ S3))) @ A3))))))))).
% 0.33/5.22 thf(fact_4831_Nats__diff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => ((((ord_less_eq @ A) @ B2) @ A3) => (((member @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ (semiring_1_Nats @ A))))))))))).
% 0.33/5.22 thf(fact_4832_sum_OUnion__comp, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B7:(set @ (set @ B)),G:(B > A)]: (((! [X3:(set @ B)]: (((((member @ (set @ B)) @ X3) @ B7) => ((finite_finite2 @ B) @ X3)))) => ((! [A14:(set @ B)]: (((((member @ (set @ B)) @ A14) @ B7) => (! [A25:(set @ B)]: (((((member @ (set @ B)) @ A25) @ B7) => ((A14 != A25) => (! [X3:B]: (((((member @ B) @ X3) @ A14) => ((((member @ B) @ X3) @ A25) => ((G @ X3) = (zero_zero @ A))))))))))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ B7)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7311177749621191930dd_sum @ (set @ B)) @ A)) @ ((groups7311177749621191930dd_sum @ B) @ A)) @ G) @ B7)))))))))).
% 0.33/5.22 thf(fact_4833_prod_OUnion__comp, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B7:(set @ (set @ B)),G:(B > A)]: (((! [X3:(set @ B)]: (((((member @ (set @ B)) @ X3) @ B7) => ((finite_finite2 @ B) @ X3)))) => ((! [A14:(set @ B)]: (((((member @ (set @ B)) @ A14) @ B7) => (! [A25:(set @ B)]: (((((member @ (set @ B)) @ A25) @ B7) => ((A14 != A25) => (! [X3:B]: (((((member @ B) @ X3) @ A14) => ((((member @ B) @ X3) @ A25) => ((G @ X3) = (one_one @ A))))))))))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ B7)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7121269368397514597t_prod @ (set @ B)) @ A)) @ ((groups7121269368397514597t_prod @ B) @ A)) @ G) @ B7)))))))))).
% 0.33/5.22 thf(fact_4834_card__Union__le__sum__card__weak, axiom, (! [A:$tType,U4:(set @ (set @ A))]: (((! [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ U4) => ((finite_finite2 @ A) @ X3)))) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ U4))) @ ((((groups7311177749621191930dd_sum @ (set @ A)) @ nat) @ (finite_card @ A)) @ U4)))))).
% 0.33/5.22 thf(fact_4835_Nats__subset__Ints, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ord_less_eq @ (set @ A)) @ (semiring_1_Nats @ A)) @ (ring_1_Ints @ A)))))).
% 0.33/5.22 thf(fact_4836_UN__UN__finite__eq, axiom, (! [A:$tType,A6:(nat > (set @ A))]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ (^ [N3:nat]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))))) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (top_top @ (set @ nat)))))))).
% 0.33/5.22 thf(fact_4837_cSup__asclose, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),L:A,E:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ L))) @ E)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ L))) @ E))))))))).
% 0.33/5.22 thf(fact_4838_card__UN__le, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A6:(A > (set @ B))]: ((((finite_finite2 @ A) @ I6) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A6) @ I6)))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I5:A]: (((finite_card @ B) @ (A6 @ I5))))) @ I6)))))).
% 0.33/5.22 thf(fact_4839_UN__finite__subset, axiom, (! [A:$tType,A6:(nat > (set @ A)),C6:(set @ A)]: (((! [N4:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N4)))) @ C6))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (top_top @ (set @ nat))))) @ C6))))).
% 0.33/5.22 thf(fact_4840_UN__finite2__eq, axiom, (! [A:$tType,A6:(nat > (set @ A)),B7:(nat > (set @ A)),K:nat]: (((! [N4:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N4))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N4) @ K))))))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B7) @ (top_top @ (set @ nat))))))))).
% 0.33/5.22 thf(fact_4841_Nats__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_Nats @ A) = ((((image @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (top_top @ (set @ nat)))))))).
% 0.33/5.22 thf(fact_4842_card__partition, axiom, (! [A:$tType,C6:(set @ (set @ A)),K:nat]: ((((finite_finite2 @ (set @ A)) @ C6) => (((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C6)) => ((! [C3:(set @ A)]: (((((member @ (set @ A)) @ C3) @ C6) => (((finite_card @ A) @ C3) = K)))) => ((! [C1:(set @ A),C22:(set @ A)]: (((((member @ (set @ A)) @ C1) @ C6) => ((((member @ (set @ A)) @ C22) @ C6) => ((C1 != C22) => ((((inf_inf @ (set @ A)) @ C1) @ C22) = (bot_bot @ (set @ A)))))))) => ((((times_times @ nat) @ K) @ ((finite_card @ (set @ A)) @ C6)) = ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C6)))))))))).
% 0.33/5.22 thf(fact_4843_Nats__altdef2, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((semiring_1_Nats @ A) = ((collect @ A) @ (^ [N3:A]: (((((member @ A) @ N3) @ (ring_1_Ints @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ N3)))))))))).
% 0.33/5.22 thf(fact_4844_UN__finite2__subset, axiom, (! [A:$tType,A6:(nat > (set @ A)),B7:(nat > (set @ A)),K:nat]: (((! [N4:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N4)))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N4) @ K))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A6) @ (top_top @ (set @ nat))))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B7) @ (top_top @ (set @ nat))))))))).
% 0.33/5.22 thf(fact_4845_UN__le__eq__Un0, axiom, (! [A:$tType,M7:(nat > (set @ A)),N:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ ((set_ord_atMost @ nat) @ N))) = (((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N)))) @ (M7 @ (zero_zero @ nat))))))).
% 0.33/5.22 thf(fact_4846_SUP__eq__top__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F:(B > A),A6:(set @ B)]: (((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)) = (top_top @ A)) = (! [X2:A]: (((((ord_less @ A) @ X2) @ (top_top @ A)) => (? [Y6:B]: (((((member @ B) @ Y6) @ A6) & (((ord_less @ A) @ X2) @ (F @ Y6))))))))))))))).
% 0.33/5.22 thf(fact_4847_Sup__eq__top__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A6:(set @ A)]: (((((complete_Sup_Sup @ A) @ A6) = (top_top @ A)) = (! [X2:A]: (((((ord_less @ A) @ X2) @ (top_top @ A)) => (? [Y6:A]: (((((member @ A) @ Y6) @ A6) & (((ord_less @ A) @ X2) @ Y6)))))))))))))).
% 0.33/5.22 thf(fact_4848_mono__SUP, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F:(A > B),A6:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ C) @ B) @ (^ [X2:C]: ((F @ (A6 @ X2))))) @ I6))) @ (F @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ A6) @ I6))))))))))).
% 0.33/5.22 thf(fact_4849_Sup__nat__empty, axiom, (((complete_Sup_Sup @ nat) @ (bot_bot @ (set @ nat))) = (zero_zero @ nat))).
% 0.33/5.22 thf(fact_4850_Sup__upper2, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,A6:(set @ A),V:A]: (((((member @ A) @ U) @ A6) => ((((ord_less_eq @ A) @ V) @ U) => (((ord_less_eq @ A) @ V) @ ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_4851_Sup__le__iff, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),B2:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ B2) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X2) @ B2))))))))))).
% 0.33/5.22 thf(fact_4852_Sup__upper, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,A6:(set @ A)]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_4853_Sup__least, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),Z3:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ X3) @ Z3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ Z3)))))))).
% 0.33/5.22 thf(fact_4854_Sup__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((! [A8:A]: (((((member @ A) @ A8) @ A6) => (? [X4:A]: (((((member @ A) @ X4) @ B7) & (((ord_less_eq @ A) @ A8) @ X4))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ B7))))))))).
% 0.33/5.22 thf(fact_4855_Sup__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),X:A]: (((! [Y5:A]: (((((member @ A) @ Y5) @ A6) => (((ord_less_eq @ A) @ Y5) @ X)))) => ((! [Y5:A]: (((! [Z5:A]: (((((member @ A) @ Z5) @ A6) => (((ord_less_eq @ A) @ Z5) @ Y5)))) => (((ord_less_eq @ A) @ X) @ Y5)))) => (((complete_Sup_Sup @ A) @ A6) = X))))))))).
% 0.33/5.22 thf(fact_4856_less__Sup__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:A,S3:(set @ A)]: (((((ord_less @ A) @ A3) @ ((complete_Sup_Sup @ A) @ S3)) = (? [X2:A]: (((((member @ A) @ X2) @ S3) & (((ord_less @ A) @ A3) @ X2))))))))))).
% 0.33/5.22 thf(fact_4857_le__Sup__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [X:A,A6:(set @ A)]: (((((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A6)) = (! [Y6:A]: (((((ord_less @ A) @ Y6) @ X) => (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less @ A) @ Y6) @ X2)))))))))))))).
% 0.33/5.22 thf(fact_4858_SUP__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),B7:(set @ C),F:(B > A),G:(C > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (? [X4:C]: (((((member @ C) @ X4) @ B7) & (((ord_less_eq @ A) @ (F @ I4)) @ (G @ X4)))))))) => ((! [J2:C]: (((((member @ C) @ J2) @ B7) => (? [X4:B]: (((((member @ B) @ X4) @ A6) & (((ord_less_eq @ A) @ (G @ J2)) @ (F @ X4)))))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B7))))))))))).
% 0.33/5.22 thf(fact_4859_less__eq__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),U:A]: (((! [V4:A]: (((((member @ A) @ V4) @ A6) => (((ord_less_eq @ A) @ U) @ V4)))) => ((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_4860_Sup__subset__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ B7))))))))).
% 0.33/5.22 thf(fact_4861_SUP__upper2, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A6:(set @ B),U:A,F:(B > A)]: (((((member @ B) @ I) @ A6) => ((((ord_less_eq @ A) @ U) @ (F @ I)) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))))).
% 0.33/5.22 thf(fact_4862_SUP__le__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(B > A),A6:(set @ B),U:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ U) = (! [X2:B]: (((((member @ B) @ X2) @ A6) => (((ord_less_eq @ A) @ (F @ X2)) @ U))))))))))).
% 0.33/5.22 thf(fact_4863_SUP__upper, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A6:(set @ B),F:(B > A)]: (((((member @ B) @ I) @ A6) => (((ord_less_eq @ A) @ (F @ I)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)))))))))).
% 0.33/5.22 thf(fact_4864_SUP__mono_H, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(B > A),G:(B > A),A6:(set @ B)]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ A6)))))))))).
% 0.33/5.22 thf(fact_4865_SUP__least, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),F:(B > A),U:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ (F @ I4)) @ U)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ U)))))))).
% 0.33/5.22 thf(fact_4866_SUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),B7:(set @ C),F:(B > A),G:(C > A)]: (((! [N4:B]: (((((member @ B) @ N4) @ A6) => (? [X4:C]: (((((member @ C) @ X4) @ B7) & (((ord_less_eq @ A) @ (F @ N4)) @ (G @ X4)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B7)))))))))).
% 0.33/5.22 thf(fact_4867_SUP__eqI, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),F:(B > A),X:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ (F @ I4)) @ X)))) => ((! [Y5:A]: (((! [I3:B]: (((((member @ B) @ I3) @ A6) => (((ord_less_eq @ A) @ (F @ I3)) @ Y5)))) => (((ord_less_eq @ A) @ X) @ Y5)))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)) = X))))))))).
% 0.33/5.22 thf(fact_4868_SUP__lessD, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(B > A),A6:(set @ B),Y3:A,I:B]: (((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ Y3) => ((((member @ B) @ I) @ A6) => (((ord_less @ A) @ (F @ I)) @ Y3))))))))).
% 0.33/5.22 thf(fact_4869_less__SUP__iff, axiom, (! [A:$tType,B:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:A,F:(B > A),A6:(set @ B)]: (((((ord_less @ A) @ A3) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) = (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ A3) @ (F @ X2)))))))))))).
% 0.33/5.22 thf(fact_4870_le__SUP__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [X:A,F:(B > A),A6:(set @ B)]: (((((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) = (! [Y6:A]: (((((ord_less @ A) @ Y6) @ X) => (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ Y6) @ (F @ X2))))))))))))))).
% 0.33/5.22 thf(fact_4871_SUP__eq__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),C2:A,F:(B > A)]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ C2) @ (F @ I4))))) => ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ I6)) = C2) = (! [X2:B]: (((((member @ B) @ X2) @ I6) => ((F @ X2) = C2))))))))))))).
% 0.33/5.22 thf(fact_4872_Sup__inter__less__eq, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ (((inf_inf @ (set @ A)) @ A6) @ B7))) @ (((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ B7))))))))).
% 0.33/5.22 thf(fact_4873_SUP__subset__mono, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),B7:(set @ B),F:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ A6) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B7))))))))))).
% 0.33/5.22 thf(fact_4874_mono__Sup, axiom, (! [B:$tType,A:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F) @ A6))) @ (F @ ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_4875_card__UNION, axiom, (! [A:$tType,A6:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ A6) => ((! [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ A6) => ((finite_finite2 @ A) @ X3)))) => (((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A6)) = (nat2 @ ((((groups7311177749621191930dd_sum @ (set @ (set @ A))) @ int) @ (^ [I8:(set @ (set @ A))]: ((((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((plus_plus @ nat) @ ((finite_card @ (set @ A)) @ I8)) @ (one_one @ nat)))) @ ((semiring_1_of_nat @ int) @ ((finite_card @ A) @ ((complete_Inf_Inf @ (set @ A)) @ I8))))))) @ ((collect @ (set @ (set @ A))) @ (^ [I8:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ I8) @ A6) & (I8 != (bot_bot @ (set @ (set @ A)))))))))))))))).
% 0.33/5.22 thf(fact_4876_surj__int__encode, axiom, (((((image @ int) @ nat) @ nat_int_encode) @ (top_top @ (set @ int))) = (top_top @ (set @ nat)))).
% 0.33/5.22 thf(fact_4877_horner__sum__eq__sum__funpow, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F2:(B > A),A5:A,Xs3:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N3:nat]: (((((compow @ (A > A)) @ N3) @ ((times_times @ A) @ A5)) @ (F2 @ (((nth @ B) @ Xs3) @ N3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs3)))))))))).
% 0.33/5.22 thf(fact_4878_Suc__funpow, axiom, (! [N:nat]: (((((compow @ (nat > nat)) @ N) @ suc) = ((plus_plus @ nat) @ N))))).
% 0.33/5.22 thf(fact_4879_funpow__0, axiom, (! [A:$tType,F:(A > A),X:A]: ((((((compow @ (A > A)) @ (zero_zero @ nat)) @ F) @ X) = X)))).
% 0.33/5.22 thf(fact_4880_Inf__eq__bot__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A6:(set @ A)]: (((((complete_Inf_Inf @ A) @ A6) = (bot_bot @ A)) = (! [X2:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X2) => (? [Y6:A]: (((((member @ A) @ Y6) @ A6) & (((ord_less @ A) @ Y6) @ X2)))))))))))))).
% 0.33/5.22 thf(fact_4881_Inf__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((complete_Inf_Inf @ A) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y3)) = X)))))))).
% 0.33/5.22 thf(fact_4882_cInf__atLeastAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => (((complete_Inf_Inf @ A) @ (((set_or1337092689740270186AtMost @ A) @ Y3) @ X)) = Y3)))))))).
% 0.33/5.22 thf(fact_4883_cInf__atLeastLessThan, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Inf_Inf @ A) @ (((set_or7035219750837199246ssThan @ A) @ Y3) @ X)) = Y3)))))))).
% 0.33/5.22 thf(fact_4884_Inf__atLeastLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Inf_Inf @ A) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y3)) = X)))))))).
% 0.33/5.22 thf(fact_4885_surj__fn, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((image @ A) @ A) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ A))) => (((((image @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_4886_cInf__greaterThanAtMost, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Inf_Inf @ A) @ (((set_or3652927894154168847AtMost @ A) @ Y3) @ X)) = Y3)))))))).
% 0.33/5.22 thf(fact_4887_Inf__greaterThanAtMost, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Inf_Inf @ A) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y3)) = X)))))))).
% 0.33/5.22 thf(fact_4888_Inf__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (((complete_Inf_Inf @ A) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y3)) = X)))))))).
% 0.33/5.22 thf(fact_4889_cInf__greaterThanLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y3:A,X:A]: (((((ord_less @ A) @ Y3) @ X) => (((complete_Inf_Inf @ A) @ (((set_or5935395276787703475ssThan @ A) @ Y3) @ X)) = Y3)))))))).
% 0.33/5.22 thf(fact_4890_INF__eq__bot__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F:(B > A),A6:(set @ B)]: (((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6)) = (bot_bot @ A)) = (! [X2:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X2) => (? [Y6:B]: (((((member @ B) @ Y6) @ A6) & (((ord_less @ A) @ (F @ Y6)) @ X2)))))))))))))).
% 0.33/5.22 thf(fact_4891_Compl__UN, axiom, (! [A:$tType,B:$tType,B7:(B > (set @ A)),A6:(set @ B)]: ((((uminus_uminus @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B7) @ A6))) = ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X2:B]: (((uminus_uminus @ (set @ A)) @ (B7 @ X2))))) @ A6)))))).
% 0.33/5.22 thf(fact_4892_Compl__INT, axiom, (! [A:$tType,B:$tType,B7:(B > (set @ A)),A6:(set @ B)]: ((((uminus_uminus @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B7) @ A6))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X2:B]: (((uminus_uminus @ (set @ A)) @ (B7 @ X2))))) @ A6)))))).
% 0.33/5.22 thf(fact_4893_Inf__atMostLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: (((((ord_less @ A) @ (top_top @ A)) @ X) => (((complete_Inf_Inf @ A) @ ((set_ord_lessThan @ A) @ X)) = (bot_bot @ A))))))))).
% 0.33/5.22 thf(fact_4894_comp__funpow, axiom, (! [B:$tType,A:$tType,N:nat,F:(A > A)]: (((((compow @ ((B > A) > (B > A))) @ N) @ ((((comp @ A) @ A) @ B) @ F)) = ((((comp @ A) @ A) @ B) @ (((compow @ (A > A)) @ N) @ F)))))).
% 0.33/5.22 thf(fact_4895_Inf__greatest, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),Z3:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ Z3) @ X3)))) => (((ord_less_eq @ A) @ Z3) @ ((complete_Inf_Inf @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_4896_le__Inf__iff, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B2:A,A6:(set @ A)]: (((((ord_less_eq @ A) @ B2) @ ((complete_Inf_Inf @ A) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ B2) @ X2))))))))))).
% 0.33/5.22 thf(fact_4897_Inf__lower2, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,A6:(set @ A),V:A]: (((((member @ A) @ U) @ A6) => ((((ord_less_eq @ A) @ U) @ V) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ V))))))))).
% 0.33/5.22 thf(fact_4898_Inf__lower, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,A6:(set @ A)]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ X)))))))).
% 0.33/5.22 thf(fact_4899_Inf__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B7:(set @ A),A6:(set @ A)]: (((! [B5:A]: (((((member @ A) @ B5) @ B7) => (? [X4:A]: (((((member @ A) @ X4) @ A6) & (((ord_less_eq @ A) @ X4) @ B5))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Inf_Inf @ A) @ B7))))))))).
% 0.33/5.22 thf(fact_4900_Inf__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),X:A]: (((! [I4:A]: (((((member @ A) @ I4) @ A6) => (((ord_less_eq @ A) @ X) @ I4)))) => ((! [Y5:A]: (((! [I3:A]: (((((member @ A) @ I3) @ A6) => (((ord_less_eq @ A) @ Y5) @ I3)))) => (((ord_less_eq @ A) @ Y5) @ X)))) => (((complete_Inf_Inf @ A) @ A6) = X))))))))).
% 0.33/5.22 thf(fact_4901_cInf__eq__minimum, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Z3:A,X8:(set @ A)]: (((((member @ A) @ Z3) @ X8) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ Z3) @ X3)))) => (((complete_Inf_Inf @ A) @ X8) = Z3))))))))).
% 0.33/5.22 thf(fact_4902_cInf__eq, axiom, (! [A:$tType]: ((((condit1219197933456340205attice @ A) & (no_top @ A)) => (! [X8:(set @ A),A3:A]: (((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ A3) @ X3)))) => ((! [Y5:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X8) => (((ord_less_eq @ A) @ Y5) @ X4)))) => (((ord_less_eq @ A) @ Y5) @ A3)))) => (((complete_Inf_Inf @ A) @ X8) = A3))))))))).
% 0.33/5.22 thf(fact_4903_Inf__less__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [S3:(set @ A),A3:A]: (((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ A3) = (? [X2:A]: (((((member @ A) @ X2) @ S3) & (((ord_less @ A) @ X2) @ A3))))))))))).
% 0.33/5.22 thf(fact_4904_mono__pow, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),N:nat]: (((((order_mono @ A) @ A) @ F) => (((order_mono @ A) @ A) @ (((compow @ (A > A)) @ N) @ F))))))))).
% 0.33/5.22 thf(fact_4905_funpow__mult, axiom, (! [A:$tType,N:nat,M:nat,F:(A > A)]: (((((compow @ (A > A)) @ N) @ (((compow @ (A > A)) @ M) @ F)) = (((compow @ (A > A)) @ (((times_times @ nat) @ M) @ N)) @ F))))).
% 0.33/5.22 thf(fact_4906_int__encode__eq, axiom, (! [X:int,Y3:int]: ((((nat_int_encode @ X) = (nat_int_encode @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_4907_funpow__swap1, axiom, (! [A:$tType,F:(A > A),N:nat,X:A]: (((F @ ((((compow @ (A > A)) @ N) @ F) @ X)) = ((((compow @ (A > A)) @ N) @ F) @ (F @ X)))))).
% 0.33/5.22 thf(fact_4908_funpow__mod__eq, axiom, (! [A:$tType,N:nat,F:(A > A),X:A,M:nat]: (((((((compow @ (A > A)) @ N) @ F) @ X) = X) => (((((compow @ (A > A)) @ (((modulo_modulo @ nat) @ M) @ N)) @ F) @ X) = ((((compow @ (A > A)) @ M) @ F) @ X)))))).
% 0.33/5.22 thf(fact_4909_Inf__le__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A6:(set @ A),X:A]: (((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ X) = (! [Y6:A]: (((((ord_less @ A) @ X) @ Y6) => (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less @ A) @ X2) @ Y6)))))))))))))).
% 0.33/5.22 thf(fact_4910_INF__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),B7:(set @ C),G:(C > A),F:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (? [X4:C]: (((((member @ C) @ X4) @ B7) & (((ord_less_eq @ A) @ (G @ X4)) @ (F @ I4)))))))) => ((! [J2:C]: (((((member @ C) @ J2) @ B7) => (? [X4:B]: (((((member @ B) @ X4) @ A6) & (((ord_less_eq @ A) @ (F @ X4)) @ (G @ J2)))))))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6)) = ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ G) @ B7))))))))))).
% 0.33/5.22 thf(fact_4911_cInf__eq__non__empty, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A3:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ A3) @ X3)))) => ((! [Y5:A]: (((! [X4:A]: (((((member @ A) @ X4) @ X8) => (((ord_less_eq @ A) @ Y5) @ X4)))) => (((ord_less_eq @ A) @ Y5) @ A3)))) => (((complete_Inf_Inf @ A) @ X8) = A3)))))))))).
% 0.33/5.22 thf(fact_4912_cInf__greatest, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),Z3:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ Z3) @ X3)))) => (((ord_less_eq @ A) @ Z3) @ ((complete_Inf_Inf @ A) @ X8)))))))))).
% 0.33/5.22 thf(fact_4913_Inf__less__eq, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),U:A]: (((! [V4:A]: (((((member @ A) @ V4) @ A6) => (((ord_less_eq @ A) @ V4) @ U)))) => ((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ U))))))))).
% 0.33/5.22 thf(fact_4914_cInf__le__finite, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),X:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ X))))))))).
% 0.33/5.22 thf(fact_4915_cInf__lessD, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Z3:A]: (((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Z3) => (? [X3:A]: (((((member @ A) @ X3) @ X8) & (((ord_less @ A) @ X3) @ Z3)))))))))))).
% 0.33/5.22 thf(fact_4916_Inf__superset__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B7:(set @ A),A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B7) @ A6) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Inf_Inf @ A) @ B7))))))))).
% 0.33/5.22 thf(fact_4917_finite__imp__less__Inf, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),X:A,A3:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X) @ X8) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less @ A) @ A3) @ X3)))) => (((ord_less @ A) @ A3) @ ((complete_Inf_Inf @ A) @ X8))))))))))).
% 0.33/5.22 thf(fact_4918_funpow__times__power, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [F:(A > nat),X:A]: (((((compow @ (A > A)) @ (F @ X)) @ ((times_times @ A) @ X)) = ((times_times @ A) @ (((power_power @ A) @ X) @ (F @ X)))))))))).
% 0.33/5.22 thf(fact_4919_funpow__Suc__right, axiom, (! [A:$tType,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (suc @ N)) @ F) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ F))))).
% 0.33/5.22 thf(fact_4920_funpow_Osimps_I2_J, axiom, (! [A:$tType,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (suc @ N)) @ F) = (((((comp @ A) @ A) @ A) @ F) @ (((compow @ (A > A)) @ N) @ F)))))).
% 0.33/5.22 thf(fact_4921_sup__Inf, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [A3:A,B7:(set @ A)]: (((((sup_sup @ A) @ A3) @ ((complete_Inf_Inf @ A) @ B7)) = ((complete_Inf_Inf @ A) @ ((((image @ A) @ A) @ ((sup_sup @ A) @ A3)) @ B7))))))))).
% 0.33/5.22 thf(fact_4922_Inf__sup__eq__top__iff, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [B7:(set @ A),A3:A]: ((((((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ B7)) @ A3) = (top_top @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ B7) => ((((sup_sup @ A) @ X2) @ A3) = (top_top @ A)))))))))))).
% 0.33/5.22 thf(fact_4923_funpow__add, axiom, (! [A:$tType,M:nat,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (((plus_plus @ nat) @ M) @ N)) @ F) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ M) @ F)) @ (((compow @ (A > A)) @ N) @ F)))))).
% 0.33/5.22 thf(fact_4924_funpow__mono, axiom, (! [A:$tType]: (((order @ A) => (! [F:(A > A),A6:A,B7:A,N:nat]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ A6) @ B7) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ N) @ F) @ A6)) @ ((((compow @ (A > A)) @ N) @ F) @ B7)))))))))).
% 0.33/5.22 thf(fact_4925_INF__greatest, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),U:A,F:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ U) @ (F @ I4))))) => (((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6)))))))))).
% 0.33/5.22 thf(fact_4926_le__INF__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,F:(B > A),A6:(set @ B)]: (((((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) = (! [X2:B]: (((((member @ B) @ X2) @ A6) => (((ord_less_eq @ A) @ U) @ (F @ X2)))))))))))).
% 0.33/5.22 thf(fact_4927_INF__lower2, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A6:(set @ B),F:(B > A),U:A]: (((((member @ B) @ I) @ A6) => ((((ord_less_eq @ A) @ (F @ I)) @ U) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ U))))))))).
% 0.33/5.22 thf(fact_4928_INF__mono_H, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(B > A),G:(B > A),A6:(set @ B)]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ A6)))))))))).
% 0.33/5.22 thf(fact_4929_INF__lower, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A6:(set @ B),F:(B > A)]: (((((member @ B) @ I) @ A6) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ (F @ I))))))))).
% 0.33/5.22 thf(fact_4930_INF__mono, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B7:(set @ B),A6:(set @ C),F:(C > A),G:(B > A)]: (((! [M6:B]: (((((member @ B) @ M6) @ B7) => (? [X4:C]: (((((member @ C) @ X4) @ A6) & (((ord_less_eq @ A) @ (F @ X4)) @ (G @ M6)))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ F) @ A6))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B7)))))))))).
% 0.33/5.22 thf(fact_4931_INF__eqI, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),X:A,F:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A6) => (((ord_less_eq @ A) @ X) @ (F @ I4))))) => ((! [Y5:A]: (((! [I3:B]: (((((member @ B) @ I3) @ A6) => (((ord_less_eq @ A) @ Y5) @ (F @ I3))))) => (((ord_less_eq @ A) @ Y5) @ X)))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6)) = X))))))))).
% 0.33/5.22 thf(fact_4932_less__INF__D, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [Y3:A,F:(B > A),A6:(set @ B),I:B]: (((((ord_less @ A) @ Y3) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) => ((((member @ B) @ I) @ A6) => (((ord_less @ A) @ Y3) @ (F @ I)))))))))).
% 0.33/5.22 thf(fact_4933_INF__less__iff, axiom, (! [A:$tType,B:$tType]: (((comple5582772986160207858norder @ A) => (! [F:(B > A),A6:(set @ B),A3:A]: (((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ A3) = (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ (F @ X2)) @ A3))))))))))).
% 0.33/5.22 thf(fact_4934_INF__sup, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [F:(B > A),B7:(set @ B),A3:A]: (((((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ B7))) @ A3) = ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [B3:B]: ((((sup_sup @ A) @ (F @ B3)) @ A3)))) @ B7))))))))).
% 0.33/5.22 thf(fact_4935_Inf__sup, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [B7:(set @ A),A3:A]: (((((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ B7)) @ A3) = ((complete_Inf_Inf @ A) @ ((((image @ A) @ A) @ (^ [B3:A]: ((((sup_sup @ A) @ B3) @ A3)))) @ B7))))))))).
% 0.33/5.22 thf(fact_4936_sup__INF, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [A3:A,F:(B > A),B7:(set @ B)]: (((((sup_sup @ A) @ A3) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ B7))) = ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [B3:B]: ((((sup_sup @ A) @ A3) @ (F @ B3))))) @ B7))))))))).
% 0.33/5.22 thf(fact_4937_INF__sup__distrib2, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [F:(B > A),A6:(set @ B),G:(C > A),B7:(set @ C)]: (((((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ G) @ B7))) = ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [A5:B]: (((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ (^ [B3:C]: ((((sup_sup @ A) @ (F @ A5)) @ (G @ B3))))) @ B7))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4938_INF__le__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F:(B > A),A6:(set @ B),X:A]: (((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ X) = (! [Y6:A]: (((((ord_less @ A) @ X) @ Y6) => (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ (F @ X2)) @ Y6)))))))))))))).
% 0.33/5.22 thf(fact_4939_cINF__greatest, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),M:A,F:(B > A)]: (((A6 != (bot_bot @ (set @ B))) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ M) @ (F @ X3))))) => (((ord_less_eq @ A) @ M) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))))).
% 0.33/5.22 thf(fact_4940_INF__eq__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),F:(B > A),C2:A]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ (F @ I4)) @ C2)))) => ((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ I6)) = C2) = (! [X2:B]: (((((member @ B) @ X2) @ I6) => ((F @ X2) = C2))))))))))))).
% 0.33/5.22 thf(fact_4941_finite__less__Inf__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),A3:A]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ A3) @ ((complete_Inf_Inf @ A) @ X8)) = (! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less @ A) @ A3) @ X2))))))))))))).
% 0.33/5.22 thf(fact_4942_Inf__le__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A)]: (((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_4943_cInf__abs__ge, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),A3:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ A3)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ ((complete_Inf_Inf @ A) @ S3))) @ A3))))))))).
% 0.33/5.22 thf(fact_4944_uminus__Sup, axiom, (! [A:$tType]: (((comple489889107523837845lgebra @ A) => (! [A6:(set @ A)]: ((((uminus_uminus @ A) @ ((complete_Sup_Sup @ A) @ A6)) = ((complete_Inf_Inf @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ A6))))))))).
% 0.33/5.22 thf(fact_4945_uminus__Inf, axiom, (! [A:$tType]: (((comple489889107523837845lgebra @ A) => (! [A6:(set @ A)]: ((((uminus_uminus @ A) @ ((complete_Inf_Inf @ A) @ A6)) = ((complete_Sup_Sup @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ A6))))))))).
% 0.33/5.22 thf(fact_4946_less__eq__Inf__inter, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Inf_Inf @ A) @ B7))) @ ((complete_Inf_Inf @ A) @ (((inf_inf @ (set @ A)) @ A6) @ B7))))))))).
% 0.33/5.22 thf(fact_4947_Kleene__iter__lpfp, axiom, (! [A:$tType]: (((order_bot @ A) => (! [F:(A > A),P5:A,K:nat]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ (F @ P5)) @ P5) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ K) @ F) @ (bot_bot @ A))) @ P5))))))))).
% 0.33/5.22 thf(fact_4948_Kleene__iter__gpfp, axiom, (! [A:$tType]: (((order_top @ A) => (! [F:(A > A),P5:A,K:nat]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ P5) @ (F @ P5)) => (((ord_less_eq @ A) @ P5) @ ((((compow @ (A > A)) @ K) @ F) @ (top_top @ A))))))))))).
% 0.33/5.22 thf(fact_4949_funpow__mono2, axiom, (! [A:$tType]: (((order @ A) => (! [F:(A > A),I:nat,J:nat,X:A,Y3:A]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ X) @ (F @ X)) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ I) @ F) @ X)) @ ((((compow @ (A > A)) @ J) @ F) @ Y3)))))))))))).
% 0.33/5.22 thf(fact_4950_INF__superset__mono, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B7:(set @ B),A6:(set @ B),F:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B7) @ A6) => ((! [X3:B]: (((((member @ B) @ X3) @ B7) => (((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B7))))))))))).
% 0.33/5.22 thf(fact_4951_uminus__INF, axiom, (! [A:$tType,B:$tType]: (((comple489889107523837845lgebra @ A) => (! [B7:(B > A),A6:(set @ B)]: ((((uminus_uminus @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ B7) @ A6))) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [X2:B]: (((uminus_uminus @ A) @ (B7 @ X2))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4952_uminus__SUP, axiom, (! [A:$tType,B:$tType]: (((comple489889107523837845lgebra @ A) => (! [B7:(B > A),A6:(set @ B)]: ((((uminus_uminus @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ B7) @ A6))) = ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [X2:B]: (((uminus_uminus @ A) @ (B7 @ X2))))) @ A6))))))))).
% 0.33/5.22 thf(fact_4953_mono__INF, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F:(A > B),A6:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ (F @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ A6) @ I6)))) @ ((complete_Inf_Inf @ B) @ ((((image @ C) @ B) @ (^ [X2:C]: ((F @ (A6 @ X2))))) @ I6)))))))))).
% 0.33/5.22 thf(fact_4954_mono__Inf, axiom, (! [B:$tType,A:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((ord_less_eq @ B) @ (F @ ((complete_Inf_Inf @ A) @ A6))) @ ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F) @ A6)))))))))).
% 0.33/5.22 thf(fact_4955_of__nat__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N3:nat]: (((((compow @ (A > A)) @ N3) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4956_numeral__add__unfold__funpow, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [K:num,A3:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ K)) @ A3) = ((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K)) @ ((plus_plus @ A) @ (one_one @ A))) @ A3)))))))).
% 0.33/5.22 thf(fact_4957_INF__le__SUP, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ B),F:(B > A)]: (((A6 != (bot_bot @ (set @ B))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)))))))))).
% 0.33/5.22 thf(fact_4958_mono__funpow, axiom, (! [A:$tType]: ((((lattice @ A) & (order_bot @ A)) => (! [Q:(A > A)]: (((((order_mono @ A) @ A) @ Q) => (((order_mono @ nat) @ A) @ (^ [I5:nat]: (((((compow @ (A > A)) @ I5) @ Q) @ (bot_bot @ A)))))))))))).
% 0.33/5.22 thf(fact_4959_cInf__asclose, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),L:A,E:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ L))) @ E)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ L))) @ E))))))))).
% 0.33/5.22 thf(fact_4960_funpow__decreasing, axiom, (! [A:$tType]: ((((lattice @ A) & (order_bot @ A)) => (! [M:nat,N:nat,F:(A > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((order_mono @ A) @ A) @ F) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ M) @ F) @ (bot_bot @ A))) @ ((((compow @ (A > A)) @ N) @ F) @ (bot_bot @ A))))))))))).
% 0.33/5.22 thf(fact_4961_funpow__increasing, axiom, (! [A:$tType]: ((((lattice @ A) & (order_top @ A)) => (! [M:nat,N:nat,F:(A > A)]: (((((ord_less_eq @ nat) @ M) @ N) => ((((order_mono @ A) @ A) @ F) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ N) @ F) @ (top_top @ A))) @ ((((compow @ (A > A)) @ M) @ F) @ (top_top @ A))))))))))).
% 0.33/5.22 thf(fact_4962_numeral__unfold__funpow, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((numeral_numeral @ A) = (^ [K3:num]: (((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K3)) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_4963_SUP__INF, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [P:(C > (B > A))]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [Y6:B]: (((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ (^ [X2:C]: (((P @ X2) @ Y6)))) @ (top_top @ (set @ C))))))) @ (top_top @ (set @ B)))) = ((complete_Inf_Inf @ A) @ ((((image @ (B > C)) @ A) @ (^ [X2:(B > C)]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [Y6:B]: (((P @ (X2 @ Y6)) @ Y6)))) @ (top_top @ (set @ B))))))) @ (top_top @ (set @ (B > C)))))))))))).
% 0.33/5.22 thf(fact_4964_INF__SUP, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [P:(C > (B > A))]: ((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [Y6:B]: (((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ (^ [X2:C]: (((P @ X2) @ Y6)))) @ (top_top @ (set @ C))))))) @ (top_top @ (set @ B)))) = ((complete_Sup_Sup @ A) @ ((((image @ (B > C)) @ A) @ (^ [F2:(B > C)]: (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [X2:B]: (((P @ (F2 @ X2)) @ X2)))) @ (top_top @ (set @ B))))))) @ (top_top @ (set @ (B > C)))))))))))).
% 0.33/5.22 thf(fact_4965_relpowp__fun__conv, axiom, (! [A:$tType]: (((compow @ (A > (A > $o))) = (^ [N3:nat,P4:(A > (A > $o)),X2:A,Y6:A]: ((? [F2:(nat > A)]: ((((F2 @ (zero_zero @ nat)) = X2) & (((F2 @ N3) = Y6) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ N3) => ((P4 @ (F2 @ I5)) @ (F2 @ (suc @ I5)))))))))))))))).
% 0.33/5.22 thf(fact_4966_relpowp__1, axiom, (! [A:$tType,P:(A > (A > $o))]: (((((compow @ (A > (A > $o))) @ (one_one @ nat)) @ P) = P)))).
% 0.33/5.22 thf(fact_4967_Nat_Ofunpow__code__def, axiom, (! [A:$tType]: (((funpow @ A) = (compow @ (A > A)))))).
% 0.33/5.22 thf(fact_4968_Inf__int__def, axiom, ((complete_Inf_Inf @ int) = (^ [X7:(set @ int)]: (((uminus_uminus @ int) @ ((complete_Sup_Sup @ int) @ ((((image @ int) @ int) @ (uminus_uminus @ int)) @ X7))))))).
% 0.33/5.22 thf(fact_4969_relpowp__Suc__I2, axiom, (! [A:$tType,P:(A > (A > $o)),X:A,Y3:A,N:nat,Z3:A]: ((((P @ X) @ Y3) => ((((((compow @ (A > (A > $o))) @ N) @ P) @ Y3) @ Z3) => (((((compow @ (A > (A > $o))) @ (suc @ N)) @ P) @ X) @ Z3)))))).
% 0.33/5.22 thf(fact_4970_relpowp__Suc__E2, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N)) @ P) @ X) @ Z3) => (~ ((! [Y5:A]: ((((P @ X) @ Y5) => (~ ((((((compow @ (A > (A > $o))) @ N) @ P) @ Y5) @ Z3)))))))))))).
% 0.33/5.22 thf(fact_4971_relpowp__Suc__D2, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N)) @ P) @ X) @ Z3) => (? [Y5:A]: ((((P @ X) @ Y5) & (((((compow @ (A > (A > $o))) @ N) @ P) @ Y5) @ Z3)))))))).
% 0.33/5.22 thf(fact_4972_relpowp__Suc__I, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Y3:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ N) @ P) @ X) @ Y3) => (((P @ Y3) @ Z3) => (((((compow @ (A > (A > $o))) @ (suc @ N)) @ P) @ X) @ Z3)))))).
% 0.33/5.22 thf(fact_4973_relpowp__Suc__E, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N)) @ P) @ X) @ Z3) => (~ ((! [Y5:A]: (((((((compow @ (A > (A > $o))) @ N) @ P) @ X) @ Y5) => (~ (((P @ Y5) @ Z3)))))))))))).
% 0.33/5.22 thf(fact_4974_relpowp__0__I, axiom, (! [A:$tType,P:(A > (A > $o)),X:A]: ((((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ P) @ X) @ X)))).
% 0.33/5.22 thf(fact_4975_relpowp__0__E, axiom, (! [A:$tType,P:(A > (A > $o)),X:A,Y3:A]: (((((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ P) @ X) @ Y3) => (X = Y3))))).
% 0.33/5.22 thf(fact_4976_relpowp_Osimps_I1_J, axiom, (! [A:$tType,R2:(A > (A > $o))]: (((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ R2) = (^ [Y:A,Z2:A]: ((Y = Z2))))))).
% 0.33/5.22 thf(fact_4977_relpowp__E, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ N) @ P) @ X) @ Z3) => (((N = (zero_zero @ nat)) => (X != Z3)) => (~ ((! [Y5:A,M6:nat]: (((N = (suc @ M6)) => ((((((compow @ (A > (A > $o))) @ M6) @ P) @ X) @ Y5) => (~ (((P @ Y5) @ Z3)))))))))))))).
% 0.33/5.22 thf(fact_4978_relpowp__E2, axiom, (! [A:$tType,N:nat,P:(A > (A > $o)),X:A,Z3:A]: (((((((compow @ (A > (A > $o))) @ N) @ P) @ X) @ Z3) => (((N = (zero_zero @ nat)) => (X != Z3)) => (~ ((! [Y5:A,M6:nat]: (((N = (suc @ M6)) => (((P @ X) @ Y5) => (~ ((((((compow @ (A > (A > $o))) @ M6) @ P) @ Y5) @ Z3)))))))))))))).
% 0.33/5.22 thf(fact_4979_relpowp__bot, axiom, (! [A:$tType,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((compow @ (A > (A > $o))) @ N) @ (bot_bot @ (A > (A > $o)))) = (bot_bot @ (A > (A > $o)))))))).
% 0.33/5.22 thf(fact_4980_set__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B)]: ((((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) = ((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [I5:nat]: (((Uu = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ B) @ Ys2) @ I5))) & (((ord_less @ nat) @ I5) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2))))))))))))).
% 0.33/5.22 thf(fact_4981_measure__function__int, axiom, ((fun_is_measure @ int) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int)))).
% 0.33/5.22 thf(fact_4982_surj__int__decode, axiom, (((((image @ nat) @ int) @ nat_int_decode) @ (top_top @ (set @ nat))) = (top_top @ (set @ int)))).
% 0.33/5.22 thf(fact_4983_int__decode__inverse, axiom, (! [N:nat]: (((nat_int_encode @ (nat_int_decode @ N)) = N)))).
% 0.33/5.22 thf(fact_4984_int__encode__inverse, axiom, (! [X:int]: (((nat_int_decode @ (nat_int_encode @ X)) = X)))).
% 0.33/5.22 thf(fact_4985_measure__snd, axiom, (! [A:$tType,B:$tType,F:(A > nat)]: ((((fun_is_measure @ A) @ F) => ((fun_is_measure @ ((product_prod @ B) @ A)) @ (^ [P6:((product_prod @ B) @ A)]: ((F @ (((product_snd @ B) @ A) @ P6))))))))).
% 0.33/5.22 thf(fact_4986_measure__fst, axiom, (! [B:$tType,A:$tType,F:(A > nat)]: ((((fun_is_measure @ A) @ F) => ((fun_is_measure @ ((product_prod @ A) @ B)) @ (^ [P6:((product_prod @ A) @ B)]: ((F @ (((product_fst @ A) @ B) @ P6))))))))).
% 0.33/5.22 thf(fact_4987_lexordp_Omono, axiom, (! [A:$tType]: (((ord @ A) => (((order_mono @ ((list @ A) > ((list @ A) > $o))) @ ((list @ A) > ((list @ A) > $o))) @ (^ [P6:((list @ A) > ((list @ A) > $o)),X12:(list @ A),X24:(list @ A)]: (((? [Y6:A,Ys3:(list @ A)]: (((X12 = (nil @ A)) & (X24 = (((cons @ A) @ Y6) @ Ys3))))) | ((? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((X12 = (((cons @ A) @ X2) @ Xs3)) & ((X24 = (((cons @ A) @ Y6) @ Ys3)) & (((ord_less @ A) @ X2) @ Y6))))) | (? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((X12 = (((cons @ A) @ X2) @ Xs3)) & ((X24 = (((cons @ A) @ Y6) @ Ys3)) & ((~ ((((ord_less @ A) @ X2) @ Y6))) & ((~ ((((ord_less @ A) @ Y6) @ X2))) & ((P6 @ Xs3) @ Ys3)))))))))))))))).
% 0.33/5.22 thf(fact_4988_is__measure_Osimps, axiom, (! [A:$tType]: (((fun_is_measure @ A) = (^ [A5:(A > nat)]: ((? [X7:(A > nat)]: ((((^ [Y:(A > nat),Z2:(A > nat)]: ((Y = Z2))) @ A5) @ X7))))))))).
% 0.33/5.22 thf(fact_4989_is__measure__trivial, axiom, (! [A:$tType,F:(A > nat)]: (((fun_is_measure @ A) @ F)))).
% 0.33/5.22 thf(fact_4990_int__decode__eq, axiom, (! [X:nat,Y3:nat]: ((((nat_int_decode @ X) = (nat_int_decode @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_4991_bot2E, axiom, (! [A:$tType,B:$tType,X:A,Y3:B]: ((~ ((((bot_bot @ (A > (B > $o))) @ X) @ Y3)))))).
% 0.33/5.22 thf(fact_4992_measure__size, axiom, (! [A:$tType]: (((size @ A) => ((fun_is_measure @ A) @ (size_size @ A)))))).
% 0.33/5.22 thf(fact_4993_set__conv__nth, axiom, (! [A:$tType]: (((set2 @ A) = (^ [Xs3:(list @ A)]: (((collect @ A) @ (^ [Uu:A]: ((? [I5:nat]: (((Uu = (((nth @ A) @ Xs3) @ I5)) & (((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs3)))))))))))))).
% 0.33/5.22 thf(fact_4994_set__nths, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((set2 @ A) @ (((nths @ A) @ Xs) @ I6)) = ((collect @ A) @ (^ [Uu:A]: ((? [I5:nat]: (((Uu = (((nth @ A) @ Xs) @ I5)) & ((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & (((member @ nat) @ I5) @ I6)))))))))))).
% 0.33/5.22 thf(fact_4995_sup__bot_Osemilattice__neutr__order__axioms, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (((((semila1105856199041335345_order @ A) @ (sup_sup @ A)) @ (bot_bot @ A)) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.22 thf(fact_4996_max__nat_Osemilattice__neutr__order__axioms, axiom, (((((semila1105856199041335345_order @ nat) @ (ord_max @ nat)) @ (zero_zero @ nat)) @ (^ [X2:nat,Y6:nat]: ((((ord_less_eq @ nat) @ Y6) @ X2)))) @ (^ [X2:nat,Y6:nat]: ((((ord_less @ nat) @ Y6) @ X2))))).
% 0.33/5.22 thf(fact_4997_times__int_Oabs__eq, axiom, (! [Xa3:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((times_times @ int) @ (abs_Integ @ Xa3)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ U3)) @ (((times_times @ nat) @ Y6) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ V5)) @ (((times_times @ nat) @ Y6) @ U3))))))))) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_4998_int_Oabs__induct, axiom, (! [P:(int > $o),X:int]: (((! [Y5:((product_prod @ nat) @ nat)]: ((P @ (abs_Integ @ Y5)))) => (P @ X))))).
% 0.33/5.22 thf(fact_4999_semilattice__neutr__order_Oneutr__eq__iff, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less) => ((Z3 = ((F @ A3) @ B2)) = ((A3 = Z3) & (B2 = Z3))))))).
% 0.33/5.22 thf(fact_5000_semilattice__neutr__order_Oeq__neutr__iff, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less) => ((((F @ A3) @ B2) = Z3) = ((A3 = Z3) & (B2 = Z3))))))).
% 0.33/5.22 thf(fact_5001_eq__Abs__Integ, axiom, (! [Z3:int]: ((~ ((! [X3:nat,Y5:nat]: ((Z3 != (abs_Integ @ ((((product_Pair @ nat) @ nat) @ X3) @ Y5)))))))))).
% 0.33/5.22 thf(fact_5002_mono__compose, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((order @ A) & (order @ C)) => (! [Q:(A > (B > C)),F:(D > B)]: (((((order_mono @ A) @ (B > C)) @ Q) => (((order_mono @ A) @ (D > C)) @ (^ [I5:A,X2:D]: (((Q @ I5) @ (F @ X2)))))))))))).
% 0.33/5.22 thf(fact_5003_zero__int__def, axiom, ((zero_zero @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5004_int__def, axiom, ((semiring_1_of_nat @ int) = (^ [N3:nat]: ((abs_Integ @ ((((product_Pair @ nat) @ nat) @ N3) @ (zero_zero @ nat))))))).
% 0.33/5.22 thf(fact_5005_nat_Oabs__eq, axiom, (! [X:((product_prod @ nat) @ nat)]: (((nat2 @ (abs_Integ @ X)) = (((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat)) @ X))))).
% 0.33/5.22 thf(fact_5006_Nats__altdef1, axiom, (! [A:$tType]: (((ring_1 @ A) => ((semiring_1_Nats @ A) = ((collect @ A) @ (^ [Uu:A]: ((? [N3:int]: (((Uu = ((ring_1_of_int @ A) @ N3)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ N3)))))))))))).
% 0.33/5.22 thf(fact_5007_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)) @ (^ [X2:nat,Y6:nat]: (((((product_Pair @ nat) @ nat) @ Y6) @ X2)))) @ X)))))).
% 0.33/5.22 thf(fact_5008_one__int__def, axiom, ((one_one @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5009_of__int_Oabs__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:((product_prod @ nat) @ nat)]: ((((ring_1_of_int @ A) @ (abs_Integ @ X)) = (((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3))))) @ X)))))))).
% 0.33/5.22 thf(fact_5010_gcd__nat_Osemilattice__neutr__order__axioms, axiom, (((((semila1105856199041335345_order @ nat) @ (gcd_gcd @ nat)) @ (zero_zero @ nat)) @ (dvd_dvd @ nat)) @ (^ [M2:nat,N3:nat]: (((((dvd_dvd @ nat) @ M2) @ N3) & (M2 != N3)))))).
% 0.33/5.22 thf(fact_5011_less__int_Oabs__eq, axiom, (! [Xa3:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((ord_less @ int) @ (abs_Integ @ Xa3)) @ (abs_Integ @ X)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6)))))))) @ Xa3) @ X))))).
% 0.33/5.22 thf(fact_5012_less__eq__int_Oabs__eq, axiom, (! [Xa3:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((ord_less_eq @ int) @ (abs_Integ @ Xa3)) @ (abs_Integ @ X)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6)))))))) @ Xa3) @ X))))).
% 0.33/5.22 thf(fact_5013_plus__int_Oabs__eq, axiom, (! [Xa3:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((plus_plus @ int) @ (abs_Integ @ Xa3)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ U3)) @ (((plus_plus @ nat) @ Y6) @ V5)))))))) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_5014_minus__int_Oabs__eq, axiom, (! [Xa3:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((minus_minus @ int) @ (abs_Integ @ Xa3)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ Y6) @ U3)))))))) @ Xa3) @ X)))))).
% 0.33/5.22 thf(fact_5015_inf__top_Osemilattice__neutr__order__axioms, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (((((semila1105856199041335345_order @ A) @ (inf_inf @ A)) @ (top_top @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.22 thf(fact_5016_take__bit__numeral__minus__numeral__int, axiom, (! [M:num,N:num]: (((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (^ [Q3:num]: ((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ M)) @ (((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ M))) @ ((numeral_numeral @ int) @ Q3)))))) @ ((bit_take_bit_num @ ((numeral_numeral @ nat) @ M)) @ N)))))).
% 0.33/5.22 thf(fact_5017_prod_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),P5:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((P5 @ X2) != (one_one @ A))))))) => (((((member @ B) @ I) @ I6) => (((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (((insert @ B) @ I) @ I6)) = ((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ I6))) & ((~ ((((member @ B) @ I) @ I6))) => (((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (((insert @ B) @ I) @ I6)) = (((times_times @ A) @ (P5 @ I)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ I6)))))))))))).
% 0.33/5.22 thf(fact_5018_sorted__insort__is__snoc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),A3:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X3) @ A3)))) => ((((((linorder_insort_key @ A) @ A) @ (^ [X2:A]: (X2))) @ A3) @ Xs) = (((append @ A) @ Xs) @ (((cons @ A) @ A3) @ (nil @ A)))))))))))).
% 0.33/5.22 thf(fact_5019_length__insort, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),X:B,Xs:(list @ B)]: ((((size_size @ (list @ B)) @ (((((linorder_insort_key @ B) @ A) @ F) @ X) @ Xs)) = (suc @ ((size_size @ (list @ B)) @ Xs))))))))).
% 0.33/5.22 thf(fact_5020_prod_Oempty_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [P5:(B > A)]: ((((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))).
% 0.33/5.22 thf(fact_5021_take__bit__numeral__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:num,N:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_take_bit_num @ ((numeral_numeral @ nat) @ M)) @ N))))))))).
% 0.33/5.22 thf(fact_5022_prod_Onon__neutral_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A),I6:(set @ B)]: ((((((groups1962203154675924110t_prod @ B) @ A) @ G) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((G @ X2) != (one_one @ A))))))) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I6)))))))).
% 0.33/5.22 thf(fact_5023_insort__key_Osimps_I2_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),X:B,Y3:B,Ys2:(list @ B)]: ((((((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3)) => ((((((linorder_insort_key @ B) @ A) @ F) @ X) @ (((cons @ B) @ Y3) @ Ys2)) = (((cons @ B) @ X) @ (((cons @ B) @ Y3) @ Ys2)))) & ((~ ((((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3)))) => ((((((linorder_insort_key @ B) @ A) @ F) @ X) @ (((cons @ B) @ Y3) @ Ys2)) = (((cons @ B) @ Y3) @ (((((linorder_insort_key @ B) @ A) @ F) @ X) @ Ys2))))))))))).
% 0.33/5.22 thf(fact_5024_sorted__insort, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((((linorder_insort_key @ A) @ A) @ (^ [X2:A]: (X2))) @ X) @ Xs)) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))).
% 0.33/5.22 thf(fact_5025_prod_Odistrib__triv_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ I6) => (((((groups1962203154675924110t_prod @ B) @ A) @ (^ [I5:B]: ((((times_times @ A) @ (G @ I5)) @ (H2 @ I5))))) @ I6) = (((times_times @ A) @ ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I6)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ I6)))))))))).
% 0.33/5.22 thf(fact_5026_insort__is__Cons, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Xs:(list @ B),F:(B > A),A3:B]: (((! [X3:B]: (((((member @ B) @ X3) @ ((set2 @ B) @ Xs)) => (((ord_less_eq @ A) @ (F @ A3)) @ (F @ X3))))) => ((((((linorder_insort_key @ B) @ A) @ F) @ A3) @ Xs) = (((cons @ B) @ A3) @ Xs))))))))).
% 0.33/5.22 thf(fact_5027_sorted__insort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),X:B,Xs:(list @ B)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ (((((linorder_insort_key @ B) @ A) @ F) @ X) @ Xs))) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs))))))))).
% 0.33/5.22 thf(fact_5028_prod_Omono__neutral__cong__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A),H2:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6) = ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ S3))))))))))).
% 0.33/5.22 thf(fact_5029_prod_Omono__neutral__cong__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T6:(set @ B),H2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((H2 @ I4) = (one_one @ A))))) => ((! [X3:B]: (((((member @ B) @ X3) @ S3) => ((G @ X3) = (H2 @ X3))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3) = ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ T6))))))))))).
% 0.33/5.22 thf(fact_5030_prod_Omono__neutral__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3)))))))))).
% 0.33/5.22 thf(fact_5031_prod_Omono__neutral__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T6) => ((! [X3:B]: (((((member @ B) @ X3) @ (((minus_minus @ (set @ B)) @ T6) @ S3)) => ((G @ X3) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6)))))))))).
% 0.33/5.22 thf(fact_5032_prod_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((G @ X2) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I6) & ((H2 @ X2) != (one_one @ A))))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ (^ [I5:B]: ((((times_times @ A) @ (G @ I5)) @ (H2 @ I5))))) @ I6) = (((times_times @ A) @ ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I6)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ I6))))))))))).
% 0.33/5.22 thf(fact_5033_prod_OG__def, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (((groups1962203154675924110t_prod @ B) @ A) = (^ [P6:(B > A),I8:(set @ B)]: (((((if @ A) @ ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I8) & ((P6 @ X2) != (one_one @ A)))))))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ P6) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ I8) & ((P6 @ X2) != (one_one @ A)))))))) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5034_filter__insort, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),P:(B > $o),X:B]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => ((P @ X) => ((((filter @ B) @ P) @ (((((linorder_insort_key @ B) @ A) @ F) @ X) @ Xs)) = (((((linorder_insort_key @ B) @ A) @ F) @ X) @ (((filter @ B) @ P) @ Xs))))))))))).
% 0.33/5.22 thf(fact_5035_and__minus__numerals_I3_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M) @ (bitM @ N))))))).
% 0.33/5.22 thf(fact_5036_and__minus__numerals_I7_J, axiom, (! [N:num,M:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) @ ((numeral_numeral @ int) @ M)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M) @ (bitM @ N))))))).
% 0.33/5.22 thf(fact_5037_and__minus__numerals_I8_J, axiom, (! [N:num,M:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) @ ((numeral_numeral @ int) @ M)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M) @ (bit0 @ N))))))).
% 0.33/5.22 thf(fact_5038_and__minus__numerals_I4_J, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M) @ (bit0 @ N))))))).
% 0.33/5.22 thf(fact_5039_int__numeral__not__and__num, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ N) @ M)))))).
% 0.33/5.22 thf(fact_5040_int__numeral__and__not__num, axiom, (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M) @ N)))))).
% 0.33/5.22 thf(fact_5041_insort__key__remove1, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [A3:B,Xs:(list @ B),F:(B > A)]: (((((member @ B) @ A3) @ ((set2 @ B) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => ((((hd @ B) @ (((filter @ B) @ (^ [X2:B]: (((F @ A3) = (F @ X2))))) @ Xs)) = A3) => ((((((linorder_insort_key @ B) @ A) @ F) @ A3) @ (((remove1 @ B) @ A3) @ Xs)) = Xs)))))))))).
% 0.33/5.22 thf(fact_5042_less__eq__int_Orep__eq, axiom, ((ord_less_eq @ int) = (^ [X2:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y6:nat,Z7:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ Y6) @ V5)) @ (((plus_plus @ nat) @ U3) @ Z7)))))))) @ (rep_Integ @ X2)) @ (rep_Integ @ Xa4)))))).
% 0.33/5.22 thf(fact_5043_less__int_Orep__eq, axiom, ((ord_less @ int) = (^ [X2:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y6:nat,Z7:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ Y6) @ V5)) @ (((plus_plus @ nat) @ U3) @ Z7)))))))) @ (rep_Integ @ X2)) @ (rep_Integ @ Xa4)))))).
% 0.33/5.22 thf(fact_5044_sorted__remove1, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),A3:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((remove1 @ A) @ A3) @ Xs))))))))).
% 0.33/5.22 thf(fact_5045_sorted__map__remove1, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),X:B]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ (((remove1 @ B) @ X) @ Xs)))))))))).
% 0.33/5.22 thf(fact_5046_length__remove1, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))))).
% 0.33/5.22 thf(fact_5047_nat_Orep__eq, axiom, (nat2 = (^ [X2:int]: ((((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat)) @ (rep_Integ @ X2)))))).
% 0.33/5.22 thf(fact_5048_insort__remove1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,Xs:(list @ A)]: (((((member @ A) @ A3) @ ((set2 @ A) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((((((linorder_insort_key @ A) @ A) @ (^ [X2:A]: (X2))) @ A3) @ (((remove1 @ A) @ A3) @ Xs)) = Xs))))))))).
% 0.33/5.22 thf(fact_5049_of__int_Orep__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [X2:int]: ((((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3))))) @ (rep_Integ @ X2))))))))).
% 0.33/5.22 thf(fact_5050_uminus__int__def, axiom, ((uminus_uminus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X2:nat,Y6:nat]: (((((product_Pair @ nat) @ nat) @ Y6) @ X2))))))).
% 0.33/5.22 thf(fact_5051_numeral__xor__num, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_un2480387367778600638or_num @ M) @ N))))))))).
% 0.33/5.22 thf(fact_5052_numeral__and__num, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_un7362597486090784418nd_num @ M) @ N))))))))).
% 0.33/5.22 thf(fact_5053_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ U3)) @ (((times_times @ nat) @ Y6) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ V5)) @ (((times_times @ nat) @ Y6) @ U3)))))))))))).
% 0.33/5.22 thf(fact_5054_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ Y6) @ U3))))))))))).
% 0.33/5.22 thf(fact_5055_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ U3)) @ (((plus_plus @ nat) @ Y6) @ V5))))))))))).
% 0.33/5.22 thf(fact_5056_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R6:(set @ ((product_prod @ A) @ A)),S7:(set @ ((product_prod @ A) @ A)),F2:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R6) @ F2)) @ ((((inv_image @ A) @ B) @ S7) @ F2))))))))).
% 0.33/5.22 thf(fact_5057_ring__1__class_Oof__int__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ A) @ A) @ rep_Integ) @ (id @ A)) @ ((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3))))))))))).
% 0.33/5.22 thf(fact_5058_card__Min__le__sum, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ ((finite_card @ A) @ A6)) @ ((lattic643756798350308766er_Min @ nat) @ ((((image @ A) @ nat) @ F) @ A6)))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6)))))).
% 0.33/5.22 thf(fact_5059_of__nat__eq__id, axiom, ((semiring_1_of_nat @ nat) = (id @ nat))).
% 0.33/5.22 thf(fact_5060_id__funpow, axiom, (! [A:$tType,N:nat]: (((((compow @ (A > A)) @ N) @ (id @ A)) = (id @ A))))).
% 0.33/5.22 thf(fact_5061_rotate0, axiom, (! [A:$tType]: ((((rotate @ A) @ (zero_zero @ nat)) = (id @ (list @ A)))))).
% 0.33/5.22 thf(fact_5062_group__add__class_Ominus__comp__minus, axiom, (! [A:$tType]: (((group_add @ A) => ((((((comp @ A) @ A) @ A) @ (uminus_uminus @ A)) @ (uminus_uminus @ A)) = (id @ A)))))).
% 0.33/5.22 thf(fact_5063_boolean__algebra__class_Ominus__comp__minus, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => ((((((comp @ A) @ A) @ A) @ (uminus_uminus @ A)) @ (uminus_uminus @ A)) = (id @ A)))))).
% 0.33/5.22 thf(fact_5064_Min__singleton, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A]: ((((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X))))))).
% 0.33/5.22 thf(fact_5065_push__bit__0__id, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se4730199178511100633sh_bit @ A) @ (zero_zero @ nat)) = (id @ A)))))).
% 0.33/5.22 thf(fact_5066_drop__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se4197421643247451524op_bit @ A) @ (zero_zero @ nat)) = (id @ A)))))).
% 0.33/5.22 thf(fact_5067_Min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5068_Min__gr__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less @ A) @ X) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5069_Min__const, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A6:(set @ B),C2:A]: ((((finite_finite2 @ B) @ A6) => ((A6 != (bot_bot @ (set @ B))) => (((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ (^ [Uu:B]: (C2))) @ A6)) = C2))))))))).
% 0.33/5.22 thf(fact_5070_Min__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_min @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5071_funpow__simps__right_I1_J, axiom, (! [A:$tType,F:(A > A)]: (((((compow @ (A > A)) @ (zero_zero @ nat)) @ F) = (id @ A))))).
% 0.33/5.22 thf(fact_5072_Min_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ A3))))))))).
% 0.33/5.22 thf(fact_5073_Min__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((! [Y5:A]: (((((member @ A) @ Y5) @ A6) => (((ord_less_eq @ A) @ X) @ Y5)))) => ((((member @ A) @ X) @ A6) => (((lattic643756798350308766er_Min @ A) @ A6) = X)))))))))).
% 0.33/5.22 thf(fact_5074_Min__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ X))))))))).
% 0.33/5.22 thf(fact_5075_Min__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ A6))))))))).
% 0.33/5.22 thf(fact_5076_Min_Oin__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((ord_min @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6)) = ((lattic643756798350308766er_Min @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5077_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)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))))).
% 0.33/5.22 thf(fact_5078_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)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))))).
% 0.33/5.22 thf(fact_5079_nat__def, axiom, (nat2 = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ nat) @ nat) @ rep_Integ) @ (id @ nat)) @ ((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat))))).
% 0.33/5.22 thf(fact_5080_Min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => (((ord_less_eq @ A) @ X) @ A8)))) => (((ord_less_eq @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5081_Min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6)) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => (((ord_less_eq @ A) @ X) @ A10))))))))))))).
% 0.33/5.22 thf(fact_5082_eq__Min__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),M:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((M = ((lattic643756798350308766er_Min @ A) @ A6)) = ((((member @ A) @ M) @ A6) & (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ M) @ X2)))))))))))))).
% 0.33/5.22 thf(fact_5083_Min__le__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ X) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less_eq @ A) @ X2) @ X))))))))))))).
% 0.33/5.22 thf(fact_5084_Min__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),M:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((lattic643756798350308766er_Min @ A) @ A6) = M) = ((((member @ A) @ M) @ A6) & (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ M) @ X2)))))))))))))).
% 0.33/5.22 thf(fact_5085_Min__less__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ X) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less @ A) @ X2) @ X))))))))))))).
% 0.33/5.22 thf(fact_5086_Min__insert2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((! [B5:A]: (((((member @ A) @ B5) @ A6) => (((ord_less_eq @ A) @ A3) @ B5)))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ A3) @ A6)) = A3))))))))).
% 0.33/5.22 thf(fact_5087_Min__Inf, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A6) = ((complete_Inf_Inf @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5088_Min_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ B7)) @ ((lattic643756798350308766er_Min @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5089_Min__antimono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M7:(set @ A),N6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M7) @ N6) => ((M7 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N6) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ N6)) @ ((lattic643756798350308766er_Min @ A) @ M7))))))))))).
% 0.33/5.22 thf(fact_5090_hom__Min__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [H2:(A > A),N6:(set @ A)]: (((! [X3:A,Y5:A]: (((H2 @ (((ord_min @ A) @ X3) @ Y5)) = (((ord_min @ A) @ (H2 @ X3)) @ (H2 @ Y5))))) => (((finite_finite2 @ A) @ N6) => ((N6 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic643756798350308766er_Min @ A) @ N6)) = ((lattic643756798350308766er_Min @ A) @ ((((image @ A) @ A) @ H2) @ N6)))))))))))).
% 0.33/5.22 thf(fact_5091_Min_Osubset, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((B7 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => ((((ord_min @ A) @ ((lattic643756798350308766er_Min @ A) @ B7)) @ ((lattic643756798350308766er_Min @ A) @ A6)) = ((lattic643756798350308766er_Min @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5092_Min_Oclosed, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ (((ord_min @ A) @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ A6)))))))))).
% 0.33/5.22 thf(fact_5093_Min_Oinsert__not__elem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_min @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5094_mono__Min__commute, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((F @ ((lattic643756798350308766er_Min @ A) @ A6)) = ((lattic643756798350308766er_Min @ B) @ ((((image @ A) @ B) @ F) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5095_Min_Ounion, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = (((ord_min @ A) @ ((lattic643756798350308766er_Min @ A) @ A6)) @ ((lattic643756798350308766er_Min @ A) @ B7))))))))))))).
% 0.33/5.22 thf(fact_5096_Min__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S3:(set @ B),F:(B > A),K:A]: ((((finite_finite2 @ B) @ S3) => ((S3 != (bot_bot @ (set @ B))) => (((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ (^ [X2:B]: ((((plus_plus @ A) @ (F @ X2)) @ K)))) @ S3)) = (((plus_plus @ A) @ ((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ F) @ S3))) @ K)))))))))).
% 0.33/5.22 thf(fact_5097_Min_Oinsert__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ A6)) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_min @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))))))).
% 0.33/5.22 thf(fact_5098_Min_Oremove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A6) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A6) = (((ord_min @ A) @ X) @ ((lattic643756798350308766er_Min @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))))))).
% 0.33/5.22 thf(fact_5099_positive__def, axiom, (positive = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ $o) @ $o) @ rep_Rat) @ (id @ $o)) @ (^ [X2:((product_prod @ int) @ int)]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ X2)))))))).
% 0.33/5.22 thf(fact_5100_dual__Max, axiom, (! [A:$tType]: (((linorder @ A) => (((lattices_Max @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) = (lattic643756798350308766er_Min @ A)))))).
% 0.33/5.22 thf(fact_5101_dual__max, axiom, (! [A:$tType]: (((linorder @ A) => (((max @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) = (ord_min @ A)))))).
% 0.33/5.22 thf(fact_5102_euclidean__size__nat__def, axiom, ((euclid6346220572633701492n_size @ nat) = (id @ nat))).
% 0.33/5.22 thf(fact_5103_linorder_OMax_Ocong, axiom, (! [A:$tType]: (((lattices_Max @ A) = (lattices_Max @ A))))).
% 0.33/5.22 thf(fact_5104_ord_Omax__def, axiom, (! [A:$tType]: (((max @ A) = (^ [Less_eq2:(A > (A > $o)),A5:A,B3:A]: (((((if @ A) @ ((Less_eq2 @ A5) @ B3)) @ B3) @ A5))))))).
% 0.33/5.22 thf(fact_5105_ord_Omax_Ocong, axiom, (! [A:$tType]: (((max @ A) = (max @ A))))).
% 0.33/5.22 thf(fact_5106_dual__min, axiom, (! [A:$tType]: (((linorder @ A) => (((min @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) = (ord_max @ A)))))).
% 0.33/5.22 thf(fact_5107_sum__le__card__Max, axiom, (! [A:$tType,A6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A6) => (((ord_less_eq @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A6)) @ (((times_times @ nat) @ ((finite_card @ A) @ A6)) @ ((lattic643756798349783984er_Max @ nat) @ ((((image @ A) @ nat) @ F) @ A6)))))))).
% 0.33/5.22 thf(fact_5108_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)))).
% 0.33/5.22 thf(fact_5109_Max__singleton, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A]: ((((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X))))))).
% 0.33/5.22 thf(fact_5110_Max__divisors__self__nat, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ N))))) = N))))).
% 0.33/5.22 thf(fact_5111_Max_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ X) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X2) @ X))))))))))))).
% 0.33/5.22 thf(fact_5112_Max__less__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ X) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less @ A) @ X2) @ X))))))))))))).
% 0.33/5.22 thf(fact_5113_Max__const, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A6:(set @ B),C2:A]: ((((finite_finite2 @ B) @ A6) => ((A6 != (bot_bot @ (set @ B))) => (((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ (^ [Uu:B]: (C2))) @ A6)) = C2))))))))).
% 0.33/5.22 thf(fact_5114_Max__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_max @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5115_minus__Max__eq__Min, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((uminus_uminus @ A) @ ((lattic643756798349783984er_Max @ A) @ S3)) = ((lattic643756798350308766er_Min @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ S3))))))))))).
% 0.33/5.22 thf(fact_5116_minus__Min__eq__Max, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((uminus_uminus @ A) @ ((lattic643756798350308766er_Min @ A) @ S3)) = ((lattic643756798349783984er_Max @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ S3))))))))))).
% 0.33/5.22 thf(fact_5117_ord_Omin_Ocong, axiom, (! [A:$tType]: (((min @ A) = (min @ A))))).
% 0.33/5.22 thf(fact_5118_ord_Omin__def, axiom, (! [A:$tType]: (((min @ A) = (^ [Less_eq2:(A > (A > $o)),A5:A,B3:A]: (((((if @ A) @ ((Less_eq2 @ A5) @ B3)) @ A5) @ B3))))))).
% 0.33/5.22 thf(fact_5119_Max_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (((ord_less_eq @ A) @ A3) @ ((lattic643756798349783984er_Max @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5120_Max__eq__if, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ A) @ B7) => ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (? [Xa:A]: (((((member @ A) @ Xa) @ B7) & (((ord_less_eq @ A) @ X3) @ Xa))))))) => ((! [X3:A]: (((((member @ A) @ X3) @ B7) => (? [Xa:A]: (((((member @ A) @ Xa) @ A6) & (((ord_less_eq @ A) @ X3) @ Xa))))))) => (((lattic643756798349783984er_Max @ A) @ A6) = ((lattic643756798349783984er_Max @ A) @ B7)))))))))))).
% 0.33/5.22 thf(fact_5121_Max__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((! [Y5:A]: (((((member @ A) @ Y5) @ A6) => (((ord_less_eq @ A) @ Y5) @ X)))) => ((((member @ A) @ X) @ A6) => (((lattic643756798349783984er_Max @ A) @ A6) = X)))))))))).
% 0.33/5.22 thf(fact_5122_Max__ge, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5123_Max__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ A6))))))))).
% 0.33/5.22 thf(fact_5124_Max_Oin__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((ord_max @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6)) = ((lattic643756798349783984er_Max @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5125_Max_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => (((ord_less_eq @ A) @ A8) @ X)))) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ X)))))))))).
% 0.33/5.22 thf(fact_5126_Max_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ X) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => (((ord_less_eq @ A) @ A10) @ X))))))))))))).
% 0.33/5.22 thf(fact_5127_eq__Max__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),M:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((M = ((lattic643756798349783984er_Max @ A) @ A6)) = ((((member @ A) @ M) @ A6) & (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X2) @ M)))))))))))))).
% 0.33/5.22 thf(fact_5128_Max__ge__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6)) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less_eq @ A) @ X) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5129_Max__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),M:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((lattic643756798349783984er_Max @ A) @ A6) = M) = ((((member @ A) @ M) @ A6) & (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X2) @ M)))))))))))))).
% 0.33/5.22 thf(fact_5130_Max__gr__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6)) = (? [X2:A]: (((((member @ A) @ X2) @ A6) & (((ord_less @ A) @ X) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5131_Max__insert2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((! [B5:A]: (((((member @ A) @ B5) @ A6) => (((ord_less_eq @ A) @ B5) @ A3)))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ A3) @ A6)) = A3))))))))).
% 0.33/5.22 thf(fact_5132_Max__Sup, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A6) = ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5133_Max_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ ((lattic643756798349783984er_Max @ A) @ B7))))))))))).
% 0.33/5.22 thf(fact_5134_Max__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M7:(set @ A),N6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M7) @ N6) => ((M7 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N6) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ M7)) @ ((lattic643756798349783984er_Max @ A) @ N6))))))))))).
% 0.33/5.22 thf(fact_5135_hom__Max__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [H2:(A > A),N6:(set @ A)]: (((! [X3:A,Y5:A]: (((H2 @ (((ord_max @ A) @ X3) @ Y5)) = (((ord_max @ A) @ (H2 @ X3)) @ (H2 @ Y5))))) => (((finite_finite2 @ A) @ N6) => ((N6 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic643756798349783984er_Max @ A) @ N6)) = ((lattic643756798349783984er_Max @ A) @ ((((image @ A) @ A) @ H2) @ N6)))))))))))).
% 0.33/5.22 thf(fact_5136_Max_Osubset, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((B7 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => ((((ord_max @ A) @ ((lattic643756798349783984er_Max @ A) @ B7)) @ ((lattic643756798349783984er_Max @ A) @ A6)) = ((lattic643756798349783984er_Max @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5137_Max_Oclosed, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ (((ord_max @ A) @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ A6)))))))))).
% 0.33/5.22 thf(fact_5138_Max_Oinsert__not__elem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => ((A6 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_max @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5139_mono__Max__commute, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((F @ ((lattic643756798349783984er_Max @ A) @ A6)) = ((lattic643756798349783984er_Max @ B) @ ((((image @ A) @ B) @ F) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5140_Max_Ounion, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = (((ord_max @ A) @ ((lattic643756798349783984er_Max @ A) @ A6)) @ ((lattic643756798349783984er_Max @ A) @ B7))))))))))))).
% 0.33/5.22 thf(fact_5141_Sup__nat__def, axiom, ((complete_Sup_Sup @ nat) = (^ [X7:(set @ nat)]: (((((if @ nat) @ (X7 = (bot_bot @ (set @ nat)))) @ (zero_zero @ nat)) @ ((lattic643756798349783984er_Max @ nat) @ X7)))))).
% 0.33/5.22 thf(fact_5142_card__le__Suc__Max, axiom, (! [S3:(set @ nat)]: ((((finite_finite2 @ nat) @ S3) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ S3)) @ (suc @ ((lattic643756798349783984er_Max @ nat) @ S3))))))).
% 0.33/5.22 thf(fact_5143_Max__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S3:(set @ B),F:(B > A),K:A]: ((((finite_finite2 @ B) @ S3) => ((S3 != (bot_bot @ (set @ B))) => (((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ (^ [X2:B]: ((((plus_plus @ A) @ (F @ X2)) @ K)))) @ S3)) = (((plus_plus @ A) @ ((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ F) @ S3))) @ K)))))))))).
% 0.33/5.22 thf(fact_5144_divide__nat__def, axiom, ((divide_divide @ nat) = (^ [M2:nat,N3:nat]: (((((if @ nat) @ (N3 = (zero_zero @ nat))) @ (zero_zero @ nat)) @ ((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [K3:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ K3) @ N3)) @ M2)))))))))).
% 0.33/5.22 thf(fact_5145_gcd__is__Max__divisors__nat, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((gcd_gcd @ nat) @ M) @ N) = ((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [D5:nat]: (((((dvd_dvd @ nat) @ D5) @ M) & (((dvd_dvd @ nat) @ D5) @ N))))))))))).
% 0.33/5.22 thf(fact_5146_Max_Oinsert__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ A6)) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ A6)) = (((ord_max @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))))))).
% 0.33/5.22 thf(fact_5147_Max_Oremove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A6) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A6) = (((ord_max @ A) @ X) @ ((lattic643756798349783984er_Max @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))))))).
% 0.33/5.22 thf(fact_5148_nat__of__integer__def, axiom, (code_nat_of_integer = (((((((map_fun @ code_integer) @ int) @ nat) @ nat) @ code_int_of_integer) @ (id @ nat)) @ nat2))).
% 0.33/5.22 thf(fact_5149_Gcd__eq__Max, axiom, (! [M7:(set @ nat)]: ((((finite_finite2 @ nat) @ M7) => ((M7 != (bot_bot @ (set @ nat))) => ((~ ((((member @ nat) @ (zero_zero @ nat)) @ M7))) => (((gcd_Gcd @ nat) @ M7) = ((lattic643756798349783984er_Max @ nat) @ ((complete_Inf_Inf @ (set @ nat)) @ ((((image @ nat) @ (set @ nat)) @ (^ [M2:nat]: (((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ M2))))))) @ M7)))))))))).
% 0.33/5.22 thf(fact_5150_dual__Min, axiom, (! [A:$tType]: (((linorder @ A) => (((lattices_Min @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) = (lattic643756798349783984er_Max @ A)))))).
% 0.33/5.22 thf(fact_5151_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)))).
% 0.33/5.22 thf(fact_5152_positive_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ $o) @ $o) @ ratrel) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))) @ (^ [X2:((product_prod @ int) @ int)]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ X2)))))) @ (^ [X2:((product_prod @ int) @ int)]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ X2))))))).
% 0.33/5.22 thf(fact_5153_Max__divisors__self__int, axiom, (! [N:int]: (((N != (zero_zero @ int)) => (((lattic643756798349783984er_Max @ int) @ ((collect @ int) @ (^ [D5:int]: ((((dvd_dvd @ int) @ D5) @ N))))) = ((abs_abs @ int) @ N)))))).
% 0.33/5.22 thf(fact_5154_power__transfer, axiom, (! [A:$tType,B:$tType]: ((((power @ B) & (power @ A)) => (! [R2:(A > (B > $o))]: ((((R2 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R2) @ R2)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ A) @ B) @ (nat > A)) @ (nat > B)) @ R2) @ ((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ R2)) @ (power_power @ A)) @ (power_power @ B)))))))))).
% 0.33/5.22 thf(fact_5155_linorder_OMin_Ocong, axiom, (! [A:$tType]: (((lattices_Min @ A) = (lattices_Min @ A))))).
% 0.33/5.22 thf(fact_5156_transfer__rule__numeral, axiom, (! [A:$tType,B:$tType]: ((((monoid_add @ B) & ((semiring_numeral @ B) & ((monoid_add @ A) & (semiring_numeral @ A)))) => (! [R2:(A > (B > $o))]: ((((R2 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R2 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R2) @ R2)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ R2) @ (numeral_numeral @ A)) @ (numeral_numeral @ B))))))))))).
% 0.33/5.22 thf(fact_5157_transfer__rule__of__int, axiom, (! [A:$tType,B:$tType]: ((((ring_1 @ B) & (ring_1 @ A)) => (! [R2:(A > (B > $o))]: ((((R2 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R2 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R2) @ R2)) @ (plus_plus @ A)) @ (plus_plus @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R2) @ R2) @ (uminus_uminus @ A)) @ (uminus_uminus @ B)) => ((((((((bNF_rel_fun @ int) @ int) @ A) @ B) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ R2) @ (ring_1_of_int @ A)) @ (ring_1_of_int @ B)))))))))))).
% 0.33/5.22 thf(fact_5158_gcd__is__Max__divisors__int, axiom, (! [N:int,M:int]: (((N != (zero_zero @ int)) => ((((gcd_gcd @ int) @ M) @ N) = ((lattic643756798349783984er_Max @ int) @ ((collect @ int) @ (^ [D5:int]: (((((dvd_dvd @ int) @ D5) @ M) & (((dvd_dvd @ int) @ D5) @ N))))))))))).
% 0.33/5.22 thf(fact_5159_transfer__rule__of__nat, axiom, (! [A:$tType,B:$tType]: ((((semiring_1 @ B) & (semiring_1 @ A)) => (! [R2:(A > (B > $o))]: ((((R2 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R2 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R2) @ R2)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ R2) @ (semiring_1_of_nat @ A)) @ (semiring_1_of_nat @ B))))))))))).
% 0.33/5.22 thf(fact_5160_integer__of__nat__def, axiom, (code_integer_of_nat = (((((((map_fun @ nat) @ nat) @ int) @ code_integer) @ (id @ nat)) @ code_integer_of_int) @ (semiring_1_of_nat @ int)))).
% 0.33/5.22 thf(fact_5161_antimono__funpow, axiom, (! [A:$tType]: ((((lattice @ A) & (order_top @ A)) => (! [Q:(A > A)]: (((((order_mono @ A) @ A) @ Q) => (((order_antimono @ nat) @ A) @ (^ [I5:nat]: (((((compow @ (A > A)) @ I5) @ Q) @ (top_top @ A)))))))))))).
% 0.33/5.22 thf(fact_5162_nat__of__integer__integer__of__nat, axiom, (! [N:nat]: (((code_nat_of_integer @ (code_integer_of_nat @ N)) = N)))).
% 0.33/5.22 thf(fact_5163_integer__of__nat_Orep__eq, axiom, (! [X:nat]: (((code_int_of_integer @ (code_integer_of_nat @ X)) = ((semiring_1_of_nat @ int) @ X))))).
% 0.33/5.22 thf(fact_5164_int__of__integer__integer__of__nat, axiom, (! [N:nat]: (((code_int_of_integer @ (code_integer_of_nat @ N)) = ((semiring_1_of_nat @ int) @ N))))).
% 0.33/5.22 thf(fact_5165_euclidean__size__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ nat) @ nat) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (euclid6346220572633701492n_size @ int)) @ (euclid6346220572633701492n_size @ int))).
% 0.33/5.22 thf(fact_5166_euclidean__size__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (euclid6346220572633701492n_size @ nat)) @ (euclid6346220572633701492n_size @ nat))).
% 0.33/5.22 thf(fact_5167_dup_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [K3:int]: ((((plus_plus @ int) @ K3) @ K3)))) @ (^ [K3:int]: ((((plus_plus @ int) @ K3) @ K3))))).
% 0.33/5.22 thf(fact_5168_sub_Orsp, axiom, ((((((((bNF_rel_fun @ num) @ num) @ (num > int)) @ (num > int)) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ num) @ num) @ int) @ int) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (^ [M2:num,N3:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N3))))) @ (^ [M2:num,N3:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N3)))))).
% 0.33/5.22 thf(fact_5169_less__eq__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > $o)) @ (int > $o)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ $o) @ $o) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ int)) @ (ord_less_eq @ int))).
% 0.33/5.22 thf(fact_5170_less__eq__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ $o) @ $o) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ nat)) @ (ord_less_eq @ nat))).
% 0.33/5.22 thf(fact_5171_less__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > $o)) @ (int > $o)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ $o) @ $o) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less @ int)) @ (ord_less @ int))).
% 0.33/5.22 thf(fact_5172_less__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ $o) @ $o) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less @ nat)) @ (ord_less @ nat))).
% 0.33/5.22 thf(fact_5173_uminus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (uminus_uminus @ int)) @ (uminus_uminus @ int))).
% 0.33/5.22 thf(fact_5174_times__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (times_times @ nat)) @ (times_times @ nat))).
% 0.33/5.22 thf(fact_5175_times__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (times_times @ int)) @ (times_times @ int))).
% 0.33/5.22 thf(fact_5176_natural__of__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ nat) @ nat) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ nat2) @ nat2)).
% 0.33/5.22 thf(fact_5177_Suc_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ suc) @ suc)).
% 0.33/5.22 thf(fact_5178_modulo__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (modulo_modulo @ int)) @ (modulo_modulo @ int))).
% 0.33/5.22 thf(fact_5179_modulo__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (modulo_modulo @ nat)) @ (modulo_modulo @ nat))).
% 0.33/5.22 thf(fact_5180_sgn__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (sgn_sgn @ int)) @ (sgn_sgn @ int))).
% 0.33/5.22 thf(fact_5181_plus__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (plus_plus @ nat)) @ (plus_plus @ nat))).
% 0.33/5.22 thf(fact_5182_plus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (plus_plus @ int)) @ (plus_plus @ int))).
% 0.33/5.22 thf(fact_5183_divide__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (divide_divide @ nat)) @ (divide_divide @ nat))).
% 0.33/5.22 thf(fact_5184_divide__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (divide_divide @ int)) @ (divide_divide @ int))).
% 0.33/5.22 thf(fact_5185_abs__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (abs_abs @ int)) @ (abs_abs @ int))).
% 0.33/5.22 thf(fact_5186_minus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (minus_minus @ int)) @ (minus_minus @ int))).
% 0.33/5.22 thf(fact_5187_minus__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (minus_minus @ nat)) @ (minus_minus @ nat))).
% 0.33/5.22 thf(fact_5188_integer__of__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ int) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (semiring_1_of_nat @ int)) @ (semiring_1_of_nat @ int))).
% 0.33/5.22 thf(fact_5189_push__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se4730199178511100633sh_bit @ nat)) @ (bit_se4730199178511100633sh_bit @ nat))).
% 0.33/5.22 thf(fact_5190_push__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se4730199178511100633sh_bit @ int)) @ (bit_se4730199178511100633sh_bit @ int))).
% 0.33/5.22 thf(fact_5191_flip__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se8732182000553998342ip_bit @ int)) @ (bit_se8732182000553998342ip_bit @ int))).
% 0.33/5.22 thf(fact_5192_flip__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se8732182000553998342ip_bit @ nat)) @ (bit_se8732182000553998342ip_bit @ nat))).
% 0.33/5.22 thf(fact_5193_take__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se2584673776208193580ke_bit @ nat)) @ (bit_se2584673776208193580ke_bit @ nat))).
% 0.33/5.22 thf(fact_5194_take__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se2584673776208193580ke_bit @ int)) @ (bit_se2584673776208193580ke_bit @ int))).
% 0.33/5.22 thf(fact_5195_and__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se5824344872417868541ns_and @ nat)) @ (bit_se5824344872417868541ns_and @ nat))).
% 0.33/5.22 thf(fact_5196_and__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se5824344872417868541ns_and @ int)) @ (bit_se5824344872417868541ns_and @ int))).
% 0.33/5.22 thf(fact_5197_unset__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se2638667681897837118et_bit @ nat)) @ (bit_se2638667681897837118et_bit @ nat))).
% 0.33/5.22 thf(fact_5198_unset__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se2638667681897837118et_bit @ int)) @ (bit_se2638667681897837118et_bit @ int))).
% 0.33/5.22 thf(fact_5199_xor__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se5824344971392196577ns_xor @ int)) @ (bit_se5824344971392196577ns_xor @ int))).
% 0.33/5.22 thf(fact_5200_xor__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se5824344971392196577ns_xor @ nat)) @ (bit_se5824344971392196577ns_xor @ nat))).
% 0.33/5.22 thf(fact_5201_set__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se5668285175392031749et_bit @ int)) @ (bit_se5668285175392031749et_bit @ int))).
% 0.33/5.22 thf(fact_5202_set__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se5668285175392031749et_bit @ nat)) @ (bit_se5668285175392031749et_bit @ nat))).
% 0.33/5.22 thf(fact_5203_drop__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se4197421643247451524op_bit @ nat)) @ (bit_se4197421643247451524op_bit @ nat))).
% 0.33/5.22 thf(fact_5204_drop__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se4197421643247451524op_bit @ int)) @ (bit_se4197421643247451524op_bit @ int))).
% 0.33/5.22 thf(fact_5205_or__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2))))) @ (bit_se1065995026697491101ons_or @ nat)) @ (bit_se1065995026697491101ons_or @ nat))).
% 0.33/5.22 thf(fact_5206_or__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2))))) @ (bit_se1065995026697491101ons_or @ int)) @ (bit_se1065995026697491101ons_or @ int))).
% 0.33/5.22 thf(fact_5207_bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (nat > $o)) @ (nat > $o)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:(nat > $o),Z2:(nat > $o)]: ((Y = Z2)))) @ (bit_se5641148757651400278ts_bit @ int)) @ (bit_se5641148757651400278ts_bit @ int))).
% 0.33/5.22 thf(fact_5208_not__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (bit_ri4277139882892585799ns_not @ int)) @ (bit_ri4277139882892585799ns_not @ int))).
% 0.33/5.22 thf(fact_5209_bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:(nat > $o),Z2:(nat > $o)]: ((Y = Z2)))) @ (bit_se5641148757651400278ts_bit @ nat)) @ (bit_se5641148757651400278ts_bit @ nat))).
% 0.33/5.22 thf(fact_5210_mask__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ int) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (bit_se2239418461657761734s_mask @ int)) @ (bit_se2239418461657761734s_mask @ int))).
% 0.33/5.22 thf(fact_5211_mask__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (bit_se2239418461657761734s_mask @ nat)) @ (bit_se2239418461657761734s_mask @ nat))).
% 0.33/5.22 thf(fact_5212_division__segment__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (euclid7384307370059645450egment @ int)) @ (euclid7384307370059645450egment @ int))).
% 0.33/5.22 thf(fact_5213_division__segment__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (euclid7384307370059645450egment @ nat)) @ (euclid7384307370059645450egment @ nat))).
% 0.33/5.22 thf(fact_5214_Fract_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > ((product_prod @ int) @ int))) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ratrel)) @ (^ [A5:int,B3:int]: (((((if @ ((product_prod @ int) @ int)) @ (B3 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A5) @ B3))))) @ (^ [A5:int,B3:int]: (((((if @ ((product_prod @ int) @ int)) @ (B3 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A5) @ B3)))))).
% 0.33/5.22 thf(fact_5215_num__of__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ num) @ num) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2))).
% 0.33/5.22 thf(fact_5216_antimonoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X)))))))))).
% 0.33/5.22 thf(fact_5217_antimonoE, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X)))))))))).
% 0.33/5.22 thf(fact_5218_antimonoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B)]: (((! [X3:A,Y5:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => (((ord_less_eq @ B) @ (F @ Y5)) @ (F @ X3))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 0.33/5.22 thf(fact_5219_antimono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) => (((ord_less_eq @ B) @ (F2 @ Y6)) @ (F2 @ X2)))))))))))).
% 0.33/5.22 thf(fact_5220_transfer__rule__of__bool, axiom, (! [A:$tType,B:$tType]: ((((zero_neq_one @ B) & (zero_neq_one @ A)) => (! [R2:(A > (B > $o))]: ((((R2 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R2 @ (one_one @ A)) @ (one_one @ B)) => ((((((((bNF_rel_fun @ $o) @ $o) @ A) @ B) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))) @ R2) @ (zero_neq_one_of_bool @ A)) @ (zero_neq_one_of_bool @ B)))))))))).
% 0.33/5.22 thf(fact_5221_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))))))).
% 0.33/5.22 thf(fact_5222_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))))))).
% 0.33/5.22 thf(fact_5223_integer__of__nat__eq__of__nat, axiom, (code_integer_of_nat = (semiring_1_of_nat @ code_integer))).
% 0.33/5.22 thf(fact_5224_antimono__iff__le__Suc, axiom, (! [A:$tType]: (((order @ A) => (((order_antimono @ nat) @ A) = (^ [F2:(nat > A)]: ((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ (suc @ N3))) @ (F2 @ N3))))))))))).
% 0.33/5.22 thf(fact_5225_integer__of__nat__0, axiom, ((code_integer_of_nat @ (zero_zero @ nat)) = (zero_zero @ code_integer))).
% 0.33/5.22 thf(fact_5226_integer__of__nat_Oabs__eq, axiom, (code_integer_of_nat = (^ [X2:nat]: ((code_integer_of_int @ ((semiring_1_of_nat @ int) @ X2)))))).
% 0.33/5.22 thf(fact_5227_integer__of__nat__numeral, axiom, (! [N:num]: (((code_integer_of_nat @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ code_integer) @ N))))).
% 0.33/5.22 thf(fact_5228_min__of__antimono, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_min @ B) @ (F @ X)) @ (F @ Y3)) = (F @ (((ord_max @ A) @ X) @ Y3)))))))))).
% 0.33/5.22 thf(fact_5229_max__of__antimono, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B),X:A,Y3:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_max @ B) @ (F @ X)) @ (F @ Y3)) = (F @ (((ord_min @ A) @ X) @ Y3)))))))))).
% 0.33/5.22 thf(fact_5230_integer__of__nat__1, axiom, ((code_integer_of_nat @ (one_one @ nat)) = (one_one @ code_integer))).
% 0.33/5.22 thf(fact_5231_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) @ (^ [X2:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X2))) @ (((product_snd @ int) @ int) @ X2))))) @ (^ [X2:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X2))) @ (((product_snd @ int) @ int) @ X2)))))).
% 0.33/5.22 thf(fact_5232_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) @ (^ [X2:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X2) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ X2)))))) @ (^ [X2:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X2) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ X2))))))).
% 0.33/5.22 thf(fact_5233_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y6)) @ (((product_snd @ int) @ int) @ X2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))) @ (plus_plus @ rat))).
% 0.33/5.22 thf(fact_5234_inverse__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat) @ (^ [X2:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X2) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ X2)))))) @ (inverse_inverse @ rat))).
% 0.33/5.22 thf(fact_5235_positive_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ $o) @ $o) @ pcr_rat) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))) @ (^ [X2:((product_prod @ int) @ int)]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ X2)))))) @ positive)).
% 0.33/5.22 thf(fact_5236_one__rat_Otransfer, axiom, ((pcr_rat @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) @ (one_one @ rat))).
% 0.33/5.22 thf(fact_5237_zero__rat_Otransfer, axiom, ((pcr_rat @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ (zero_zero @ rat))).
% 0.33/5.22 thf(fact_5238_uminus__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat) @ (^ [X2:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X2))) @ (((product_snd @ int) @ int) @ X2))))) @ (uminus_uminus @ rat))).
% 0.33/5.22 thf(fact_5239_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)) @ (^ [X2:((product_prod @ int) @ int),Y6:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X2)) @ (((product_fst @ int) @ int) @ Y6))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X2)) @ (((product_snd @ int) @ int) @ Y6)))))) @ (times_times @ rat))).
% 0.33/5.22 thf(fact_5240_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ U3)) @ (((times_times @ nat) @ Y6) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ V5)) @ (((times_times @ nat) @ Y6) @ U3)))))))))) @ (times_times @ int))).
% 0.33/5.22 thf(fact_5241_num_Orec__transfer, axiom, (! [A:$tType,B:$tType,S3:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((num > (A > A)) > ((num > (A > A)) > (num > A)))) @ ((num > (B > B)) > ((num > (B > B)) > (num > B)))) @ S3) @ ((((((bNF_rel_fun @ (num > (A > A))) @ (num > (B > B))) @ ((num > (A > A)) > (num > A))) @ ((num > (B > B)) > (num > B))) @ ((((((bNF_rel_fun @ num) @ num) @ (A > A)) @ (B > B)) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ S3) @ S3))) @ ((((((bNF_rel_fun @ (num > (A > A))) @ (num > (B > B))) @ (num > A)) @ (num > B)) @ ((((((bNF_rel_fun @ num) @ num) @ (A > A)) @ (B > B)) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ S3) @ S3))) @ ((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ S3)))) @ (rec_num @ A)) @ (rec_num @ B))))).
% 0.33/5.22 thf(fact_5242_num__of__integer__def, axiom, (code_num_of_integer = (((((((map_fun @ code_integer) @ int) @ num) @ num) @ code_int_of_integer) @ (id @ num)) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)))).
% 0.33/5.22 thf(fact_5243_verit__eq__simplify_I20_J, axiom, (! [A:$tType,F1:A,F22:(num > (A > A)),F32:(num > (A > A)),X22:num]: (((((((rec_num @ A) @ F1) @ F22) @ F32) @ (bit0 @ X22)) = ((F22 @ X22) @ (((((rec_num @ A) @ F1) @ F22) @ F32) @ X22)))))).
% 0.33/5.22 thf(fact_5244_verit__eq__simplify_I19_J, axiom, (! [A:$tType,F1:A,F22:(num > (A > A)),F32:(num > (A > A))]: (((((((rec_num @ A) @ F1) @ F22) @ F32) @ one2) = F1)))).
% 0.33/5.22 thf(fact_5245_verit__eq__simplify_I21_J, axiom, (! [A:$tType,F1:A,F22:(num > (A > A)),F32:(num > (A > A)),X32:num]: (((((((rec_num @ A) @ F1) @ F22) @ F32) @ (bit1 @ X32)) = ((F32 @ X32) @ (((((rec_num @ A) @ F1) @ F22) @ F32) @ X32)))))).
% 0.33/5.22 thf(fact_5246_zero__int_Otransfer, axiom, ((pcr_int @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))) @ (zero_zero @ int))).
% 0.33/5.22 thf(fact_5247_int__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ int) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ pcr_int) @ (^ [N3:nat]: (((((product_Pair @ nat) @ nat) @ N3) @ (zero_zero @ nat))))) @ (semiring_1_of_nat @ int))).
% 0.33/5.22 thf(fact_5248_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)) @ (^ [X2:nat,Y6:nat]: (((((product_Pair @ nat) @ nat) @ Y6) @ X2))))) @ (uminus_uminus @ int))).
% 0.33/5.22 thf(fact_5249_nat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ nat) @ nat) @ pcr_int) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat))) @ nat2)).
% 0.33/5.22 thf(fact_5250_one__int_Otransfer, axiom, ((pcr_int @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))) @ (one_one @ int))).
% 0.33/5.22 thf(fact_5251_of__int_Otransfer, axiom, (! [A:$tType]: (((ring_1 @ A) => ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ A) @ A) @ pcr_int) @ (^ [Y:A,Z2:A]: ((Y = Z2)))) @ ((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3)))))) @ (ring_1_of_int @ A)))))).
% 0.33/5.22 thf(fact_5252_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) @ $o) @ pcr_int) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))) @ (ord_less @ int))).
% 0.33/5.22 thf(fact_5253_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) @ $o) @ pcr_int) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))) @ (ord_less_eq @ int))).
% 0.33/5.22 thf(fact_5254_plus__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ U3)) @ (((plus_plus @ nat) @ Y6) @ V5))))))))) @ (plus_plus @ int))).
% 0.33/5.22 thf(fact_5255_minus__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ Y6) @ U3))))))))) @ (minus_minus @ int))).
% 0.33/5.22 thf(fact_5256_num_Ocase__transfer, axiom, (! [A:$tType,B:$tType,S3:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((num > A) > ((num > A) > (num > A)))) @ ((num > B) > ((num > B) > (num > B)))) @ S3) @ ((((((bNF_rel_fun @ (num > A)) @ (num > B)) @ ((num > A) > (num > A))) @ ((num > B) > (num > B))) @ ((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ S3)) @ ((((((bNF_rel_fun @ (num > A)) @ (num > B)) @ (num > A)) @ (num > B)) @ ((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ S3)) @ ((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ S3)))) @ (case_num @ A)) @ (case_num @ B))))).
% 0.33/5.22 thf(fact_5257_times__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ intrel) @ intrel)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ U3)) @ (((times_times @ nat) @ Y6) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ V5)) @ (((times_times @ nat) @ Y6) @ U3)))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ U3)) @ (((times_times @ nat) @ Y6) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X2) @ V5)) @ (((times_times @ nat) @ Y6) @ U3))))))))))).
% 0.33/5.22 thf(fact_5258_list_Osize__gen_I2_J, axiom, (! [A:$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))))))).
% 0.33/5.22 thf(fact_5259_intrel__iff, axiom, (! [X:nat,Y3:nat,U:nat,V:nat]: ((((intrel @ ((((product_Pair @ nat) @ nat) @ X) @ Y3)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) = ((((plus_plus @ nat) @ X) @ V) = (((plus_plus @ nat) @ U) @ Y3)))))).
% 0.33/5.22 thf(fact_5260_int_Orel__eq__transfer, 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) @ $o) @ pcr_int) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ intrel) @ (^ [Y:int,Z2:int]: ((Y = Z2))))).
% 0.33/5.22 thf(fact_5261_int_Oabs__eq__iff, axiom, (! [X:((product_prod @ nat) @ nat),Y3:((product_prod @ nat) @ nat)]: ((((abs_Integ @ X) = (abs_Integ @ Y3)) = ((intrel @ X) @ Y3))))).
% 0.33/5.22 thf(fact_5262_num_Ocase__distrib, axiom, (! [B:$tType,A:$tType,H2:(A > B),F1:A,F22:(num > A),F32:(num > A),Num:num]: (((H2 @ (((((case_num @ A) @ F1) @ F22) @ F32) @ Num)) = (((((case_num @ B) @ (H2 @ F1)) @ (^ [X2:num]: ((H2 @ (F22 @ X2))))) @ (^ [X2:num]: ((H2 @ (F32 @ X2))))) @ Num))))).
% 0.33/5.22 thf(fact_5263_verit__eq__simplify_I17_J, axiom, (! [A:$tType,F1:A,F22:(num > A),F32:(num > A),X22:num]: (((((((case_num @ A) @ F1) @ F22) @ F32) @ (bit0 @ X22)) = (F22 @ X22))))).
% 0.33/5.22 thf(fact_5264_verit__eq__simplify_I16_J, axiom, (! [A:$tType,F1:A,F22:(num > A),F32:(num > A)]: (((((((case_num @ A) @ F1) @ F22) @ F32) @ one2) = F1)))).
% 0.33/5.22 thf(fact_5265_zero__int_Orsp, axiom, ((intrel @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))) @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat)))).
% 0.33/5.22 thf(fact_5266_verit__eq__simplify_I18_J, axiom, (! [A:$tType,F1:A,F22:(num > A),F32:(num > A),X32:num]: (((((((case_num @ A) @ F1) @ F22) @ F32) @ (bit1 @ X32)) = (F32 @ X32))))).
% 0.33/5.22 thf(fact_5267_list_Osize__gen_I1_J, axiom, (! [A:$tType,X:(A > nat)]: (((((size_list @ A) @ X) @ (nil @ A)) = (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5268_size__list__estimation, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y3:nat,F:(A > nat)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => ((((ord_less @ nat) @ Y3) @ (F @ X)) => (((ord_less @ nat) @ Y3) @ (((size_list @ A) @ F) @ Xs))))))).
% 0.33/5.22 thf(fact_5269_size__list__pointwise, axiom, (! [A:$tType,Xs:(list @ A),F:(A > nat),G:(A > nat)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ nat) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ nat) @ (((size_list @ A) @ F) @ Xs)) @ (((size_list @ A) @ G) @ Xs)))))).
% 0.33/5.22 thf(fact_5270_size__list__estimation_H, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y3:nat,F:(A > nat)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => ((((ord_less_eq @ nat) @ Y3) @ (F @ X)) => (((ord_less_eq @ nat) @ Y3) @ (((size_list @ A) @ F) @ Xs))))))).
% 0.33/5.22 thf(fact_5271_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)) @ (^ [X2:nat,Y6:nat]: (((((product_Pair @ nat) @ nat) @ Y6) @ X2))))) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X2:nat,Y6:nat]: (((((product_Pair @ nat) @ nat) @ Y6) @ X2)))))).
% 0.33/5.22 thf(fact_5272_nat_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ nat) @ nat) @ intrel) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat))) @ ((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat)))).
% 0.33/5.22 thf(fact_5273_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)))).
% 0.33/5.22 thf(fact_5274_of__int_Orsp, axiom, (! [A:$tType]: (((ring_1 @ A) => ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ A) @ A) @ intrel) @ (^ [Y:A,Z2:A]: ((Y = Z2)))) @ ((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3)))))) @ ((((product_case_prod @ nat) @ nat) @ A) @ (^ [I5:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I5)) @ ((semiring_1_of_nat @ A) @ J3)))))))))).
% 0.33/5.22 thf(fact_5275_intrel__def, axiom, (intrel = ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: (((((plus_plus @ nat) @ X2) @ V5) = (((plus_plus @ nat) @ U3) @ Y6)))))))))).
% 0.33/5.22 thf(fact_5276_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) @ $o) @ intrel) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6)))))))))).
% 0.33/5.22 thf(fact_5277_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) @ $o) @ intrel) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U3:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ U3) @ Y6)))))))))).
% 0.33/5.22 thf(fact_5278_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ U3)) @ (((plus_plus @ nat) @ Y6) @ V5))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ U3)) @ (((plus_plus @ nat) @ Y6) @ V5)))))))))).
% 0.33/5.22 thf(fact_5279_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))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ Y6) @ U3))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X2:nat,Y6:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U3:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X2) @ V5)) @ (((plus_plus @ nat) @ Y6) @ U3)))))))))).
% 0.33/5.22 thf(fact_5280_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))))))))))).
% 0.33/5.22 thf(fact_5281_execute__make, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H2) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H2)))))))))).
% 0.33/5.22 thf(fact_5282_relpow__finite__bounded1, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),K:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ K) @ R2)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N3:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N3) @ R2)))) @ ((collect @ nat) @ (^ [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((ord_less_eq @ nat) @ N3) @ ((finite_card @ ((product_prod @ A) @ A)) @ R2)))))))))))))).
% 0.33/5.22 thf(fact_5283_tl__upt, axiom, (! [M:nat,N:nat]: ((((tl @ nat) @ ((upt @ M) @ N)) = ((upt @ (suc @ M)) @ N))))).
% 0.33/5.22 thf(fact_5284_relpow__1, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (one_one @ nat)) @ R2) = R2)))).
% 0.33/5.22 thf(fact_5285_length__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((tl @ A) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))).
% 0.33/5.22 thf(fact_5286_finite__relpow, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),N:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((finite_finite2 @ ((product_prod @ A) @ A)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2))))))).
% 0.33/5.22 thf(fact_5287_tl__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((tl @ A) @ (((replicate @ A) @ N) @ X)) = (((replicate @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ X))))).
% 0.33/5.22 thf(fact_5288_execute__heap, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((heap_Time_execute @ A) @ ((heap_Time_heap @ A) @ F)) = (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ F))))).
% 0.33/5.22 thf(fact_5289_execute__guard_I2_J, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H2:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((P @ H2) => ((((heap_Time_execute @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H2) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (F @ H2))))))).
% 0.33/5.22 thf(fact_5290_Heap__eqI, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),G:(heap_Time_Heap @ A)]: (((! [H6:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ F) @ H6) = (((heap_Time_execute @ A) @ G) @ H6)))) => (F = G))))).
% 0.33/5.22 thf(fact_5291_effectI, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),R:A,H:(heap_ext @ product_unit),N:nat]: ((((((heap_Time_execute @ A) @ C2) @ H2) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ N)))) => ((((((heap_Time_effect @ A) @ C2) @ H2) @ H) @ R) @ N))))).
% 0.33/5.22 thf(fact_5292_effect__def, axiom, (! [A:$tType]: (((heap_Time_effect @ A) = (^ [C4:(heap_Time_Heap @ A),H4:(heap_ext @ product_unit),H5:(heap_ext @ product_unit),R5:A,N3:nat]: (((((heap_Time_execute @ A) @ C4) @ H4) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ N3)))))))))).
% 0.33/5.22 thf(fact_5293_sorted__tl, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((tl @ A) @ Xs))))))))).
% 0.33/5.22 thf(fact_5294_relpow__Suc__E, axiom, (! [A:$tType,X:A,Z3:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2)) => (~ ((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ R2)))))))))))).
% 0.33/5.22 thf(fact_5295_relpow__Suc__I, axiom, (! [A:$tType,X:A,Y3:A,N:nat,R2:(set @ ((product_prod @ A) @ A)),Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2))))))).
% 0.33/5.22 thf(fact_5296_relpow__Suc__D2, axiom, (! [A:$tType,X:A,Z3:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2)) => (? [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ R2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2))))))))).
% 0.33/5.22 thf(fact_5297_relpow__Suc__E2, axiom, (! [A:$tType,X:A,Z3:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2)) => (~ ((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2))))))))))))).
% 0.33/5.22 thf(fact_5298_relpow__Suc__I2, axiom, (! [A:$tType,X:A,Y3:A,R2:(set @ ((product_prod @ A) @ A)),Z3:A,N:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2))))))).
% 0.33/5.22 thf(fact_5299_relpow__0__I, axiom, (! [A:$tType,X:A,R2:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R2))))).
% 0.33/5.22 thf(fact_5300_relpow__0__E, axiom, (! [A:$tType,X:A,Y3:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R2)) => (X = Y3))))).
% 0.33/5.22 thf(fact_5301_take__tl, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((take @ A) @ N) @ ((tl @ A) @ Xs)) = ((tl @ A) @ (((take @ A) @ (suc @ N)) @ Xs)))))).
% 0.33/5.22 thf(fact_5302_drop__Suc, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((drop @ A) @ (suc @ N)) @ Xs) = (((drop @ A) @ N) @ ((tl @ A) @ Xs)))))).
% 0.33/5.22 thf(fact_5303_execute__assert_I1_J, axiom, (! [A:$tType,P:(A > $o),X:A,H2:(heap_ext @ product_unit)]: (((P @ X) => ((((heap_Time_execute @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H2) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ (one_one @ nat))))))))).
% 0.33/5.22 thf(fact_5304_execute__tap, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ ((heap_Time_tap @ A) @ F)) @ H2) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H2)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ (one_one @ nat)))))))).
% 0.33/5.22 thf(fact_5305_execute__return, axiom, (! [A:$tType,X:A]: ((((heap_Time_execute @ A) @ ((heap_Time_return @ A) @ X)) = (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (one_one @ nat)))))))))).
% 0.33/5.22 thf(fact_5306_execute__ureturn, axiom, (! [A:$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)))) @ (^ [H4:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (zero_zero @ nat)))))))))).
% 0.33/5.22 thf(fact_5307_relpow__E, axiom, (! [A:$tType,X:A,Z3:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) => (((N = (zero_zero @ nat)) => (X != Z3)) => (~ ((! [Y5:A,M6:nat]: (((N = (suc @ M6)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M6) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ R2)))))))))))))).
% 0.33/5.22 thf(fact_5308_relpow__E2, axiom, (! [A:$tType,X:A,Z3:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) => (((N = (zero_zero @ nat)) => (X != Z3)) => (~ ((! [Y5:A,M6:nat]: (((N = (suc @ M6)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M6) @ R2))))))))))))))).
% 0.33/5.22 thf(fact_5309_tl__take, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((tl @ A) @ (((take @ A) @ N) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ ((tl @ A) @ Xs)))))).
% 0.33/5.22 thf(fact_5310_relpow__empty, axiom, (! [A:$tType,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 0.33/5.22 thf(fact_5311_relpow__finite__bounded, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),K:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ K) @ R2)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N3:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N3) @ R2)))) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ ((finite_card @ ((product_prod @ A) @ A)) @ R2)))))))))))).
% 0.33/5.22 thf(fact_5312_Nitpick_Osize__list__simp_I2_J, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = (^ [Xs3:(list @ A)]: (((((if @ nat) @ (Xs3 = (nil @ A))) @ (zero_zero @ nat)) @ (suc @ ((size_size @ (list @ A)) @ ((tl @ A) @ Xs3)))))))))).
% 0.33/5.22 thf(fact_5313_nth__tl, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ ((tl @ A) @ Xs))) => ((((nth @ A) @ ((tl @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (suc @ N))))))).
% 0.33/5.22 thf(fact_5314_relpow__fun__conv, axiom, (! [A:$tType,A3:A,B2:A,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) = (? [F2:(nat > A)]: ((((F2 @ (zero_zero @ nat)) = A3) & (((F2 @ N) = B2) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ N) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F2 @ I5)) @ (F2 @ (suc @ I5)))) @ R2)))))))))))).
% 0.33/5.22 thf(fact_5315_execute__of__list, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H2) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))) @ (((array_alloc @ A) @ Xs) @ H2)))))))))).
% 0.33/5.22 thf(fact_5316_execute__new, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H2) = ((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))) @ (^ [R5:(array @ A),H5:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H2)))))))))).
% 0.33/5.22 thf(fact_5317_take__Suc, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((Xs != (nil @ A)) => ((((take @ A) @ (suc @ N)) @ Xs) = (((cons @ A) @ ((hd @ A) @ Xs)) @ (((take @ A) @ N) @ ((tl @ A) @ Xs)))))))).
% 0.33/5.22 thf(fact_5318_ntrancl__def, axiom, (! [A:$tType]: (((transitive_ntrancl @ A) = (^ [N3:nat,R6:(set @ ((product_prod @ A) @ A))]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [I5:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ I5) @ R6)))) @ ((collect @ nat) @ (^ [I5:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I5) & (((ord_less_eq @ nat) @ I5) @ (suc @ N3)))))))))))))).
% 0.33/5.22 thf(fact_5319_option_Osize_I4_J, axiom, (! [A:$tType,X22:A]: ((((size_size @ (option @ A)) @ ((some @ A) @ X22)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5320_trancl__finite__eq__relpow, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => (((transitive_trancl @ A) @ R2) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N3:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N3) @ R2)))) @ ((collect @ nat) @ (^ [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((ord_less_eq @ nat) @ N3) @ ((finite_card @ ((product_prod @ A) @ A)) @ R2))))))))))))).
% 0.33/5.22 thf(fact_5321_ntrancl__Zero, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((transitive_ntrancl @ A) @ (zero_zero @ nat)) @ R2) = R2)))).
% 0.33/5.22 thf(fact_5322_take__bit__num__simps_I2_J, axiom, (! [N:nat]: ((((bit_take_bit_num @ (suc @ N)) @ one2) = ((some @ num) @ one2))))).
% 0.33/5.22 thf(fact_5323_take__bit__num__simps_I4_J, axiom, (! [N:nat,M:num]: ((((bit_take_bit_num @ (suc @ N)) @ (bit1 @ M)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ N) @ M))))))).
% 0.33/5.22 thf(fact_5324_execute__len, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ nat) @ ((array_len @ A) @ A3)) @ H2) = ((some @ ((product_prod @ nat) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ nat) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((array_length @ A) @ H2) @ A3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ (one_one @ nat))))))))))).
% 0.33/5.22 thf(fact_5325_finite__trancl__ntranl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => (((transitive_trancl @ A) @ R2) = (((transitive_ntrancl @ A) @ (((minus_minus @ nat) @ ((finite_card @ ((product_prod @ A) @ A)) @ R2)) @ (one_one @ nat))) @ R2)))))).
% 0.33/5.22 thf(fact_5326_trancl__set__ntrancl, axiom, (! [A:$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)))))).
% 0.33/5.22 thf(fact_5327_trancl__power, axiom, (! [A:$tType,P5:((product_prod @ A) @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ P5) @ ((transitive_trancl @ A) @ R2)) = (? [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((member @ ((product_prod @ A) @ A)) @ P5) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N3) @ R2))))))))).
% 0.33/5.22 thf(fact_5328_option_Osize__neq, axiom, (! [A:$tType,X:(option @ A)]: ((((size_size @ (option @ A)) @ X) != (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5329_option_Osize__gen_I2_J, axiom, (! [A:$tType,X:(A > nat),X22:A]: (((((size_option @ A) @ X) @ ((some @ A) @ X22)) = (((plus_plus @ nat) @ (X @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.22 thf(fact_5330_sorted__find__Min, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((? [X4:A]: (((((member @ A) @ X4) @ ((set2 @ A) @ Xs)) & (P @ X4)))) => ((((find @ A) @ P) @ Xs) = ((some @ A) @ ((lattic643756798350308766er_Min @ A) @ ((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))))))))))))))).
% 0.33/5.22 thf(fact_5331_timeFrame_Osimps_I1_J, axiom, (! [A:$tType,N:nat,R:A,H2:(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) @ H2) @ 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) @ H2) @ (((plus_plus @ nat) @ N) @ N2)))))))).
% 0.33/5.22 thf(fact_5332_timeFrame__zero, axiom, (! [A:$tType,H2:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((((heap_Time_timeFrame @ A) @ (zero_zero @ nat)) @ H2) = H2)))).
% 0.33/5.22 thf(fact_5333_timeFrame__assoc, axiom, (! [A:$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))))).
% 0.33/5.22 thf(fact_5334_find__Some__iff2, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(list @ A)]: (((((some @ A) @ X) = (((find @ A) @ P) @ Xs)) = (? [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I5)) & ((X = (((nth @ A) @ Xs) @ I5)) & (! [J3:nat]: (((((ord_less @ nat) @ J3) @ I5) => (~ ((P @ (((nth @ A) @ Xs) @ J3)))))))))))))))).
% 0.33/5.22 thf(fact_5335_find__Some__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),X:A]: ((((((find @ A) @ P) @ Xs) = ((some @ A) @ X)) = (? [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I5)) & ((X = (((nth @ A) @ Xs) @ I5)) & (! [J3:nat]: (((((ord_less @ nat) @ J3) @ I5) => (~ ((P @ (((nth @ A) @ Xs) @ J3)))))))))))))))).
% 0.33/5.22 thf(fact_5336_tap__def, axiom, (! [A:$tType]: (((heap_Time_tap @ A) = (^ [F2:((heap_ext @ product_unit) > A)]: (((heap_Time_Heap2 @ A) @ (^ [H4:(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 @ H4)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ (one_one @ nat))))))))))))).
% 0.33/5.22 thf(fact_5337_execute__bind_I1_J, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),X:A,H:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H2) = ((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) @ H) @ N)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2) = (((heap_Time_timeFrame @ B) @ N) @ (((heap_Time_execute @ B) @ (G @ X)) @ H))))))).
% 0.33/5.22 thf(fact_5338_rtrancl__finite__eq__relpow, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => (((transitive_rtrancl @ A) @ R2) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N3:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N3) @ R2)))) @ ((collect @ nat) @ (^ [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ ((finite_card @ ((product_prod @ A) @ A)) @ R2)))))))))))).
% 0.33/5.22 thf(fact_5339_Heap_Oinject, axiom, (! [A:$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))))).
% 0.33/5.22 thf(fact_5340_Heap__Time__Monad_Obind__bind, axiom, (! [C:$tType,B:$tType,A:$tType,F:(heap_Time_Heap @ A),G:(A > (heap_Time_Heap @ C)),K:(C > (heap_Time_Heap @ B))]: ((((((heap_Time_bind @ C) @ B) @ ((((heap_Time_bind @ A) @ C) @ F) @ G)) @ K) = ((((heap_Time_bind @ A) @ B) @ F) @ (^ [X2:A]: (((((heap_Time_bind @ C) @ B) @ (G @ X2)) @ K)))))))).
% 0.33/5.22 thf(fact_5341_Heap__execute, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((heap_Time_Heap2 @ A) @ ((heap_Time_execute @ A) @ F)) = F)))).
% 0.33/5.22 thf(fact_5342_execute_Osimps, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((heap_Time_execute @ A) @ ((heap_Time_Heap2 @ A) @ F)) = F)))).
% 0.33/5.22 thf(fact_5343_distrib__if__bind, axiom, (! [A:$tType,B:$tType,B2:$o,C2:(heap_Time_Heap @ B),D2:(heap_Time_Heap @ B),F:(B > (heap_Time_Heap @ A))]: (((B2 => (((((heap_Time_bind @ B) @ A) @ ((((if @ (heap_Time_Heap @ B)) @ B2) @ C2) @ D2)) @ F) = ((((heap_Time_bind @ B) @ A) @ C2) @ F))) & ((~ (B2)) => (((((heap_Time_bind @ B) @ A) @ ((((if @ (heap_Time_Heap @ B)) @ B2) @ C2) @ D2)) @ F) = ((((heap_Time_bind @ B) @ A) @ D2) @ F))))))).
% 0.33/5.22 thf(fact_5344_Heap_Oexhaust, axiom, (! [A:$tType,Y3:(heap_Time_Heap @ A)]: ((~ ((! [X3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((Y3 != ((heap_Time_Heap2 @ A) @ X3))))))))).
% 0.33/5.22 thf(fact_5345_assert__cong, axiom, (! [B:$tType,A:$tType,P:(A > $o),P2:(A > $o),F:(A > (heap_Time_Heap @ B)),F5:(A > (heap_Time_Heap @ B)),X:A]: (((P = P2) => ((! [X3:A]: (((P2 @ X3) => ((F @ X3) = (F5 @ X3))))) => (((((heap_Time_bind @ A) @ B) @ (((heap_Time_assert @ A) @ P) @ X)) @ F) = ((((heap_Time_bind @ A) @ B) @ (((heap_Time_assert @ A) @ P2) @ X)) @ F5))))))).
% 0.33/5.22 thf(fact_5346_bind__ureturn, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((((heap_Time_bind @ A) @ A) @ F) @ (heap_Time_ureturn @ A)) = F)))).
% 0.33/5.22 thf(fact_5347_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))))).
% 0.33/5.22 thf(fact_5348_effect__bindE, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ B),G:(B > (heap_Time_Heap @ A)),H2:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),R4:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((((heap_Time_bind @ B) @ A) @ F) @ G)) @ H2) @ H3) @ R4) @ N) => (~ ((! [H7:(heap_ext @ product_unit),R3:B,N1:nat]: ((((((((heap_Time_effect @ B) @ F) @ H2) @ H7) @ R3) @ N1) => (! [N22:nat]: ((((((((heap_Time_effect @ A) @ (G @ R3)) @ H7) @ H3) @ R4) @ N22) => (N != (((plus_plus @ nat) @ N1) @ N22)))))))))))))).
% 0.33/5.22 thf(fact_5349_effect__bindI, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat,G:(A > (heap_Time_Heap @ B)),H3:(heap_ext @ product_unit),R4:B,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H2) @ H) @ R) @ N) => (((((((heap_Time_effect @ B) @ (G @ R)) @ H) @ H3) @ R4) @ N2) => ((((((heap_Time_effect @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2) @ H3) @ R4) @ (((plus_plus @ nat) @ N) @ N2))))))).
% 0.33/5.22 thf(fact_5350_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) @ (^ [X2:B]: (((heap_Time_return @ A) @ (G @ X2))))))))).
% 0.33/5.22 thf(fact_5351_Heap_Osize_I2_J, axiom, (! [A:$tType,X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((size_size @ (heap_Time_Heap @ A)) @ ((heap_Time_Heap2 @ A) @ X)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5352_heap__def, axiom, (! [A:$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)))))))).
% 0.33/5.22 thf(fact_5353_execute__bind__eq__SomeI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),X:A,H:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B)),Y3:B,H3:(heap_ext @ product_unit),N2:nat]: ((((((heap_Time_execute @ A) @ F) @ H2) = ((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) @ H) @ N)))) => (((((heap_Time_execute @ B) @ (G @ X)) @ H) = ((some @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Y3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ N2)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2) = ((some @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Y3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (((plus_plus @ nat) @ N) @ N2)))))))))).
% 0.33/5.22 thf(fact_5354_bind__return, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((((heap_Time_bind @ A) @ A) @ F) @ (heap_Time_return @ A)) = ((((heap_Time_bind @ product_unit) @ A) @ (heap_Time_wait @ (one_one @ nat))) @ (^ [Uu:product_unit]: (F))))))).
% 0.33/5.22 thf(fact_5355_return__bind, axiom, (! [B:$tType,A:$tType,X:B,F:(B > (heap_Time_Heap @ A))]: ((((((heap_Time_bind @ B) @ A) @ ((heap_Time_return @ B) @ X)) @ F) = ((((heap_Time_bind @ product_unit) @ A) @ (heap_Time_wait @ (one_one @ nat))) @ (^ [Uu:product_unit]: ((F @ X)))))))).
% 0.33/5.22 thf(fact_5356_Heap_Osize__gen, axiom, (! [A:$tType,Xa3:(A > nat),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((((heap_Time_size_Heap @ A) @ Xa3) @ ((heap_Time_Heap2 @ A) @ X)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5357_success__bind__executeI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),X:A,H:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H2) = ((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) @ H) @ N)))) => ((((heap_Time_success @ B) @ (G @ X)) @ H) => (((heap_Time_success @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2)))))).
% 0.33/5.22 thf(fact_5358_take__bit__num__def, axiom, (bit_take_bit_num = (^ [N3:nat,M2:num]: (((((if @ (option @ num)) @ ((((bit_se2584673776208193580ke_bit @ nat) @ N3) @ ((numeral_numeral @ nat) @ M2)) = (zero_zero @ nat))) @ (none @ num)) @ ((some @ num) @ (num_of_nat @ (((bit_se2584673776208193580ke_bit @ nat) @ N3) @ ((numeral_numeral @ nat) @ M2))))))))).
% 0.33/5.22 thf(fact_5359_power__int__numeral__neg__numeral, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [M:num,N:num]: (((((power_int @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = ((inverse_inverse @ A) @ ((numeral_numeral @ A) @ ((pow @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_5360_power__int__1__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int]: (((((power_int @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 0.33/5.22 thf(fact_5361_power__int__1__right, axiom, (! [A:$tType]: ((((inverse @ A) & (monoid_mult @ A)) => (! [Y3:A]: (((((power_int @ A) @ Y3) @ (one_one @ int)) = Y3))))))).
% 0.33/5.22 thf(fact_5362_power__int__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,N:int]: ((((sgn_sgn @ A) @ (((power_int @ A) @ A3) @ N)) = (((power_int @ A) @ ((sgn_sgn @ A) @ A3)) @ N)))))))).
% 0.33/5.22 thf(fact_5363_power__int__mult__distrib__numeral2, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,W2:num,M:int]: (((((power_int @ A) @ (((times_times @ A) @ X) @ ((numeral_numeral @ A) @ W2))) @ M) = (((times_times @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ ((numeral_numeral @ A) @ W2)) @ M))))))))).
% 0.33/5.22 thf(fact_5364_power__int__mult__distrib__numeral1, axiom, (! [A:$tType]: (((field @ A) => (! [W2:num,Y3:A,M:int]: (((((power_int @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ Y3)) @ M) = (((times_times @ A) @ (((power_int @ A) @ ((numeral_numeral @ A) @ W2)) @ M)) @ (((power_int @ A) @ Y3) @ M))))))))).
% 0.33/5.22 thf(fact_5365_power__int__eq__0__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: ((((((power_int @ A) @ X) @ N) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (N != (zero_zero @ int)))))))))).
% 0.33/5.22 thf(fact_5366_power__int__0__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int]: (((M != (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5367_power__int__0__right, axiom, (! [B:$tType]: ((((inverse @ B) & (power @ B)) => (! [X:B]: (((((power_int @ B) @ X) @ (zero_zero @ int)) = (one_one @ B)))))))).
% 0.33/5.22 thf(fact_5368_abs__power__int__minus, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,N:int]: ((((abs_abs @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N)) = ((abs_abs @ A) @ (((power_int @ A) @ A3) @ N))))))))).
% 0.33/5.22 thf(fact_5369_power__int__of__nat, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => (! [X:A,N:nat]: (((((power_int @ A) @ X) @ ((semiring_1_of_nat @ int) @ N)) = (((power_power @ A) @ X) @ N)))))))).
% 0.33/5.22 thf(fact_5370_power__int__mult__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:num,N:num]: (((((power_int @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N)) = (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ (((times_times @ num) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_5371_take__bit__num__simps_I1_J, axiom, (! [M:num]: ((((bit_take_bit_num @ (zero_zero @ nat)) @ M) = (none @ num))))).
% 0.33/5.22 thf(fact_5372_power__int__minus__one__mult__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int]: (((((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M)) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M)) = (one_one @ A)))))))).
% 0.33/5.22 thf(fact_5373_power__int__minus__one__mult__self_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int,B2:A]: (((((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M)) @ (((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M)) @ B2)) = B2))))))).
% 0.33/5.22 thf(fact_5374_power__int__numeral, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => (! [X:A,N:num]: (((((power_int @ A) @ X) @ ((numeral_numeral @ int) @ N)) = (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ N))))))))).
% 0.33/5.22 thf(fact_5375_power__int__add__numeral2, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:num,N:num,B2:A]: (((((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ M))) @ (((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) @ M) @ N)))) @ B2)))))))).
% 0.33/5.22 thf(fact_5376_power__int__add__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:num,N:num]: (((((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ M))) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ N))) = (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ (((plus_plus @ num) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_5377_power__int__minus1__right, axiom, (! [A:$tType]: ((((inverse @ A) & (monoid_mult @ A)) => (! [Y3:A]: (((((power_int @ A) @ Y3) @ ((uminus_uminus @ int) @ (one_one @ int))) = ((inverse_inverse @ A) @ Y3)))))))).
% 0.33/5.22 thf(fact_5378_power__int__mono__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,N:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => ((((ord_less_eq @ A) @ (((power_int @ A) @ A3) @ N)) @ (((power_int @ A) @ B2) @ N)) = (((ord_less_eq @ A) @ A3) @ B2))))))))))).
% 0.33/5.22 thf(fact_5379_power__int__minus__left__even, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int,A3:A]: (((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N) => ((((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((power_int @ A) @ A3) @ N))))))))).
% 0.33/5.22 thf(fact_5380_power__int__minus__left__odd, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int,A3:A]: (((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) => ((((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((uminus_uminus @ A) @ (((power_int @ A) @ A3) @ N)))))))))).
% 0.33/5.22 thf(fact_5381_take__bit__num__simps_I3_J, axiom, (! [N:nat,M:num]: ((((bit_take_bit_num @ (suc @ N)) @ (bit0 @ M)) = (((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q3:num]: (((some @ num) @ (bit0 @ Q3))))) @ ((bit_take_bit_num @ N) @ M)))))).
% 0.33/5.22 thf(fact_5382_power__int__not__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: ((((X != (zero_zero @ A)) | (N = (zero_zero @ int))) => ((((power_int @ A) @ X) @ N) != (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5383_zero__le__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,N:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5384_success__LetI, axiom, (! [A:$tType,B:$tType,X:A,T2:A,F:(A > (heap_Time_Heap @ B)),H2:(heap_ext @ product_unit)]: (((X = T2) => ((((heap_Time_success @ B) @ (F @ X)) @ H2) => (((heap_Time_success @ B) @ (F @ T2)) @ H2)))))).
% 0.33/5.22 thf(fact_5385_power__int__commutes, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: (((((times_times @ A) @ (((power_int @ A) @ X) @ N)) @ X) = (((times_times @ A) @ X) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5386_power__int__mult__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,Y3:A,M:int]: (((((power_int @ A) @ (((times_times @ A) @ X) @ Y3)) @ M) = (((times_times @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ Y3) @ M))))))))).
% 0.33/5.22 thf(fact_5387_power__int__mult, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:int,N:int]: (((((power_int @ A) @ X) @ (((times_times @ int) @ M) @ N)) = (((power_int @ A) @ (((power_int @ A) @ X) @ M)) @ N)))))))).
% 0.33/5.22 thf(fact_5388_power__int__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: (((((power_int @ A) @ ((inverse_inverse @ A) @ X)) @ N) = ((inverse_inverse @ A) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5389_success__ifI, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),E:(heap_Time_Heap @ A)]: (((C2 => (((heap_Time_success @ A) @ T2) @ H2)) => (((~ (C2)) => (((heap_Time_success @ A) @ E) @ H2)) => (((heap_Time_success @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E)) @ H2)))))).
% 0.33/5.22 thf(fact_5390_power__int__divide__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,Y3:A,M:int]: (((((power_int @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ M) = (((divide_divide @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ Y3) @ M))))))))).
% 0.33/5.22 thf(fact_5391_power__int__abs, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,N:int]: ((((abs_abs @ A) @ (((power_int @ A) @ A3) @ N)) = (((power_int @ A) @ ((abs_abs @ A) @ A3)) @ N)))))))).
% 0.33/5.22 thf(fact_5392_effect__success, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ C2) @ H2) @ H) @ R) @ N) => (((heap_Time_success @ A) @ C2) @ H2))))).
% 0.33/5.22 thf(fact_5393_success__effectE, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit)]: (((((heap_Time_success @ A) @ C2) @ H2) => (~ ((! [R3:A,H7:(heap_ext @ product_unit),N4:nat]: ((~ (((((((heap_Time_effect @ A) @ C2) @ H2) @ H7) @ R3) @ N4))))))))))).
% 0.33/5.22 thf(fact_5394_zero__less__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,N:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5395_power__int__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: (((((power_int @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ X)) @ N) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5396_power__int__minus, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: (((((power_int @ A) @ X) @ ((uminus_uminus @ int) @ N)) = ((inverse_inverse @ A) @ (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5397_success__returnI, axiom, (! [A:$tType,X:A,H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_return @ A) @ X)) @ H2)))).
% 0.33/5.22 thf(fact_5398_success__heapI, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_heap @ A) @ F)) @ H2)))).
% 0.33/5.22 thf(fact_5399_success__guardI, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H2:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((P @ H2) => (((heap_Time_success @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H2))))).
% 0.33/5.22 thf(fact_5400_success__assertI, axiom, (! [A:$tType,P:(A > $o),X:A,H2:(heap_ext @ product_unit)]: (((P @ X) => (((heap_Time_success @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H2))))).
% 0.33/5.22 thf(fact_5401_success__ureturnI, axiom, (! [A:$tType,X:A,H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H2)))).
% 0.33/5.22 thf(fact_5402_success__makeI, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H2))))))).
% 0.33/5.22 thf(fact_5403_success__of__listI, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H2))))))).
% 0.33/5.22 thf(fact_5404_success__newI, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H2))))))).
% 0.33/5.22 thf(fact_5405_success__tapI, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_tap @ A) @ F)) @ H2)))).
% 0.33/5.22 thf(fact_5406_success__lenI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H2:(heap_ext @ product_unit)]: ((((heap_Time_success @ nat) @ ((array_len @ A) @ A3)) @ H2))))))).
% 0.33/5.22 thf(fact_5407_power__int__0__left__If, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int]: ((((M = (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (one_one @ A))) & ((M != (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_5408_power__int__increasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less_eq @ int) @ N) @ N6) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A3) => (((ord_less_eq @ A) @ (((power_int @ A) @ A3) @ N)) @ (((power_int @ A) @ A3) @ N6)))))))))).
% 0.33/5.22 thf(fact_5409_power__int__strict__increasing, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_5410_power__int__minus__one__minus, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int]: (((((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ int) @ N)) = (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)))))))).
% 0.33/5.22 thf(fact_5411_power__int__diff, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,M:int,N:int]: ((((X != (zero_zero @ A)) | (M != N)) => ((((power_int @ A) @ X) @ (((minus_minus @ int) @ M) @ N)) = (((divide_divide @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ X) @ N)))))))))).
% 0.33/5.22 thf(fact_5412_power__int__minus__one__diff__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:int,B2:int]: (((((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ int) @ A3) @ B2)) = (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ int) @ B2) @ A3))))))))).
% 0.33/5.22 thf(fact_5413_success__bind__effectI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),X:A,N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((((heap_Time_effect @ A) @ F) @ H2) @ H) @ X) @ N) => ((((heap_Time_success @ B) @ (G @ X)) @ H) => (((heap_Time_success @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2)))))).
% 0.33/5.22 thf(fact_5414_power__int__strict__decreasing, axiom, (! [A:$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))))))))))).
% 0.33/5.22 thf(fact_5415_power__int__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,N:int]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => (((ord_less_eq @ A) @ (((power_int @ A) @ X) @ N)) @ (((power_int @ A) @ Y3) @ N))))))))))).
% 0.33/5.22 thf(fact_5416_power__int__strict__antimono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,N:int]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ int) @ N) @ (zero_zero @ int)) => (((ord_less @ A) @ (((power_int @ A) @ B2) @ N)) @ (((power_int @ A) @ A3) @ N))))))))))).
% 0.33/5.22 thf(fact_5417_one__le__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,N:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_int @ A) @ X) @ N)))))))))).
% 0.33/5.22 thf(fact_5418_one__less__power__int, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_5419_power__int__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:int,N:int]: ((((X != (zero_zero @ A)) | ((((plus_plus @ int) @ M) @ N) != (zero_zero @ int))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M) @ N)) = (((times_times @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ X) @ N)))))))))).
% 0.33/5.22 thf(fact_5420_option_Osize__gen_I1_J, axiom, (! [A:$tType,X:(A > nat)]: (((((size_option @ A) @ X) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5421_option_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (option @ A)) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5422_power__int__minus__left__distrib, axiom, (! [C:$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)))))))))).
% 0.33/5.22 thf(fact_5423_take__bit__num__eq__None__imp, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N:num]: (((((bit_take_bit_num @ M) @ N) = (none @ num)) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ ((numeral_numeral @ A) @ N)) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5424_and__num__eq__None__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:num]: (((((bit_un7362597486090784418nd_num @ M) @ N) = (none @ num)) = ((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5425_xor__num__eq__None__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N:num]: (((((bit_un2480387367778600638or_num @ M) @ N) = (none @ num)) = ((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5426_power__int__antimono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,N:int]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ int) @ N) @ (zero_zero @ int)) => (((ord_less_eq @ A) @ (((power_int @ A) @ B2) @ N)) @ (((power_int @ A) @ A3) @ N))))))))))).
% 0.33/5.22 thf(fact_5427_power__int__strict__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,N:int]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((ord_less @ A) @ (((power_int @ A) @ A3) @ N)) @ (((power_int @ A) @ B2) @ N))))))))))).
% 0.33/5.22 thf(fact_5428_power__int__le__one, axiom, (! [A:$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))))))))))).
% 0.33/5.22 thf(fact_5429_power__int__decreasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less_eq @ int) @ N) @ N6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((A3 != (zero_zero @ A)) | ((N6 != (zero_zero @ int)) | (N = (zero_zero @ int)))) => (((ord_less_eq @ A) @ (((power_int @ A) @ A3) @ N6)) @ (((power_int @ A) @ A3) @ N)))))))))))).
% 0.33/5.22 thf(fact_5430_power__int__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,M:int,N:int]: (((((ord_less @ A) @ (one_one @ A)) @ X) => ((((ord_less_eq @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ X) @ N)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ int) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_5431_power__int__le__imp__less__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,M:int,N:int]: (((((ord_less @ A) @ (one_one @ A)) @ X) => ((((ord_less @ A) @ (((power_int @ A) @ X) @ M)) @ (((power_int @ A) @ X) @ N)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less @ int) @ M) @ N)))))))))).
% 0.33/5.22 thf(fact_5432_power__int__minus__mult, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,N:int]: ((((X != (zero_zero @ A)) | (N != (zero_zero @ int))) => ((((times_times @ A) @ (((power_int @ A) @ X) @ (((minus_minus @ int) @ N) @ (one_one @ int)))) @ X) = (((power_int @ A) @ X) @ N))))))))).
% 0.33/5.22 thf(fact_5433_power__int__add__1_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:int]: ((((X != (zero_zero @ A)) | (M != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M) @ (one_one @ int))) = (((times_times @ A) @ X) @ (((power_int @ A) @ X) @ M)))))))))).
% 0.33/5.22 thf(fact_5434_power__int__add__1, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M:int]: ((((X != (zero_zero @ A)) | (M != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M) @ (one_one @ int))) = (((times_times @ A) @ (((power_int @ A) @ X) @ M)) @ X))))))))).
% 0.33/5.22 thf(fact_5435_power__int__minus__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int,A3:A]: ((((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N) => ((((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((power_int @ A) @ A3) @ N))) & ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) => ((((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((uminus_uminus @ A) @ (((power_int @ A) @ A3) @ N))))))))))).
% 0.33/5.22 thf(fact_5436_successE, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit)]: (((((heap_Time_success @ A) @ F) @ H2) => (~ ((! [R3:A,H7:((product_prod @ (heap_ext @ product_unit)) @ nat)]: (((((heap_Time_execute @ A) @ F) @ H2) != ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ H7))))))))))).
% 0.33/5.22 thf(fact_5437_and__not__num__eq__None__iff, axiom, (! [M:num,N:num]: (((((bit_and_not_num @ M) @ N) = (none @ num)) = ((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))) = (zero_zero @ int)))))).
% 0.33/5.22 thf(fact_5438_power__int__def, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => ((power_int @ A) = (^ [X2:A,N3:int]: (((((if @ A) @ (((ord_less_eq @ int) @ (zero_zero @ int)) @ N3)) @ (((power_power @ A) @ X2) @ (nat2 @ N3))) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X2)) @ (nat2 @ ((uminus_uminus @ int) @ N3))))))))))).
% 0.33/5.22 thf(fact_5439_execute__bind__success, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),G:(A > (heap_Time_Heap @ B))]: (((((heap_Time_success @ A) @ F) @ H2) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2) = (((heap_Time_timeFrame @ B) @ (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H2))))) @ (((heap_Time_execute @ B) @ (G @ (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H2))))) @ (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H2))))))))))).
% 0.33/5.22 thf(fact_5440_effectE, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ C2) @ H2) @ H) @ 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) @ H2)))) => ((H = (((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) @ H2))))) => ((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) @ H2))))) => (~ ((((heap_Time_success @ A) @ C2) @ H2)))))))))))).
% 0.33/5.22 thf(fact_5441_in__finite__psubset, axiom, (! [A:$tType,A6:(set @ A),B7:(set @ A)]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A6) @ B7)) @ (finite_psubset @ A)) = ((((ord_less @ (set @ A)) @ A6) @ B7) & ((finite_finite2 @ A) @ B7)))))).
% 0.33/5.22 thf(fact_5442_timeFrame_Osimps_I2_J, axiom, (! [A:$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))))))).
% 0.33/5.22 thf(fact_5443_execute__bind_I2_J, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit),G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H2) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H2) = (none @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.22 thf(fact_5444_successI, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit)]: ((((((heap_Time_execute @ A) @ F) @ H2) != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => (((heap_Time_success @ A) @ F) @ H2))))).
% 0.33/5.22 thf(fact_5445_success__def, axiom, (! [A:$tType]: (((heap_Time_success @ A) = (^ [F2:(heap_Time_Heap @ A),H4:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ F2) @ H4) != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 0.33/5.22 thf(fact_5446_execute__guard_I1_J, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H2:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((~ ((P @ H2))) => ((((heap_Time_execute @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H2) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.22 thf(fact_5447_execute__assert_I2_J, axiom, (! [A:$tType,P:(A > $o),X:A,H2:(heap_ext @ product_unit)]: (((~ ((P @ X))) => ((((heap_Time_execute @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H2) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.22 thf(fact_5448_finite__psubset__def, axiom, (! [A:$tType]: (((finite_psubset @ A) = ((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [A7:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B6) & ((finite_finite2 @ A) @ B6)))))))))).
% 0.33/5.22 thf(fact_5449_Heap__cases, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit)]: (((! [X3:A,H7:((product_prod @ (heap_ext @ product_unit)) @ nat)]: (((((heap_Time_execute @ A) @ F) @ H2) != ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X3) @ H7))))) => ((((heap_Time_execute @ A) @ F) @ H2) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 0.33/5.22 thf(fact_5450_timeFrame_Ocases, axiom, (! [A:$tType,X:((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((! [N4:nat,R3:A,H6:(heap_ext @ product_unit),N10:nat]: ((X != ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ N4) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H6) @ N10))))))) => (~ ((! [N4:nat]: ((X != ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ N4) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))))).
% 0.33/5.22 thf(fact_5451_Max_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((lattic643756798349783984er_Max @ A) @ A6) = ((the2 @ A) @ (none @ A)))))))))).
% 0.33/5.22 thf(fact_5452_Min_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((lattic643756798350308766er_Min @ A) @ A6) = ((the2 @ A) @ (none @ A)))))))))).
% 0.33/5.22 thf(fact_5453_guard__def, axiom, (! [A:$tType]: (((heap_Time_guard @ A) = (^ [P4:((heap_ext @ product_unit) > $o),F2:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((heap_Time_Heap2 @ A) @ (^ [H4:(heap_ext @ product_unit)]: (((((if @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (P4 @ H4)) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (F2 @ H4))) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))))).
% 0.33/5.22 thf(fact_5454_timeFrame_Oelims, axiom, (! [A:$tType,X:nat,Xa3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),Y3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((((heap_Time_timeFrame @ A) @ X) @ Xa3) = Y3) => ((! [R3:A,H6:(heap_ext @ product_unit),N10:nat]: (((Xa3 = ((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) @ H6) @ N10)))) => (Y3 != ((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) @ H6) @ (((plus_plus @ nat) @ X) @ N10)))))))) => (~ (((Xa3 = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => (Y3 != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))))).
% 0.33/5.22 thf(fact_5455_in__measure, axiom, (! [A:$tType,X:A,Y3:A,F:(A > nat)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((measure @ A) @ F)) = (((ord_less @ nat) @ (F @ X)) @ (F @ Y3)))))).
% 0.33/5.22 thf(fact_5456_pred__nat__def, axiom, (pred_nat = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [M2:nat,N3:nat]: ((N3 = (suc @ M2)))))))).
% 0.33/5.22 thf(fact_5457_Min_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798350308766er_Min @ A) = (^ [A7:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X2:A,Y6:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X2) @ ((ord_min @ A) @ X2)) @ Y6))))) @ (none @ A)) @ A7))))))))).
% 0.33/5.22 thf(fact_5458_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))))))))).
% 0.33/5.22 thf(fact_5459_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))))))))).
% 0.33/5.22 thf(fact_5460_Max_Oeq__fold, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X) @ A6)) = (((((finite_fold @ A) @ A) @ (ord_max @ A)) @ X) @ A6))))))))).
% 0.33/5.22 thf(fact_5461_Min_Oeq__fold, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X) @ A6)) = (((((finite_fold @ A) @ A) @ (ord_min @ A)) @ X) @ A6))))))))).
% 0.33/5.22 thf(fact_5462_execute__bind__case, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ B),G:(B > (heap_Time_Heap @ A)),H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ ((((heap_Time_bind @ B) @ A) @ F) @ G)) @ H2) = (((((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)))) @ (^ [X2:B]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H5:(heap_ext @ product_unit),N3:nat]: ((((heap_Time_timeFrame @ A) @ N3) @ (((heap_Time_execute @ A) @ (G @ X2)) @ H5))))))))) @ (((heap_Time_execute @ B) @ F) @ H2)))))).
% 0.33/5.22 thf(fact_5463_less__eq, axiom, (! [M:nat,N:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M) @ N)) @ ((transitive_trancl @ nat) @ pred_nat)) = (((ord_less @ nat) @ M) @ N))))).
% 0.33/5.22 thf(fact_5464_pred__nat__trancl__eq__le, axiom, (! [M:nat,N:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M) @ N)) @ ((transitive_rtrancl @ nat) @ pred_nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.33/5.22 thf(fact_5465_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) @ (^ [H4:(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)))) @ (^ [R5:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H5:(heap_ext @ product_unit),N3:nat]: ((((heap_Time_timeFrame @ B) @ N3) @ (((heap_Time_execute @ B) @ (G2 @ R5)) @ H5))))))))) @ (((heap_Time_execute @ A) @ F2) @ H4))))))))))).
% 0.33/5.22 thf(fact_5466_Max_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798349783984er_Max @ A) = (^ [A7:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X2:A,Y6:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X2) @ ((ord_max @ A) @ X2)) @ Y6))))) @ (none @ A)) @ A7))))))))).
% 0.33/5.22 thf(fact_5467_Sup__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((lattic5882676163264333800up_fin @ A) = (^ [A7:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X2:A,Y6:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X2) @ ((sup_sup @ A) @ X2)) @ Y6))))) @ (none @ A)) @ A7))))))))).
% 0.33/5.22 thf(fact_5468_Inf__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((lattic7752659483105999362nf_fin @ A) = (^ [A7:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X2:A,Y6:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X2) @ ((inf_inf @ A) @ X2)) @ Y6))))) @ (none @ A)) @ A7))))))))).
% 0.33/5.22 thf(fact_5469_timeFrame_Opelims, axiom, (! [A:$tType,X:nat,Xa3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),Y3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((((heap_Time_timeFrame @ A) @ X) @ Xa3) = Y3) => ((((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) @ Xa3)) => ((! [R3:A,H6:(heap_ext @ product_unit),N10:nat]: (((Xa3 = ((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) @ H6) @ N10)))) => ((Y3 = ((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) @ H6) @ (((plus_plus @ nat) @ X) @ N10))))) => (~ ((((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)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H6) @ N10))))))))))) => (~ (((Xa3 = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => ((Y3 = (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))))))))))))))))).
% 0.33/5.22 thf(fact_5470_Inf__fin_Osingleton, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A]: ((((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X))))))).
% 0.33/5.22 thf(fact_5471_Sup__fin_Osingleton, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A]: ((((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X))))))).
% 0.33/5.22 thf(fact_5472_inf__Sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => ((((inf_inf @ A) @ A3) @ ((lattic5882676163264333800up_fin @ A) @ A6)) = A3))))))))).
% 0.33/5.22 thf(fact_5473_sup__Inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => ((((sup_sup @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ A3) = A3))))))))).
% 0.33/5.22 thf(fact_5474_Inf__fin_Oinsert, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((inf_inf @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5475_Sup__fin_Oinsert, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((sup_sup @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5476_Inf__fin__Min, axiom, (! [A:$tType]: ((((semilattice_inf @ A) & (linorder @ A)) => ((lattic7752659483105999362nf_fin @ A) = (lattic643756798350308766er_Min @ A)))))).
% 0.33/5.22 thf(fact_5477_Sup__fin__Max, axiom, (! [A:$tType]: ((((semilattice_sup @ A) & (linorder @ A)) => ((lattic5882676163264333800up_fin @ A) = (lattic643756798349783984er_Max @ A)))))).
% 0.33/5.22 thf(fact_5478_Inf__fin__le__Sup__fin, axiom, (! [A:$tType]: (((lattice @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ ((lattic5882676163264333800up_fin @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5479_Sup__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (((ord_less_eq @ A) @ A3) @ ((lattic5882676163264333800up_fin @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5480_Inf__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ A3))))))))).
% 0.33/5.22 thf(fact_5481_Inf__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((inf_inf @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) = ((lattic7752659483105999362nf_fin @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5482_Sup__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((sup_sup @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ A6)) = ((lattic5882676163264333800up_fin @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5483_Sup__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ X) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X2) @ X))))))))))))).
% 0.33/5.22 thf(fact_5484_Inf__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((ord_less_eq @ A) @ X) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5485_Sup__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => (((ord_less_eq @ A) @ A8) @ X)))) => (((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ X)))))))))).
% 0.33/5.22 thf(fact_5486_Sup__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ X) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => (((ord_less_eq @ A) @ A10) @ X))))))))))))).
% 0.33/5.22 thf(fact_5487_Inf__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => (((ord_less_eq @ A) @ X) @ A8)))) => (((ord_less_eq @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5488_Inf__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => (((ord_less_eq @ A) @ X) @ A10))))))))))))).
% 0.33/5.22 thf(fact_5489_Sup__fin__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A6) = ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5490_card_Oeq__fold, axiom, (! [A:$tType]: (((finite_card @ A) = ((((finite_fold @ A) @ nat) @ (^ [Uu:A]: (suc))) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5491_Inf__fin__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A6) = ((complete_Inf_Inf @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_5492_Inf__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((lattic7752659483105999362nf_fin @ A) @ A6) = ((the2 @ A) @ (none @ A)))))))))).
% 0.33/5.22 thf(fact_5493_Sup__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((lattic5882676163264333800up_fin @ A) @ A6) = ((the2 @ A) @ (none @ A)))))))))).
% 0.33/5.22 thf(fact_5494_Sup__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => (((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ ((lattic5882676163264333800up_fin @ A) @ B7))))))))))).
% 0.33/5.22 thf(fact_5495_Inf__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ B7)) @ ((lattic7752659483105999362nf_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5496_Inf__fin_Ohom__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [H2:(A > A),N6:(set @ A)]: (((! [X3:A,Y5:A]: (((H2 @ (((inf_inf @ A) @ X3) @ Y5)) = (((inf_inf @ A) @ (H2 @ X3)) @ (H2 @ Y5))))) => (((finite_finite2 @ A) @ N6) => ((N6 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic7752659483105999362nf_fin @ A) @ N6)) = ((lattic7752659483105999362nf_fin @ A) @ ((((image @ A) @ A) @ H2) @ N6)))))))))))).
% 0.33/5.22 thf(fact_5497_Sup__fin_Ohom__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [H2:(A > A),N6:(set @ A)]: (((! [X3:A,Y5:A]: (((H2 @ (((sup_sup @ A) @ X3) @ Y5)) = (((sup_sup @ A) @ (H2 @ X3)) @ (H2 @ Y5))))) => (((finite_finite2 @ A) @ N6) => ((N6 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic5882676163264333800up_fin @ A) @ N6)) = ((lattic5882676163264333800up_fin @ A) @ ((((image @ A) @ A) @ H2) @ N6)))))))))))).
% 0.33/5.22 thf(fact_5498_Inf__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((B7 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => ((((inf_inf @ A) @ ((lattic7752659483105999362nf_fin @ A) @ B7)) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) = ((lattic7752659483105999362nf_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5499_Sup__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((B7 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => ((((sup_sup @ A) @ ((lattic5882676163264333800up_fin @ A) @ B7)) @ ((lattic5882676163264333800up_fin @ A) @ A6)) = ((lattic5882676163264333800up_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5500_Inf__fin_Oinsert__not__elem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => ((A6 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((inf_inf @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5501_Inf__fin_Oclosed, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ (((inf_inf @ A) @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ A6)))))))))).
% 0.33/5.22 thf(fact_5502_Sup__fin_Oclosed, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ (((sup_sup @ A) @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ A6)))))))))).
% 0.33/5.22 thf(fact_5503_Sup__fin_Oinsert__not__elem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => ((A6 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((sup_sup @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5504_Inf__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = (((inf_inf @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ ((lattic7752659483105999362nf_fin @ A) @ B7))))))))))))).
% 0.33/5.22 thf(fact_5505_Sup__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = (((sup_sup @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ ((lattic5882676163264333800up_fin @ A) @ B7))))))))))))).
% 0.33/5.22 thf(fact_5506_Inf__fin_Oeq__fold, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((((finite_fold @ A) @ A) @ (inf_inf @ A)) @ X) @ A6))))))))).
% 0.33/5.22 thf(fact_5507_Sup__fin_Oeq__fold, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((((finite_fold @ A) @ A) @ (sup_sup @ A)) @ X) @ A6))))))))).
% 0.33/5.22 thf(fact_5508_inf__Sup1__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((inf_inf @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ A6)) = ((lattic5882676163264333800up_fin @ A) @ ((collect @ A) @ (^ [Uu:A]: ((? [A5:A]: (((Uu = (((inf_inf @ A) @ X) @ A5)) & (((member @ A) @ A5) @ A6))))))))))))))))).
% 0.33/5.22 thf(fact_5509_inf__Sup2__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => ((((inf_inf @ A) @ ((lattic5882676163264333800up_fin @ A) @ A6)) @ ((lattic5882676163264333800up_fin @ A) @ B7)) = ((lattic5882676163264333800up_fin @ A) @ ((collect @ A) @ (^ [Uu:A]: ((? [A5:A,B3:A]: (((Uu = (((inf_inf @ A) @ A5) @ B3)) & ((((member @ A) @ A5) @ A6) & (((member @ A) @ B3) @ B7)))))))))))))))))))).
% 0.33/5.22 thf(fact_5510_sup__Inf2__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => ((((sup_sup @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) @ ((lattic7752659483105999362nf_fin @ A) @ B7)) = ((lattic7752659483105999362nf_fin @ A) @ ((collect @ A) @ (^ [Uu:A]: ((? [A5:A,B3:A]: (((Uu = (((sup_sup @ A) @ A5) @ B3)) & ((((member @ A) @ A5) @ A6) & (((member @ A) @ B3) @ B7)))))))))))))))))))).
% 0.33/5.22 thf(fact_5511_sup__Inf1__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((sup_sup @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ A6)) = ((lattic7752659483105999362nf_fin @ A) @ ((collect @ A) @ (^ [Uu:A]: ((? [A5:A]: (((Uu = (((sup_sup @ A) @ X) @ A5)) & (((member @ A) @ A5) @ A6))))))))))))))))).
% 0.33/5.22 thf(fact_5512_Inf__fin_Oinsert__remove, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ A6)) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((inf_inf @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))))))).
% 0.33/5.22 thf(fact_5513_Inf__fin_Oremove, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A6) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A6) = (((inf_inf @ A) @ X) @ ((lattic7752659483105999362nf_fin @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))))))).
% 0.33/5.22 thf(fact_5514_Sup__fin_Oinsert__remove, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ A6)) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X) @ A6)) = (((sup_sup @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))))))).
% 0.33/5.22 thf(fact_5515_Sup__fin_Oremove, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A6:(set @ A),X:A]: ((((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A6) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A6) = (((sup_sup @ A) @ X) @ ((lattic5882676163264333800up_fin @ A) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))))))).
% 0.33/5.22 thf(fact_5516_raise__def, axiom, (! [A:$tType]: (((heap_Time_raise @ A) = (^ [S6:(list @ char)]: (((heap_Time_Heap2 @ A) @ (^ [Uu:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))))).
% 0.33/5.22 thf(fact_5517_Gcd__fin_Oeq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Gcd_fin @ A) = (^ [A7:(set @ A)]: (((((if @ A) @ ((finite_finite2 @ A) @ A7)) @ (((((finite_fold @ A) @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ A7)) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5518_funpow__inj__finite, axiom, (! [A:$tType,P5:(A > A),X:A]: ((((((inj_on @ A) @ A) @ P5) @ (top_top @ (set @ A))) => (((finite_finite2 @ A) @ ((collect @ A) @ (^ [Y6:A]: ((? [N3:nat]: ((Y6 = ((((compow @ (A > A)) @ N3) @ P5) @ X)))))))) => (~ ((! [N4:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N4) => (((((compow @ (A > A)) @ N4) @ P5) @ X) != X))))))))))).
% 0.33/5.22 thf(fact_5519_inj__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A6:(set @ A)]: (((((inj_on @ A) @ A) @ (uminus_uminus @ A)) @ A6))))))).
% 0.33/5.22 thf(fact_5520_raise__bind, axiom, (! [B:$tType,A:$tType,E:(list @ char),F:(B > (heap_Time_Heap @ A))]: ((((((heap_Time_bind @ B) @ A) @ ((heap_Time_raise @ B) @ E)) @ F) = ((heap_Time_raise @ A) @ E))))).
% 0.33/5.22 thf(fact_5521_Gcd__fin_Oempty, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((semiring_gcd_Gcd_fin @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_5522_Gcd__fin_Oinfinite, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((semiring_gcd_Gcd_fin @ A) @ A6) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5523_is__unit__Gcd__fin__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((((dvd_dvd @ A) @ ((semiring_gcd_Gcd_fin @ A) @ A6)) @ (one_one @ A)) = (((semiring_gcd_Gcd_fin @ A) @ A6) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5524_inj__mult__left, axiom, (! [A:$tType]: (((idom @ A) => (! [A3:A]: ((((((inj_on @ A) @ A) @ ((times_times @ A) @ A3)) @ (top_top @ (set @ A))) = (A3 != (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5525_inj__divide__right, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A]: ((((((inj_on @ A) @ A) @ (^ [B3:A]: ((((divide_divide @ A) @ B3) @ A3)))) @ (top_top @ (set @ A))) = (A3 != (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_5526_inj__fn, axiom, (! [A:$tType,F:(A > A),N:nat]: ((((((inj_on @ A) @ A) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_5527_subset__image__inj, axiom, (! [A:$tType,B:$tType,S3:(set @ A),F:(B > A),T6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ S3) @ ((((image @ B) @ A) @ F) @ T6)) = (? [U5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ U5) @ T6) & (((((inj_on @ B) @ A) @ F) @ U5) & (S3 = ((((image @ B) @ A) @ F) @ U5)))))))))).
% 0.33/5.22 thf(fact_5528_inj__list__encode, axiom, (! [A6:(set @ (list @ nat))]: (((((inj_on @ (list @ nat)) @ nat) @ nat_list_encode) @ A6)))).
% 0.33/5.22 thf(fact_5529_inj__prod__encode, axiom, (! [A6:(set @ ((product_prod @ nat) @ nat))]: (((((inj_on @ ((product_prod @ nat) @ nat)) @ nat) @ nat_prod_encode) @ A6)))).
% 0.33/5.22 thf(fact_5530_inj__Suc, axiom, (! [N6:(set @ nat)]: (((((inj_on @ nat) @ nat) @ suc) @ N6)))).
% 0.33/5.22 thf(fact_5531_inj__on__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N6:(set @ nat)]: (((((inj_on @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6))))))).
% 0.33/5.22 thf(fact_5532_inj__int__encode, axiom, (! [A6:(set @ int)]: (((((inj_on @ int) @ nat) @ nat_int_encode) @ A6)))).
% 0.33/5.22 thf(fact_5533_inj__int__decode, axiom, (! [A6:(set @ nat)]: (((((inj_on @ nat) @ int) @ nat_int_decode) @ A6)))).
% 0.33/5.22 thf(fact_5534_linorder__inj__onI, axiom, (! [B:$tType,A:$tType]: (((order @ A) => (! [A6:(set @ A),F:(A > B)]: (((! [X3:A,Y5:A]: (((((ord_less @ A) @ X3) @ Y5) => ((((member @ A) @ X3) @ A6) => ((((member @ A) @ Y5) @ A6) => ((F @ X3) != (F @ Y5))))))) => ((! [X3:A,Y5:A]: (((((member @ A) @ X3) @ A6) => ((((member @ A) @ Y5) @ A6) => ((((ord_less_eq @ A) @ X3) @ Y5) | (((ord_less_eq @ A) @ Y5) @ X3)))))) => ((((inj_on @ A) @ B) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_5535_inj__on__mult, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A,A6:(set @ A)]: (((A3 != (zero_zero @ A)) => ((((inj_on @ A) @ A) @ ((times_times @ A) @ A3)) @ A6)))))))).
% 0.33/5.22 thf(fact_5536_linorder__injI, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [F:(A > B)]: (((! [X3:A,Y5:A]: (((((ord_less @ A) @ X3) @ Y5) => ((F @ X3) != (F @ Y5))))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A)))))))))).
% 0.33/5.22 thf(fact_5537_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),B7:(set @ A),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B7) => ((((ord_less @ (set @ A)) @ A6) @ B7) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A6)) @ ((((image @ A) @ B) @ F) @ B7))))))).
% 0.33/5.22 thf(fact_5538_inj__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((((inj_on @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (top_top @ (set @ nat))))))).
% 0.33/5.22 thf(fact_5539_inj__on__diff__nat, axiom, (! [N6:(set @ nat),K:nat]: (((! [N4:nat]: (((((member @ nat) @ N4) @ N6) => (((ord_less_eq @ nat) @ K) @ N4)))) => ((((inj_on @ nat) @ nat) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ K)))) @ N6))))).
% 0.33/5.22 thf(fact_5540_inj__on__set__encode, axiom, ((((inj_on @ (set @ nat)) @ nat) @ nat_set_encode) @ ((collect @ (set @ nat)) @ (finite_finite2 @ nat)))).
% 0.33/5.22 thf(fact_5541_inj__on__iff__surj, axiom, (! [A:$tType,B:$tType,A6:(set @ A),A11:(set @ B)]: (((A6 != (bot_bot @ (set @ A))) => ((? [F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ A6) & (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A6)) @ A11)))) = (? [G2:(B > A)]: ((((((image @ B) @ A) @ G2) @ A11) = A6)))))))).
% 0.33/5.22 thf(fact_5542_pigeonhole, axiom, (! [A:$tType,B:$tType,F:(B > A),A6:(set @ B)]: (((((ord_less @ nat) @ ((finite_card @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((finite_card @ B) @ A6)) => (~ (((((inj_on @ B) @ A) @ F) @ A6))))))).
% 0.33/5.22 thf(fact_5543_effect__raiseE, axiom, (! [A:$tType,X:(list @ char),H2:(heap_ext @ product_unit),H:(heap_ext @ product_unit),R:A,N:nat]: ((~ (((((((heap_Time_effect @ A) @ ((heap_Time_raise @ A) @ X)) @ H2) @ H) @ R) @ N)))))).
% 0.33/5.22 thf(fact_5544_finite__imp__nat__seg__image__inj__on, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (? [N4:nat,F3:(nat > A)]: (((A6 = ((((image @ nat) @ A) @ F3) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N4)))))) & ((((inj_on @ nat) @ A) @ F3) @ ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N4)))))))))))).
% 0.33/5.22 thf(fact_5545_finite__imp__inj__to__nat__seg, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => (? [F3:(A > nat),N4:nat]: (((((((image @ A) @ nat) @ F3) @ A6) = ((collect @ nat) @ (^ [I5:nat]: ((((ord_less @ nat) @ I5) @ N4))))) & ((((inj_on @ A) @ nat) @ F3) @ A6)))))))).
% 0.33/5.22 thf(fact_5546_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A6))) @ ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A6))))))).
% 0.33/5.22 thf(fact_5547_inj__on__nth, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((distinct @ A) @ Xs) => ((! [X3:nat]: (((((member @ nat) @ X3) @ I6) => (((ord_less @ nat) @ X3) @ ((size_size @ (list @ A)) @ Xs))))) => ((((inj_on @ nat) @ A) @ ((nth @ A) @ Xs)) @ I6)))))).
% 0.33/5.22 thf(fact_5548_infinite__iff__countable__subset, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) = (? [F2:(nat > A)]: ((((((inj_on @ nat) @ A) @ F2) @ (top_top @ (set @ nat))) & (((ord_less_eq @ (set @ A)) @ ((((image @ nat) @ A) @ F2) @ (top_top @ (set @ nat)))) @ S3)))))))).
% 0.33/5.22 thf(fact_5549_infinite__countable__subset, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (? [F3:(nat > A)]: ((((((inj_on @ nat) @ A) @ F3) @ (top_top @ (set @ nat))) & (((ord_less_eq @ (set @ A)) @ ((((image @ nat) @ A) @ F3) @ (top_top @ (set @ nat)))) @ S3)))))))).
% 0.33/5.22 thf(fact_5550_inj__on__funpow__least, axiom, (! [A:$tType,N:nat,F:(A > A),S:A]: (((((((compow @ (A > A)) @ N) @ F) @ S) = S) => ((! [M6:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M6) => ((((ord_less @ nat) @ M6) @ N) => (((((compow @ (A > A)) @ M6) @ F) @ S) != S))))) => ((((inj_on @ nat) @ A) @ (^ [K3:nat]: (((((compow @ (A > A)) @ K3) @ F) @ S)))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.22 thf(fact_5551_inj__on__iff__card__le, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ B) @ B7) => ((? [F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ A6) & (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A6)) @ B7)))) = (((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ B) @ B7)))))))).
% 0.33/5.22 thf(fact_5552_card__inj__on__le, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ A),B7:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A6) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A6)) @ B7) => (((finite_finite2 @ B) @ B7) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ B) @ B7)))))))).
% 0.33/5.22 thf(fact_5553_card__le__inj, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ B) @ B7) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ B) @ B7)) => (? [F3:(A > B)]: (((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A6)) @ B7) & ((((inj_on @ A) @ B) @ F3) @ A6)))))))))).
% 0.33/5.22 thf(fact_5554_execute__raise, axiom, (! [A:$tType,S:(list @ char)]: ((((heap_Time_execute @ A) @ ((heap_Time_raise @ A) @ S)) = (^ [Uu:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))).
% 0.33/5.22 thf(fact_5555_map__sorted__distinct__set__unique, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),Ys2:(list @ B)]: ((((((inj_on @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ ((set2 @ B) @ Xs)) @ ((set2 @ B) @ Ys2))) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => (((distinct @ A) @ ((((map @ B) @ A) @ F) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Ys2)) => (((distinct @ A) @ ((((map @ B) @ A) @ F) @ Ys2)) => ((((set2 @ B) @ Xs) = ((set2 @ B) @ Ys2)) => (Xs = Ys2))))))))))))).
% 0.33/5.22 thf(fact_5556_Gcd__fin__0__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((((semiring_gcd_Gcd_fin @ A) @ A6) = (zero_zero @ A)) = ((((ord_less_eq @ (set @ A)) @ A6) @ (((insert @ A) @ (zero_zero @ A)) @ (bot_bot @ (set @ A)))) & ((finite_finite2 @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_5557_greaterThan__Suc, axiom, (! [K:nat]: ((((set_ord_greaterThan @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_greaterThan @ nat) @ K)) @ (((insert @ nat) @ (suc @ K)) @ (bot_bot @ (set @ nat)))))))).
% 0.33/5.22 thf(fact_5558_last__list__update, axiom, (! [A:$tType,Xs:(list @ A),K:nat,X:A]: (((Xs != (nil @ A)) => (((K = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((last @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = X)) & ((K != (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((last @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((last @ A) @ Xs)))))))).
% 0.33/5.22 thf(fact_5559_last__upt, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((last @ nat) @ ((upt @ I) @ J)) = (((minus_minus @ nat) @ J) @ (one_one @ nat))))))).
% 0.33/5.22 thf(fact_5560_greaterThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_greaterThan @ A) @ K)) = (((ord_less @ A) @ K) @ I)))))))).
% 0.33/5.22 thf(fact_5561_greaterThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ X)) @ ((set_ord_greaterThan @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.22 thf(fact_5562_last__replicate, axiom, (! [A:$tType,N:nat,X:A]: (((N != (zero_zero @ nat)) => (((last @ A) @ (((replicate @ A) @ N) @ X)) = X))))).
% 0.33/5.22 thf(fact_5563_Compl__greaterThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_greaterThan @ A) @ K)) = ((set_ord_atMost @ A) @ K)))))))).
% 0.33/5.22 thf(fact_5564_Compl__atMost, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atMost @ A) @ K)) = ((set_ord_greaterThan @ A) @ K)))))))).
% 0.33/5.22 thf(fact_5565_Sup__greaterThanAtLeast, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (((complete_Sup_Sup @ A) @ ((set_ord_greaterThan @ A) @ X)) = (top_top @ A))))))))).
% 0.33/5.22 thf(fact_5566_image__uminus__greaterThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_greaterThan @ A) @ X)) = ((set_ord_lessThan @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.22 thf(fact_5567_image__uminus__lessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_lessThan @ A) @ X)) = ((set_ord_greaterThan @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.22 thf(fact_5568_last__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((last @ A) @ (((drop @ A) @ N) @ Xs)) = ((last @ A) @ Xs)))))).
% 0.33/5.22 thf(fact_5569_greaterThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_greaterThan @ A) = (^ [L3:A]: (((collect @ A) @ ((ord_less @ A) @ L3))))))))).
% 0.33/5.22 thf(fact_5570_ivl__disj__un__one_I5_J, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5571_greaterThan__0, axiom, (((set_ord_greaterThan @ nat) @ (zero_zero @ nat)) = ((((image @ nat) @ nat) @ suc) @ (top_top @ (set @ nat))))).
% 0.33/5.22 thf(fact_5572_last__conv__nth, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((last @ A) @ Xs) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))))).
% 0.33/5.22 thf(fact_5573_interval__cases, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [S3:(set @ A)]: (((! [A8:A,B5:A,X3:A]: (((((member @ A) @ A8) @ S3) => ((((member @ A) @ B5) @ S3) => ((((ord_less_eq @ A) @ A8) @ X3) => ((((ord_less_eq @ A) @ X3) @ B5) => (((member @ A) @ X3) @ S3))))))) => (? [A8:A,B5:A]: (((S3 = (bot_bot @ (set @ A))) | ((S3 = (top_top @ (set @ A))) | ((S3 = ((set_ord_lessThan @ A) @ B5)) | ((S3 = ((set_ord_atMost @ A) @ B5)) | ((S3 = ((set_ord_greaterThan @ A) @ A8)) | ((S3 = ((set_ord_atLeast @ A) @ A8)) | ((S3 = (((set_or5935395276787703475ssThan @ A) @ A8) @ B5)) | ((S3 = (((set_or3652927894154168847AtMost @ A) @ A8) @ B5)) | ((S3 = (((set_or7035219750837199246ssThan @ A) @ A8) @ B5)) | (S3 = (((set_or1337092689740270186AtMost @ A) @ A8) @ B5)))))))))))))))))))).
% 0.33/5.22 thf(fact_5574_sum__mult__sum__if__inj, axiom, (! [A:$tType,C:$tType,B:$tType]: (((semiring_0 @ B) => (! [F:(A > B),G:(C > B),A6:(set @ A),B7:(set @ C)]: ((((((inj_on @ ((product_prod @ A) @ C)) @ B) @ ((((product_case_prod @ A) @ C) @ B) @ (^ [A5:A,B3:C]: ((((times_times @ B) @ (F @ A5)) @ (G @ B3)))))) @ ((((product_Sigma @ A) @ C) @ A6) @ (^ [Uu:A]: (B7)))) => ((((times_times @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ A6)) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ B7)) = ((((groups7311177749621191930dd_sum @ B) @ B) @ (id @ B)) @ ((collect @ B) @ (^ [Uu:B]: ((? [A5:A,B3:C]: (((Uu = (((times_times @ B) @ (F @ A5)) @ (G @ B3))) & ((((member @ A) @ A5) @ A6) & (((member @ C) @ B3) @ B7))))))))))))))))).
% 0.33/5.22 thf(fact_5575_arg__min__inj__eq, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [F:(A > B),P:(A > $o),A3:A]: ((((((inj_on @ A) @ B) @ F) @ ((collect @ A) @ P)) => ((P @ A3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (F @ A3)) @ (F @ Y5))))) => (((((lattices_ord_arg_min @ A) @ B) @ F) @ P) = A3)))))))))).
% 0.33/5.22 thf(fact_5576_atLeast__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atLeast @ A) @ K)) = (((ord_less_eq @ A) @ K) @ I)))))))).
% 0.33/5.22 thf(fact_5577_atLeast__0, axiom, (((set_ord_atLeast @ nat) @ (zero_zero @ nat)) = (top_top @ (set @ nat)))).
% 0.33/5.22 thf(fact_5578_atLeast__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ X)) @ ((set_ord_atLeast @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 0.33/5.22 thf(fact_5579_Compl__atLeast, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atLeast @ A) @ K)) = ((set_ord_lessThan @ A) @ K)))))))).
% 0.33/5.22 thf(fact_5580_Compl__lessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) = ((set_ord_atLeast @ A) @ K)))))))).
% 0.33/5.22 thf(fact_5581_Icc__subset__Ici__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H2:A,L4:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H2)) @ ((set_ord_atLeast @ A) @ L4)) = ((~ ((((ord_less_eq @ A) @ L) @ H2))) | (((ord_less_eq @ A) @ L4) @ L))))))))).
% 0.33/5.22 thf(fact_5582_image__uminus__atMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atMost @ A) @ X)) = ((set_ord_atLeast @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.22 thf(fact_5583_image__uminus__atLeast, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atLeast @ A) @ X)) = ((set_ord_atMost @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.22 thf(fact_5584_arg__min__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => ((P @ ((((lattices_ord_arg_min @ A) @ nat) @ M) @ P)) & (! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ nat) @ (M @ ((((lattices_ord_arg_min @ A) @ nat) @ M) @ P))) @ (M @ Y4)))))))))).
% 0.33/5.22 thf(fact_5585_arg__min__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M @ ((((lattices_ord_arg_min @ A) @ nat) @ M) @ P))) @ (M @ X)))))).
% 0.33/5.22 thf(fact_5586_atLeast__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atLeast @ A) = (^ [L3:A]: (((collect @ A) @ ((ord_less_eq @ A) @ L3))))))))).
% 0.33/5.22 thf(fact_5587_arg__min__equality, axiom, (! [A:$tType,C:$tType]: (((order @ A) => (! [P:(C > $o),K:C,F:(C > A)]: (((P @ K) => ((! [X3:C]: (((P @ X3) => (((ord_less_eq @ A) @ (F @ K)) @ (F @ X3))))) => ((F @ ((((lattices_ord_arg_min @ C) @ A) @ F) @ P)) = (F @ K)))))))))).
% 0.33/5.22 thf(fact_5588_arg__minI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [P:(A > $o),X:A,F:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (~ ((((ord_less @ B) @ (F @ Y5)) @ (F @ X))))))) => ((! [X3:A]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (~ ((((ord_less @ B) @ (F @ Y4)) @ (F @ X3))))))) => (Q @ X3))))) => (Q @ ((((lattices_ord_arg_min @ A) @ B) @ F) @ P))))))))))).
% 0.33/5.22 thf(fact_5589_atLeast__Suc__greaterThan, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = ((set_ord_greaterThan @ nat) @ K))))).
% 0.33/5.22 thf(fact_5590_Ex__inj__on__UNION__Sigma, axiom, (! [A:$tType,B:$tType,A6:(B > (set @ A)),I6:(set @ B)]: ((? [F3:(A > ((product_prod @ B) @ A))]: ((((((inj_on @ A) @ ((product_prod @ B) @ A)) @ F3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A6) @ I6))) & (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((image @ A) @ ((product_prod @ B) @ A)) @ F3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A6) @ I6)))) @ ((((product_Sigma @ B) @ A) @ I6) @ A6)))))))).
% 0.33/5.22 thf(fact_5591_ivl__disj__un__one_I8_J, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5592_Ici__subset__Ioi__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ A3)) @ ((set_ord_greaterThan @ A) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))).
% 0.33/5.22 thf(fact_5593_card__cartesian__product, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B7:(set @ B)]: ((((finite_card @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B7)))) = (((times_times @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ B) @ B7)))))).
% 0.33/5.22 thf(fact_5594_ivl__disj__un__one_I7_J, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5595_ivl__disj__un__one_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = ((set_ord_greaterThan @ A) @ L))))))))).
% 0.33/5.22 thf(fact_5596_lists__length__Suc__eq, axiom, (! [A:$tType,A6:(set @ A),N:nat]: ((((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A6) & (((size_size @ (list @ A)) @ Xs3) = (suc @ N)))))) = ((((image @ ((product_prod @ (list @ A)) @ A)) @ (list @ A)) @ ((((product_case_prod @ (list @ A)) @ A) @ (list @ A)) @ (^ [Xs3:(list @ A),N3:A]: ((((cons @ A) @ N3) @ Xs3))))) @ ((((product_Sigma @ (list @ A)) @ A) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A6) & (((size_size @ (list @ A)) @ Xs3) = N)))))) @ (^ [Uu:(list @ A)]: (A6)))))))).
% 0.33/5.22 thf(fact_5597_arg__min__on__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattic7623131987881927897min_on @ B) @ A) = (^ [F2:(B > A),S7:(set @ B)]: (((((lattices_ord_arg_min @ B) @ A) @ F2) @ (^ [X2:B]: ((((member @ B) @ X2) @ S7))))))))))).
% 0.33/5.22 thf(fact_5598_atLeast__Suc, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_atLeast @ nat) @ K)) @ (((insert @ nat) @ K) @ (bot_bot @ (set @ nat)))))))).
% 0.33/5.22 thf(fact_5599_Compl__Times__UNIV1, axiom, (! [B:$tType,A:$tType,A6:(set @ B)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (A6)))) = ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (((uminus_uminus @ (set @ B)) @ A6)))))))).
% 0.33/5.22 thf(fact_5600_Compl__Times__UNIV2, axiom, (! [B:$tType,A:$tType,A6:(set @ A)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: ((top_top @ (set @ B)))))) = ((((product_Sigma @ A) @ B) @ ((uminus_uminus @ (set @ A)) @ A6)) @ (^ [Uu:A]: ((top_top @ (set @ B))))))))).
% 0.33/5.22 thf(fact_5601_mult__inj__if__coprime__nat, axiom, (! [B:$tType,A:$tType,F:(A > nat),A6:(set @ A),G:(B > nat),B7:(set @ B)]: ((((((inj_on @ A) @ nat) @ F) @ A6) => (((((inj_on @ B) @ nat) @ G) @ B7) => ((! [A8:A,B5:B]: (((((member @ A) @ A8) @ A6) => ((((member @ B) @ B5) @ B7) => (((algebr8660921524188924756oprime @ nat) @ (F @ A8)) @ (G @ B5)))))) => ((((inj_on @ ((product_prod @ A) @ B)) @ nat) @ ((((product_case_prod @ A) @ B) @ nat) @ (^ [A5:A,B3:B]: ((((times_times @ nat) @ (F @ A5)) @ (G @ B3)))))) @ ((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B7)))))))))).
% 0.33/5.22 thf(fact_5602_coprime__mult__left__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((algebr8660921524188924756oprime @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = ((((algebr8660921524188924756oprime @ A) @ A3) @ C2) & (((algebr8660921524188924756oprime @ A) @ B2) @ C2))))))))).
% 0.33/5.22 thf(fact_5603_coprime__mult__right__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [C2:A,A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ C2) @ (((times_times @ A) @ A3) @ B2)) = ((((algebr8660921524188924756oprime @ A) @ C2) @ A3) & (((algebr8660921524188924756oprime @ A) @ C2) @ B2))))))))).
% 0.33/5.22 thf(fact_5604_coprime__minus__left__iff, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5605_coprime__minus__right__iff, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5606_coprime__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ A3) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5607_coprime__mod__right__iff, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((algebr8660921524188924756oprime @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ A3)) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5608_coprime__mod__left__iff, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((algebr8660921524188924756oprime @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ B2) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5609_coprime__power__right__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,N:nat]: (((((algebr8660921524188924756oprime @ A) @ A3) @ (((power_power @ A) @ B2) @ N)) = ((((algebr8660921524188924756oprime @ A) @ A3) @ B2) | (N = (zero_zero @ nat)))))))))).
% 0.33/5.22 thf(fact_5610_coprime__power__left__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,N:nat,B2:A]: (((((algebr8660921524188924756oprime @ A) @ (((power_power @ A) @ A3) @ N)) @ B2) = ((((algebr8660921524188924756oprime @ A) @ A3) @ B2) | (N = (zero_zero @ nat)))))))))).
% 0.33/5.22 thf(fact_5611_coprime__imp__gcd__eq__1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5612_coprime__0__right__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ (zero_zero @ A)) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5613_coprime__0__left__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ (zero_zero @ A)) @ A3) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5614_coprime__mult__self__right__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) & (((algebr8660921524188924756oprime @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5615_coprime__mult__self__left__iff, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5616_is__unit__gcd, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (one_one @ A)) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5617_arg__min__natI, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (P @ ((((lattices_ord_arg_min @ A) @ nat) @ M) @ P)))))).
% 0.33/5.22 thf(fact_5618_coprime__add__one__left, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ A3))))))).
% 0.33/5.22 thf(fact_5619_coprime__add__one__right, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (((plus_plus @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5620_coprime__diff__one__left, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ A3))))))).
% 0.33/5.22 thf(fact_5621_coprime__doff__one__right, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (((minus_minus @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5622_coprime__dvd__mult__right__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5623_coprime__dvd__mult__left__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5624_divides__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ B2) @ C2) => ((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => (((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2)))))))))).
% 0.33/5.22 thf(fact_5625_coprime__Suc__0__left, axiom, (! [N:nat]: ((((algebr8660921524188924756oprime @ nat) @ (suc @ (zero_zero @ nat))) @ N)))).
% 0.33/5.22 thf(fact_5626_coprime__Suc__0__right, axiom, (! [N:nat]: ((((algebr8660921524188924756oprime @ nat) @ N) @ (suc @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_5627_coprime__1__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (one_one @ A)) @ A3))))))).
% 0.33/5.22 thf(fact_5628_coprime__1__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (one_one @ A)))))))).
% 0.33/5.22 thf(fact_5629_coprime__Suc__left__nat, axiom, (! [N:nat]: ((((algebr8660921524188924756oprime @ nat) @ (suc @ N)) @ N)))).
% 0.33/5.22 thf(fact_5630_coprime__Suc__right__nat, axiom, (! [N:nat]: ((((algebr8660921524188924756oprime @ nat) @ N) @ (suc @ N))))).
% 0.33/5.22 thf(fact_5631_coprime__commute, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => ((algebr8660921524188924756oprime @ A) = (^ [B3:A,A5:A]: ((((algebr8660921524188924756oprime @ A) @ A5) @ B3)))))))).
% 0.33/5.22 thf(fact_5632_coprime__divisors, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A,B2:A,D2:A]: (((((dvd_dvd @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ B2) @ D2) => ((((algebr8660921524188924756oprime @ A) @ C2) @ D2) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_5633_coprime__crossproduct__nat, axiom, (! [A3:nat,D2:nat,B2:nat,C2:nat]: (((((algebr8660921524188924756oprime @ nat) @ A3) @ D2) => ((((algebr8660921524188924756oprime @ nat) @ B2) @ C2) => (((((times_times @ nat) @ A3) @ C2) = (((times_times @ nat) @ B2) @ D2)) = ((A3 = B2) & (C2 = D2)))))))).
% 0.33/5.22 thf(fact_5634_is__unit__right__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5635_is__unit__left__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5636_coprime__common__divisor, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => (((dvd_dvd @ A) @ C2) @ (one_one @ A))))))))))).
% 0.33/5.22 thf(fact_5637_coprime__absorb__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [Y3:A,X:A]: (((((dvd_dvd @ A) @ Y3) @ X) => ((((algebr8660921524188924756oprime @ A) @ X) @ Y3) = (((dvd_dvd @ A) @ Y3) @ (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_5638_coprime__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,D2:A,A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ C2) @ D2) => ((! [E2:A]: (((~ ((((dvd_dvd @ A) @ E2) @ (one_one @ A)))) => ((((dvd_dvd @ A) @ E2) @ A3) => ((((dvd_dvd @ A) @ E2) @ B2) => (((dvd_dvd @ A) @ E2) @ C2)))))) => ((! [E2:A]: (((~ ((((dvd_dvd @ A) @ E2) @ (one_one @ A)))) => ((((dvd_dvd @ A) @ E2) @ A3) => ((((dvd_dvd @ A) @ E2) @ B2) => (((dvd_dvd @ A) @ E2) @ D2)))))) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_5639_coprime__absorb__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [X:A,Y3:A]: (((((dvd_dvd @ A) @ X) @ Y3) => ((((algebr8660921524188924756oprime @ A) @ X) @ Y3) = (((dvd_dvd @ A) @ X) @ (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_5640_not__coprimeI, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => ((~ ((((dvd_dvd @ A) @ C2) @ (one_one @ A)))) => (~ ((((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))))))).
% 0.33/5.22 thf(fact_5641_not__coprimeE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((~ ((((algebr8660921524188924756oprime @ A) @ A3) @ B2))) => (~ ((! [C3:A]: (((((dvd_dvd @ A) @ C3) @ A3) => ((((dvd_dvd @ A) @ C3) @ B2) => (((dvd_dvd @ A) @ C3) @ (one_one @ A))))))))))))))).
% 0.33/5.22 thf(fact_5642_coprime__def, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => ((algebr8660921524188924756oprime @ A) = (^ [A5:A,B3:A]: ((! [C4:A]: (((((dvd_dvd @ A) @ C4) @ A5) => ((((dvd_dvd @ A) @ C4) @ B3) => (((dvd_dvd @ A) @ C4) @ (one_one @ A))))))))))))).
% 0.33/5.22 thf(fact_5643_coprimeI, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((! [C3:A]: (((((dvd_dvd @ A) @ C3) @ A3) => ((((dvd_dvd @ A) @ C3) @ B2) => (((dvd_dvd @ A) @ C3) @ (one_one @ A)))))) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5644_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)))))))).
% 0.33/5.22 thf(fact_5645_gcd__eq__1__imp__coprime, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_5646_coprime__iff__gcd__eq__1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((algebr8660921524188924756oprime @ A) = (^ [A5:A,B3:A]: (((((gcd_gcd @ A) @ A5) @ B3) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5647_gcd__mult__right__right__cancel, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ C2) => ((((gcd_gcd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((gcd_gcd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5648_gcd__mult__right__left__cancel, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ C2) => ((((gcd_gcd @ A) @ A3) @ (((times_times @ A) @ C2) @ B2)) = (((gcd_gcd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5649_gcd__mult__left__right__cancel, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [B2:A,C2:A,A3:A]: (((((algebr8660921524188924756oprime @ A) @ B2) @ C2) => ((((gcd_gcd @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2) = (((gcd_gcd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5650_gcd__mult__left__left__cancel, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [B2:A,C2:A,A3:A]: (((((algebr8660921524188924756oprime @ A) @ B2) @ C2) => ((((gcd_gcd @ A) @ (((times_times @ A) @ C2) @ A3)) @ B2) = (((gcd_gcd @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_5651_mult__mod__cancel__right, axiom, (! [A:$tType]: ((((euclid8851590272496341667cancel @ A) & (semiring_gcd @ A)) => (! [A3:A,N:A,M:A,B2:A]: ((((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ N)) @ M) = (((modulo_modulo @ A) @ (((times_times @ A) @ B2) @ N)) @ M)) => ((((algebr8660921524188924756oprime @ A) @ M) @ N) => ((((modulo_modulo @ A) @ A3) @ M) = (((modulo_modulo @ A) @ B2) @ M)))))))))).
% 0.33/5.22 thf(fact_5652_mult__mod__cancel__left, axiom, (! [A:$tType]: ((((euclid8851590272496341667cancel @ A) & (semiring_gcd @ A)) => (! [N:A,A3:A,M:A,B2:A]: ((((((modulo_modulo @ A) @ (((times_times @ A) @ N) @ A3)) @ M) = (((modulo_modulo @ A) @ (((times_times @ A) @ N) @ B2)) @ M)) => ((((algebr8660921524188924756oprime @ A) @ M) @ N) => ((((modulo_modulo @ A) @ A3) @ M) = (((modulo_modulo @ A) @ B2) @ M)))))))))).
% 0.33/5.22 thf(fact_5653_invertible__coprime, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A,C2:A]: ((((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ C2)))))))).
% 0.33/5.22 thf(fact_5654_gcd__coprime__exists, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_gcd @ A) @ A3) @ B2) != (zero_zero @ A)) => (? [A15:A,B8:A]: (((A3 = (((times_times @ A) @ A15) @ (((gcd_gcd @ A) @ A3) @ B2))) & ((B2 = (((times_times @ A) @ B8) @ (((gcd_gcd @ A) @ A3) @ B2))) & (((algebr8660921524188924756oprime @ A) @ A15) @ B8)))))))))))).
% 0.33/5.22 thf(fact_5655_gcd__coprime, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,A4:A,B4:A]: ((((((gcd_gcd @ A) @ A3) @ B2) != (zero_zero @ A)) => ((A3 = (((times_times @ A) @ A4) @ (((gcd_gcd @ A) @ A3) @ B2))) => ((B2 = (((times_times @ A) @ B4) @ (((gcd_gcd @ A) @ A3) @ B2))) => (((algebr8660921524188924756oprime @ A) @ A4) @ B4)))))))))).
% 0.33/5.22 thf(fact_5656_div__gcd__coprime, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((A3 != (zero_zero @ A)) | (B2 != (zero_zero @ A))) => (((algebr8660921524188924756oprime @ A) @ (((divide_divide @ A) @ A3) @ (((gcd_gcd @ A) @ A3) @ B2))) @ (((divide_divide @ A) @ B2) @ (((gcd_gcd @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_5657_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))))))).
% 0.33/5.22 thf(fact_5658_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))))).
% 0.33/5.22 thf(fact_5659_set__encode__vimage__Suc, axiom, (! [A6:(set @ nat)]: (((nat_set_encode @ ((((vimage @ nat) @ nat) @ suc) @ A6)) = (((divide_divide @ nat) @ (nat_set_encode @ A6)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 0.33/5.22 thf(fact_5660_floor__rat__def, axiom, ((archim6421214686448440834_floor @ rat) = (^ [X2:rat]: (((the @ int) @ (^ [Z7:int]: (((((ord_less_eq @ rat) @ ((ring_1_of_int @ rat) @ Z7)) @ X2) & (((ord_less @ rat) @ X2) @ ((ring_1_of_int @ rat) @ (((plus_plus @ int) @ Z7) @ (one_one @ int)))))))))))).
% 0.33/5.22 thf(fact_5661_sorted__insort__insert__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B),X:B]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ Xs)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ (((((linord329482645794927042rt_key @ B) @ A) @ F) @ X) @ Xs)))))))))).
% 0.33/5.22 thf(fact_5662_coprime__abs__left__iff, axiom, (! [K:int,L:int]: (((((algebr8660921524188924756oprime @ int) @ ((abs_abs @ int) @ K)) @ L) = (((algebr8660921524188924756oprime @ int) @ K) @ L))))).
% 0.33/5.22 thf(fact_5663_coprime__abs__right__iff, axiom, (! [K:int,L:int]: (((((algebr8660921524188924756oprime @ int) @ K) @ ((abs_abs @ int) @ L)) = (((algebr8660921524188924756oprime @ int) @ K) @ L))))).
% 0.33/5.22 thf(fact_5664_coprime__int__iff, axiom, (! [M:nat,N:nat]: (((((algebr8660921524188924756oprime @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((algebr8660921524188924756oprime @ nat) @ M) @ N))))).
% 0.33/5.22 thf(fact_5665_normalize__stable, axiom, (! [Q4:int,P5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Q4) => ((((algebr8660921524188924756oprime @ int) @ P5) @ Q4) => ((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q4)) = ((((product_Pair @ int) @ int) @ P5) @ Q4))))))).
% 0.33/5.22 thf(fact_5666_coprime__nat__abs__left__iff, axiom, (! [K:int,N:nat]: (((((algebr8660921524188924756oprime @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N) = (((algebr8660921524188924756oprime @ int) @ K) @ ((semiring_1_of_nat @ int) @ N)))))).
% 0.33/5.22 thf(fact_5667_coprime__nat__abs__right__iff, axiom, (! [N:nat,K:int]: (((((algebr8660921524188924756oprime @ nat) @ N) @ (nat2 @ ((abs_abs @ int) @ K))) = (((algebr8660921524188924756oprime @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K))))).
% 0.33/5.22 thf(fact_5668_vimage__if, axiom, (! [B:$tType,A:$tType,C2:B,A6:(set @ B),D2:B,B7:(set @ A)]: ((((((member @ B) @ C2) @ A6) => (((((member @ B) @ D2) @ A6) => (((((vimage @ A) @ B) @ (^ [X2:A]: (((((if @ B) @ (((member @ A) @ X2) @ B7)) @ C2) @ D2)))) @ A6) = (top_top @ (set @ A)))) & ((~ ((((member @ B) @ D2) @ A6))) => (((((vimage @ A) @ B) @ (^ [X2:A]: (((((if @ B) @ (((member @ A) @ X2) @ B7)) @ C2) @ D2)))) @ A6) = B7)))) & ((~ ((((member @ B) @ C2) @ A6))) => (((((member @ B) @ D2) @ A6) => (((((vimage @ A) @ B) @ (^ [X2:A]: (((((if @ B) @ (((member @ A) @ X2) @ B7)) @ C2) @ D2)))) @ A6) = ((uminus_uminus @ (set @ A)) @ B7))) & ((~ ((((member @ B) @ D2) @ A6))) => (((((vimage @ A) @ B) @ (^ [X2:A]: (((((if @ B) @ (((member @ A) @ X2) @ B7)) @ C2) @ D2)))) @ A6) = (bot_bot @ (set @ A)))))))))).
% 0.33/5.22 thf(fact_5669_vimage__Compl, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ ((uminus_uminus @ (set @ B)) @ A6)) = ((uminus_uminus @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A6)))))).
% 0.33/5.22 thf(fact_5670_coprime__crossproduct__int, axiom, (! [A3:int,D2:int,B2:int,C2:int]: (((((algebr8660921524188924756oprime @ int) @ A3) @ D2) => ((((algebr8660921524188924756oprime @ int) @ B2) @ C2) => (((((times_times @ int) @ ((abs_abs @ int) @ A3)) @ ((abs_abs @ int) @ C2)) = (((times_times @ int) @ ((abs_abs @ int) @ B2)) @ ((abs_abs @ int) @ D2))) = ((((abs_abs @ int) @ A3) = ((abs_abs @ int) @ B2)) & (((abs_abs @ int) @ C2) = ((abs_abs @ int) @ D2))))))))).
% 0.33/5.22 thf(fact_5671_finite__vimage__Suc__iff, axiom, (! [F4:(set @ nat)]: ((((finite_finite2 @ nat) @ ((((vimage @ nat) @ nat) @ suc) @ F4)) = ((finite_finite2 @ nat) @ F4))))).
% 0.33/5.22 thf(fact_5672_Nitpick_OThe__psimp, axiom, (! [A:$tType,P:(A > $o),X:A]: (((P = ((^ [Y:A,Z2:A]: ((Y = Z2))) @ X)) => (((the @ A) @ P) = X))))).
% 0.33/5.22 thf(fact_5673_vimage__Suc__insert__Suc, axiom, (! [N:nat,A6:(set @ nat)]: ((((((vimage @ nat) @ nat) @ suc) @ (((insert @ nat) @ (suc @ N)) @ A6)) = (((insert @ nat) @ N) @ ((((vimage @ nat) @ nat) @ suc) @ A6)))))).
% 0.33/5.22 thf(fact_5674_vimage__Suc__insert__0, axiom, (! [A6:(set @ nat)]: ((((((vimage @ nat) @ nat) @ suc) @ (((insert @ nat) @ (zero_zero @ nat)) @ A6)) = ((((vimage @ nat) @ nat) @ suc) @ A6))))).
% 0.33/5.22 thf(fact_5675_coprime__common__divisor__int, axiom, (! [A3:int,B2:int,X:int]: (((((algebr8660921524188924756oprime @ int) @ A3) @ B2) => ((((dvd_dvd @ int) @ X) @ A3) => ((((dvd_dvd @ int) @ X) @ B2) => (((abs_abs @ int) @ X) = (one_one @ int)))))))).
% 0.33/5.22 thf(fact_5676_sorted__insort__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),X:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((((linord329482645794927042rt_key @ A) @ A) @ (^ [X2:A]: (X2))) @ X) @ Xs))))))))).
% 0.33/5.22 thf(fact_5677_card__vimage__inj__on__le, axiom, (! [A:$tType,B:$tType,F:(A > B),D4:(set @ A),A6:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ D4) => (((finite_finite2 @ B) @ A6) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A6)) @ D4))) @ ((finite_card @ B) @ A6))))))).
% 0.33/5.22 thf(fact_5678_set__decode__div__2, axiom, (! [X:nat]: (((nat_set_decode @ (((divide_divide @ nat) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((((vimage @ nat) @ nat) @ suc) @ (nat_set_decode @ X)))))).
% 0.33/5.22 thf(fact_5679_old_Orec__nat__def, axiom, (! [T:$tType]: (((rec_nat @ T) = (^ [F12:T,F23:(nat > (T > T)),X2:nat]: (((the @ T) @ ((((rec_set_nat @ T) @ F12) @ F23) @ X2)))))))).
% 0.33/5.22 thf(fact_5680_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)))))))).
% 0.33/5.22 thf(fact_5681_sgn__rat, axiom, (! [A3:int,B2:int]: ((((sgn_sgn @ rat) @ ((fract @ A3) @ B2)) = ((ring_1_of_int @ rat) @ (((times_times @ int) @ ((sgn_sgn @ int) @ A3)) @ ((sgn_sgn @ int) @ B2))))))).
% 0.33/5.22 thf(fact_5682_minus__rat__cancel, axiom, (! [A3:int,B2:int]: ((((fract @ ((uminus_uminus @ int) @ A3)) @ ((uminus_uminus @ int) @ B2)) = ((fract @ A3) @ B2))))).
% 0.33/5.22 thf(fact_5683_minus__rat, axiom, (! [A3:int,B2:int]: ((((uminus_uminus @ rat) @ ((fract @ A3) @ B2)) = ((fract @ ((uminus_uminus @ int) @ A3)) @ B2))))).
% 0.33/5.22 thf(fact_5684_divide__rat, axiom, (! [A3:int,B2:int,C2:int,D2:int]: (((((divide_divide @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = ((fract @ (((times_times @ int) @ A3) @ D2)) @ (((times_times @ int) @ B2) @ C2)))))).
% 0.33/5.22 thf(fact_5685_mult__rat, axiom, (! [A3:int,B2:int,C2:int,D2:int]: (((((times_times @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = ((fract @ (((times_times @ int) @ A3) @ C2)) @ (((times_times @ int) @ B2) @ D2)))))).
% 0.33/5.22 thf(fact_5686_less__rat, axiom, (! [B2:int,D2:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((ord_less @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = (((ord_less @ int) @ (((times_times @ int) @ (((times_times @ int) @ A3) @ D2)) @ (((times_times @ int) @ B2) @ D2))) @ (((times_times @ int) @ (((times_times @ int) @ C2) @ B2)) @ (((times_times @ int) @ B2) @ D2))))))))).
% 0.33/5.22 thf(fact_5687_le__rat, axiom, (! [B2:int,D2:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((ord_less_eq @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = (((ord_less_eq @ int) @ (((times_times @ int) @ (((times_times @ int) @ A3) @ D2)) @ (((times_times @ int) @ B2) @ D2))) @ (((times_times @ int) @ (((times_times @ int) @ C2) @ B2)) @ (((times_times @ int) @ B2) @ D2))))))))).
% 0.33/5.22 thf(fact_5688_add__rat, axiom, (! [B2:int,D2:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((plus_plus @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = ((fract @ (((plus_plus @ int) @ (((times_times @ int) @ A3) @ D2)) @ (((times_times @ int) @ C2) @ B2))) @ (((times_times @ int) @ B2) @ D2)))))))).
% 0.33/5.22 thf(fact_5689_diff__rat, axiom, (! [B2:int,D2:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((minus_minus @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D2)) = ((fract @ (((minus_minus @ int) @ (((times_times @ int) @ A3) @ D2)) @ (((times_times @ int) @ C2) @ B2))) @ (((times_times @ int) @ B2) @ D2)))))))).
% 0.33/5.22 thf(fact_5690_Rat__cases, axiom, (! [Q4:rat]: ((~ ((! [A8:int,B5:int]: (((Q4 = ((fract @ A8) @ B5)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B5) => (~ ((((algebr8660921524188924756oprime @ int) @ A8) @ B5)))))))))))).
% 0.33/5.22 thf(fact_5691_Rat__induct, axiom, (! [P:(rat > $o),Q4:rat]: (((! [A8:int,B5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B5) => ((((algebr8660921524188924756oprime @ int) @ A8) @ B5) => (P @ ((fract @ A8) @ B5)))))) => (P @ Q4))))).
% 0.33/5.22 thf(fact_5692_Fract__of__nat__eq, axiom, (! [K:nat]: ((((fract @ ((semiring_1_of_nat @ int) @ K)) @ (one_one @ int)) = ((semiring_1_of_nat @ rat) @ K))))).
% 0.33/5.22 thf(fact_5693_eq__rat_I3_J, axiom, (! [A3:int,C2:int]: ((((fract @ (zero_zero @ int)) @ A3) = ((fract @ (zero_zero @ int)) @ C2))))).
% 0.33/5.22 thf(fact_5694_eq__rat_I1_J, axiom, (! [B2:int,D2:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((fract @ A3) @ B2) = ((fract @ C2) @ D2)) = ((((times_times @ int) @ A3) @ D2) = (((times_times @ int) @ C2) @ B2)))))))).
% 0.33/5.22 thf(fact_5695_mult__rat__cancel, axiom, (! [C2:int,A3:int,B2:int]: (((C2 != (zero_zero @ int)) => (((fract @ (((times_times @ int) @ C2) @ A3)) @ (((times_times @ int) @ C2) @ B2)) = ((fract @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_5696_eq__rat_I2_J, axiom, (! [A3:int]: ((((fract @ A3) @ (zero_zero @ int)) = ((fract @ (zero_zero @ int)) @ (one_one @ int)))))).
% 0.33/5.22 thf(fact_5697_Rat__induct__pos, axiom, (! [P:(rat > $o),Q4:rat]: (((! [A8:int,B5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B5) => (P @ ((fract @ A8) @ B5))))) => (P @ Q4))))).
% 0.33/5.22 thf(fact_5698_rat__number__collapse_I6_J, axiom, (! [K:int]: ((((fract @ K) @ (zero_zero @ int)) = (zero_zero @ rat))))).
% 0.33/5.22 thf(fact_5699_rat__number__collapse_I1_J, axiom, (! [K:int]: ((((fract @ (zero_zero @ int)) @ K) = (zero_zero @ rat))))).
% 0.33/5.22 thf(fact_5700_One__rat__def, axiom, ((one_one @ rat) = ((fract @ (one_one @ int)) @ (one_one @ int)))).
% 0.33/5.22 thf(fact_5701_Fract__of__int__eq, axiom, (! [K:int]: ((((fract @ K) @ (one_one @ int)) = ((ring_1_of_int @ rat) @ K))))).
% 0.33/5.22 thf(fact_5702_quotient__of__def, axiom, (quotient_of = (^ [X2:rat]: (((the @ ((product_prod @ int) @ int)) @ (^ [Pair:((product_prod @ int) @ int)]: (((X2 = ((fract @ (((product_fst @ int) @ int) @ Pair)) @ (((product_snd @ int) @ int) @ Pair))) & ((((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ Pair)) & (((algebr8660921524188924756oprime @ int) @ (((product_fst @ int) @ int) @ Pair)) @ (((product_snd @ int) @ int) @ Pair))))))))))).
% 0.33/5.22 thf(fact_5703_Rat__cases__nonzero, axiom, (! [Q4:rat]: (((! [A8:int,B5:int]: (((Q4 = ((fract @ A8) @ B5)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B5) => ((A8 != (zero_zero @ int)) => (~ ((((algebr8660921524188924756oprime @ int) @ A8) @ B5)))))))) => (Q4 = (zero_zero @ rat)))))).
% 0.33/5.22 thf(fact_5704_Zero__rat__def, axiom, ((zero_zero @ rat) = ((fract @ (zero_zero @ int)) @ (one_one @ int)))).
% 0.33/5.22 thf(fact_5705_rat__number__expand_I3_J, axiom, ((numeral_numeral @ rat) = (^ [K3:num]: (((fract @ ((numeral_numeral @ int) @ K3)) @ (one_one @ int)))))).
% 0.33/5.22 thf(fact_5706_rat__number__collapse_I3_J, axiom, (! [W2:num]: ((((fract @ ((numeral_numeral @ int) @ W2)) @ (one_one @ int)) = ((numeral_numeral @ rat) @ W2))))).
% 0.33/5.22 thf(fact_5707_quotient__of__unique, axiom, (! [R:rat]: ((? [X3:((product_prod @ int) @ int)]: (((R = ((fract @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ X3))) & ((((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ X3)) & ((((algebr8660921524188924756oprime @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ X3)) & (! [Y4:((product_prod @ int) @ int)]: ((((R = ((fract @ (((product_fst @ int) @ int) @ Y4)) @ (((product_snd @ int) @ int) @ Y4))) & ((((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ Y4)) & (((algebr8660921524188924756oprime @ int) @ (((product_fst @ int) @ int) @ Y4)) @ (((product_snd @ int) @ int) @ Y4)))) => (Y4 = X3)))))))))))).
% 0.33/5.22 thf(fact_5708_Fract_Oabs__eq, axiom, (fract = (^ [Xa4:int,X2:int]: ((abs_Rat @ ((((if @ ((product_prod @ int) @ int)) @ (X2 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ Xa4) @ X2))))))).
% 0.33/5.22 thf(fact_5709_positive__rat, axiom, (! [A3:int,B2:int]: (((positive @ ((fract @ A3) @ B2)) = (((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_5710_Fract__less__zero__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ ((fract @ A3) @ B2)) @ (zero_zero @ rat)) = (((ord_less @ int) @ A3) @ (zero_zero @ int))))))).
% 0.33/5.22 thf(fact_5711_zero__less__Fract__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ (zero_zero @ rat)) @ ((fract @ A3) @ B2)) = (((ord_less @ int) @ (zero_zero @ int)) @ A3)))))).
% 0.33/5.22 thf(fact_5712_one__less__Fract__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ (one_one @ rat)) @ ((fract @ A3) @ B2)) = (((ord_less @ int) @ B2) @ A3)))))).
% 0.33/5.22 thf(fact_5713_Fract__less__one__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ ((fract @ A3) @ B2)) @ (one_one @ rat)) = (((ord_less @ int) @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_5714_rat__number__collapse_I5_J, axiom, (((fract @ ((uminus_uminus @ int) @ (one_one @ int))) @ (one_one @ int)) = ((uminus_uminus @ rat) @ (one_one @ rat)))).
% 0.33/5.22 thf(fact_5715_Fract_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > rat)) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ rat) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ pcr_rat)) @ (^ [A5:int,B3:int]: (((((if @ ((product_prod @ int) @ int)) @ (B3 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A5) @ B3))))) @ fract)).
% 0.33/5.22 thf(fact_5716_Fract__add__one, axiom, (! [N:int,M:int]: (((N != (zero_zero @ int)) => (((fract @ (((plus_plus @ int) @ M) @ N)) @ N) = (((plus_plus @ rat) @ ((fract @ M) @ N)) @ (one_one @ rat))))))).
% 0.33/5.22 thf(fact_5717_Fract__le__zero__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ ((fract @ A3) @ B2)) @ (zero_zero @ rat)) = (((ord_less_eq @ int) @ A3) @ (zero_zero @ int))))))).
% 0.33/5.22 thf(fact_5718_zero__le__Fract__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ (zero_zero @ rat)) @ ((fract @ A3) @ B2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A3)))))).
% 0.33/5.22 thf(fact_5719_one__le__Fract__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ (one_one @ rat)) @ ((fract @ A3) @ B2)) = (((ord_less_eq @ int) @ B2) @ A3)))))).
% 0.33/5.22 thf(fact_5720_Fract__le__one__iff, axiom, (! [B2:int,A3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ ((fract @ A3) @ B2)) @ (one_one @ rat)) = (((ord_less_eq @ int) @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_5721_rat__number__expand_I5_J, axiom, (! [K:num]: ((((uminus_uminus @ rat) @ ((numeral_numeral @ rat) @ K)) = ((fract @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))))).
% 0.33/5.22 thf(fact_5722_rat__number__collapse_I4_J, axiom, (! [W2:num]: ((((fract @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ W2))) @ (one_one @ int)) = ((uminus_uminus @ rat) @ ((numeral_numeral @ rat) @ W2)))))).
% 0.33/5.22 thf(fact_5723_Rats__cases_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [X:A]: (((((member @ A) @ X) @ (field_char_0_Rats @ A)) => (~ ((! [A8:int,B5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B5) => ((((algebr8660921524188924756oprime @ int) @ A8) @ B5) => (X != (((divide_divide @ A) @ ((ring_1_of_int @ A) @ A8)) @ ((ring_1_of_int @ A) @ B5)))))))))))))))).
% 0.33/5.22 thf(fact_5724_horner__sum__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType]: ((((comm_semiring_0 @ B) & (comm_semiring_0 @ A)) => (! [A6:(A > (B > $o)),B7:(C > (D > $o))]: ((((A6 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A6) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A6) @ A6)) @ (plus_plus @ A)) @ (plus_plus @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A6) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A6) @ A6)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ (C > A)) @ (D > B)) @ (A > ((list @ C) > A))) @ (B > ((list @ D) > B))) @ ((((((bNF_rel_fun @ C) @ D) @ A) @ B) @ B7) @ A6)) @ ((((((bNF_rel_fun @ A) @ B) @ ((list @ C) > A)) @ ((list @ D) > B)) @ A6) @ ((((((bNF_rel_fun @ (list @ C)) @ (list @ D)) @ A) @ B) @ (((list_all2 @ C) @ D) @ B7)) @ A6))) @ ((groups4207007520872428315er_sum @ C) @ A)) @ ((groups4207007520872428315er_sum @ D) @ B))))))))))).
% 0.33/5.22 thf(fact_5725_of__rat__neg__numeral__eq, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [W2:num]: ((((field_char_0_of_rat @ A) @ ((uminus_uminus @ rat) @ ((numeral_numeral @ rat) @ W2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))))))).
% 0.33/5.22 thf(fact_5726_Rats__minus__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A]: (((((member @ A) @ ((uminus_uminus @ A) @ A3)) @ (field_char_0_Rats @ A)) = (((member @ A) @ A3) @ (field_char_0_Rats @ A))))))))).
% 0.33/5.22 thf(fact_5727_of__rat__0, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((field_char_0_of_rat @ A) @ (zero_zero @ rat)) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_5728_of__rat__eq__0__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: (((((field_char_0_of_rat @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ rat))))))))).
% 0.33/5.22 thf(fact_5729_zero__eq__of__rat__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: ((((zero_zero @ A) = ((field_char_0_of_rat @ A) @ A3)) = ((zero_zero @ rat) = A3)))))))).
% 0.33/5.22 thf(fact_5730_one__eq__of__rat__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: ((((one_one @ A) = ((field_char_0_of_rat @ A) @ A3)) = ((one_one @ rat) = A3)))))))).
% 0.33/5.22 thf(fact_5731_of__rat__eq__1__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: (((((field_char_0_of_rat @ A) @ A3) = (one_one @ A)) = (A3 = (one_one @ rat))))))))).
% 0.33/5.22 thf(fact_5732_of__rat__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((field_char_0_of_rat @ A) @ (one_one @ rat)) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_5733_of__rat__of__nat__eq, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat]: ((((field_char_0_of_rat @ A) @ ((semiring_1_of_nat @ rat) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 0.33/5.22 thf(fact_5734_of__rat__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less @ A) @ ((field_char_0_of_rat @ A) @ R)) @ (zero_zero @ A)) = (((ord_less @ rat) @ R) @ (zero_zero @ rat))))))))).
% 0.33/5.22 thf(fact_5735_zero__less__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less @ rat) @ (zero_zero @ rat)) @ R)))))))).
% 0.33/5.22 thf(fact_5736_of__rat__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less_eq @ A) @ ((field_char_0_of_rat @ A) @ R)) @ (zero_zero @ A)) = (((ord_less_eq @ rat) @ R) @ (zero_zero @ rat))))))))).
% 0.33/5.22 thf(fact_5737_zero__le__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less_eq @ rat) @ (zero_zero @ rat)) @ R)))))))).
% 0.33/5.22 thf(fact_5738_of__rat__neg__one, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((field_char_0_of_rat @ A) @ ((uminus_uminus @ rat) @ (one_one @ rat))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_5739_of__rat__less__1__iff, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5740_one__less__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less @ A) @ (one_one @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less @ rat) @ (one_one @ rat)) @ R)))))))).
% 0.33/5.22 thf(fact_5741_of__rat__le__1__iff, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_5742_one__le__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less_eq @ rat) @ (one_one @ rat)) @ R)))))))).
% 0.33/5.22 thf(fact_5743_Rats__mult, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (field_char_0_Rats @ A)) => ((((member @ A) @ B2) @ (field_char_0_Rats @ A)) => (((member @ A) @ (((times_times @ A) @ A3) @ B2)) @ (field_char_0_Rats @ A)))))))))).
% 0.33/5.22 thf(fact_5744_Rats__0, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((member @ A) @ (zero_zero @ A)) @ (field_char_0_Rats @ A)))))).
% 0.33/5.22 thf(fact_5745_Rats__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((member @ A) @ (one_one @ A)) @ (field_char_0_Rats @ A)))))).
% 0.33/5.22 thf(fact_5746_Rats__of__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat]: ((((member @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (field_char_0_Rats @ A)))))))).
% 0.33/5.22 thf(fact_5747_of__rat__minus, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: ((((field_char_0_of_rat @ A) @ ((uminus_uminus @ rat) @ A3)) = ((uminus_uminus @ A) @ ((field_char_0_of_rat @ A) @ A3))))))))).
% 0.33/5.22 thf(fact_5748_of__rat__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat,S:rat]: (((((ord_less @ A) @ ((field_char_0_of_rat @ A) @ R)) @ ((field_char_0_of_rat @ A) @ S)) = (((ord_less @ rat) @ R) @ S)))))))).
% 0.33/5.22 thf(fact_5749_of__rat__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat,S:rat]: (((((ord_less_eq @ A) @ ((field_char_0_of_rat @ A) @ R)) @ ((field_char_0_of_rat @ A) @ S)) = (((ord_less_eq @ rat) @ R) @ S)))))))).
% 0.33/5.22 thf(fact_5750_of__rat__mult, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat,B2:rat]: ((((field_char_0_of_rat @ A) @ (((times_times @ rat) @ A3) @ B2)) = (((times_times @ A) @ ((field_char_0_of_rat @ A) @ A3)) @ ((field_char_0_of_rat @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_5751_list__all2__conv__all__nth, axiom, (! [B:$tType,A:$tType]: ((((list_all2 @ A) @ B) = (^ [P4:(A > (B > $o)),Xs3:(list @ A),Ys3:(list @ B)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B)) @ Ys3)) & (! [I5:nat]: (((((ord_less @ nat) @ I5) @ ((size_size @ (list @ A)) @ Xs3)) => ((P4 @ (((nth @ A) @ Xs3) @ I5)) @ (((nth @ B) @ Ys3) @ I5)))))))))))).
% 0.33/5.22 thf(fact_5752_list__all2__all__nthI, axiom, (! [A:$tType,B:$tType,A3:(list @ A),B2:(list @ B),P:(A > (B > $o))]: (((((size_size @ (list @ A)) @ A3) = ((size_size @ (list @ B)) @ B2)) => ((! [N4:nat]: (((((ord_less @ nat) @ N4) @ ((size_size @ (list @ A)) @ A3)) => ((P @ (((nth @ A) @ A3) @ N4)) @ (((nth @ B) @ B2) @ N4))))) => (((((list_all2 @ A) @ B) @ P) @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_5753_list__all2__nthD2, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(list @ A),Ys2:(list @ B),P5:nat]: (((((((list_all2 @ A) @ B) @ P) @ Xs) @ Ys2) => ((((ord_less @ nat) @ P5) @ ((size_size @ (list @ B)) @ Ys2)) => ((P @ (((nth @ A) @ Xs) @ P5)) @ (((nth @ B) @ Ys2) @ P5))))))).
% 0.33/5.22 thf(fact_5754_list__all2__nthD, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Xs:(list @ A),Ys2:(list @ B),P5:nat]: (((((((list_all2 @ A) @ B) @ P) @ Xs) @ Ys2) => ((((ord_less @ nat) @ P5) @ ((size_size @ (list @ A)) @ Xs)) => ((P @ (((nth @ A) @ Xs) @ P5)) @ (((nth @ B) @ Ys2) @ P5))))))).
% 0.33/5.22 thf(fact_5755_of__rat__rat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [B2:int,A3:int]: (((B2 != (zero_zero @ int)) => (((field_char_0_of_rat @ A) @ ((fract @ A3) @ B2)) = (((divide_divide @ A) @ ((ring_1_of_int @ A) @ A3)) @ ((ring_1_of_int @ A) @ B2)))))))))).
% 0.33/5.22 thf(fact_5756_sum__list__transfer, axiom, (! [A:$tType,B:$tType]: ((((monoid_add @ B) & (monoid_add @ A)) => (! [A6:(A > (B > $o))]: ((((A6 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A6) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A6) @ A6)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ (list @ A)) @ (list @ B)) @ A) @ B) @ (((list_all2 @ A) @ B) @ A6)) @ A6) @ (groups8242544230860333062m_list @ A)) @ (groups8242544230860333062m_list @ B)))))))))).
% 0.33/5.22 thf(fact_5757_prod__list__transfer, axiom, (! [A:$tType,B:$tType]: ((((monoid_mult @ B) & (monoid_mult @ A)) => (! [A6:(A > (B > $o))]: ((((A6 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A6) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A6) @ A6)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ (list @ A)) @ (list @ B)) @ A) @ B) @ (((list_all2 @ A) @ B) @ A6)) @ A6) @ (groups5270119922927024881d_list @ A)) @ (groups5270119922927024881d_list @ B)))))))))).
% 0.33/5.22 thf(fact_5758_bot_Oordering__top__axioms, axiom, (! [A:$tType]: (((order_bot @ A) => ((((ordering_top @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))) @ (bot_bot @ A)))))).
% 0.33/5.22 thf(fact_5759_dropWhile__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) => ((((nth @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ J) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))))))))).
% 0.33/5.22 thf(fact_5760_prod__list_OCons, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A,Xs:(list @ A)]: ((((groups5270119922927024881d_list @ A) @ (((cons @ A) @ X) @ Xs)) = (((times_times @ A) @ X) @ ((groups5270119922927024881d_list @ A) @ Xs))))))))).
% 0.33/5.22 thf(fact_5761_prod__list_ONil, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((groups5270119922927024881d_list @ A) @ (nil @ A)) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_5762_prod__list_Oappend, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: ((((groups5270119922927024881d_list @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((times_times @ A) @ ((groups5270119922927024881d_list @ A) @ Xs)) @ ((groups5270119922927024881d_list @ A) @ Ys2))))))))).
% 0.33/5.22 thf(fact_5763_ordering__top_Oextremum__uniqueI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A3) => (A3 = Top)))))).
% 0.33/5.22 thf(fact_5764_ordering__top_Onot__eq__extremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((A3 != Top) = ((Less @ A3) @ Top)))))).
% 0.33/5.22 thf(fact_5765_ordering__top_Oextremum__unique, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A3) = (A3 = Top)))))).
% 0.33/5.22 thf(fact_5766_ordering__top_Oextremum__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (~ (((Less @ Top) @ A3))))))).
% 0.33/5.22 thf(fact_5767_ordering__top_Oextremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A3:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((Less_eq @ A3) @ Top))))).
% 0.33/5.22 thf(fact_5768_length__dropWhile__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.22 thf(fact_5769_sorted__dropWhile, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((dropWhile @ A) @ P) @ Xs))))))))).
% 0.33/5.22 thf(fact_5770_prod__list__zero__iff, axiom, (! [A:$tType]: ((((semiring_1 @ A) & (semiri3467727345109120633visors @ A)) => (! [Xs:(list @ A)]: (((((groups5270119922927024881d_list @ A) @ Xs) = (zero_zero @ A)) = (((member @ A) @ (zero_zero @ A)) @ ((set2 @ A) @ Xs))))))))).
% 0.33/5.22 thf(fact_5771_gcd__nat_Oordering__top__axioms, axiom, ((((ordering_top @ nat) @ (dvd_dvd @ nat)) @ (^ [M2:nat,N3:nat]: (((((dvd_dvd @ nat) @ M2) @ N3) & (M2 != N3))))) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_5772_prod__list_Oeq__foldr, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((groups5270119922927024881d_list @ A) = (^ [Xs3:(list @ A)]: ((((((foldr @ A) @ A) @ (times_times @ A)) @ Xs3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_5773_top_Oordering__top__axioms, axiom, (! [A:$tType]: (((order_top @ A) => ((((ordering_top @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)) @ (top_top @ A)))))).
% 0.33/5.22 thf(fact_5774_bot__nat__0_Oordering__top__axioms, axiom, ((((ordering_top @ nat) @ (^ [X2:nat,Y6:nat]: ((((ord_less_eq @ nat) @ Y6) @ X2)))) @ (^ [X2:nat,Y6:nat]: ((((ord_less @ nat) @ Y6) @ X2)))) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_5775_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) @ $o) @ (^ [X2:A,Y6:A]: (((((ord_less @ nat) @ (F2 @ X2)) @ (F2 @ Y6)) | ((((ord_less_eq @ nat) @ (F2 @ X2)) @ (F2 @ Y6)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y6)) @ R6))))))))))))).
% 0.33/5.22 thf(fact_5776_arg__min__SOME__Min, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S3:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ S3) => (((((lattic7623131987881927897min_on @ A) @ B) @ F) @ S3) = ((fChoice @ A) @ (^ [Y6:A]: (((((member @ A) @ Y6) @ S3) & ((F @ Y6) = ((lattic643756798350308766er_Min @ B) @ ((((image @ A) @ B) @ F) @ S3))))))))))))))).
% 0.33/5.22 thf(fact_5777_prod__decode__triangle__add, axiom, (! [K:nat,M:nat]: (((nat_prod_decode @ (((plus_plus @ nat) @ (nat_triangle @ K)) @ M)) = ((nat_prod_decode_aux @ K) @ M))))).
% 0.33/5.22 thf(fact_5778_prod__decode__eq, axiom, (! [X:nat,Y3:nat]: ((((nat_prod_decode @ X) = (nat_prod_decode @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_5779_some__equality, axiom, (! [A:$tType,P:(A > $o),A3:A]: (((P @ A3) => ((! [X3:A]: (((P @ X3) => (X3 = A3)))) => (((fChoice @ A) @ P) = A3)))))).
% 0.33/5.22 thf(fact_5780_some__eq__trivial, axiom, (! [A:$tType,X:A]: ((((fChoice @ A) @ (^ [Y6:A]: ((Y6 = X)))) = X)))).
% 0.33/5.22 thf(fact_5781_some__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((fChoice @ A) @ ((^ [Y:A,Z2:A]: ((Y = Z2))) @ X)) = X)))).
% 0.33/5.22 thf(fact_5782_Eps__case__prod__eq, axiom, (! [A:$tType,B:$tType,X:A,Y3:B]: ((((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X9:A,Y8:B]: (((X = X9) & (Y3 = Y8)))))) = ((((product_Pair @ A) @ B) @ X) @ Y3))))).
% 0.33/5.22 thf(fact_5783_prod__decode__inverse, axiom, (! [N:nat]: (((nat_prod_encode @ (nat_prod_decode @ N)) = N)))).
% 0.33/5.22 thf(fact_5784_prod__encode__inverse, axiom, (! [X:((product_prod @ nat) @ nat)]: (((nat_prod_decode @ (nat_prod_encode @ X)) = X)))).
% 0.33/5.22 thf(fact_5785_inj__prod__decode, axiom, (! [A6:(set @ nat)]: (((((inj_on @ nat) @ ((product_prod @ nat) @ nat)) @ nat_prod_decode) @ A6)))).
% 0.33/5.22 thf(fact_5786_someI2, axiom, (! [A:$tType,P:(A > $o),A3:A,Q:(A > $o)]: (((P @ A3) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((fChoice @ A) @ P))))))).
% 0.33/5.22 thf(fact_5787_someI__ex, axiom, (! [A:$tType,P:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => (P @ ((fChoice @ A) @ P)))))).
% 0.33/5.22 thf(fact_5788_someI2__ex, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((fChoice @ A) @ P))))))).
% 0.33/5.22 thf(fact_5789_someI2__bex, axiom, (! [A:$tType,A6:(set @ A),P:(A > $o),Q:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ A6) & (P @ X4)))) => ((! [X3:A]: ((((((member @ A) @ X3) @ A6) & (P @ X3)) => (Q @ X3)))) => (Q @ ((fChoice @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A6) & (P @ X2))))))))))).
% 0.33/5.22 thf(fact_5790_some__eq__ex, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((fChoice @ A) @ P)) = (? [X7:A]: ((P @ X7))))))).
% 0.33/5.22 thf(fact_5791_some1__equality, axiom, (! [A:$tType,P:(A > $o),A3:A]: (((? [X4:A]: (((P @ X4) & (! [Y5:A]: (((P @ Y5) => (Y5 = X4))))))) => ((P @ A3) => (((fChoice @ A) @ P) = A3)))))).
% 0.33/5.22 thf(fact_5792_verit__sko__ex_H, axiom, (! [A:$tType,P:(A > $o),A6:$o]: ((((P @ ((fChoice @ A) @ P)) = A6) => ((? [X7:A]: ((P @ X7))) = A6))))).
% 0.33/5.22 thf(fact_5793_verit__sko__forall, axiom, (! [A:$tType]: (((^ [P3:(A > $o)]: ((! [X6:A]: ((P3 @ X6))))) = (^ [P4:(A > $o)]: ((P4 @ ((fChoice @ A) @ (^ [X2:A]: ((~ ((P4 @ X2))))))))))))).
% 0.33/5.22 thf(fact_5794_verit__sko__forall_H, axiom, (! [A:$tType,P:(A > $o),A6:$o]: ((((P @ ((fChoice @ A) @ (^ [X2:A]: ((~ ((P @ X2))))))) = A6) => ((! [X7:A]: ((P @ X7))) = A6))))).
% 0.33/5.22 thf(fact_5795_verit__sko__forall_H_H, axiom, (! [A:$tType,B7:A,A6:A,P:(A > $o)]: (((B7 = A6) => ((((fChoice @ A) @ P) = A6) = (((fChoice @ A) @ P) = B7)))))).
% 0.33/5.22 thf(fact_5796_verit__sko__ex__indirect, axiom, (! [A:$tType,X:A,P:(A > $o)]: (((X = ((fChoice @ A) @ P)) => ((? [X7:A]: ((P @ X7))) = (P @ X)))))).
% 0.33/5.22 thf(fact_5797_verit__sko__ex__indirect2, axiom, (! [A:$tType,X:A,P:(A > $o),P2:(A > $o)]: (((X = ((fChoice @ A) @ P)) => ((! [X3:A]: (((P @ X3) = (P2 @ X3)))) => ((? [X7:A]: ((P2 @ X7))) = (P @ X))))))).
% 0.33/5.22 thf(fact_5798_verit__sko__forall__indirect, axiom, (! [A:$tType,X:A,P:(A > $o)]: (((X = ((fChoice @ A) @ (^ [X2:A]: ((~ ((P @ X2))))))) => ((! [X7:A]: ((P @ X7))) = (P @ X)))))).
% 0.33/5.22 thf(fact_5799_verit__sko__forall__indirect2, axiom, (! [A:$tType,X:A,P:(A > $o),P2:(A > $o)]: (((X = ((fChoice @ A) @ (^ [X2:A]: ((~ ((P @ X2))))))) => ((! [X3:A]: (((P @ X3) = (P2 @ X3)))) => ((! [X7:A]: ((P2 @ X7))) = (P @ X))))))).
% 0.33/5.22 thf(fact_5800_Nitpick_OEps__psimp, axiom, (! [A:$tType,P:(A > $o),X:A,Y3:A]: (((P @ X) => ((~ ((P @ Y3))) => ((((fChoice @ A) @ P) = Y3) => (((fChoice @ A) @ P) = X))))))).
% 0.33/5.22 thf(fact_5801_someI, axiom, (! [A:$tType,P:(A > $o),X:A]: (((P @ X) => (P @ ((fChoice @ A) @ P)))))).
% 0.33/5.22 thf(fact_5802_Eps__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((fChoice @ A) @ P) = ((fChoice @ A) @ Q)))))).
% 0.33/5.22 thf(fact_5803_tfl__some, axiom, (! [A:$tType,P7:(A > $o),X4:A]: (((P7 @ X4) => (P7 @ ((fChoice @ A) @ P7)))))).
% 0.33/5.22 thf(fact_5804_some__eq__imp, axiom, (! [A:$tType,P:(A > $o),A3:A,B2:A]: (((((fChoice @ A) @ P) = A3) => ((P @ B2) => (P @ A3)))))).
% 0.33/5.22 thf(fact_5805_some__in__eq, axiom, (! [A:$tType,A6:(set @ A)]: (((((member @ A) @ ((fChoice @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A6))))) @ A6) = (A6 != (bot_bot @ (set @ A))))))).
% 0.33/5.22 thf(fact_5806_exE__some, axiom, (! [A:$tType,P:(A > $o),C2:A]: (((? [X_1:A]: ((P @ X_1))) => ((C2 = ((fChoice @ A) @ P)) => (P @ C2)))))).
% 0.33/5.22 thf(fact_5807_split__paired__Eps, axiom, (! [B:$tType,A:$tType]: (((fChoice @ ((product_prod @ A) @ B)) = (^ [P4:(((product_prod @ A) @ B) > $o)]: (((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A5:A,B3:B]: ((P4 @ ((((product_Pair @ A) @ B) @ A5) @ B3)))))))))))).
% 0.33/5.22 thf(fact_5808_prod__decode__def, axiom, (nat_prod_decode = (nat_prod_decode_aux @ (zero_zero @ nat)))).
% 0.33/5.22 thf(fact_5809_Eps__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: ((((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P)) = ((fChoice @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))).
% 0.33/5.22 thf(fact_5810_surj__prod__decode, axiom, (((((image @ nat) @ ((product_prod @ nat) @ nat)) @ nat_prod_decode) @ (top_top @ (set @ nat))) = (top_top @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.22 thf(fact_5811_mlex__less, axiom, (! [A:$tType,F:(A > nat),X:A,Y3:A,R2:(set @ ((product_prod @ A) @ A))]: (((((ord_less @ nat) @ (F @ X)) @ (F @ Y3)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ (((mlex_prod @ A) @ F) @ R2)))))).
% 0.33/5.22 thf(fact_5812_mlex__iff, axiom, (! [A:$tType,X:A,Y3:A,F:(A > nat),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ (((mlex_prod @ A) @ F) @ R2)) = ((((ord_less @ nat) @ (F @ X)) @ (F @ Y3)) | (((F @ X) = (F @ Y3)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ R2))))))).
% 0.33/5.22 thf(fact_5813_mlex__leq, axiom, (! [A:$tType,F:(A > nat),X:A,Y3:A,R2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ nat) @ (F @ X)) @ (F @ Y3)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ (((mlex_prod @ A) @ F) @ R2))))))).
% 0.33/5.22 thf(fact_5814_list__decode_Oelims, axiom, (! [X:nat,Y3:(list @ nat)]: ((((nat_list_decode @ X) = Y3) => (((X = (zero_zero @ nat)) => (Y3 != (nil @ nat))) => (~ ((! [N4:nat]: (((X = (suc @ N4)) => (Y3 != (((((product_case_prod @ nat) @ nat) @ (list @ nat)) @ (^ [X2:nat,Y6:nat]: ((((cons @ nat) @ X2) @ (nat_list_decode @ Y6))))) @ (nat_prod_decode @ N4))))))))))))).
% 0.33/5.22 thf(fact_5815_arg__min__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattices_ord_arg_min @ B) @ A) = (^ [F2:(B > A),P4:(B > $o)]: (((fChoice @ B) @ ((((lattic501386751177426532rg_min @ B) @ A) @ F2) @ P4))))))))).
% 0.33/5.22 thf(fact_5816_list__decode_Osimps_I2_J, axiom, (! [N:nat]: (((nat_list_decode @ (suc @ N)) = (((((product_case_prod @ nat) @ nat) @ (list @ nat)) @ (^ [X2:nat,Y6:nat]: ((((cons @ nat) @ X2) @ (nat_list_decode @ Y6))))) @ (nat_prod_decode @ N)))))).
% 0.33/5.22 thf(fact_5817_list__encode__inverse, axiom, (! [X:(list @ nat)]: (((nat_list_decode @ (nat_list_encode @ X)) = X)))).
% 0.33/5.22 thf(fact_5818_list__decode__inverse, axiom, (! [N:nat]: (((nat_list_encode @ (nat_list_decode @ N)) = N)))).
% 0.33/5.22 thf(fact_5819_list__decode__eq, axiom, (! [X:nat,Y3:nat]: ((((nat_list_decode @ X) = (nat_list_decode @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_5820_inj__list__decode, axiom, (! [A6:(set @ nat)]: (((((inj_on @ nat) @ (list @ nat)) @ nat_list_decode) @ A6)))).
% 0.33/5.22 thf(fact_5821_is__arg__min__antimono, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [F:(A > B),P:(A > $o),X:A,Y3:A]: (((((((lattic501386751177426532rg_min @ A) @ B) @ F) @ P) @ X) => ((((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X)) => ((P @ Y3) => (((((lattic501386751177426532rg_min @ A) @ B) @ F) @ P) @ Y3)))))))))).
% 0.33/5.22 thf(fact_5822_is__arg__min__linorder, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (((lattic501386751177426532rg_min @ A) @ B) = (^ [F2:(A > B),P4:(A > $o),X2:A]: (((P4 @ X2) & (! [Y6:A]: (((P4 @ Y6) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (F2 @ Y6))))))))))))).
% 0.33/5.22 thf(fact_5823_is__arg__min__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattic501386751177426532rg_min @ B) @ A) = (^ [F2:(B > A),P4:(B > $o),X2:B]: (((P4 @ X2) & (~ ((? [Y6:B]: (((P4 @ Y6) & (((ord_less @ A) @ (F2 @ Y6)) @ (F2 @ X2))))))))))))))).
% 0.33/5.22 thf(fact_5824_list__decode_Osimps_I1_J, axiom, ((nat_list_decode @ (zero_zero @ nat)) = (nil @ nat))).
% 0.33/5.22 thf(fact_5825_is__arg__min__arg__min__nat, axiom, (! [A:$tType,P:(A > $o),X:A,M:(A > nat)]: (((P @ X) => (((((lattic501386751177426532rg_min @ A) @ nat) @ M) @ P) @ ((((lattices_ord_arg_min @ A) @ nat) @ M) @ P)))))).
% 0.33/5.22 thf(fact_5826_surj__list__decode, axiom, (((((image @ nat) @ (list @ nat)) @ nat_list_decode) @ (top_top @ (set @ nat))) = (top_top @ (set @ (list @ nat))))).
% 0.33/5.22 thf(fact_5827_ex__is__arg__min__if__finite, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S3:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (? [X_12:A]: ((((((lattic501386751177426532rg_min @ A) @ B) @ F) @ (^ [X2:A]: ((((member @ A) @ X2) @ S3)))) @ X_12))))))))))).
% 0.33/5.22 thf(fact_5828_list__decode_Opelims, axiom, (! [X:nat,Y3:(list @ nat)]: ((((nat_list_decode @ X) = Y3) => ((((accp @ nat) @ nat_list_decode_rel) @ X) => (((X = (zero_zero @ nat)) => ((Y3 = (nil @ nat)) => (~ ((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)))))) => (~ ((! [N4:nat]: (((X = (suc @ N4)) => ((Y3 = (((((product_case_prod @ nat) @ nat) @ (list @ nat)) @ (^ [X2:nat,Y6:nat]: ((((cons @ nat) @ X2) @ (nat_list_decode @ Y6))))) @ (nat_prod_decode @ N4))) => (~ ((((accp @ nat) @ nat_list_decode_rel) @ (suc @ N4)))))))))))))))).
% 0.33/5.22 thf(fact_5829_mono__cSUP, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F:(A > B),A6:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F) => (((condit941137186595557371_above @ A) @ ((((image @ C) @ A) @ A6) @ I6)) => ((I6 != (bot_bot @ (set @ C))) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ C) @ B) @ (^ [X2:C]: ((F @ (A6 @ X2))))) @ I6))) @ (F @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ A6) @ I6))))))))))))).
% 0.33/5.22 thf(fact_5830_mono__cSup, axiom, (! [B:$tType,A:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((condit941137186595557371_above @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F) @ A6))) @ (F @ ((complete_Sup_Sup @ A) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5831_bdd__above_OI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A6:(set @ A),M7:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ X3) @ M7)))) => ((condit941137186595557371_above @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_5832_bdd__above_OI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A6:(set @ B),F:(B > A),M7:A]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ M7)))) => ((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_5833_bdd__above_OE, axiom, (! [A:$tType]: (((preorder @ A) => (! [A6:(set @ A)]: ((((condit941137186595557371_above @ A) @ A6) => (~ ((! [M8:A]: ((~ ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((ord_less_eq @ A) @ X4) @ M8))))))))))))))))).
% 0.33/5.22 thf(fact_5834_bdd__above_Ounfold, axiom, (! [A:$tType]: (((preorder @ A) => ((condit941137186595557371_above @ A) = (^ [A7:(set @ A)]: ((? [M9:A]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((ord_less_eq @ A) @ X2) @ M9))))))))))))).
% 0.33/5.22 thf(fact_5835_cSup__upper2, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X:A,X8:(set @ A),Y3:A]: (((((member @ A) @ X) @ X8) => ((((ord_less_eq @ A) @ Y3) @ X) => (((condit941137186595557371_above @ A) @ X8) => (((ord_less_eq @ A) @ Y3) @ ((complete_Sup_Sup @ A) @ X8))))))))))).
% 0.33/5.22 thf(fact_5836_cSup__upper, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X:A,X8:(set @ A)]: (((((member @ A) @ X) @ X8) => (((condit941137186595557371_above @ A) @ X8) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ X8)))))))))).
% 0.33/5.22 thf(fact_5837_list__decode_Opsimps_I1_J, axiom, ((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)) => ((nat_list_decode @ (zero_zero @ nat)) = (nil @ nat)))).
% 0.33/5.22 thf(fact_5838_list__decode_Opinduct, axiom, (! [A0:nat,P:(nat > $o)]: (((((accp @ nat) @ nat_list_decode_rel) @ A0) => (((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)) => (P @ (zero_zero @ nat))) => ((! [N4:nat]: (((((accp @ nat) @ nat_list_decode_rel) @ (suc @ N4)) => ((! [X4:nat,Y4:nat]: (((((((product_Pair @ nat) @ nat) @ X4) @ Y4) = (nat_prod_decode @ N4)) => (P @ Y4)))) => (P @ (suc @ N4)))))) => (P @ A0))))))).
% 0.33/5.22 thf(fact_5839_cSUP__upper, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [X:B,A6:(set @ B),F:(B > A)]: (((((member @ B) @ X) @ A6) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6)) => (((ord_less_eq @ A) @ (F @ X)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))))).
% 0.33/5.22 thf(fact_5840_cSUP__upper2, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F:(B > A),A6:(set @ B),X:B,U:A]: ((((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((member @ B) @ X) @ A6) => ((((ord_less_eq @ A) @ U) @ (F @ X)) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5841_cSup__le__iff, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A),A3:A]: (((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ A3) = (! [X2:A]: (((((member @ A) @ X2) @ S3) => (((ord_less_eq @ A) @ X2) @ A3))))))))))))).
% 0.33/5.22 thf(fact_5842_cSup__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [B7:(set @ A),A6:(set @ A)]: (((B7 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ A6) => ((! [B5:A]: (((((member @ A) @ B5) @ B7) => (? [X4:A]: (((((member @ A) @ X4) @ A6) & (((ord_less_eq @ A) @ B5) @ X4))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ B7)) @ ((complete_Sup_Sup @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5843_less__cSup__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Y3:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ X8) => ((((ord_less @ A) @ Y3) @ ((complete_Sup_Sup @ A) @ X8)) = (? [X2:A]: (((((member @ A) @ X2) @ X8) & (((ord_less @ A) @ Y3) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5844_cSUP__lessD, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [F:(B > A),A6:(set @ B),Y3:A,I:B]: ((((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ Y3) => ((((member @ B) @ I) @ A6) => (((ord_less @ A) @ (F @ I)) @ Y3)))))))))).
% 0.33/5.22 thf(fact_5845_cSUP__le__iff, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),F:(B > A),U:A]: (((A6 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ U) = (! [X2:B]: (((((member @ B) @ X2) @ A6) => (((ord_less_eq @ A) @ (F @ X2)) @ U))))))))))))).
% 0.33/5.22 thf(fact_5846_cSUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),G:(C > A),B7:(set @ C),F:(B > A)]: (((A6 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ C) @ A) @ G) @ B7)) => ((! [N4:B]: (((((member @ B) @ N4) @ A6) => (? [X4:C]: (((((member @ C) @ X4) @ B7) & (((ord_less_eq @ A) @ (F @ N4)) @ (G @ X4)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B7)))))))))))).
% 0.33/5.22 thf(fact_5847_cSup__subset__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((A6 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ B7))))))))))).
% 0.33/5.22 thf(fact_5848_less__cSUP__iff, axiom, (! [A:$tType,B:$tType]: (((condit6923001295902523014norder @ A) => (! [A6:(set @ B),F:(B > A),A3:A]: (((A6 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less @ A) @ A3) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) = (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ A3) @ (F @ X2)))))))))))))).
% 0.33/5.22 thf(fact_5849_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)) @ (^ [X2:nat,Y6:nat]: ((((cons @ nat) @ X2) @ (nat_list_decode @ Y6))))) @ (nat_prod_decode @ N))))))).
% 0.33/5.22 thf(fact_5850_cSUP__subset__mono, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),G:(B > A),B7:(set @ B),F:(B > A)]: (((A6 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ G) @ B7)) => ((((ord_less_eq @ (set @ B)) @ A6) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ (F @ X3)) @ (G @ X3))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B7))))))))))))).
% 0.33/5.22 thf(fact_5851_cSup__inter__less__eq, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((condit941137186595557371_above @ A) @ A6) => (((condit941137186595557371_above @ A) @ B7) => (((((inf_inf @ (set @ A)) @ A6) @ B7) != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ (((inf_inf @ (set @ A)) @ A6) @ B7))) @ (((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ B7)))))))))))).
% 0.33/5.22 thf(fact_5852_division__segment__integer__def, axiom, ((euclid7384307370059645450egment @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (euclid7384307370059645450egment @ int)))).
% 0.33/5.22 thf(fact_5853_mono__cInf, axiom, (! [B:$tType,A:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => (((condit1013018076250108175_below @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((ord_less_eq @ B) @ (F @ ((complete_Inf_Inf @ A) @ A6))) @ ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F) @ A6)))))))))))).
% 0.33/5.22 thf(fact_5854_mono__cINF, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F:(A > B),A6:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F) => (((condit1013018076250108175_below @ A) @ ((((image @ C) @ A) @ A6) @ I6)) => ((I6 != (bot_bot @ (set @ C))) => (((ord_less_eq @ B) @ (F @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ A6) @ I6)))) @ ((complete_Inf_Inf @ B) @ ((((image @ C) @ B) @ (^ [X2:C]: ((F @ (A6 @ X2))))) @ I6)))))))))))).
% 0.33/5.22 thf(fact_5855_bdd__belowI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A6:(set @ A),M:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ M) @ X3)))) => ((condit1013018076250108175_below @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_5856_bdd__below_OI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A6:(set @ A),M7:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ M7) @ X3)))) => ((condit1013018076250108175_below @ A) @ A6)))))))).
% 0.33/5.22 thf(fact_5857_bdd__below__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X8:(set @ A)]: ((((condit1013018076250108175_below @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ X8)) = ((condit941137186595557371_above @ A) @ X8)))))))).
% 0.33/5.22 thf(fact_5858_bdd__above__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X8:(set @ A)]: ((((condit941137186595557371_above @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ X8)) = ((condit1013018076250108175_below @ A) @ X8)))))))).
% 0.33/5.22 thf(fact_5859_bdd__below_OI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A6:(set @ B),M7:A,F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ M7) @ (F @ X3))))) => ((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_5860_bdd__belowI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A6:(set @ B),M:A,F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A6) => (((ord_less_eq @ A) @ M) @ (F @ X3))))) => ((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_5861_cInf__lower2, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X:A,X8:(set @ A),Y3:A]: (((((member @ A) @ X) @ X8) => ((((ord_less_eq @ A) @ X) @ Y3) => (((condit1013018076250108175_below @ A) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Y3)))))))))).
% 0.33/5.22 thf(fact_5862_cInf__lower, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X:A,X8:(set @ A)]: (((((member @ A) @ X) @ X8) => (((condit1013018076250108175_below @ A) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ X))))))))).
% 0.33/5.22 thf(fact_5863_bdd__below_Ounfold, axiom, (! [A:$tType]: (((preorder @ A) => ((condit1013018076250108175_below @ A) = (^ [A7:(set @ A)]: ((? [M9:A]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((ord_less_eq @ A) @ M9) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5864_bdd__below_OE, axiom, (! [A:$tType]: (((preorder @ A) => (! [A6:(set @ A)]: ((((condit1013018076250108175_below @ A) @ A6) => (~ ((! [M8:A]: ((~ ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((ord_less_eq @ A) @ M8) @ X4))))))))))))))))).
% 0.33/5.22 thf(fact_5865_cINF__lower2, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [F:(B > A),A6:(set @ B),X:B,U:A]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((member @ B) @ X) @ A6) => ((((ord_less_eq @ A) @ (F @ X)) @ U) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ U)))))))))).
% 0.33/5.22 thf(fact_5866_cINF__lower, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F:(B > A),A6:(set @ B),X:B]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((member @ B) @ X) @ A6) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ (F @ X)))))))))).
% 0.33/5.22 thf(fact_5867_le__cInf__iff, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A),A3:A]: (((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => ((((ord_less_eq @ A) @ A3) @ ((complete_Inf_Inf @ A) @ S3)) = (! [X2:A]: (((((member @ A) @ X2) @ S3) => (((ord_less_eq @ A) @ A3) @ X2))))))))))))).
% 0.33/5.22 thf(fact_5868_cInf__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [B7:(set @ A),A6:(set @ A)]: (((B7 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ A6) => ((! [B5:A]: (((((member @ A) @ B5) @ B7) => (? [X4:A]: (((((member @ A) @ X4) @ A6) & (((ord_less_eq @ A) @ X4) @ B5))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Inf_Inf @ A) @ B7))))))))))).
% 0.33/5.22 thf(fact_5869_cInf__less__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Y3:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ X8) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Y3) = (? [X2:A]: (((((member @ A) @ X2) @ X8) & (((ord_less @ A) @ X2) @ Y3))))))))))))).
% 0.33/5.22 thf(fact_5870_less__cINF__D, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F:(B > A),A6:(set @ B),Y3:A,I:B]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less @ A) @ Y3) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) => ((((member @ B) @ I) @ A6) => (((ord_less @ A) @ Y3) @ (F @ I))))))))))).
% 0.33/5.22 thf(fact_5871_cINF__mono, axiom, (! [C:$tType,A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [B7:(set @ B),F:(C > A),A6:(set @ C),G:(B > A)]: (((B7 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ C) @ A) @ F) @ A6)) => ((! [M6:B]: (((((member @ B) @ M6) @ B7) => (? [X4:C]: (((((member @ C) @ X4) @ A6) & (((ord_less_eq @ A) @ (F @ X4)) @ (G @ M6)))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ F) @ A6))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B7)))))))))))).
% 0.33/5.22 thf(fact_5872_le__cINF__iff, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),F:(B > A),U:A]: (((A6 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) = (! [X2:B]: (((((member @ B) @ X2) @ A6) => (((ord_less_eq @ A) @ U) @ (F @ X2)))))))))))))).
% 0.33/5.22 thf(fact_5873_cInf__superset__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: (((A6 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ B7)) @ ((complete_Inf_Inf @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5874_cINF__less__iff, axiom, (! [A:$tType,B:$tType]: (((condit6923001295902523014norder @ A) => (! [A6:(set @ B),F:(B > A),A3:A]: (((A6 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F) @ A6)) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))) @ A3) = (? [X2:B]: (((((member @ B) @ X2) @ A6) & (((ord_less @ A) @ (F @ X2)) @ A3))))))))))))).
% 0.33/5.22 thf(fact_5875_uminus__integer__def, axiom, ((uminus_uminus @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (uminus_uminus @ int)))).
% 0.33/5.22 thf(fact_5876_abs__integer__def, axiom, ((abs_abs @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (abs_abs @ int)))).
% 0.33/5.22 thf(fact_5877_sgn__integer__def, axiom, ((sgn_sgn @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (sgn_sgn @ int)))).
% 0.33/5.22 thf(fact_5878_cINF__superset__mono, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ B),G:(B > A),B7:(set @ B),F:(B > A)]: (((A6 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ G) @ B7)) => ((((ord_less_eq @ (set @ B)) @ A6) @ B7) => ((! [X3:B]: (((((member @ B) @ X3) @ B7) => (((ord_less_eq @ A) @ (G @ X3)) @ (F @ X3))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B7))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F) @ A6))))))))))))).
% 0.33/5.22 thf(fact_5879_less__eq__cInf__inter, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ A),B7:(set @ A)]: ((((condit1013018076250108175_below @ A) @ A6) => (((condit1013018076250108175_below @ A) @ B7) => (((((inf_inf @ (set @ A)) @ A6) @ B7) != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Inf_Inf @ A) @ B7))) @ ((complete_Inf_Inf @ A) @ (((inf_inf @ (set @ A)) @ A6) @ B7)))))))))))).
% 0.33/5.22 thf(fact_5880_cInf__le__cSup, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A6:(set @ A)]: (((A6 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ A6) => (((condit1013018076250108175_below @ A) @ A6) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A6)) @ ((complete_Sup_Sup @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_5881_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)))).
% 0.33/5.22 thf(fact_5882_Code__Numeral_Odup__def, axiom, (code_dup = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (^ [K3:int]: ((((plus_plus @ int) @ K3) @ K3)))))).
% 0.33/5.22 thf(fact_5883_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)))).
% 0.33/5.22 thf(fact_5884_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)))).
% 0.33/5.22 thf(fact_5885_Code__Numeral_Odup__code_I1_J, axiom, ((code_dup @ (zero_zero @ code_integer)) = (zero_zero @ code_integer))).
% 0.33/5.22 thf(fact_5886_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)))))).
% 0.33/5.22 thf(fact_5887_dup_Oabs__eq, axiom, (! [X:int]: (((code_dup @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((plus_plus @ int) @ X) @ X)))))).
% 0.33/5.22 thf(fact_5888_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)))).
% 0.33/5.22 thf(fact_5889_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)))).
% 0.33/5.22 thf(fact_5890_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)))).
% 0.33/5.22 thf(fact_5891_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)))).
% 0.33/5.22 thf(fact_5892_Code__Numeral_Osub__code_I9_J, axiom, (! [M:num,N:num]: ((((code_sub @ (bit0 @ M)) @ (bit1 @ N)) = (((minus_minus @ code_integer) @ (code_dup @ ((code_sub @ M) @ N))) @ (one_one @ code_integer)))))).
% 0.33/5.22 thf(fact_5893_Code__Numeral_Osub__code_I8_J, axiom, (! [M:num,N:num]: ((((code_sub @ (bit1 @ M)) @ (bit0 @ N)) = (((plus_plus @ code_integer) @ (code_dup @ ((code_sub @ M) @ N))) @ (one_one @ code_integer)))))).
% 0.33/5.22 thf(fact_5894_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)))).
% 0.33/5.22 thf(fact_5895_Code__Numeral_Osub__code_I1_J, axiom, (((code_sub @ one2) @ one2) = (zero_zero @ code_integer))).
% 0.33/5.22 thf(fact_5896_Code__Numeral_Osub__code_I6_J, axiom, (! [M:num,N:num]: ((((code_sub @ (bit0 @ M)) @ (bit0 @ N)) = (code_dup @ ((code_sub @ M) @ N)))))).
% 0.33/5.22 thf(fact_5897_Code__Numeral_Osub__code_I7_J, axiom, (! [M:num,N:num]: ((((code_sub @ (bit1 @ M)) @ (bit1 @ N)) = (code_dup @ ((code_sub @ M) @ N)))))).
% 0.33/5.22 thf(fact_5898_sub_Orep__eq, axiom, (! [X:num,Xa3:num]: (((code_int_of_integer @ ((code_sub @ X) @ Xa3)) = (((minus_minus @ int) @ ((numeral_numeral @ int) @ X)) @ ((numeral_numeral @ int) @ Xa3)))))).
% 0.33/5.22 thf(fact_5899_sub_Oabs__eq, axiom, (code_sub = (^ [Xa4:num,X2:num]: ((code_integer_of_int @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ Xa4)) @ ((numeral_numeral @ int) @ X2))))))).
% 0.33/5.22 thf(fact_5900_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)))).
% 0.33/5.22 thf(fact_5901_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)))).
% 0.33/5.22 thf(fact_5902_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)))).
% 0.33/5.22 thf(fact_5903_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)))).
% 0.33/5.22 thf(fact_5904_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)))).
% 0.33/5.22 thf(fact_5905_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)))).
% 0.33/5.22 thf(fact_5906_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)))).
% 0.33/5.22 thf(fact_5907_Sup__int__def, axiom, ((complete_Sup_Sup @ int) = (^ [X7:(set @ int)]: (((the @ int) @ (^ [X2:int]: (((((member @ int) @ X2) @ X7) & (! [Y6:int]: (((((member @ int) @ Y6) @ X7) => (((ord_less_eq @ int) @ Y6) @ X2)))))))))))).
% 0.33/5.22 thf(fact_5908_lexordp__conv__lexord, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp @ A) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys3)) @ ((lexord @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (ord_less @ A)))))))))))).
% 0.33/5.22 thf(fact_5909_sorted__wrt__iff__nth__Suc__transp, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A)]: ((((transp @ A) @ P) => ((((sorted_wrt @ A) @ P) @ Xs) = (! [I5:nat]: (((((ord_less @ nat) @ (suc @ I5)) @ ((size_size @ (list @ A)) @ Xs)) => ((P @ (((nth @ A) @ Xs) @ I5)) @ (((nth @ A) @ Xs) @ (suc @ I5))))))))))).
% 0.33/5.22 thf(fact_5910_lexordp__simps_I3_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Xs:(list @ A),Y3:A,Ys2:(list @ A)]: (((((ord_lexordp @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2)) = ((((ord_less @ A) @ X) @ Y3) | ((~ ((((ord_less @ A) @ Y3) @ X))) & (((ord_lexordp @ A) @ Xs) @ Ys2)))))))))).
% 0.33/5.22 thf(fact_5911_lexordp__irreflexive, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A)]: (((! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))) => (~ ((((ord_lexordp @ A) @ Xs) @ Xs)))))))))).
% 0.33/5.22 thf(fact_5912_lexordp_OCons, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => (((ord_lexordp @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2))))))))).
% 0.33/5.22 thf(fact_5913_lexordp_OCons__eq, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A,Xs:(list @ A),Ys2:(list @ A)]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => ((((ord_lexordp @ A) @ Xs) @ Ys2) => (((ord_lexordp @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2))))))))))).
% 0.33/5.22 thf(fact_5914_lexordp__append__leftD, axiom, (! [A:$tType]: (((ord @ A) => (! [Xs:(list @ A),Us:(list @ A),Vs:(list @ A)]: (((((ord_lexordp @ A) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Xs) @ Vs)) => ((! [A8:A]: ((~ ((((ord_less @ A) @ A8) @ A8))))) => (((ord_lexordp @ A) @ Us) @ Vs))))))))).
% 0.33/5.22 thf(fact_5915_Sup__eq__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Sup_Sup @ A) = (^ [A7:(set @ A)]: (((complete_Inf_Inf @ A) @ ((collect @ A) @ (^ [B3:A]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((ord_less_eq @ A) @ X2) @ B3))))))))))))))).
% 0.33/5.22 thf(fact_5916_Inf__eq__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Inf_Inf @ A) = (^ [A7:(set @ A)]: (((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [B3:A]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (((ord_less_eq @ A) @ B3) @ X2))))))))))))))).
% 0.33/5.22 thf(fact_5917_lexordp__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => ((! [Y5:A,Ys4:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ Y5) @ Ys4)))) => ((! [X3:A,Xs2:(list @ A),Y5:A,Ys4:(list @ A)]: (((((ord_less @ A) @ X3) @ Y5) => ((P @ (((cons @ A) @ X3) @ Xs2)) @ (((cons @ A) @ Y5) @ Ys4))))) => ((! [X3:A,Xs2:(list @ A),Ys4:(list @ A)]: (((((ord_lexordp @ A) @ Xs2) @ Ys4) => (((P @ Xs2) @ Ys4) => ((P @ (((cons @ A) @ X3) @ Xs2)) @ (((cons @ A) @ X3) @ Ys4)))))) => ((P @ Xs) @ Ys2))))))))))).
% 0.33/5.22 thf(fact_5918_lexordp__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: (((((ord_lexordp @ A) @ Xs) @ Ys2) => (((Xs = (nil @ A)) => (! [Y5:A,Ys5:(list @ A)]: ((Ys2 != (((cons @ A) @ Y5) @ Ys5))))) => ((! [X3:A]: (((? [Xs4:(list @ A)]: ((Xs = (((cons @ A) @ X3) @ Xs4)))) => (! [Y5:A]: (((? [Ys5:(list @ A)]: ((Ys2 = (((cons @ A) @ Y5) @ Ys5)))) => (~ ((((ord_less @ A) @ X3) @ Y5))))))))) => (~ ((! [X3:A,Xs4:(list @ A)]: (((Xs = (((cons @ A) @ X3) @ Xs4)) => (! [Ys5:(list @ A)]: (((Ys2 = (((cons @ A) @ X3) @ Ys5)) => (~ ((((ord_lexordp @ A) @ Xs4) @ Ys5)))))))))))))))))))).
% 0.33/5.22 thf(fact_5919_lexordp_Osimps, axiom, (! [A:$tType]: (((ord @ A) => ((ord_lexordp @ A) = (^ [A12:(list @ A),A23:(list @ A)]: (((? [Y6:A,Ys3:(list @ A)]: (((A12 = (nil @ A)) & (A23 = (((cons @ A) @ Y6) @ Ys3))))) | ((? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((A12 = (((cons @ A) @ X2) @ Xs3)) & ((A23 = (((cons @ A) @ Y6) @ Ys3)) & (((ord_less @ A) @ X2) @ Y6))))) | (? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((A12 = (((cons @ A) @ X2) @ Xs3)) & ((A23 = (((cons @ A) @ Y6) @ Ys3)) & ((~ ((((ord_less @ A) @ X2) @ Y6))) & ((~ ((((ord_less @ A) @ Y6) @ X2))) & (((ord_lexordp @ A) @ Xs3) @ Ys3)))))))))))))))).
% 0.33/5.22 thf(fact_5920_lexordp_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [A1:(list @ A),A22:(list @ A)]: (((((ord_lexordp @ A) @ A1) @ A22) => (((A1 = (nil @ A)) => (! [Y5:A,Ys4:(list @ A)]: ((A22 != (((cons @ A) @ Y5) @ Ys4))))) => ((! [X3:A]: (((? [Xs2:(list @ A)]: ((A1 = (((cons @ A) @ X3) @ Xs2)))) => (! [Y5:A]: (((? [Ys4:(list @ A)]: ((A22 = (((cons @ A) @ Y5) @ Ys4)))) => (~ ((((ord_less @ A) @ X3) @ Y5))))))))) => (~ ((! [X3:A,Y5:A,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X3) @ Xs2)) => (! [Ys4:(list @ A)]: (((A22 = (((cons @ A) @ Y5) @ Ys4)) => ((~ ((((ord_less @ A) @ X3) @ Y5))) => ((~ ((((ord_less @ A) @ Y5) @ X3))) => (~ ((((ord_lexordp @ A) @ Xs2) @ Ys4)))))))))))))))))))))).
% 0.33/5.22 thf(fact_5921_lexordp__append__left__rightI, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A,Us:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => (((ord_lexordp @ A) @ (((append @ A) @ Us) @ (((cons @ A) @ X) @ Xs))) @ (((append @ A) @ Us) @ (((cons @ A) @ Y3) @ Ys2)))))))))).
% 0.33/5.22 thf(fact_5922_lexordp__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp @ A) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((? [X2:A,Vs2:(list @ A)]: ((Ys3 = (((append @ A) @ Xs3) @ (((cons @ A) @ X2) @ Vs2))))) | (? [Us2:(list @ A),A5:A,B3:A,Vs2:(list @ A),Ws:(list @ A)]: (((((ord_less @ A) @ A5) @ B3) & ((Xs3 = (((append @ A) @ Us2) @ (((cons @ A) @ A5) @ Vs2))) & (Ys3 = (((append @ A) @ Us2) @ (((cons @ A) @ B3) @ Ws))))))))))))))).
% 0.33/5.22 thf(fact_5923_cInf__cSup, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A)]: (((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => (((complete_Inf_Inf @ A) @ S3) = ((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [X2:A]: ((! [Y6:A]: (((((member @ A) @ Y6) @ S3) => (((ord_less_eq @ A) @ X2) @ Y6))))))))))))))))).
% 0.33/5.22 thf(fact_5924_cSup__cInf, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A)]: (((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => (((complete_Sup_Sup @ A) @ S3) = ((complete_Inf_Inf @ A) @ ((collect @ A) @ (^ [X2:A]: ((! [Y6:A]: (((((member @ A) @ Y6) @ S3) => (((ord_less_eq @ A) @ Y6) @ X2))))))))))))))))).
% 0.33/5.22 thf(fact_5925_transp__gr, axiom, (! [A:$tType]: (((preorder @ A) => ((transp @ A) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.22 thf(fact_5926_transp__ge, axiom, (! [A:$tType]: (((preorder @ A) => ((transp @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))))))).
% 0.33/5.22 thf(fact_5927_ord__class_Olexordp__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_lexordp @ A) = ((complete_lattice_lfp @ ((list @ A) > ((list @ A) > $o))) @ (^ [P6:((list @ A) > ((list @ A) > $o)),X12:(list @ A),X24:(list @ A)]: (((? [Y6:A,Ys3:(list @ A)]: (((X12 = (nil @ A)) & (X24 = (((cons @ A) @ Y6) @ Ys3))))) | ((? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((X12 = (((cons @ A) @ X2) @ Xs3)) & ((X24 = (((cons @ A) @ Y6) @ Ys3)) & (((ord_less @ A) @ X2) @ Y6))))) | (? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((X12 = (((cons @ A) @ X2) @ Xs3)) & ((X24 = (((cons @ A) @ Y6) @ Ys3)) & ((~ ((((ord_less @ A) @ X2) @ Y6))) & ((~ ((((ord_less @ A) @ Y6) @ X2))) & ((P6 @ Xs3) @ Ys3))))))))))))))))).
% 0.33/5.22 thf(fact_5928_Sup__Inf, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [A6:(set @ (set @ A))]: ((((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ A6)) = ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [F2:((set @ A) > A)]: (((Uu = ((((image @ (set @ A)) @ A) @ F2) @ A6)) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A6) => (((member @ A) @ (F2 @ X2)) @ X2))))))))))))))))))).
% 0.33/5.22 thf(fact_5929_Inf__Sup__le, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [A6:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [F2:((set @ A) > A)]: (((Uu = ((((image @ (set @ A)) @ A) @ F2) @ A6)) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A6) => (((member @ A) @ (F2 @ X2)) @ X2))))))))))))))))))).
% 0.33/5.22 thf(fact_5930_Sup__Inf__le, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [F2:((set @ A) > A)]: (((Uu = ((((image @ (set @ A)) @ A) @ F2) @ A6)) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A6) => (((member @ A) @ (F2 @ X2)) @ X2))))))))))))) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A6))))))))).
% 0.33/5.22 thf(fact_5931_finite__Inf__Sup, axiom, (! [A:$tType]: (((finite8700451911770168679attice @ A) => (! [A6:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A6))) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [F2:((set @ A) > A)]: (((Uu = ((((image @ (set @ A)) @ A) @ F2) @ A6)) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A6) => (((member @ A) @ (F2 @ X2)) @ X2))))))))))))))))))).
% 0.33/5.22 thf(fact_5932_SUP__INF__set, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [G:(B > A),A6:(set @ (set @ B))]: ((((complete_Sup_Sup @ A) @ ((((image @ (set @ B)) @ A) @ (^ [X2:(set @ B)]: (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ X2))))) @ A6)) = ((complete_Inf_Inf @ A) @ ((((image @ (set @ B)) @ A) @ (^ [X2:(set @ B)]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ X2))))) @ ((collect @ (set @ B)) @ (^ [Uu:(set @ B)]: ((? [F2:((set @ B) > B)]: (((Uu = ((((image @ (set @ B)) @ B) @ F2) @ A6)) & (! [X2:(set @ B)]: (((((member @ (set @ B)) @ X2) @ A6) => (((member @ B) @ (F2 @ X2)) @ X2))))))))))))))))))).
% 0.33/5.22 thf(fact_5933_INF__SUP__set, axiom, (! [A:$tType,B:$tType]: (((comple592849572758109894attice @ A) => (! [G:(B > A),A6:(set @ (set @ B))]: ((((complete_Inf_Inf @ A) @ ((((image @ (set @ B)) @ A) @ (^ [B6:(set @ B)]: (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B6))))) @ A6)) = ((complete_Sup_Sup @ A) @ ((((image @ (set @ B)) @ A) @ (^ [B6:(set @ B)]: (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B6))))) @ ((collect @ (set @ B)) @ (^ [Uu:(set @ B)]: ((? [F2:((set @ B) > B)]: (((Uu = ((((image @ (set @ B)) @ B) @ F2) @ A6)) & (! [X2:(set @ B)]: (((((member @ (set @ B)) @ X2) @ A6) => (((member @ B) @ (F2 @ X2)) @ X2))))))))))))))))))).
% 0.33/5.22 thf(fact_5934_lfp__funpow, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),N:nat]: (((((order_mono @ A) @ A) @ F) => (((complete_lattice_lfp @ A) @ (((compow @ (A > A)) @ (suc @ N)) @ F)) = ((complete_lattice_lfp @ A) @ F))))))))).
% 0.33/5.22 thf(fact_5935_lfp__Kleene__iter, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),K:nat]: (((((order_mono @ A) @ A) @ F) => ((((((compow @ (A > A)) @ (suc @ K)) @ F) @ (bot_bot @ A)) = ((((compow @ (A > A)) @ K) @ F) @ (bot_bot @ A))) => (((complete_lattice_lfp @ A) @ F) = ((((compow @ (A > A)) @ K) @ F) @ (bot_bot @ A))))))))))).
% 0.33/5.22 thf(fact_5936_transp__le, axiom, (! [A:$tType]: (((preorder @ A) => ((transp @ A) @ (ord_less_eq @ A)))))).
% 0.33/5.22 thf(fact_5937_transp__less, axiom, (! [A:$tType]: (((preorder @ A) => ((transp @ A) @ (ord_less @ A)))))).
% 0.33/5.22 thf(fact_5938_lfp__ordinal__induct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),P:(A > $o)]: (((((order_mono @ A) @ A) @ F) => ((! [S4:A]: (((P @ S4) => ((((ord_less_eq @ A) @ S4) @ ((complete_lattice_lfp @ A) @ F)) => (P @ (F @ S4)))))) => ((! [M8:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ M8) => (P @ X4)))) => (P @ ((complete_Sup_Sup @ A) @ M8))))) => (P @ ((complete_lattice_lfp @ A) @ F))))))))))).
% 0.33/5.22 thf(fact_5939_def__lfp__induct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:A,F:(A > A),P:A]: (((A6 = ((complete_lattice_lfp @ A) @ F)) => ((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ (F @ (((inf_inf @ A) @ A6) @ P))) @ P) => (((ord_less_eq @ A) @ A6) @ P)))))))))).
% 0.33/5.22 thf(fact_5940_lfp__induct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),P:A]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ (F @ (((inf_inf @ A) @ ((complete_lattice_lfp @ A) @ F)) @ P))) @ P) => (((ord_less_eq @ A) @ ((complete_lattice_lfp @ A) @ F)) @ P))))))))).
% 0.33/5.22 thf(fact_5941_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X7:($o > A),Y9:($o > A)]: (((((ord_less_eq @ A) @ (X7 @ $false)) @ (Y9 @ $false)) & (((ord_less_eq @ A) @ (X7 @ $true)) @ (Y9 @ $true)))))))))).
% 0.33/5.22 thf(fact_5942_lfp__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),G:(A > A)]: (((! [Z9:A]: ((((ord_less_eq @ A) @ (F @ Z9)) @ (G @ Z9)))) => (((ord_less_eq @ A) @ ((complete_lattice_lfp @ A) @ F)) @ ((complete_lattice_lfp @ A) @ G))))))))).
% 0.33/5.22 thf(fact_5943_lfp__lowerbound, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),A6:A]: (((((ord_less_eq @ A) @ (F @ A6)) @ A6) => (((ord_less_eq @ A) @ ((complete_lattice_lfp @ A) @ F)) @ A6)))))))).
% 0.33/5.22 thf(fact_5944_lfp__greatest, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),A6:A]: (((! [U2:A]: (((((ord_less_eq @ A) @ (F @ U2)) @ U2) => (((ord_less_eq @ A) @ A6) @ U2)))) => (((ord_less_eq @ A) @ A6) @ ((complete_lattice_lfp @ A) @ F))))))))).
% 0.33/5.22 thf(fact_5945_lfp__lfp, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > (A > A))]: (((! [X3:A,Y5:A,W:A,Z4:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => ((((ord_less_eq @ A) @ W) @ Z4) => (((ord_less_eq @ A) @ ((F @ X3) @ W)) @ ((F @ Y5) @ Z4)))))) => (((complete_lattice_lfp @ A) @ (^ [X2:A]: (((complete_lattice_lfp @ A) @ (F @ X2))))) = ((complete_lattice_lfp @ A) @ (^ [X2:A]: (((F @ X2) @ X2)))))))))))).
% 0.33/5.22 thf(fact_5946_lfp__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F4:(A > A),X:A]: (((((order_mono @ A) @ A) @ F4) => (((F4 @ X) = X) => ((! [Z4:A]: ((((F4 @ Z4) = Z4) => (((ord_less_eq @ A) @ X) @ Z4)))) => (((complete_lattice_lfp @ A) @ F4) = X)))))))))).
% 0.33/5.22 thf(fact_5947_lfp__def, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_lattice_lfp @ A) = (^ [F2:(A > A)]: (((complete_Inf_Inf @ A) @ ((collect @ A) @ (^ [U3:A]: ((((ord_less_eq @ A) @ (F2 @ U3)) @ U3)))))))))))).
% 0.33/5.22 thf(fact_5948_char__of__take__bit__eq, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,M:A]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ (bit0 @ one2))))) @ N) => (((unique5772411509450598832har_of @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ M)) = ((unique5772411509450598832har_of @ A) @ M))))))))).
% 0.33/5.22 thf(fact_5949_num__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ num) @ num) @ code_pcr_integer) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)) @ code_num_of_integer)).
% 0.33/5.22 thf(fact_5950_Max_Osemilattice__order__set__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((lattic4895041142388067077er_set @ A) @ (ord_max @ A)) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.22 thf(fact_5951_char__of__nat, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: ((((unique5772411509450598832har_of @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((unique5772411509450598832har_of @ nat) @ N)))))))).
% 0.33/5.22 thf(fact_5952_integer_Oid__abs__transfer, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ code_integer) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ code_pcr_integer) @ (^ [X2:int]: (X2))) @ code_integer_of_int)).
% 0.33/5.22 thf(fact_5953_integer_Orep__transfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ int) @ code_pcr_integer) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ (^ [X2:int]: (X2))) @ code_int_of_integer)).
% 0.33/5.22 thf(fact_5954_zero__integer_Otransfer, axiom, ((code_pcr_integer @ (zero_zero @ int)) @ (zero_zero @ code_integer))).
% 0.33/5.22 thf(fact_5955_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) @ $o) @ code_pcr_integer) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less @ int)) @ (ord_less @ code_integer))).
% 0.33/5.22 thf(fact_5956_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))).
% 0.33/5.22 thf(fact_5957_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))).
% 0.33/5.22 thf(fact_5958_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))).
% 0.33/5.22 thf(fact_5959_one__integer_Otransfer, axiom, ((code_pcr_integer @ (one_one @ int)) @ (one_one @ code_integer))).
% 0.33/5.22 thf(fact_5960_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) @ $o) @ code_pcr_integer) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ int)) @ (ord_less_eq @ code_integer))).
% 0.33/5.22 thf(fact_5961_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))).
% 0.33/5.22 thf(fact_5962_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))).
% 0.33/5.22 thf(fact_5963_integer__of__nat_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ code_integer) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ code_pcr_integer) @ (semiring_1_of_nat @ int)) @ code_integer_of_nat)).
% 0.33/5.22 thf(fact_5964_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))).
% 0.33/5.22 thf(fact_5965_nat__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ nat) @ code_pcr_integer) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ nat2) @ code_nat_of_integer)).
% 0.33/5.22 thf(fact_5966_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))).
% 0.33/5.22 thf(fact_5967_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))).
% 0.33/5.22 thf(fact_5968_dup_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (^ [K3:int]: ((((plus_plus @ int) @ K3) @ K3)))) @ code_dup)).
% 0.33/5.22 thf(fact_5969_take__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se2584673776208193580ke_bit @ int)) @ (bit_se2584673776208193580ke_bit @ code_integer))).
% 0.33/5.22 thf(fact_5970_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))).
% 0.33/5.22 thf(fact_5971_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))).
% 0.33/5.22 thf(fact_5972_bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (nat > $o)) @ (nat > $o)) @ code_pcr_integer) @ (^ [Y:(nat > $o),Z2:(nat > $o)]: ((Y = Z2)))) @ (bit_se5641148757651400278ts_bit @ int)) @ (bit_se5641148757651400278ts_bit @ code_integer))).
% 0.33/5.22 thf(fact_5973_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))).
% 0.33/5.22 thf(fact_5974_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))).
% 0.33/5.22 thf(fact_5975_push__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se4730199178511100633sh_bit @ int)) @ (bit_se4730199178511100633sh_bit @ code_integer))).
% 0.33/5.22 thf(fact_5976_drop__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se4197421643247451524op_bit @ int)) @ (bit_se4197421643247451524op_bit @ code_integer))).
% 0.33/5.22 thf(fact_5977_mask__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ code_integer) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ code_pcr_integer) @ (bit_se2239418461657761734s_mask @ int)) @ (bit_se2239418461657761734s_mask @ code_integer))).
% 0.33/5.22 thf(fact_5978_unset__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se2638667681897837118et_bit @ int)) @ (bit_se2638667681897837118et_bit @ code_integer))).
% 0.33/5.22 thf(fact_5979_set__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se5668285175392031749et_bit @ int)) @ (bit_se5668285175392031749et_bit @ code_integer))).
% 0.33/5.22 thf(fact_5980_flip__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se8732182000553998342ip_bit @ int)) @ (bit_se8732182000553998342ip_bit @ code_integer))).
% 0.33/5.22 thf(fact_5981_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))).
% 0.33/5.22 thf(fact_5982_euclidean__size__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ nat) @ code_pcr_integer) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (euclid6346220572633701492n_size @ int)) @ (euclid6346220572633701492n_size @ code_integer))).
% 0.33/5.22 thf(fact_5983_sub_Otransfer, axiom, ((((((((bNF_rel_fun @ num) @ num) @ (num > int)) @ (num > code_integer)) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ num) @ num) @ int) @ code_integer) @ (^ [Y:num,Z2:num]: ((Y = Z2)))) @ code_pcr_integer)) @ (^ [M2:num,N3:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N3))))) @ code_sub)).
% 0.33/5.22 thf(fact_5984_Inf__fin_Osemilattice__order__set__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((((lattic4895041142388067077er_set @ A) @ (inf_inf @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.22 thf(fact_5985_Min_Osemilattice__order__set__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((lattic4895041142388067077er_set @ A) @ (ord_min @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.22 thf(fact_5986_UNIV__char__of__nat, axiom, ((top_top @ (set @ char)) = ((((image @ nat) @ char) @ (unique5772411509450598832har_of @ nat)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ one2))))))))))))).
% 0.33/5.22 thf(fact_5987_inj__on__char__of__nat, axiom, ((((inj_on @ nat) @ char) @ (unique5772411509450598832har_of @ nat)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ one2)))))))))))).
% 0.33/5.22 thf(fact_5988_Sup__fin_Osemilattice__order__set__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((((lattic4895041142388067077er_set @ A) @ (sup_sup @ A)) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.22 thf(fact_5989_char__of__def, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => ((unique5772411509450598832har_of @ A) = (^ [N3:A]: (((((((((char2 @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N3)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ (one_one @ nat))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit1 @ one2)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit1 @ (bit0 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ (bit1 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N3) @ ((numeral_numeral @ nat) @ (bit1 @ (bit1 @ one2)))))))))))).
% 0.33/5.22 thf(fact_5990_range__nat__of__char, axiom, (((((image @ char) @ nat) @ (comm_s6883823935334413003f_char @ nat)) @ (top_top @ (set @ char))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ one2)))))))))))).
% 0.33/5.22 thf(fact_5991_Gcd__fin__def, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Gcd_fin @ A) = ((((bounde2362111253966948842tice_F @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_5992_char_Osize_I2_J, axiom, (! [X13:$o,X22:$o,X32:$o,X42:$o,X52:$o,X62:$o,X72:$o,X82:$o]: ((((size_size @ char) @ ((((((((char2 @ X13) @ X22) @ X32) @ X42) @ X52) @ X62) @ X72) @ X82)) = (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5993_nat__of__char__less__256, axiom, (! [C2:char]: ((((ord_less @ nat) @ ((comm_s6883823935334413003f_char @ nat) @ C2)) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ one2))))))))))))).
% 0.33/5.22 thf(fact_5994_assert__def, axiom, (! [A:$tType]: (((heap_Time_assert @ A) = (^ [P4:(A > $o),X2:A]: (((((if @ (heap_Time_Heap @ A)) @ (P4 @ X2)) @ ((heap_Time_return @ A) @ X2)) @ ((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) @ $false)) @ (nil @ char))))))))))))))).
% 0.33/5.22 thf(fact_5995_char_Osize__gen, axiom, (! [X13:$o,X22:$o,X32:$o,X42:$o,X52:$o,X62:$o,X72:$o,X82:$o]: (((size_char @ ((((((((char2 @ X13) @ X22) @ X32) @ X42) @ X52) @ X62) @ X72) @ X82)) = (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_5996_n__lists_Osimps_I2_J, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((n_lists @ A) @ (suc @ N)) @ Xs) = ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ (^ [Ys3:(list @ A)]: (((((map @ A) @ (list @ A)) @ (^ [Y6:A]: ((((cons @ A) @ Y6) @ Ys3)))) @ Xs)))) @ (((n_lists @ A) @ N) @ Xs))))))).
% 0.33/5.22 thf(fact_5997_comm__append__is__replicate, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs != (nil @ A)) => ((Ys2 != (nil @ A)) => (((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Ys2) @ Xs)) => (? [N4:nat,Zs2:(list @ A)]: (((((ord_less @ nat) @ (one_one @ nat)) @ N4) & (((concat @ A) @ (((replicate @ (list @ A)) @ N4) @ Zs2)) = (((append @ A) @ Xs) @ Ys2))))))))))).
% 0.33/5.22 thf(fact_5998_strict__sorted__equal__Uniq, axiom, (! [A:$tType]: (((linorder @ A) => (! [A6:(set @ A)]: (((uniq @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ Xs3) & (((set2 @ A) @ Xs3) = A6))))))))))).
% 0.33/5.22 thf(fact_5999_integer__of__natural__of__nat, axiom, (! [N:nat]: (((code_i5400310926305786745atural @ ((semiring_1_of_nat @ code_natural) @ N)) = ((semiring_1_of_nat @ code_integer) @ N))))).
% 0.33/5.22 thf(fact_6000_integer__of__natural__of__integer, axiom, (! [K:code_integer]: (((code_i5400310926305786745atural @ (code_n4118661773612635043nteger @ K)) = (((ord_max @ code_integer) @ (zero_zero @ code_integer)) @ K))))).
% 0.33/5.22 thf(fact_6001_natural__of__integer__of__natural, axiom, (! [N:code_natural]: (((code_n4118661773612635043nteger @ (code_i5400310926305786745atural @ N)) = N)))).
% 0.33/5.22 thf(fact_6002_card__def, axiom, (! [B:$tType]: (((finite_card @ B) = ((((finite_folding_F @ B) @ nat) @ (^ [Uu:B]: (suc))) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6003_length__code, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = ((gen_length @ A) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6004_gen__length__code_I2_J, axiom, (! [B:$tType,N:nat,X:B,Xs:(list @ B)]: (((((gen_length @ B) @ N) @ (((cons @ B) @ X) @ Xs)) = (((gen_length @ B) @ (suc @ N)) @ Xs))))).
% 0.33/5.22 thf(fact_6005_less__than__iff, axiom, (! [X:nat,Y3:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X) @ Y3)) @ less_than) = (((ord_less @ nat) @ X) @ Y3))))).
% 0.33/5.22 thf(fact_6006_Code__Numeral_Osub__code_I4_J, axiom, (! [N:num]: ((((code_sub @ one2) @ (bit0 @ N)) = (code_Neg @ (bitM @ N)))))).
% 0.33/5.22 thf(fact_6007_Code__Numeral_Odup__code_I3_J, axiom, (! [N:num]: (((code_dup @ (code_Neg @ N)) = (code_Neg @ (bit0 @ N)))))).
% 0.33/5.22 thf(fact_6008_less__integer__code_I7_J, axiom, (! [K:num]: ((((ord_less @ code_integer) @ (code_Neg @ K)) @ (zero_zero @ code_integer))))).
% 0.33/5.22 thf(fact_6009_less__integer__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (code_Neg @ L))))))).
% 0.33/5.22 thf(fact_6010_plus__integer__code_I6_J, axiom, (! [M:num,N:num]: (((((plus_plus @ code_integer) @ (code_Neg @ M)) @ (code_Neg @ N)) = (code_Neg @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6011_less__eq__integer__code_I7_J, axiom, (! [K:num]: ((((ord_less_eq @ code_integer) @ (code_Neg @ K)) @ (zero_zero @ code_integer))))).
% 0.33/5.22 thf(fact_6012_less__eq__integer__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ (code_Neg @ L))))))).
% 0.33/5.22 thf(fact_6013_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))))).
% 0.33/5.22 thf(fact_6014_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))))).
% 0.33/5.22 thf(fact_6015_minus__integer__code_I6_J, axiom, (! [M:num,N:num]: (((((minus_minus @ code_integer) @ (code_Neg @ M)) @ (code_Neg @ N)) = ((code_sub @ N) @ M))))).
% 0.33/5.22 thf(fact_6016_divmod__abs__code_I2_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Neg @ K)) @ (code_Neg @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 0.33/5.22 thf(fact_6017_Code__Numeral_Osub__code_I5_J, axiom, (! [N:num]: ((((code_sub @ one2) @ (bit1 @ N)) = (code_Neg @ (bit0 @ N)))))).
% 0.33/5.22 thf(fact_6018_pair__less__def, axiom, (fun_pair_less = ((((lex_prod @ nat) @ nat) @ less_than) @ less_than))).
% 0.33/5.22 thf(fact_6019_Gcd__int__set__eq__fold, axiom, (! [Xs:(list @ int)]: ((((gcd_Gcd @ int) @ ((set2 @ int) @ Xs)) = (((((fold @ int) @ int) @ (gcd_gcd @ int)) @ Xs) @ (zero_zero @ int)))))).
% 0.33/5.22 thf(fact_6020_Gcd__set__eq__fold, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [Xs:(list @ A)]: ((((gcd_Gcd @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_gcd @ A)) @ Xs) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6021_Gcd__fin_Oset__eq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [Xs:(list @ A)]: ((((semiring_gcd_Gcd_fin @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_gcd @ A)) @ Xs) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6022_mono__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType]: ((((order @ B) & ((order @ D) & ((order @ C) & (order @ A)))) => (! [A6:(A > (B > $o)),B7:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A6) => (((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A6) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A6) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ A)) @ (ord_less_eq @ B)) => (((((((((bNF_rel_fun @ C) @ D) @ (C > $o)) @ (D > $o)) @ B7) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B7) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ C)) @ (ord_less_eq @ D)) => ((((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A6) @ B7)) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))) @ ((order_mono @ A) @ C)) @ ((order_mono @ B) @ D))))))))))).
% 0.33/5.22 thf(fact_6023_binomial__def, axiom, (binomial = (^ [N3:nat,K3:nat]: (((finite_card @ (set @ nat)) @ ((collect @ (set @ nat)) @ (^ [K6:(set @ nat)]: (((((member @ (set @ nat)) @ K6) @ ((pow2 @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3))) & (((finite_card @ nat) @ K6) = K3)))))))))).
% 0.33/5.22 thf(fact_6024_Pow__Compl, axiom, (! [A:$tType,A6:(set @ A)]: ((((pow2 @ A) @ ((uminus_uminus @ (set @ A)) @ A6)) = ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [B6:(set @ A)]: (((Uu = ((uminus_uminus @ (set @ A)) @ B6)) & (((member @ (set @ A)) @ A6) @ ((pow2 @ A) @ B6)))))))))))).
% 0.33/5.22 thf(fact_6025_int_Obi__total, axiom, (((bi_total @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int)).
% 0.33/5.22 thf(fact_6026_integer_Obi__total, axiom, (((bi_total @ int) @ code_integer) @ code_pcr_integer)).
% 0.33/5.22 thf(fact_6027_Gcd__nat__set__eq__fold, axiom, (! [Xs:(list @ nat)]: ((((gcd_Gcd @ nat) @ ((set2 @ nat) @ Xs)) = (((((fold @ nat) @ nat) @ (gcd_gcd @ nat)) @ Xs) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6028_Heap__lub__empty, axiom, (! [A:$tType]: ((((heap_Time_Heap_lub @ A) @ (bot_bot @ (set @ (heap_Time_Heap @ A)))) = ((heap_Time_Heap2 @ A) @ (^ [X2:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 0.33/5.22 thf(fact_6029_gfp__Kleene__iter, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),K:nat]: (((((order_mono @ A) @ A) @ F) => ((((((compow @ (A > A)) @ (suc @ K)) @ F) @ (top_top @ A)) = ((((compow @ (A > A)) @ K) @ F) @ (top_top @ A))) => (((complete_lattice_gfp @ A) @ F) = ((((compow @ (A > A)) @ K) @ F) @ (top_top @ A))))))))))).
% 0.33/5.22 thf(fact_6030_gfp__gfp, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > (A > A))]: (((! [X3:A,Y5:A,W:A,Z4:A]: (((((ord_less_eq @ A) @ X3) @ Y5) => ((((ord_less_eq @ A) @ W) @ Z4) => (((ord_less_eq @ A) @ ((F @ X3) @ W)) @ ((F @ Y5) @ Z4)))))) => (((complete_lattice_gfp @ A) @ (^ [X2:A]: (((complete_lattice_gfp @ A) @ (F @ X2))))) = ((complete_lattice_gfp @ A) @ (^ [X2:A]: (((F @ X2) @ X2)))))))))))).
% 0.33/5.22 thf(fact_6031_gfp__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),G:(A > A)]: (((! [Z9:A]: ((((ord_less_eq @ A) @ (F @ Z9)) @ (G @ Z9)))) => (((ord_less_eq @ A) @ ((complete_lattice_gfp @ A) @ F)) @ ((complete_lattice_gfp @ A) @ G))))))))).
% 0.33/5.22 thf(fact_6032_gfp__upperbound, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X8:A,F:(A > A)]: (((((ord_less_eq @ A) @ X8) @ (F @ X8)) => (((ord_less_eq @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F))))))))).
% 0.33/5.22 thf(fact_6033_gfp__least, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),X8:A]: (((! [U2:A]: (((((ord_less_eq @ A) @ U2) @ (F @ U2)) => (((ord_less_eq @ A) @ U2) @ X8)))) => (((ord_less_eq @ A) @ ((complete_lattice_gfp @ A) @ F)) @ X8)))))))).
% 0.33/5.22 thf(fact_6034_gfp__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F4:(A > A),X:A]: (((((order_mono @ A) @ A) @ F4) => (((F4 @ X) = X) => ((! [Z4:A]: ((((F4 @ Z4) = Z4) => (((ord_less_eq @ A) @ Z4) @ X)))) => (((complete_lattice_gfp @ A) @ F4) = X)))))))))).
% 0.33/5.22 thf(fact_6035_gfp__def, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_lattice_gfp @ A) = (^ [F2:(A > A)]: (((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [U3:A]: ((((ord_less_eq @ A) @ U3) @ (F2 @ U3))))))))))))).
% 0.33/5.22 thf(fact_6036_coinduct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),X8:A]: (((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ X8) @ (F @ (((sup_sup @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F)))) => (((ord_less_eq @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F)))))))))).
% 0.33/5.22 thf(fact_6037_def__coinduct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A6:A,F:(A > A),X8:A]: (((A6 = ((complete_lattice_gfp @ A) @ F)) => ((((order_mono @ A) @ A) @ F) => ((((ord_less_eq @ A) @ X8) @ (F @ (((sup_sup @ A) @ X8) @ A6))) => (((ord_less_eq @ A) @ X8) @ A6)))))))))).
% 0.33/5.22 thf(fact_6038_coinduct__lemma, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X8:A,F:(A > A)]: (((((ord_less_eq @ A) @ X8) @ (F @ (((sup_sup @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F)))) => ((((order_mono @ A) @ A) @ F) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F))) @ (F @ (((sup_sup @ A) @ X8) @ ((complete_lattice_gfp @ A) @ F)))))))))))).
% 0.33/5.22 thf(fact_6039_gfp__ordinal__induct, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),P:(A > $o)]: (((((order_mono @ A) @ A) @ F) => ((! [S4:A]: (((P @ S4) => ((((ord_less_eq @ A) @ ((complete_lattice_gfp @ A) @ F)) @ S4) => (P @ (F @ S4)))))) => ((! [M8:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ M8) => (P @ X4)))) => (P @ ((complete_Inf_Inf @ A) @ M8))))) => (P @ ((complete_lattice_gfp @ A) @ F))))))))))).
% 0.33/5.22 thf(fact_6040_gfp__funpow, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A),N:nat]: (((((order_mono @ A) @ A) @ F) => (((complete_lattice_gfp @ A) @ (((compow @ (A > A)) @ (suc @ N)) @ F)) = ((complete_lattice_gfp @ A) @ F))))))))).
% 0.33/5.22 thf(fact_6041_lfp__le__gfp, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(A > A)]: (((((order_mono @ A) @ A) @ F) => (((ord_less_eq @ A) @ ((complete_lattice_lfp @ A) @ F)) @ ((complete_lattice_gfp @ A) @ F))))))))).
% 0.33/5.22 thf(fact_6042_Code__Numeral_Osub__code_I2_J, axiom, (! [M:num]: ((((code_sub @ (bit0 @ M)) @ one2) = (code_Pos @ (bitM @ M)))))).
% 0.33/5.22 thf(fact_6043_is__singleton__altdef, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((((finite_card @ A) @ A7) = (one_one @ nat)))))))).
% 0.33/5.22 thf(fact_6044_Code__Numeral_OPos__def, axiom, (code_Pos = (numeral_numeral @ code_integer))).
% 0.33/5.22 thf(fact_6045_Pos__fold_I2_J, axiom, (! [K:num]: ((((numeral_numeral @ code_integer) @ (bit0 @ K)) = (code_Pos @ (bit0 @ K)))))).
% 0.33/5.22 thf(fact_6046_Pos__fold_I1_J, axiom, (((numeral_numeral @ code_integer) @ one2) = (code_Pos @ one2))).
% 0.33/5.22 thf(fact_6047_Pos__fold_I3_J, axiom, (! [K:num]: ((((numeral_numeral @ code_integer) @ (bit1 @ K)) = (code_Pos @ (bit1 @ K)))))).
% 0.33/5.22 thf(fact_6048_one__integer__code, axiom, ((one_one @ code_integer) = (code_Pos @ one2))).
% 0.33/5.22 thf(fact_6049_Code__Numeral_Odup__code_I2_J, axiom, (! [N:num]: (((code_dup @ (code_Pos @ N)) = (code_Pos @ (bit0 @ N)))))).
% 0.33/5.22 thf(fact_6050_less__integer__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less @ code_integer) @ (code_Pos @ K)) @ (zero_zero @ code_integer))))))).
% 0.33/5.22 thf(fact_6051_less__integer__code_I2_J, axiom, (! [L:num]: ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (code_Pos @ L))))).
% 0.33/5.22 thf(fact_6052_plus__integer__code_I3_J, axiom, (! [M:num,N:num]: (((((plus_plus @ code_integer) @ (code_Pos @ M)) @ (code_Pos @ N)) = (code_Pos @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6053_less__eq__integer__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less_eq @ code_integer) @ (code_Pos @ K)) @ (zero_zero @ code_integer))))))).
% 0.33/5.22 thf(fact_6054_less__eq__integer__code_I2_J, axiom, (! [L:num]: ((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ (code_Pos @ L))))).
% 0.33/5.22 thf(fact_6055_Code__Numeral_ONeg__def, axiom, (code_Neg = (^ [N3:num]: (((uminus_uminus @ code_integer) @ (code_Pos @ N3)))))).
% 0.33/5.22 thf(fact_6056_uminus__integer__code_I2_J, axiom, (! [M:num]: ((((uminus_uminus @ code_integer) @ (code_Pos @ M)) = (code_Neg @ M))))).
% 0.33/5.22 thf(fact_6057_uminus__integer__code_I3_J, axiom, (! [M:num]: ((((uminus_uminus @ code_integer) @ (code_Neg @ M)) = (code_Pos @ M))))).
% 0.33/5.22 thf(fact_6058_less__integer__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less @ code_integer) @ (code_Neg @ K)) @ (code_Pos @ L))))).
% 0.33/5.22 thf(fact_6059_less__integer__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less @ code_integer) @ (code_Pos @ K)) @ (code_Neg @ L))))))).
% 0.33/5.22 thf(fact_6060_times__integer__code_I3_J, axiom, (! [M:num,N:num]: (((((times_times @ code_integer) @ (code_Pos @ M)) @ (code_Pos @ N)) = (code_Pos @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6061_less__eq__integer__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less_eq @ code_integer) @ (code_Neg @ K)) @ (code_Pos @ L))))).
% 0.33/5.22 thf(fact_6062_less__eq__integer__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less_eq @ code_integer) @ (code_Pos @ K)) @ (code_Neg @ L))))))).
% 0.33/5.22 thf(fact_6063_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))))).
% 0.33/5.22 thf(fact_6064_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))))).
% 0.33/5.22 thf(fact_6065_minus__integer__code_I3_J, axiom, (! [M:num,N:num]: (((((minus_minus @ code_integer) @ (code_Pos @ M)) @ (code_Pos @ N)) = ((code_sub @ M) @ N))))).
% 0.33/5.22 thf(fact_6066_divmod__abs__code_I1_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Pos @ K)) @ (code_Pos @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 0.33/5.22 thf(fact_6067_minus__integer__code_I5_J, axiom, (! [M:num,N:num]: (((((minus_minus @ code_integer) @ (code_Neg @ M)) @ (code_Pos @ N)) = (code_Neg @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6068_minus__integer__code_I4_J, axiom, (! [M:num,N:num]: (((((minus_minus @ code_integer) @ (code_Pos @ M)) @ (code_Neg @ N)) = (code_Pos @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6069_times__integer__code_I6_J, axiom, (! [M:num,N:num]: (((((times_times @ code_integer) @ (code_Neg @ M)) @ (code_Neg @ N)) = (code_Pos @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6070_times__integer__code_I5_J, axiom, (! [M:num,N:num]: (((((times_times @ code_integer) @ (code_Neg @ M)) @ (code_Pos @ N)) = (code_Neg @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6071_times__integer__code_I4_J, axiom, (! [M:num,N:num]: (((((times_times @ code_integer) @ (code_Pos @ M)) @ (code_Neg @ N)) = (code_Neg @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6072_plus__integer__code_I4_J, axiom, (! [M:num,N:num]: (((((plus_plus @ code_integer) @ (code_Pos @ M)) @ (code_Neg @ N)) = ((code_sub @ M) @ N))))).
% 0.33/5.22 thf(fact_6073_plus__integer__code_I5_J, axiom, (! [M:num,N:num]: (((((plus_plus @ code_integer) @ (code_Neg @ M)) @ (code_Pos @ N)) = ((code_sub @ N) @ M))))).
% 0.33/5.22 thf(fact_6074_divmod__abs__code_I3_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Neg @ K)) @ (code_Pos @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 0.33/5.22 thf(fact_6075_divmod__abs__code_I4_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Pos @ K)) @ (code_Neg @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 0.33/5.22 thf(fact_6076_Code__Numeral_Osub__code_I3_J, axiom, (! [M:num]: ((((code_sub @ (bit1 @ M)) @ one2) = (code_Pos @ (bit0 @ M)))))).
% 0.33/5.22 thf(fact_6077_prod__list__def, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((groups5270119922927024881d_list @ A) = (((groups_monoid_F @ A) @ (times_times @ A)) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_6078_semilattice__order__set_Osubset__imp, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),B7:(set @ A)]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((Less_eq @ (((lattic1715443433743089157tice_F @ A) @ F) @ B7)) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_6079_semilattice__set_OF_Ocong, axiom, (! [A:$tType]: (((lattic1715443433743089157tice_F @ A) = (lattic1715443433743089157tice_F @ A))))).
% 0.33/5.22 thf(fact_6080_Inf__fin__def, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((lattic7752659483105999362nf_fin @ A) = ((lattic1715443433743089157tice_F @ A) @ (inf_inf @ A))))))).
% 0.33/5.22 thf(fact_6081_Sup__fin__def, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((lattic5882676163264333800up_fin @ A) = ((lattic1715443433743089157tice_F @ A) @ (sup_sup @ A))))))).
% 0.33/5.22 thf(fact_6082_Max__def, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798349783984er_Max @ A) = ((lattic1715443433743089157tice_F @ A) @ (ord_max @ A))))))).
% 0.33/5.22 thf(fact_6083_Min__def, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798350308766er_Min @ A) = ((lattic1715443433743089157tice_F @ A) @ (ord_min @ A))))))).
% 0.33/5.22 thf(fact_6084_semilattice__order__set_OcoboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),A3:A]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => ((Less_eq @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) @ A3))))))).
% 0.33/5.22 thf(fact_6085_semilattice__order__set_OboundedE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((Less_eq @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => ((Less_eq @ X) @ A10))))))))))).
% 0.33/5.22 thf(fact_6086_semilattice__order__set_OboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => ((Less_eq @ X) @ A8)))) => ((Less_eq @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_6087_semilattice__order__set_Obounded__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((Less_eq @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => ((Less_eq @ X) @ X2))))))))))).
% 0.33/5.22 thf(fact_6088_sum__list__def, axiom, (! [A:$tType]: (((monoid_add @ A) => ((groups8242544230860333062m_list @ A) = (((groups_monoid_F @ A) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 0.33/5.22 thf(fact_6089_semilattice__set_Oeq__fold_H, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A6) = ((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X2:A,Y6:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X2) @ (F @ X2)) @ Y6))))) @ (none @ A)) @ A6))))))).
% 0.33/5.22 thf(fact_6090_semilattice__set_Oinsert__remove, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ A6)) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ A6)) = ((F @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))))).
% 0.33/5.22 thf(fact_6091_Min_Osemilattice__set__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic149705377957585745ce_set @ A) @ (ord_min @ A)))))).
% 0.33/5.22 thf(fact_6092_Max_Osemilattice__set__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic149705377957585745ce_set @ A) @ (ord_max @ A)))))).
% 0.33/5.22 thf(fact_6093_Inf__fin_Osemilattice__set__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((lattic149705377957585745ce_set @ A) @ (inf_inf @ A)))))).
% 0.33/5.22 thf(fact_6094_Sup__fin_Osemilattice__set__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((lattic149705377957585745ce_set @ A) @ (sup_sup @ A)))))).
% 0.33/5.22 thf(fact_6095_semilattice__order__set_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => ((lattic149705377957585745ce_set @ A) @ F))))).
% 0.33/5.22 thf(fact_6096_semilattice__set_Oin__idem, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((F @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) = (((lattic1715443433743089157tice_F @ A) @ F) @ A6)))))))).
% 0.33/5.22 thf(fact_6097_semilattice__set_Osingleton, axiom, (! [A:$tType,F:(A > (A > A)),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X))))).
% 0.33/5.22 thf(fact_6098_semilattice__set_Ohom__commute, axiom, (! [A:$tType,F:(A > (A > A)),H2:(A > A),N6:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => ((! [X3:A,Y5:A]: (((H2 @ ((F @ X3) @ Y5)) = ((F @ (H2 @ X3)) @ (H2 @ Y5))))) => (((finite_finite2 @ A) @ N6) => ((N6 != (bot_bot @ (set @ A))) => ((H2 @ (((lattic1715443433743089157tice_F @ A) @ F) @ N6)) = (((lattic1715443433743089157tice_F @ A) @ F) @ ((((image @ A) @ A) @ H2) @ N6)))))))))).
% 0.33/5.22 thf(fact_6099_semilattice__set_Osubset, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),B7:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((B7 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => (((F @ (((lattic1715443433743089157tice_F @ A) @ F) @ B7)) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) = (((lattic1715443433743089157tice_F @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_6100_semilattice__set_Oinsert__not__elem, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((~ ((((member @ A) @ X) @ A6))) => ((A6 != (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ A6)) = ((F @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)))))))))).
% 0.33/5.22 thf(fact_6101_semilattice__set_Oinsert, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ A6)) = ((F @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6))))))))).
% 0.33/5.22 thf(fact_6102_semilattice__set_Oclosed, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ ((F @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) @ A6)))))))).
% 0.33/5.22 thf(fact_6103_semilattice__set_Ounion, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),B7:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => ((B7 != (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = ((F @ (((lattic1715443433743089157tice_F @ A) @ F) @ A6)) @ (((lattic1715443433743089157tice_F @ A) @ F) @ B7))))))))))).
% 0.33/5.22 thf(fact_6104_semilattice__set_Oeq__fold, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((((lattic1715443433743089157tice_F @ A) @ F) @ (((insert @ A) @ X) @ A6)) = (((((finite_fold @ A) @ A) @ F) @ X) @ A6))))))).
% 0.33/5.22 thf(fact_6105_semilattice__set_Oinfinite, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => ((~ (((finite_finite2 @ A) @ A6))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A6) = ((the2 @ A) @ (none @ A)))))))).
% 0.33/5.22 thf(fact_6106_semilattice__set_Oremove, axiom, (! [A:$tType,F:(A > (A > A)),A6:(set @ A),X:A]: ((((lattic149705377957585745ce_set @ A) @ F) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => ((((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A6) = X)) & (((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A6) = ((F @ X) @ (((lattic1715443433743089157tice_F @ A) @ F) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))))).
% 0.33/5.22 thf(fact_6107_Least__mono, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),S3:(set @ A)]: (((((order_mono @ A) @ B) @ F) => ((? [X4:A]: (((((member @ A) @ X4) @ S3) & (! [Xa2:A]: (((((member @ A) @ Xa2) @ S3) => (((ord_less_eq @ A) @ X4) @ Xa2))))))) => (((ord_Least @ B) @ (^ [Y6:B]: ((((member @ B) @ Y6) @ ((((image @ A) @ B) @ F) @ S3))))) = (F @ ((ord_Least @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ S3)))))))))))))).
% 0.33/5.22 thf(fact_6108_cr__int__def, axiom, (cr_int = (^ [X2:((product_prod @ nat) @ nat)]: (((^ [Y:int,Z2:int]: ((Y = Z2))) @ (abs_Integ @ X2)))))).
% 0.33/5.22 thf(fact_6109_Least__eq__0, axiom, (! [P:(nat > $o)]: (((P @ (zero_zero @ nat)) => (((ord_Least @ nat) @ P) = (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6110_Least__le, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),K:A]: (((P @ K) => (((ord_less_eq @ A) @ ((ord_Least @ A) @ P)) @ K)))))))).
% 0.33/5.22 thf(fact_6111_Least1I, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o)]: (((? [X4:A]: (((P @ X4) & ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X4) @ Y5)))) & (! [Y5:A]: ((((P @ Y5) & (! [Ya2:A]: (((P @ Ya2) => (((ord_less_eq @ A) @ Y5) @ Ya2))))) => (Y5 = X4)))))))) => (P @ ((ord_Least @ A) @ P))))))))).
% 0.33/5.22 thf(fact_6112_Least1__le, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),Z3:A]: (((? [X4:A]: (((P @ X4) & ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X4) @ Y5)))) & (! [Y5:A]: ((((P @ Y5) & (! [Ya2:A]: (((P @ Ya2) => (((ord_less_eq @ A) @ Y5) @ Ya2))))) => (Y5 = X4)))))))) => ((P @ Z3) => (((ord_less_eq @ A) @ ((ord_Least @ A) @ P)) @ Z3))))))))).
% 0.33/5.22 thf(fact_6113_LeastI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X) @ Y5)))) => ((! [X3:A]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ X3) @ Y4)))) => (Q @ X3))))) => (Q @ ((ord_Least @ A) @ P))))))))))).
% 0.33/5.22 thf(fact_6114_Least__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X) @ Y5)))) => (((ord_Least @ A) @ P) = X))))))))).
% 0.33/5.22 thf(fact_6115_LeastI2__wellorder, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A,Q:(A > $o)]: (((P @ A3) => ((! [A8:A]: (((P @ A8) => ((! [B11:A]: (((P @ B11) => (((ord_less_eq @ A) @ A8) @ B11)))) => (Q @ A8))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 0.33/5.22 thf(fact_6116_LeastI2__wellorder__ex, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),Q:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => ((! [A8:A]: (((P @ A8) => ((! [B11:A]: (((P @ B11) => (((ord_less_eq @ A) @ A8) @ B11)))) => (Q @ A8))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 0.33/5.22 thf(fact_6117_not__less__Least, axiom, (! [A:$tType]: (((wellorder @ A) => (! [K:A,P:(A > $o)]: (((((ord_less @ A) @ K) @ ((ord_Least @ A) @ P)) => (~ ((P @ K)))))))))).
% 0.33/5.22 thf(fact_6118_LeastI, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),K:A]: (((P @ K) => (P @ ((ord_Least @ A) @ P))))))))).
% 0.33/5.22 thf(fact_6119_LeastI2, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A,Q:(A > $o)]: (((P @ A3) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 0.33/5.22 thf(fact_6120_LeastI__ex, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => (P @ ((ord_Least @ A) @ P))))))))).
% 0.33/5.22 thf(fact_6121_LeastI2__ex, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),Q:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 0.33/5.22 thf(fact_6122_Least__Suc2, axiom, (! [P:(nat > $o),N:nat,Q:(nat > $o),M:nat]: (((P @ N) => ((Q @ M) => ((~ ((P @ (zero_zero @ nat)))) => ((! [K2:nat]: (((P @ (suc @ K2)) = (Q @ K2)))) => (((ord_Least @ nat) @ P) = (suc @ ((ord_Least @ nat) @ Q)))))))))).
% 0.33/5.22 thf(fact_6123_Least__Suc, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (((ord_Least @ nat) @ P) = (suc @ ((ord_Least @ nat) @ (^ [M2:nat]: ((P @ (suc @ M2)))))))))))).
% 0.33/5.22 thf(fact_6124_Least__Min, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => ((? [X_1:A]: ((P @ X_1))) => (((ord_Least @ A) @ P) = ((lattic643756798350308766er_Min @ A) @ ((collect @ A) @ P))))))))))).
% 0.33/5.22 thf(fact_6125_int_Opcr__cr__eq, axiom, (pcr_int = cr_int)).
% 0.33/5.22 thf(fact_6126_Quotient__int, axiom, ((((((quotient @ ((product_prod @ nat) @ nat)) @ int) @ intrel) @ abs_Integ) @ rep_Integ) @ cr_int)).
% 0.33/5.22 thf(fact_6127_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((? [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3)) & (P4 @ (((nth @ A) @ Xs3) @ N3))))))))))).
% 0.33/5.22 thf(fact_6128_map__of__zip__nth, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),I:nat]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys2)) => (((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ (((nth @ A) @ Xs) @ I)) = ((some @ B) @ (((nth @ B) @ Ys2) @ I))))))))).
% 0.33/5.22 thf(fact_6129_and_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((semilattice_neutr @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_6130_semilattice__neutr__order_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((semilattice_neutr @ A) @ F) @ Z3))))).
% 0.33/5.22 thf(fact_6131_sup__bot_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (((semilattice_neutr @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 0.33/5.22 thf(fact_6132_inf__top_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (((semilattice_neutr @ A) @ (inf_inf @ A)) @ (top_top @ A)))))).
% 0.33/5.22 thf(fact_6133_gcd__nat_Osemilattice__neutr__axioms, axiom, (((semilattice_neutr @ nat) @ (gcd_gcd @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6134_or_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((semilattice_neutr @ A) @ (bit_se1065995026697491101ons_or @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6135_max__nat_Osemilattice__neutr__axioms, axiom, (((semilattice_neutr @ nat) @ (ord_max @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6136_map__upds__list__update2__drop, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),I:nat,M:(A > (option @ B)),Ys2:(list @ B),Y3:B]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I) => ((((((map_upds @ A) @ B) @ M) @ Xs) @ ((((list_update @ B) @ Ys2) @ I) @ Y3)) = (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys2)))))).
% 0.33/5.22 thf(fact_6137_natLess__def, axiom, (bNF_Ca8459412986667044542atLess = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less @ nat))))).
% 0.33/5.22 thf(fact_6138_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca3754400796208372196lChain @ A) @ B) = (^ [R5:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I5:A,J3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I5) @ J3)) @ R5) => (((ord_less_eq @ B) @ (As @ I5)) @ (As @ J3)))))))))))).
% 0.33/5.22 thf(fact_6139_map__upds__append1, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B),M:(A > (option @ B)),X:A]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2)) => ((((((map_upds @ A) @ B) @ M) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) @ Ys2) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys2)) @ X) @ ((some @ B) @ (((nth @ B) @ Ys2) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 0.33/5.22 thf(fact_6140_restrict__upd__same, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B)),X:A,Y3:B]: ((((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ X) @ ((some @ B) @ Y3))) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) = ((((restrict_map @ A) @ B) @ M) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 0.33/5.22 thf(fact_6141_restrict__complement__singleton__eq, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B)),X:A]: ((((((restrict_map @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) = (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ (none @ B)))))).
% 0.33/5.22 thf(fact_6142_Restr__natLeq, axiom, (! [N:nat]: (((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ bNF_Ca8665028551170535155natLeq) @ ((((product_Sigma @ nat) @ nat) @ ((collect @ nat) @ (^ [X2:nat]: ((((ord_less @ nat) @ X2) @ N))))) @ (^ [Uu:nat]: (((collect @ nat) @ (^ [X2:nat]: ((((ord_less @ nat) @ X2) @ N)))))))) = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))))).
% 0.33/5.22 thf(fact_6143_sorted__list__of__set_Ofolding__insort__key__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((((folding_insort_key @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)) @ (top_top @ (set @ A))) @ (^ [X2:A]: (X2))))))).
% 0.33/5.22 thf(fact_6144_natLeq__def, axiom, (bNF_Ca8665028551170535155natLeq = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less_eq @ nat))))).
% 0.33/5.22 thf(fact_6145_Restr__natLeq2, axiom, (! [N:nat]: (((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ bNF_Ca8665028551170535155natLeq) @ ((((product_Sigma @ nat) @ nat) @ (((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N)) @ (^ [Uu:nat]: ((((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N))))) = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))))).
% 0.33/5.22 thf(fact_6146_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: (((zero @ B) => (! [F:((A > B) > C),G:C]: (((F = (^ [X2:(A > B)]: (G))) => ((F @ (^ [X2:A]: ((zero_zero @ B)))) = G)))))))).
% 0.33/5.22 thf(fact_6147_natLeq__underS__less, axiom, (! [N:nat]: (((((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N) = ((collect @ nat) @ (^ [X2:nat]: ((((ord_less @ nat) @ X2) @ N)))))))).
% 0.33/5.22 thf(fact_6148_add_Ogroup__axioms, axiom, (! [A:$tType]: (((group_add @ A) => ((((group @ A) @ (plus_plus @ A)) @ (zero_zero @ A)) @ (uminus_uminus @ A)))))).
% 0.33/5.22 thf(fact_6149_inv__o__cancel, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((comp @ B) @ A) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ F) = (id @ A)))))).
% 0.33/5.22 thf(fact_6150_inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F) @ A6) => ((((member @ A) @ X) @ A6) => ((((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ (F @ X)) = X)))))).
% 0.33/5.22 thf(fact_6151_inv__identity, axiom, (! [A:$tType]: ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (^ [A5:A]: (A5))) = (^ [A5:A]: (A5)))))).
% 0.33/5.22 thf(fact_6152_inv__id, axiom, (! [A:$tType]: ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (id @ A)) = (id @ A))))).
% 0.33/5.22 thf(fact_6153_inv__into__image__cancel, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),S3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A6) => ((((ord_less_eq @ (set @ A)) @ S3) @ A6) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A6) @ F)) @ ((((image @ A) @ B) @ F) @ S3)) = S3)))))).
% 0.33/5.22 thf(fact_6154_o__inv__o__cancel, axiom, (! [B:$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))))).
% 0.33/5.22 thf(fact_6155_inv__into__def, axiom, (! [B:$tType,A:$tType]: ((((hilbert_inv_into @ A) @ B) = (^ [A7:(set @ A),F2:(A > B),X2:B]: (((fChoice @ A) @ (^ [Y6:A]: (((((member @ A) @ Y6) @ A7) & ((F2 @ Y6) = X2))))))))))).
% 0.33/5.22 thf(fact_6156_inv__into__def2, axiom, (! [B:$tType,A:$tType]: ((((hilbert_inv_into @ A) @ B) = (^ [A7:(set @ A),F2:(A > B),X2:B]: (((fChoice @ A) @ (^ [Y6:A]: (((((member @ A) @ Y6) @ A7) & ((F2 @ Y6) = X2))))))))))).
% 0.33/5.22 thf(fact_6157_group_Oleft__cancel, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A,B2:A,C2:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((((F @ A3) @ B2) = ((F @ A3) @ C2)) = (B2 = C2)))))).
% 0.33/5.22 thf(fact_6158_group_Oleft__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => (((F @ (Inverse @ A3)) @ A3) = Z3))))).
% 0.33/5.22 thf(fact_6159_group_Oright__cancel, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),B2:A,A3:A,C2:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((((F @ B2) @ A3) = ((F @ C2) @ A3)) = (B2 = C2)))))).
% 0.33/5.22 thf(fact_6160_group_Oright__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => (((F @ A3) @ (Inverse @ A3)) = Z3))))).
% 0.33/5.22 thf(fact_6161_group_Oinverse__unique, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A,B2:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((((F @ A3) @ B2) = Z3) => ((Inverse @ A3) = B2)))))).
% 0.33/5.22 thf(fact_6162_group_Oinverse__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((Inverse @ (Inverse @ A3)) = A3))))).
% 0.33/5.22 thf(fact_6163_group_Oinverse__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((Inverse @ Z3) = Z3))))).
% 0.33/5.22 thf(fact_6164_group_Ogroup__left__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => (((F @ Z3) @ A3) = A3))))).
% 0.33/5.22 thf(fact_6165_group_Oinverse__distrib__swap, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A),A3:A,B2:A]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((Inverse @ ((F @ A3) @ B2)) = ((F @ (Inverse @ B2)) @ (Inverse @ A3))))))).
% 0.33/5.22 thf(fact_6166_inv__into__injective, axiom, (! [A:$tType,B:$tType,A6:(set @ A),F:(A > B),X:B,Y3:B]: ((((((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ X) = (((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ Y3)) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F) @ A6)) => ((((member @ B) @ Y3) @ ((((image @ A) @ B) @ F) @ A6)) => (X = Y3))))))).
% 0.33/5.22 thf(fact_6167_inv__into__into, axiom, (! [A:$tType,B:$tType,X:A,F:(B > A),A6:(set @ B)]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A6)) => (((member @ B) @ (((((hilbert_inv_into @ B) @ A) @ A6) @ F) @ X)) @ A6))))).
% 0.33/5.22 thf(fact_6168_f__inv__into__f, axiom, (! [B:$tType,A:$tType,Y3:A,F:(B > A),A6:(set @ B)]: (((((member @ A) @ Y3) @ ((((image @ B) @ A) @ F) @ A6)) => ((F @ (((((hilbert_inv_into @ B) @ A) @ A6) @ F) @ Y3)) = Y3))))).
% 0.33/5.22 thf(fact_6169_surj__f__inv__f, axiom, (! [B:$tType,A:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((F @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F) @ Y3)) = Y3))))).
% 0.33/5.22 thf(fact_6170_surj__iff__all, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [X2:A]: (((F @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F) @ X2)) = X2))))))).
% 0.33/5.22 thf(fact_6171_image__f__inv__f, axiom, (! [B:$tType,A:$tType,F:(B > A),A6:(set @ A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F) @ ((((image @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F)) @ A6)) = A6))))).
% 0.33/5.22 thf(fact_6172_surj__imp__inv__eq, axiom, (! [B:$tType,A:$tType,F:(B > A),G:(A > B)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((! [X3:B]: (((G @ (F @ X3)) = X3))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F) = G)))))).
% 0.33/5.22 thf(fact_6173_inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),X:A,Y3:B]: ((((((inj_on @ A) @ B) @ F) @ A6) => ((((member @ A) @ X) @ A6) => (((F @ X) = Y3) => ((((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ Y3) = X))))))).
% 0.33/5.22 thf(fact_6174_inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ (F @ X)) = X))))).
% 0.33/5.22 thf(fact_6175_inv__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y3:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = Y3) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ Y3) = X)))))).
% 0.33/5.22 thf(fact_6176_inj__imp__inv__eq, axiom, (! [A:$tType,B:$tType,F:(A > B),G:(B > A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((! [X3:B]: (((F @ (G @ X3)) = X3))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) = G)))))).
% 0.33/5.22 thf(fact_6177_inv__equality, axiom, (! [A:$tType,B:$tType,G:(B > A),F:(A > B)]: (((! [X3:A]: (((G @ (F @ X3)) = X3))) => ((! [Y5:B]: (((F @ (G @ Y5)) = Y5))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) = G)))))).
% 0.33/5.22 thf(fact_6178_image__inv__into__cancel, axiom, (! [B:$tType,A:$tType,F:(B > A),A6:(set @ B),A11:(set @ A),B13:(set @ A)]: (((((((image @ B) @ A) @ F) @ A6) = A11) => ((((ord_less_eq @ (set @ A)) @ B13) @ A11) => (((((image @ B) @ A) @ F) @ ((((image @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ A6) @ F)) @ B13)) = B13)))))).
% 0.33/5.22 thf(fact_6179_inv__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F) = (^ [Y6:A]: (((fChoice @ B) @ (^ [X2:B]: (((F @ X2) = Y6)))))))))).
% 0.33/5.22 thf(fact_6180_inj__on__inv__into, axiom, (! [B:$tType,A:$tType,B7:(set @ A),F:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F) @ A6)) => ((((inj_on @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ A6) @ F)) @ B7))))).
% 0.33/5.22 thf(fact_6181_inj__transfer, axiom, (! [B:$tType,A:$tType,F:(A > B),P:(A > $o),X:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((! [Y5:B]: (((((member @ B) @ Y5) @ ((((image @ A) @ B) @ F) @ (top_top @ (set @ A)))) => (P @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ Y5))))) => (P @ X)))))).
% 0.33/5.22 thf(fact_6182_image__inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ ((((image @ A) @ B) @ F) @ A6)) = A6))))).
% 0.33/5.22 thf(fact_6183_inj__imp__surj__inv, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6184_surj__imp__inj__inv, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((inj_on @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F)) @ (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6185_inv__into__comp, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),G:(C > A),A6:(set @ C),X:B]: ((((((inj_on @ A) @ B) @ F) @ ((((image @ C) @ A) @ G) @ A6)) => (((((inj_on @ C) @ A) @ G) @ A6) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F) @ ((((image @ C) @ A) @ G) @ A6))) => ((((((hilbert_inv_into @ C) @ B) @ A6) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ X) = ((((((comp @ A) @ C) @ B) @ ((((hilbert_inv_into @ C) @ A) @ A6) @ G)) @ ((((hilbert_inv_into @ A) @ B) @ ((((image @ C) @ A) @ G) @ A6)) @ F)) @ X)))))))).
% 0.33/5.22 thf(fact_6186_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)))))).
% 0.33/5.22 thf(fact_6187_surj__iff, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ 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)))))).
% 0.33/5.22 thf(fact_6188_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)))))).
% 0.33/5.22 thf(fact_6189_bijection_Oinv__comp__left, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => ((((((comp @ A) @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) @ F) = (id @ A)))))).
% 0.33/5.22 thf(fact_6190_bijection_Oinv__comp__right, axiom, (! [A:$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)))))).
% 0.33/5.22 thf(fact_6191_bijection_Oeq__iff, axiom, (! [A:$tType,F:(A > A),A3:A,B2:A]: ((((hilbert_bijection @ A) @ F) => (((F @ A3) = (F @ B2)) = (A3 = B2)))))).
% 0.33/5.22 thf(fact_6192_bijection_OeqI, axiom, (! [A:$tType,F:(A > A),A3:A,B2:A]: ((((hilbert_bijection @ A) @ F) => (((F @ A3) = (F @ B2)) => (A3 = B2)))))).
% 0.33/5.22 thf(fact_6193_bijection_Osurj, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => (((((image @ A) @ A) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6194_bijection_Oinj, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => ((((inj_on @ A) @ A) @ F) @ (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6195_bijection_Oinv__right__eq__iff, axiom, (! [A:$tType,F:(A > A),B2:A,A3:A]: ((((hilbert_bijection @ A) @ F) => ((B2 = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ A3)) = ((F @ B2) = A3)))))).
% 0.33/5.22 thf(fact_6196_bijection_Oinv__left__eq__iff, axiom, (! [A:$tType,F:(A > A),A3:A,B2:A]: ((((hilbert_bijection @ A) @ F) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ A3) = B2) = ((F @ B2) = A3)))))).
% 0.33/5.22 thf(fact_6197_bijection_Oeq__inv__iff, axiom, (! [A:$tType,F:(A > A),A3:A,B2:A]: ((((hilbert_bijection @ A) @ F) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ A3) = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ B2)) = (A3 = B2)))))).
% 0.33/5.22 thf(fact_6198_bijection_Oinv__right, axiom, (! [A:$tType,F:(A > A),A3:A]: ((((hilbert_bijection @ A) @ F) => ((F @ (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ A3)) = A3))))).
% 0.33/5.22 thf(fact_6199_bijection_Oinv__left, axiom, (! [A:$tType,F:(A > A),A3:A]: ((((hilbert_bijection @ A) @ F) => ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ (F @ A3)) = A3))))).
% 0.33/5.22 thf(fact_6200_bijection_Oeq__invI, axiom, (! [A:$tType,F:(A > A),A3:A,B2:A]: ((((hilbert_bijection @ A) @ F) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ A3) = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F) @ B2)) => (A3 = B2)))))).
% 0.33/5.22 thf(fact_6201_bijection_Osurj__inv, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => (((((image @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6202_bijection_Oinj__inv, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => ((((inj_on @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) @ (top_top @ (set @ A))))))).
% 0.33/5.22 thf(fact_6203_iteratesp_Omono, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A)]: ((((order_mono @ (A > $o)) @ (A > $o)) @ (^ [P6:(A > $o),X2:A]: (((? [Y6:A]: (((X2 = (F @ Y6)) & (P6 @ Y6)))) | (? [M9:(set @ A)]: (((X2 = ((complete_Sup_Sup @ A) @ M9)) & ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M9) & (! [Y6:A]: (((((member @ A) @ Y6) @ M9) => (P6 @ Y6)))))))))))))))))).
% 0.33/5.22 thf(fact_6204_finite__subset__wf, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((wf @ (set @ A)) @ ((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [X7:(set @ A),Y9:(set @ A)]: (((((ord_less @ (set @ A)) @ X7) @ Y9) & (((ord_less_eq @ (set @ A)) @ Y9) @ A6))))))))))).
% 0.33/5.22 thf(fact_6205_wf, axiom, (! [A:$tType]: (((wellorder @ A) => ((wf @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (ord_less @ A)))))))).
% 0.33/5.22 thf(fact_6206_ccpo__Sup__least, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A6:(set @ A),Z3:A]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A6) => ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((ord_less_eq @ A) @ X3) @ Z3)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ Z3))))))))).
% 0.33/5.22 thf(fact_6207_ccpo__Sup__upper, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A6:(set @ A),X:A]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A6) => ((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_6208_wf__iff__no__infinite__down__chain, axiom, (! [A:$tType]: (((wf @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((~ ((? [F2:(nat > A)]: ((! [I5:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F2 @ (suc @ I5))) @ (F2 @ I5))) @ R5))))))))))))).
% 0.33/5.22 thf(fact_6209_wf__no__infinite__down__chainE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(nat > A)]: ((((wf @ A) @ R) => (~ ((! [K2:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F @ (suc @ K2))) @ (F @ K2))) @ R))))))))).
% 0.33/5.22 thf(fact_6210_chain__singleton, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [X:A]: ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))).
% 0.33/5.22 thf(fact_6211_wf__if__measure, axiom, (! [A:$tType,P:(A > $o),F:(A > nat),G:(A > A)]: (((! [X3:A]: (((P @ X3) => (((ord_less @ nat) @ (F @ (G @ X3))) @ (F @ X3))))) => ((wf @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [Y6:A,X2:A]: (((P @ X2) & (Y6 = (G @ X2)))))))))))).
% 0.33/5.22 thf(fact_6212_wf__bounded__measure, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Ub:(A > nat),F:(A > nat)]: (((! [A8:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A8)) @ R) => ((((ord_less_eq @ nat) @ (Ub @ B5)) @ (Ub @ A8)) & ((((ord_less_eq @ nat) @ (F @ B5)) @ (Ub @ A8)) & (((ord_less @ nat) @ (F @ A8)) @ (F @ B5))))))) => ((wf @ A) @ R))))).
% 0.33/5.22 thf(fact_6213_wf__linord__ex__has__least, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(B > $o),K:B,M:(B > A)]: ((((wf @ A) @ R) => ((! [X3:A,Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y5)) @ ((transitive_trancl @ A) @ R)) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X3)) @ ((transitive_rtrancl @ A) @ R))))))) => ((P @ K) => (? [X3:B]: (((P @ X3) & (! [Y4:B]: (((P @ Y4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (M @ X3)) @ (M @ Y4))) @ ((transitive_rtrancl @ A) @ R)))))))))))))).
% 0.33/5.22 thf(fact_6214_reduction__pair__lemma, axiom, (! [A:$tType,P:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),R2:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: ((((fun_reduction_pair @ A) @ P) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ (((product_fst @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P)) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S3) @ (((product_snd @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P)) => (((wf @ A) @ S3) => ((wf @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R2) @ S3))))))))).
% 0.33/5.22 thf(fact_6215_wf__bounded__set, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),Ub:(A > (set @ B)),F:(A > (set @ B))]: (((! [A8:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A8)) @ R) => (((finite_finite2 @ B) @ (Ub @ A8)) & ((((ord_less_eq @ (set @ B)) @ (Ub @ B5)) @ (Ub @ A8)) & ((((ord_less_eq @ (set @ B)) @ (F @ B5)) @ (Ub @ A8)) & (((ord_less @ (set @ B)) @ (F @ A8)) @ (F @ B5)))))))) => ((wf @ A) @ R))))).
% 0.33/5.22 thf(fact_6216_in__chain__finite, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A6:(set @ A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A6) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => (((member @ A) @ ((complete_Sup_Sup @ A) @ A6)) @ A6)))))))))).
% 0.33/5.22 thf(fact_6217_iteratesp__def, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => ((comple7512665784863727008ratesp @ A) = (^ [F2:(A > A)]: (((complete_lattice_lfp @ (A > $o)) @ (^ [P6:(A > $o),X2:A]: (((? [Y6:A]: (((X2 = (F2 @ Y6)) & (P6 @ Y6)))) | (? [M9:(set @ A)]: (((X2 = ((complete_Sup_Sup @ A) @ M9)) & ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M9) & (! [Y6:A]: (((((member @ A) @ Y6) @ M9) => (P6 @ Y6))))))))))))))))))).
% 0.33/5.22 thf(fact_6218_wf__int__ge__less__than2, axiom, (! [D2:int]: (((wf @ int) @ (int_ge_less_than2 @ D2))))).
% 0.33/5.22 thf(fact_6219_wf__int__ge__less__than, axiom, (! [D2:int]: (((wf @ int) @ (int_ge_less_than @ D2))))).
% 0.33/5.22 thf(fact_6220_wf__pair__less, axiom, ((wf @ ((product_prod @ nat) @ nat)) @ fun_pair_less)).
% 0.33/5.22 thf(fact_6221_wf__less, axiom, ((wf @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less @ nat))))).
% 0.33/5.22 thf(fact_6222_iteratesp_Osimps, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => ((comple7512665784863727008ratesp @ A) = (^ [F2:(A > A),A5:A]: (((? [X2:A]: (((A5 = (F2 @ X2)) & (((comple7512665784863727008ratesp @ A) @ F2) @ X2)))) | (? [M9:(set @ A)]: (((A5 = ((complete_Sup_Sup @ A) @ M9)) & ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M9) & (! [X2:A]: (((((member @ A) @ X2) @ M9) => (((comple7512665784863727008ratesp @ A) @ F2) @ X2)))))))))))))))).
% 0.33/5.22 thf(fact_6223_iteratesp_Ocases, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A),A3:A]: (((((comple7512665784863727008ratesp @ A) @ F) @ A3) => ((! [X3:A]: (((A3 = (F @ X3)) => (~ ((((comple7512665784863727008ratesp @ A) @ F) @ X3)))))) => (~ ((! [M8:(set @ A)]: (((A3 = ((complete_Sup_Sup @ A) @ M8)) => ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M8) => (~ ((! [X4:A]: (((((member @ A) @ X4) @ M8) => (((comple7512665784863727008ratesp @ A) @ F) @ X4)))))))))))))))))))).
% 0.33/5.22 thf(fact_6224_iteratesp_OSup, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [M7:(set @ A),F:(A > A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M7) => ((! [X3:A]: (((((member @ A) @ X3) @ M7) => (((comple7512665784863727008ratesp @ A) @ F) @ X3)))) => (((comple7512665784863727008ratesp @ A) @ F) @ ((complete_Sup_Sup @ A) @ M7)))))))))).
% 0.33/5.22 thf(fact_6225_reduction__pair__def, axiom, (! [A:$tType]: (((fun_reduction_pair @ A) = (^ [P4:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))]: ((((wf @ A) @ (((product_fst @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P4)) & (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ (((product_fst @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P4)) @ (((product_snd @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P4))) @ (((product_fst @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P4))))))))).
% 0.33/5.22 thf(fact_6226_admissible__chfin, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [P:(A > $o)]: (((! [S4:(set @ A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ S4) => ((finite_finite2 @ A) @ S4)))) => ((((comple1908693960933563346ssible @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ P)))))))).
% 0.33/5.22 thf(fact_6227_reduction__pairI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R2) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R2) @ S3)) @ R2) => ((fun_reduction_pair @ A) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R2) @ S3))))))).
% 0.33/5.22 thf(fact_6228_wf__no__loop, axiom, (! [B:$tType,R2:(set @ ((product_prod @ B) @ B))]: ((((((((relcomp @ B) @ B) @ B) @ R2) @ R2) = (bot_bot @ (set @ ((product_prod @ B) @ B)))) => ((wf @ B) @ R2))))).
% 0.33/5.22 thf(fact_6229_union__comp__emptyL, axiom, (! [A:$tType,A6:(set @ ((product_prod @ A) @ A)),C6:(set @ ((product_prod @ A) @ A)),B7:(set @ ((product_prod @ A) @ A))]: ((((((((relcomp @ A) @ A) @ A) @ A6) @ C6) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((((((relcomp @ A) @ A) @ A) @ B7) @ C6) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => ((((((relcomp @ A) @ A) @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A6) @ B7)) @ C6) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))))).
% 0.33/5.22 thf(fact_6230_union__comp__emptyR, axiom, (! [A:$tType,A6:(set @ ((product_prod @ A) @ A)),B7:(set @ ((product_prod @ A) @ A)),C6:(set @ ((product_prod @ A) @ A))]: ((((((((relcomp @ A) @ A) @ A) @ A6) @ B7) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((((((relcomp @ A) @ A) @ A) @ A6) @ C6) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => ((((((relcomp @ A) @ A) @ A) @ A6) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ B7) @ C6)) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))))).
% 0.33/5.22 thf(fact_6231_relpow_Osimps_I2_J, axiom, (! [A:$tType,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R2) = (((((relcomp @ A) @ A) @ A) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R2)) @ R2))))).
% 0.33/5.22 thf(fact_6232_admissible__disj, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [P:(A > $o),Q:(A > $o)]: ((((((comple1908693960933563346ssible @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ P) => (((((comple1908693960933563346ssible @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ Q) => ((((comple1908693960933563346ssible @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ (^ [X2:A]: (((P @ X2) | (Q @ X2))))))))))))).
% 0.33/5.22 thf(fact_6233_max__ext__compat, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R2) @ S3)) @ R2) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((max_ext @ A) @ R2)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((max_ext @ A) @ S3)) @ (((insert @ ((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) @ R2)))))).
% 0.33/5.22 thf(fact_6234_min__ext__compat, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R2) @ S3)) @ R2) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((min_ext @ A) @ R2)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((min_ext @ A) @ S3)) @ (((insert @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ A)))) @ (bot_bot @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))))))) @ ((min_ext @ A) @ R2)))))).
% 0.33/5.22 thf(fact_6235_ntrancl__Suc, axiom, (! [A:$tType,N:nat,R2:(set @ ((product_prod @ A) @ A))]: (((((transitive_ntrancl @ A) @ (suc @ N)) @ R2) = (((((relcomp @ A) @ A) @ A) @ (((transitive_ntrancl @ A) @ N) @ R2)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ (id2 @ A)) @ R2)))))).
% 0.33/5.22 thf(fact_6236_and_Omonoid__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((monoid @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_6237_monoid_Oright__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:A]: (((((monoid @ A) @ F) @ Z3) => (((F @ A3) @ Z3) = A3))))).
% 0.33/5.22 thf(fact_6238_monoid_Oleft__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:A]: (((((monoid @ A) @ F) @ Z3) => (((F @ Z3) @ A3) = A3))))).
% 0.33/5.22 thf(fact_6239_relpow_Osimps_I1_J, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R2) = (id2 @ A))))).
% 0.33/5.22 thf(fact_6240_add_Omonoid__axioms, axiom, (! [A:$tType]: (((monoid_add @ A) => (((monoid @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6241_mult_Omonoid__axioms, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((monoid @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.22 thf(fact_6242_sup__bot_Omonoid__axioms, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (((monoid @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 0.33/5.22 thf(fact_6243_inf__top_Omonoid__axioms, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (((monoid @ A) @ (inf_inf @ A)) @ (top_top @ A)))))).
% 0.33/5.22 thf(fact_6244_gcd__nat_Omonoid__axioms, axiom, (((monoid @ nat) @ (gcd_gcd @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6245_or_Omonoid__axioms, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((monoid @ A) @ (bit_se1065995026697491101ons_or @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6246_xor_Omonoid__axioms, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((monoid @ A) @ (bit_se5824344971392196577ns_xor @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6247_max__nat_Omonoid__axioms, axiom, (((monoid @ nat) @ (ord_max @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6248_pair__leq__def, axiom, (fun_pair_leq = (((sup_sup @ (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))) @ fun_pair_less) @ (id2 @ ((product_prod @ nat) @ nat))))).
% 0.33/5.22 thf(fact_6249_cr__integer__def, axiom, (code_cr_integer = (^ [X2:int,Y6:code_integer]: ((X2 = (code_int_of_integer @ Y6)))))).
% 0.33/5.22 thf(fact_6250_bit__integer__def, axiom, ((bit_se5641148757651400278ts_bit @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (nat > $o)) @ (nat > $o)) @ code_int_of_integer) @ (id @ (nat > $o))) @ (bit_se5641148757651400278ts_bit @ int)))).
% 0.33/5.22 thf(fact_6251_integer_Opcr__cr__eq, axiom, (code_pcr_integer = code_cr_integer)).
% 0.33/5.22 thf(fact_6252_arg__min__list_Osimps_I2_J, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [F:(A > B),X:A,Y3:A,Zs:(list @ A)]: ((((((arg_min_list @ A) @ B) @ F) @ (((cons @ A) @ X) @ (((cons @ A) @ Y3) @ Zs))) = ((((if @ A) @ (((ord_less_eq @ B) @ (F @ X)) @ (F @ ((((arg_min_list @ A) @ B) @ F) @ (((cons @ A) @ Y3) @ Zs))))) @ X) @ ((((arg_min_list @ A) @ B) @ F) @ (((cons @ A) @ Y3) @ Zs)))))))))).
% 0.33/5.22 thf(fact_6253_iterates_OSup, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [M7:(set @ A),F:(A > A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M7) => ((! [X3:A]: (((((member @ A) @ X3) @ M7) => (((member @ A) @ X3) @ ((comple6359979572994053840erates @ A) @ F))))) => (((member @ A) @ ((complete_Sup_Sup @ A) @ M7)) @ ((comple6359979572994053840erates @ A) @ F)))))))))).
% 0.33/5.22 thf(fact_6254_iterates_Osimps, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A3:A,F:(A > A)]: (((((member @ A) @ A3) @ ((comple6359979572994053840erates @ A) @ F)) = ((? [X2:A]: (((A3 = (F @ X2)) & (((member @ A) @ X2) @ ((comple6359979572994053840erates @ A) @ F))))) | (? [M9:(set @ A)]: (((A3 = ((complete_Sup_Sup @ A) @ M9)) & ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M9) & (! [X2:A]: (((((member @ A) @ X2) @ M9) => (((member @ A) @ X2) @ ((comple6359979572994053840erates @ A) @ F))))))))))))))))).
% 0.33/5.22 thf(fact_6255_iterates_Ocases, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A3:A,F:(A > A)]: (((((member @ A) @ A3) @ ((comple6359979572994053840erates @ A) @ F)) => ((! [X3:A]: (((A3 = (F @ X3)) => (~ ((((member @ A) @ X3) @ ((comple6359979572994053840erates @ A) @ F))))))) => (~ ((! [M8:(set @ A)]: (((A3 = ((complete_Sup_Sup @ A) @ M8)) => ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M8) => (~ ((! [X4:A]: (((((member @ A) @ X4) @ M8) => (((member @ A) @ X4) @ ((comple6359979572994053840erates @ A) @ F))))))))))))))))))))).
% 0.33/5.22 thf(fact_6256_arg__min__list_Oelims, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X:(A > B),Xa3:(list @ A),Y3:A]: (((((((arg_min_list @ A) @ B) @ X) @ Xa3) = Y3) => ((! [X3:A]: (((Xa3 = (((cons @ A) @ X3) @ (nil @ A))) => (Y3 != X3)))) => ((! [X3:A,Y5:A,Zs2:(list @ A)]: (((Xa3 = (((cons @ A) @ X3) @ (((cons @ A) @ Y5) @ Zs2))) => (Y3 != ((((if @ A) @ (((ord_less_eq @ B) @ (X @ X3)) @ (X @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y5) @ Zs2))))) @ X3) @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y5) @ Zs2))))))) => (~ (((Xa3 = (nil @ A)) => (Y3 != (undefined @ A)))))))))))))).
% 0.33/5.22 thf(fact_6257_chain__iterates, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A)]: (((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => (((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ ((comple6359979572994053840erates @ A) @ F))))))))).
% 0.33/5.22 thf(fact_6258_iterates__le__f, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [X:A,F:(A > A)]: (((((member @ A) @ X) @ ((comple6359979572994053840erates @ A) @ F)) => ((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => (((ord_less_eq @ A) @ X) @ (F @ X)))))))))).
% 0.33/5.22 thf(fact_6259_fixp__induct, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [P:(A > $o),F:(A > A)]: ((((((comple1908693960933563346ssible @ A) @ (complete_Sup_Sup @ A)) @ (ord_less_eq @ A)) @ P) => ((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => ((P @ ((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A)))) => ((! [X3:A]: (((P @ X3) => (P @ (F @ X3))))) => (P @ ((comple115746919287870866o_fixp @ A) @ F)))))))))))).
% 0.33/5.22 thf(fact_6260_iterates__fixp, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A)]: (((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => (((member @ A) @ ((comple115746919287870866o_fixp @ A) @ F)) @ ((comple6359979572994053840erates @ A) @ F))))))))).
% 0.33/5.22 thf(fact_6261_fixp__unfold, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A)]: (((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => (((comple115746919287870866o_fixp @ A) @ F) = (F @ ((comple115746919287870866o_fixp @ A) @ F)))))))))).
% 0.33/5.22 thf(fact_6262_fixp__lowerbound, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A),Z3:A]: (((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => ((((ord_less_eq @ A) @ (F @ Z3)) @ Z3) => (((ord_less_eq @ A) @ ((comple115746919287870866o_fixp @ A) @ F)) @ Z3))))))))).
% 0.33/5.22 thf(fact_6263_arg__min__list_Opelims, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X:(A > B),Xa3:(list @ A),Y3:A]: (((((((arg_min_list @ A) @ B) @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ Xa3)) => ((! [X3:A]: (((Xa3 = (((cons @ A) @ X3) @ (nil @ A))) => ((Y3 = X3) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (((cons @ A) @ X3) @ (nil @ A)))))))))) => ((! [X3:A,Y5:A,Zs2:(list @ A)]: (((Xa3 = (((cons @ A) @ X3) @ (((cons @ A) @ Y5) @ Zs2))) => ((Y3 = ((((if @ A) @ (((ord_less_eq @ B) @ (X @ X3)) @ (X @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y5) @ Zs2))))) @ X3) @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y5) @ Zs2)))) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y5) @ Zs2)))))))))) => (~ (((Xa3 = (nil @ A)) => ((Y3 = (undefined @ A)) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (nil @ A))))))))))))))))))).
% 0.33/5.22 thf(fact_6264_sorted__sort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Xs:(list @ B)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F) @ ((((linorder_sort_key @ B) @ A) @ F) @ Xs))))))))).
% 0.33/5.22 thf(fact_6265_sorted__sort, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((linorder_sort_key @ A) @ A) @ (^ [X2:A]: (X2))) @ Xs)))))))).
% 0.33/5.22 thf(fact_6266_sorted__sort__id, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((((linorder_sort_key @ A) @ A) @ (^ [X2:A]: (X2))) @ Xs) = Xs)))))))).
% 0.33/5.22 thf(fact_6267_length__removeAll__less, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((removeAll @ A) @ X) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)))))).
% 0.33/5.22 thf(fact_6268_card__Plus__conv__if, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B7:(set @ B)]: ((((((finite_finite2 @ A) @ A6) & ((finite_finite2 @ B) @ B7)) => (((finite_card @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B7)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A6)) @ ((finite_card @ B) @ B7)))) & ((~ ((((finite_finite2 @ A) @ A6) & ((finite_finite2 @ B) @ B7)))) => (((finite_card @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B7)) = (zero_zero @ nat))))))).
% 0.33/5.22 thf(fact_6269_length__removeAll__less__eq, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((removeAll @ A) @ X) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.22 thf(fact_6270_and_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((comm_monoid @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 0.33/5.22 thf(fact_6271_lcm__altdef__int, axiom, ((gcd_lcm @ int) = (^ [A5:int,B3:int]: ((((divide_divide @ int) @ (((times_times @ int) @ ((abs_abs @ int) @ A5)) @ ((abs_abs @ int) @ B3))) @ (((gcd_gcd @ int) @ A5) @ B3)))))).
% 0.33/5.22 thf(fact_6272_lcm_Obottom__left__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_6273_lcm_Obottom__right__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ A3) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.33/5.22 thf(fact_6274_lcm__neg1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((gcd_lcm @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((gcd_lcm @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6275_lcm__neg2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,B2:A]: (((((gcd_lcm @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((gcd_lcm @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6276_lcm__0__iff__int, axiom, (! [M:int,N:int]: ((((((gcd_lcm @ int) @ M) @ N) = (zero_zero @ int)) = ((M = (zero_zero @ int)) | (N = (zero_zero @ int))))))).
% 0.33/5.22 thf(fact_6277_lcm__neg__numeral__1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [N:num,A3:A]: (((((gcd_lcm @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) @ A3) = (((gcd_lcm @ A) @ ((numeral_numeral @ A) @ N)) @ A3)))))))).
% 0.33/5.22 thf(fact_6278_lcm__neg__numeral__2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A,N:num]: (((((gcd_lcm @ A) @ A3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((gcd_lcm @ A) @ A3) @ ((numeral_numeral @ A) @ N))))))))).
% 0.33/5.22 thf(fact_6279_lcm__eq__1__iff, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_6280_lcm__1__iff__int, axiom, (! [M:int,N:int]: ((((((gcd_lcm @ int) @ M) @ N) = (one_one @ int)) = (((M = (one_one @ int)) | (M = ((uminus_uminus @ int) @ (one_one @ int)))) & ((N = (one_one @ int)) | (N = ((uminus_uminus @ int) @ (one_one @ int))))))))).
% 0.33/5.22 thf(fact_6281_lcm__ge__0__int, axiom, (! [X:int,Y3:int]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((gcd_lcm @ int) @ X) @ Y3))))).
% 0.33/5.22 thf(fact_6282_semilattice__neutr_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((semilattice_neutr @ A) @ F) @ Z3) => (((comm_monoid @ A) @ F) @ Z3))))).
% 0.33/5.22 thf(fact_6283_lcm__neg1__int, axiom, (! [X:int,Y3:int]: (((((gcd_lcm @ int) @ ((uminus_uminus @ int) @ X)) @ Y3) = (((gcd_lcm @ int) @ X) @ Y3))))).
% 0.33/5.22 thf(fact_6284_lcm__neg2__int, axiom, (! [X:int,Y3:int]: (((((gcd_lcm @ int) @ X) @ ((uminus_uminus @ int) @ Y3)) = (((gcd_lcm @ int) @ X) @ Y3))))).
% 0.33/5.22 thf(fact_6285_lcm__pos__int, axiom, (! [M:int,N:int]: (((M != (zero_zero @ int)) => ((N != (zero_zero @ int)) => (((ord_less @ int) @ (zero_zero @ int)) @ (((gcd_lcm @ int) @ M) @ N))))))).
% 0.33/5.22 thf(fact_6286_comm__monoid_Ocomm__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:A]: (((((comm_monoid @ A) @ F) @ Z3) => (((F @ A3) @ Z3) = A3))))).
% 0.33/5.22 thf(fact_6287_zero__eq__lcm__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((zero_zero @ A) = (((gcd_lcm @ A) @ A3) @ B2)) = ((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_6288_lcm__eq__0__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_lcm @ A) @ A3) @ B2) = (zero_zero @ A)) = ((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_6289_add_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((comm_monoid @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6290_mult_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((comm_monoid @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.22 thf(fact_6291_lcm__mult__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 0.33/5.22 thf(fact_6292_lcm__mult__unit2, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_6293_lcm__div__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ C2) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 0.33/5.22 thf(fact_6294_lcm__div__unit2, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_6295_sup__bot_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (((comm_monoid @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 0.33/5.22 thf(fact_6296_inf__top_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (((comm_monoid @ A) @ (inf_inf @ A)) @ (top_top @ A)))))).
% 0.33/5.22 thf(fact_6297_or_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((comm_monoid @ A) @ (bit_se1065995026697491101ons_or @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6298_gcd__nat_Ocomm__monoid__axioms, axiom, (((comm_monoid @ nat) @ (gcd_gcd @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6299_xor_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((comm_monoid @ A) @ (bit_se5824344971392196577ns_xor @ A)) @ (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6300_lcm__cases__int, axiom, (! [X:int,Y3:int,P:(int > $o)]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (P @ (((gcd_lcm @ int) @ X) @ Y3)))) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ Y3) @ (zero_zero @ int)) => (P @ (((gcd_lcm @ int) @ X) @ ((uminus_uminus @ int) @ Y3))))) => (((((ord_less_eq @ int) @ X) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y3) => (P @ (((gcd_lcm @ int) @ ((uminus_uminus @ int) @ X)) @ Y3)))) => (((((ord_less_eq @ int) @ X) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ Y3) @ (zero_zero @ int)) => (P @ (((gcd_lcm @ int) @ ((uminus_uminus @ int) @ X)) @ ((uminus_uminus @ int) @ Y3))))) => (P @ (((gcd_lcm @ int) @ X) @ Y3))))))))).
% 0.33/5.22 thf(fact_6301_max__nat_Ocomm__monoid__axioms, axiom, (((comm_monoid @ nat) @ (ord_max @ nat)) @ (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6302_lcm__unique__int, axiom, (! [D2:int,A3:int,B2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ D2) & ((((dvd_dvd @ int) @ A3) @ D2) & ((((dvd_dvd @ int) @ B2) @ D2) & (! [E3:int]: ((((((dvd_dvd @ int) @ A3) @ E3) & (((dvd_dvd @ int) @ B2) @ E3)) => (((dvd_dvd @ int) @ D2) @ E3))))))) = (D2 = (((gcd_lcm @ int) @ A3) @ B2)))))).
% 0.33/5.22 thf(fact_6303_prod__gcd__lcm__int, axiom, (! [M:int,N:int]: (((((times_times @ int) @ ((abs_abs @ int) @ M)) @ ((abs_abs @ int) @ N)) = (((times_times @ int) @ (((gcd_gcd @ int) @ M) @ N)) @ (((gcd_lcm @ int) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6304_Lcm__int__set__eq__fold, axiom, (! [Xs:(list @ int)]: ((((gcd_Lcm @ int) @ ((set2 @ int) @ Xs)) = (((((fold @ int) @ int) @ (gcd_lcm @ int)) @ Xs) @ (one_one @ int)))))).
% 0.33/5.22 thf(fact_6305_Lcm__fin_Oeq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Lcm_fin @ A) = (^ [A7:(set @ A)]: (((((if @ A) @ ((finite_finite2 @ A) @ A7)) @ (((((finite_fold @ A) @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ A7)) @ (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6306_lcm__0__iff__nat, axiom, (! [M:nat,N:nat]: ((((((gcd_lcm @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N = (zero_zero @ nat))))))).
% 0.33/5.22 thf(fact_6307_lcm__1__iff__nat, axiom, (! [M:nat,N:nat]: ((((((gcd_lcm @ nat) @ M) @ N) = (suc @ (zero_zero @ nat))) = ((M = (suc @ (zero_zero @ nat))) & (N = (suc @ (zero_zero @ nat)))))))).
% 0.33/5.22 thf(fact_6308_lcm__int__int__eq, axiom, (! [M:nat,N:nat]: (((((gcd_lcm @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ int) @ (((gcd_lcm @ nat) @ M) @ N)))))).
% 0.33/5.22 thf(fact_6309_Lcm__UNIV, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Lcm @ A) @ (top_top @ (set @ A))) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6310_Lcm__empty, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Lcm @ A) @ (bot_bot @ (set @ A))) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_6311_Lcm__1__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((gcd_Lcm @ A) @ A6) = (one_one @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((dvd_dvd @ A) @ X2) @ (one_one @ A)))))))))))).
% 0.33/5.22 thf(fact_6312_Lcm__fin_Oinfinite, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (((semiring_gcd_Lcm_fin @ A) @ A6) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6313_Lcm__fin_Oempty, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((semiring_gcd_Lcm_fin @ A) @ (bot_bot @ (set @ A))) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_6314_is__unit__Lcm__fin__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((((dvd_dvd @ A) @ ((semiring_gcd_Lcm_fin @ A) @ A6)) @ (one_one @ A)) = (((semiring_gcd_Lcm_fin @ A) @ A6) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6315_lcm__nat__abs__right__eq, axiom, (! [N:nat,K:int]: (((((gcd_lcm @ nat) @ N) @ (nat2 @ ((abs_abs @ int) @ K))) = (nat2 @ (((gcd_lcm @ int) @ ((semiring_1_of_nat @ int) @ N)) @ K)))))).
% 0.33/5.22 thf(fact_6316_lcm__nat__abs__left__eq, axiom, (! [K:int,N:nat]: (((((gcd_lcm @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N) = (nat2 @ (((gcd_lcm @ int) @ K) @ ((semiring_1_of_nat @ int) @ N))))))).
% 0.33/5.22 thf(fact_6317_prod__gcd__lcm__nat, axiom, ((times_times @ nat) = (^ [M2:nat,N3:nat]: ((((times_times @ nat) @ (((gcd_gcd @ nat) @ M2) @ N3)) @ (((gcd_lcm @ nat) @ M2) @ N3)))))).
% 0.33/5.22 thf(fact_6318_lcm__pos__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((gcd_lcm @ nat) @ M) @ N))))))).
% 0.33/5.22 thf(fact_6319_Lcm__no__multiple, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((! [M6:A]: (((M6 != (zero_zero @ A)) => (? [X4:A]: (((((member @ A) @ X4) @ A6) & (~ ((((dvd_dvd @ A) @ X4) @ M6))))))))) => (((gcd_Lcm @ A) @ A6) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6320_Lcm__0__iff_H, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((gcd_Lcm @ A) @ A6) = (zero_zero @ A)) = (~ ((? [L3:A]: (((L3 != (zero_zero @ A)) & (! [X2:A]: (((((member @ A) @ X2) @ A6) => (((dvd_dvd @ A) @ X2) @ L3)))))))))))))))).
% 0.33/5.22 thf(fact_6321_Lcm__eq__0__I, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((member @ A) @ (zero_zero @ A)) @ A6) => (((gcd_Lcm @ A) @ A6) = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6322_Lcm__0__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((((gcd_Lcm @ A) @ A6) = (zero_zero @ A)) = (((member @ A) @ (zero_zero @ A)) @ A6))))))))).
% 0.33/5.22 thf(fact_6323_Lcm__int__greater__eq__0, axiom, (! [K5:(set @ int)]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((gcd_Lcm @ int) @ K5))))).
% 0.33/5.22 thf(fact_6324_Lcm__fin__0__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: ((((finite_finite2 @ A) @ A6) => ((((semiring_gcd_Lcm_fin @ A) @ A6) = (zero_zero @ A)) = (((member @ A) @ (zero_zero @ A)) @ A6))))))))).
% 0.33/5.22 thf(fact_6325_lcm__nat__def, axiom, ((gcd_lcm @ nat) = (^ [X2:nat,Y6:nat]: ((((divide_divide @ nat) @ (((times_times @ nat) @ X2) @ Y6)) @ (((gcd_gcd @ nat) @ X2) @ Y6)))))).
% 0.33/5.22 thf(fact_6326_Lcm__fin__1__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: (((((semiring_gcd_Lcm_fin @ A) @ A6) = (one_one @ A)) = ((! [X2:A]: (((((member @ A) @ X2) @ A6) => (((dvd_dvd @ A) @ X2) @ (one_one @ A))))) & ((finite_finite2 @ A) @ A6))))))))).
% 0.33/5.22 thf(fact_6327_Lcm__no__units, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => ((gcd_Lcm @ A) = (^ [A7:(set @ A)]: (((gcd_Lcm @ A) @ (((minus_minus @ (set @ A)) @ A7) @ ((collect @ A) @ (^ [A5:A]: ((((dvd_dvd @ A) @ A5) @ (one_one @ A)))))))))))))).
% 0.33/5.22 thf(fact_6328_lcm__int__def, axiom, ((gcd_lcm @ int) = (^ [X2:int,Y6:int]: (((semiring_1_of_nat @ int) @ (((gcd_lcm @ nat) @ (nat2 @ ((abs_abs @ int) @ X2))) @ (nat2 @ ((abs_abs @ int) @ Y6)))))))).
% 0.33/5.22 thf(fact_6329_Lcm__set__eq__fold, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [Xs:(list @ A)]: ((((gcd_Lcm @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_lcm @ A)) @ Xs) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6330_Lcm__fin_Oset__eq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [Xs:(list @ A)]: ((((semiring_gcd_Lcm_fin @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_lcm @ A)) @ Xs) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6331_Lcm__fin__def, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Lcm_fin @ A) = ((((bounde2362111253966948842tice_F @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A))))))).
% 0.33/5.22 thf(fact_6332_Lcm__int__def, axiom, ((gcd_Lcm @ int) = (^ [K6:(set @ int)]: (((semiring_1_of_nat @ int) @ ((gcd_Lcm @ nat) @ ((((image @ int) @ nat) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int))) @ K6))))))).
% 0.33/5.22 thf(fact_6333_Lcm__int__eq, axiom, (! [N6:(set @ nat)]: ((((gcd_Lcm @ int) @ ((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ N6)) = ((semiring_1_of_nat @ int) @ ((gcd_Lcm @ nat) @ N6)))))).
% 0.33/5.22 thf(fact_6334_Lcm__eq__0__I__nat, axiom, (! [A6:(set @ nat)]: (((((member @ nat) @ (zero_zero @ nat)) @ A6) => (((gcd_Lcm @ nat) @ A6) = (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6335_Lcm__0__iff__nat, axiom, (! [A6:(set @ nat)]: ((((finite_finite2 @ nat) @ A6) => ((((gcd_Lcm @ nat) @ A6) = (zero_zero @ nat)) = (((member @ nat) @ (zero_zero @ nat)) @ A6)))))).
% 0.33/5.22 thf(fact_6336_Lcm__nat__infinite, axiom, (! [M7:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ M7))) => (((gcd_Lcm @ nat) @ M7) = (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6337_Lcm__nat__empty, axiom, (((gcd_Lcm @ nat) @ (bot_bot @ (set @ nat))) = (one_one @ nat))).
% 0.33/5.22 thf(fact_6338_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)))))).
% 0.33/5.22 thf(fact_6339_Lcm__eq__Max__nat, axiom, (! [M7:(set @ nat)]: ((((finite_finite2 @ nat) @ M7) => ((M7 != (bot_bot @ (set @ nat))) => ((~ ((((member @ nat) @ (zero_zero @ nat)) @ M7))) => ((! [M6:nat,N4:nat]: (((((member @ nat) @ M6) @ M7) => ((((member @ nat) @ N4) @ M7) => (((member @ nat) @ (((gcd_lcm @ nat) @ M6) @ N4)) @ M7))))) => (((gcd_Lcm @ nat) @ M7) = ((lattic643756798349783984er_Max @ nat) @ M7))))))))).
% 0.33/5.22 thf(fact_6340_Lcm__nat__def, axiom, ((gcd_Lcm @ nat) = (^ [M9:(set @ nat)]: (((((if @ nat) @ ((finite_finite2 @ nat) @ M9)) @ ((((lattic5214292709420241887eutr_F @ nat) @ (gcd_lcm @ nat)) @ (one_one @ nat)) @ M9)) @ (zero_zero @ nat)))))).
% 0.33/5.22 thf(fact_6341_Lcm__coprime_H, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: (((((finite_card @ A) @ A6) != (zero_zero @ nat)) => ((! [A8:A,B5:A]: (((((member @ A) @ A8) @ A6) => ((((member @ A) @ B5) @ A6) => ((A8 != B5) => (((algebr8660921524188924756oprime @ A) @ A8) @ B5)))))) => (((gcd_Lcm @ A) @ A6) = ((normal6383669964737779283malize @ A) @ ((((groups7121269368397514597t_prod @ A) @ A) @ (^ [X2:A]: (X2))) @ A6))))))))))).
% 0.33/5.22 thf(fact_6342_normalize__idem, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: ((((normal6383669964737779283malize @ A) @ ((normal6383669964737779283malize @ A) @ A3)) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6343_lcm_Onormalize__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((normal6383669964737779283malize @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6344_normalize__eq__0__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6345_normalize__0, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (((normal6383669964737779283malize @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6346_normalize__mult__normalize__right, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: ((((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ ((normal6383669964737779283malize @ A) @ B2))) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_6347_normalize__mult__normalize__left, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: ((((normal6383669964737779283malize @ A) @ (((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ B2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_6348_gcd_Onormalize__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((normal6383669964737779283malize @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_6349_normalize__1, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (((normal6383669964737779283malize @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_6350_dvd__normalize__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ ((normal6383669964737779283malize @ A) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6351_normalize__dvd__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ B2) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6352_coprime__normalize__left__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ B2) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6353_coprime__normalize__right__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ ((normal6383669964737779283malize @ A) @ B2)) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6354_gcd_Otop__left__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ (zero_zero @ A)) @ A3) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6355_gcd_Otop__right__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ A3) @ (zero_zero @ A)) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6356_lcm_Otop__right__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ A3) @ (one_one @ A)) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6357_lcm_Otop__left__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ (one_one @ A)) @ A3) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6358_normalize__mult__unit__right, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)) = ((normal6383669964737779283malize @ A) @ A3))))))))).
% 0.33/5.22 thf(fact_6359_normalize__mult__unit__left, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)) = ((normal6383669964737779283malize @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_6360_lcm__mult__gcd, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) @ (((gcd_gcd @ A) @ A3) @ B2)) = (((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_6361_gcd__mult__lcm, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (((gcd_lcm @ A) @ A3) @ B2)) = (((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_6362_lcm__mult__distrib_H, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [C2:A,A3:A,B2:A]: (((((times_times @ A) @ ((normal6383669964737779283malize @ A) @ C2)) @ (((gcd_lcm @ A) @ A3) @ B2)) = (((gcd_lcm @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2))))))))).
% 0.33/5.22 thf(fact_6363_lcm__mult__right, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((gcd_lcm @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ (((gcd_lcm @ A) @ B2) @ A3)) @ C2))))))))).
% 0.33/5.22 thf(fact_6364_lcm__mult__left, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [C2:A,A3:A,B2:A]: (((((gcd_lcm @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ C2) @ (((gcd_lcm @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_6365_normalize__nat__def, axiom, ((normal6383669964737779283malize @ nat) = (id @ nat))).
% 0.33/5.22 thf(fact_6366_gcd__mult__distrib_H, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [C2:A,A3:A,B2:A]: (((((times_times @ A) @ ((normal6383669964737779283malize @ A) @ C2)) @ (((gcd_gcd @ A) @ A3) @ B2)) = (((gcd_gcd @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2))))))))).
% 0.33/5.22 thf(fact_6367_gcd__mult__right, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,C2:A,B2:A]: (((((gcd_gcd @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ (((gcd_gcd @ A) @ B2) @ A3)) @ C2))))))))).
% 0.33/5.22 thf(fact_6368_gcd__mult__left, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [C2:A,A3:A,B2:A]: (((((gcd_gcd @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ C2) @ (((gcd_gcd @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_6369_dvd__normalize__div, axiom, (! [A:$tType]: (((normal6328177297339901930cative @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => (((normal6383669964737779283malize @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ B2)))))))))).
% 0.33/5.22 thf(fact_6370_normalize__idem__imp__is__unit__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = A3) => ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) = (A3 = (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_6371_is__unit__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ A3) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6372_normalize__1__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = (one_one @ A)) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6373_associated__unit, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_6374_normalize__int__def, axiom, ((normal6383669964737779283malize @ int) = (abs_abs @ int))).
% 0.33/5.22 thf(fact_6375_semilattice__neutr__set_OF_Ocong, axiom, (! [A:$tType]: (((lattic5214292709420241887eutr_F @ A) = (lattic5214292709420241887eutr_F @ A))))).
% 0.33/5.22 thf(fact_6376_associated__iff__dvd, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) = ((((dvd_dvd @ A) @ A3) @ B2) & (((dvd_dvd @ A) @ B2) @ A3))))))))).
% 0.33/5.22 thf(fact_6377_associated__eqI, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ B2) @ A3) => ((((normal6383669964737779283malize @ A) @ A3) = A3) => ((((normal6383669964737779283malize @ A) @ B2) = B2) => (A3 = B2))))))))))).
% 0.33/5.22 thf(fact_6378_associatedD2, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) => (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 0.33/5.22 thf(fact_6379_associatedD1, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) => (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 0.33/5.22 thf(fact_6380_associatedI, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ B2) @ A3) => (((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)))))))))).
% 0.33/5.22 thf(fact_6381_normalize__mult, axiom, (! [A:$tType]: (((normal6328177297339901930cative @ A) => (! [A3:A,B2:A]: ((((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_6382_coprime__crossproduct, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,D2:A,B2:A,C2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ D2) => ((((algebr8660921524188924756oprime @ A) @ B2) @ C2) => (((((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ C2)) = (((times_times @ A) @ ((normal6383669964737779283malize @ A) @ B2)) @ ((normal6383669964737779283malize @ A) @ D2))) = ((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) & (((normal6383669964737779283malize @ A) @ C2) = ((normal6383669964737779283malize @ A) @ D2)))))))))))).
% 0.33/5.22 thf(fact_6383_lcm__gcd__prod, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) @ (((gcd_gcd @ A) @ A3) @ B2)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2))))))))).
% 0.33/5.22 thf(fact_6384_Gcd__mult, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [C2:A,A6:(set @ A)]: ((((gcd_Gcd @ A) @ ((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ A6)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ C2) @ ((gcd_Gcd @ A) @ A6)))))))))).
% 0.33/5.22 thf(fact_6385_lcm__coprime, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((gcd_lcm @ A) @ A3) @ B2) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)))))))))).
% 0.33/5.22 thf(fact_6386_Lcm__mult, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A),C2:A]: (((A6 != (bot_bot @ (set @ A))) => (((gcd_Lcm @ A) @ ((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ A6)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ C2) @ ((gcd_Lcm @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_6387_lcm__gcd, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((gcd_lcm @ A) = (^ [A5:A,B3:A]: (((normal6383669964737779283malize @ A) @ (((divide_divide @ A) @ (((times_times @ A) @ A5) @ B3)) @ (((gcd_gcd @ A) @ A5) @ B3)))))))))).
% 0.33/5.22 thf(fact_6388_Lcm__fin__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A),B2:A]: (((A6 != (bot_bot @ (set @ A))) => (((semiring_gcd_Lcm_fin @ A) @ ((((image @ A) @ A) @ ((times_times @ A) @ B2)) @ A6)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ B2) @ ((semiring_gcd_Lcm_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_6389_Gcd__fin__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A),B2:A]: ((((finite_finite2 @ A) @ A6) => (((semiring_gcd_Gcd_fin @ A) @ ((((image @ A) @ A) @ ((times_times @ A) @ B2)) @ A6)) = ((normal6383669964737779283malize @ A) @ (((times_times @ A) @ B2) @ ((semiring_gcd_Gcd_fin @ A) @ A6))))))))))).
% 0.33/5.22 thf(fact_6390_gcd__lcm, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((gcd_gcd @ A) @ A3) @ B2) = ((normal6383669964737779283malize @ A) @ (((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((gcd_lcm @ A) @ A3) @ B2)))))))))))).
% 0.33/5.22 thf(fact_6391_semilattice__neutr__set_Oinsert__remove, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),X:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (((insert @ A) @ X) @ A6)) = ((F @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 0.33/5.22 thf(fact_6392_semilattice__neutr__set_Oremove, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),X:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6) = ((F @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))).
% 0.33/5.22 thf(fact_6393_semilattice__neutr__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((semilattice_neutr @ A) @ F) @ Z3) => (((lattic5652469242046573047tr_set @ A) @ F) @ Z3))))).
% 0.33/5.22 thf(fact_6394_semilattice__neutr__set_Oaxioms, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((semilattice_neutr @ A) @ F) @ Z3))))).
% 0.33/5.22 thf(fact_6395_semilattice__neutr__set__def, axiom, (! [A:$tType]: (((lattic5652469242046573047tr_set @ A) = (semilattice_neutr @ A))))).
% 0.33/5.22 thf(fact_6396_semilattice__neutr__set_Oempty, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (bot_bot @ (set @ A))) = Z3))))).
% 0.33/5.22 thf(fact_6397_semilattice__neutr__set_Oin__idem, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),X:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ X) @ A6) => (((F @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) = ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)))))))).
% 0.33/5.22 thf(fact_6398_semilattice__neutr__set_Oinfinite, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A)]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => ((~ (((finite_finite2 @ A) @ A6))) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6) = Z3)))))).
% 0.33/5.22 thf(fact_6399_semilattice__neutr__set_Oeq__fold, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A)]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6) = (((((finite_fold @ A) @ A) @ F) @ Z3) @ A6)))))).
% 0.33/5.22 thf(fact_6400_semilattice__neutr__set_Osubset, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),B7:(set @ A)]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => (((F @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ B7)) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) = ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)))))))).
% 0.33/5.22 thf(fact_6401_semilattice__neutr__set_Oinsert, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),X:A]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (((insert @ A) @ X) @ A6)) = ((F @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)))))))).
% 0.33/5.22 thf(fact_6402_semilattice__neutr__set_Ounion, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A),B7:(set @ A)]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => (((finite_finite2 @ A) @ B7) => (((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ (((sup_sup @ (set @ A)) @ A6) @ B7)) = ((F @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ B7))))))))).
% 0.33/5.22 thf(fact_6403_semilattice__neutr__set_Oclosed, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A6:(set @ A)]: (((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((finite_finite2 @ A) @ A6) => ((A6 != (bot_bot @ (set @ A))) => ((! [X3:A,Y5:A]: ((((member @ A) @ ((F @ X3) @ Y5)) @ (((insert @ A) @ X3) @ (((insert @ A) @ Y5) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) @ A6)))))))).
% 0.33/5.22 thf(fact_6404_Lcm__fin_Obounded__quasi__semilattice__set__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde6485984586167503788ce_set @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A)) @ (normal6383669964737779283malize @ A)))))).
% 0.33/5.22 thf(fact_6405_Gcd__fin_Obounded__quasi__semilattice__set__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde6485984586167503788ce_set @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A)) @ (normal6383669964737779283malize @ A)))))).
% 0.33/5.22 thf(fact_6406_semilattice__order__neutr__set_Osubset__imp, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),B7:(set @ A)]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => ((((ord_less_eq @ (set @ A)) @ A6) @ B7) => (((finite_finite2 @ A) @ B7) => ((Less_eq @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ B7)) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)))))))).
% 0.33/5.22 thf(fact_6407_lcm_Obounded__quasi__semilattice__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde8507323023520639062attice @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A)) @ (normal6383669964737779283malize @ A)))))).
% 0.33/5.22 thf(fact_6408_semilattice__order__neutr__set_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((lattic5652469242046573047tr_set @ A) @ F) @ Z3))))).
% 0.33/5.22 thf(fact_6409_semilattice__order__neutr__set_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less))))).
% 0.33/5.22 thf(fact_6410_semilattice__order__neutr__set_OboundedE, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => (((Less_eq @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) => (! [A10:A]: (((((member @ A) @ A10) @ A6) => ((Less_eq @ X) @ A10)))))))))).
% 0.33/5.22 thf(fact_6411_semilattice__order__neutr__set_OboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((! [A8:A]: (((((member @ A) @ A8) @ A6) => ((Less_eq @ X) @ A8)))) => ((Less_eq @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)))))))).
% 0.33/5.22 thf(fact_6412_semilattice__order__neutr__set_OcoboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),A3:A]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => ((((member @ A) @ A3) @ A6) => ((Less_eq @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) @ A3))))))).
% 0.33/5.22 thf(fact_6413_semilattice__order__neutr__set_Obounded__iff, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A6:(set @ A),X:A]: (((((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less) => (((finite_finite2 @ A) @ A6) => (((Less_eq @ X) @ ((((lattic5214292709420241887eutr_F @ A) @ F) @ Z3) @ A6)) = (! [X2:A]: (((((member @ A) @ X2) @ A6) => ((Less_eq @ X) @ X2)))))))))).
% 0.33/5.22 thf(fact_6414_semilattice__order__neutr__set__def, axiom, (! [A:$tType]: (((lattic3600114342068043075tr_set @ A) = (^ [F2:(A > (A > A)),Z7:A,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((((((semila1105856199041335345_order @ A) @ F2) @ Z7) @ Less_eq2) @ Less2) & (((lattic5652469242046573047tr_set @ A) @ F2) @ Z7)))))))).
% 0.33/5.22 thf(fact_6415_semilattice__order__neutr__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less) => ((((lattic5652469242046573047tr_set @ A) @ F) @ Z3) => (((((lattic3600114342068043075tr_set @ A) @ F) @ Z3) @ Less_eq) @ Less)))))).
% 0.33/5.22 thf(fact_6416_gcd_Obounded__quasi__semilattice__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde8507323023520639062attice @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A)) @ (normal6383669964737779283malize @ A)))))).
% 0.33/5.22 thf(fact_6417_normalize__div, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((divide_divide @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ A3) = (((divide_divide @ A) @ (one_one @ A)) @ ((unit_f5069060285200089521factor @ A) @ A3))))))))).
% 0.33/5.22 thf(fact_6418_normalize__unit__factor, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((normal6383669964737779283malize @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6419_unit__factor__simps_I1_J, axiom, (((unit_f5069060285200089521factor @ nat) @ (zero_zero @ nat)) = (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6420_unit__factor__idem, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: ((((unit_f5069060285200089521factor @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) = ((unit_f5069060285200089521factor @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6421_unit__factor__simps_I2_J, axiom, (! [N:nat]: ((((unit_f5069060285200089521factor @ nat) @ (suc @ N)) = (one_one @ nat))))).
% 0.33/5.22 thf(fact_6422_unit__factor__eq__0__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((unit_f5069060285200089521factor @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6423_unit__factor__0, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (((unit_f5069060285200089521factor @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.33/5.22 thf(fact_6424_unit__factor__1, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (((unit_f5069060285200089521factor @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 0.33/5.22 thf(fact_6425_normalize__mult__unit__factor, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((times_times @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ ((unit_f5069060285200089521factor @ A) @ A3)) = A3))))))).
% 0.33/5.22 thf(fact_6426_unit__factor__mult__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((times_times @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ ((normal6383669964737779283malize @ A) @ A3)) = A3))))))).
% 0.33/5.22 thf(fact_6427_div__unit__factor, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ ((unit_f5069060285200089521factor @ A) @ A3)) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6428_div__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ ((normal6383669964737779283malize @ A) @ A3)) = ((unit_f5069060285200089521factor @ A) @ A3)))))))).
% 0.33/5.22 thf(fact_6429_inv__unit__factor__eq__0__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ ((unit_f5069060285200089521factor @ A) @ A3)) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.33/5.22 thf(fact_6430_unit__factor__mult__unit__right, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_6431_unit__factor__mult__unit__left, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((unit_f5069060285200089521factor @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ A3) @ ((unit_f5069060285200089521factor @ A) @ B2)))))))))).
% 0.33/5.22 thf(fact_6432_mult__one__div__unit__factor, axiom, (! [A:$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))))))))).
% 0.33/5.22 thf(fact_6433_unit__factor__lcm, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) = (zero_zero @ A))) & ((~ (((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A))))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) = (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_6434_unit__factor__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((normal6383669964737779283malize @ A) @ A3)) = (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6435_normalize__unit__factor__eqI, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) => ((((unit_f5069060285200089521factor @ A) @ A3) = ((unit_f5069060285200089521factor @ A) @ B2)) => (A3 = B2))))))))).
% 0.33/5.22 thf(fact_6436_lcm__mult__distrib, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [K:A,A3:A,B2:A]: (((((times_times @ A) @ K) @ (((gcd_lcm @ A) @ A3) @ B2)) = (((times_times @ A) @ (((gcd_lcm @ A) @ (((times_times @ A) @ K) @ A3)) @ (((times_times @ A) @ K) @ B2))) @ ((unit_f5069060285200089521factor @ A) @ K))))))))).
% 0.33/5.22 thf(fact_6437_mult__lcm__right, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) @ C2) = (((times_times @ A) @ (((gcd_lcm @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))) @ ((unit_f5069060285200089521factor @ A) @ C2))))))))).
% 0.33/5.22 thf(fact_6438_mult__lcm__left, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [C2:A,A3:A,B2:A]: (((((times_times @ A) @ C2) @ (((gcd_lcm @ A) @ A3) @ B2)) = (((times_times @ A) @ ((unit_f5069060285200089521factor @ A) @ C2)) @ (((gcd_lcm @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.22 thf(fact_6439_unit__factor__nat__def, axiom, ((unit_f5069060285200089521factor @ nat) = (^ [N3:nat]: (((((if @ nat) @ (N3 = (zero_zero @ nat))) @ (zero_zero @ nat)) @ (one_one @ nat)))))).
% 0.33/5.22 thf(fact_6440_dvd__unit__factor__div, axiom, (! [A:$tType]: (((normal6328177297339901930cative @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ A3) => (((unit_f5069060285200089521factor @ A) @ (((divide_divide @ A) @ A3) @ B2)) = (((divide_divide @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ ((unit_f5069060285200089521factor @ A) @ B2)))))))))).
% 0.33/5.22 thf(fact_6441_unit__factor__self, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ A3))))))).
% 0.33/5.22 thf(fact_6442_unit__factor__int__def, axiom, ((unit_f5069060285200089521factor @ int) = (sgn_sgn @ int))).
% 0.33/5.22 thf(fact_6443_unit__factor__mult, axiom, (! [A:$tType]: (((normal6328177297339901930cative @ A) => (! [A3:A,B2:A]: ((((unit_f5069060285200089521factor @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ ((unit_f5069060285200089521factor @ A) @ B2))))))))).
% 0.33/5.22 thf(fact_6444_unit__factor__dvd, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => (((dvd_dvd @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ B2)))))))).
% 0.33/5.22 thf(fact_6445_is__unit__unit__factor, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((unit_f5069060285200089521factor @ A) @ A3) = A3)))))))).
% 0.33/5.22 thf(fact_6446_gcd__mult__distrib, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [K:A,A3:A,B2:A]: (((((times_times @ A) @ K) @ (((gcd_gcd @ A) @ A3) @ B2)) = (((times_times @ A) @ (((gcd_gcd @ A) @ (((times_times @ A) @ K) @ A3)) @ (((times_times @ A) @ K) @ B2))) @ ((unit_f5069060285200089521factor @ A) @ K))))))))).
% 0.33/5.22 thf(fact_6447_mult__gcd__right, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ C2) = (((times_times @ A) @ (((gcd_gcd @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))) @ ((unit_f5069060285200089521factor @ A) @ C2))))))))).
% 0.33/5.22 thf(fact_6448_mult__gcd__left, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [C2:A,A3:A,B2:A]: (((((times_times @ A) @ C2) @ (((gcd_gcd @ A) @ A3) @ B2)) = (((times_times @ A) @ ((unit_f5069060285200089521factor @ A) @ C2)) @ (((gcd_gcd @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 0.33/5.22 thf(fact_6449_unit__factor__is__unit, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((dvd_dvd @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ (one_one @ A))))))))).
% 0.33/5.22 thf(fact_6450_unit__factor__gcd, axiom, (! [A:$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)))))))))).
% 0.33/5.22 thf(fact_6451_coprime__crossproduct_H, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [B2:A,D2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((unit_f5069060285200089521factor @ A) @ B2) = ((unit_f5069060285200089521factor @ A) @ D2)) => ((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((algebr8660921524188924756oprime @ A) @ C2) @ D2) => (((((times_times @ A) @ A3) @ D2) = (((times_times @ A) @ B2) @ C2)) = ((A3 = C2) & (B2 = D2))))))))))))).
% 0.33/5.22 thf(fact_6452_unit__factor__Lcm, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: ((((((gcd_Lcm @ A) @ A6) = (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Lcm @ A) @ A6)) = (zero_zero @ A))) & ((((gcd_Lcm @ A) @ A6) != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Lcm @ A) @ A6)) = (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_6453_unit__factor__Gcd, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A6:(set @ A)]: ((((((gcd_Gcd @ A) @ A6) = (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Gcd @ A) @ A6)) = (zero_zero @ A))) & ((((gcd_Gcd @ A) @ A6) != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Gcd @ A) @ A6)) = (one_one @ A)))))))))).
% 0.33/5.22 thf(fact_6454_normalize__idem__imp__unit__factor__eq, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = A3) => (((unit_f5069060285200089521factor @ A) @ A3) = ((zero_neq_one_of_bool @ A) @ (A3 != (zero_zero @ A))))))))))).
% 0.33/5.22 thf(fact_6455_unit__factor__Lcm__fin, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: ((((unit_f5069060285200089521factor @ A) @ ((semiring_gcd_Lcm_fin @ A) @ A6)) = ((zero_neq_one_of_bool @ A) @ (((semiring_gcd_Lcm_fin @ A) @ A6) != (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_6456_unit__factor__Gcd__fin, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A6:(set @ A)]: ((((unit_f5069060285200089521factor @ A) @ ((semiring_gcd_Gcd_fin @ A) @ A6)) = ((zero_neq_one_of_bool @ A) @ (((semiring_gcd_Gcd_fin @ A) @ A6) != (zero_zero @ A)))))))))).
% 0.33/5.22 thf(fact_6457_times__num__def, axiom, ((times_times @ num) = (^ [M2:num,N3:num]: ((num_of_nat @ (((times_times @ nat) @ (nat_of_num @ M2)) @ (nat_of_num @ N3))))))).
% 0.33/5.22 thf(fact_6458_nat__of__num__pos, axiom, (! [X:num]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (nat_of_num @ X))))).
% 0.33/5.22 thf(fact_6459_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)))))).
% 0.33/5.22 thf(fact_6460_is__num__normalize_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A]: ((((neg_numeral_is_num @ A) @ X) => ((neg_numeral_is_num @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 0.33/5.22 thf(fact_6461_is__num__numeral, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [K:num]: (((neg_numeral_is_num @ A) @ ((numeral_numeral @ A) @ K)))))))).
% 0.33/5.22 thf(fact_6462_nat__of__num__neq__0, axiom, (! [X:num]: (((nat_of_num @ X) != (zero_zero @ nat))))).
% 0.33/5.22 thf(fact_6463_num__eq__iff, axiom, ((^ [Y:num,Z2:num]: ((Y = Z2))) = (^ [X2:num,Y6:num]: (((nat_of_num @ X2) = (nat_of_num @ Y6)))))).
% 0.33/5.22 thf(fact_6464_nat__of__num__numeral, axiom, (nat_of_num = (numeral_numeral @ nat))).
% 0.33/5.22 thf(fact_6465_nat__of__num__inverse, axiom, (! [X:num]: (((num_of_nat @ (nat_of_num @ X)) = X)))).
% 0.33/5.22 thf(fact_6466_is__num__normalize_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_is_num @ A) @ (one_one @ A)))))).
% 0.33/5.22 thf(fact_6467_is__num__add__left__commute, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y3:A,Z3:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y3) => ((((plus_plus @ A) @ X) @ (((plus_plus @ A) @ Y3) @ Z3)) = (((plus_plus @ A) @ Y3) @ (((plus_plus @ A) @ X) @ Z3))))))))))).
% 0.33/5.22 thf(fact_6468_is__num__add__commute, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y3:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y3) => ((((plus_plus @ A) @ X) @ Y3) = (((plus_plus @ A) @ Y3) @ X)))))))))).
% 0.33/5.22 thf(fact_6469_is__num__normalize_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y3:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y3) => ((neg_numeral_is_num @ A) @ (((plus_plus @ A) @ X) @ Y3)))))))))).
% 0.33/5.22 thf(fact_6470_nat__of__num__code_I1_J, axiom, ((nat_of_num @ one2) = (one_one @ nat))).
% 0.33/5.22 thf(fact_6471_nat__of__num__add, axiom, (! [X:num,Y3:num]: (((nat_of_num @ (((plus_plus @ num) @ X) @ Y3)) = (((plus_plus @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ Y3)))))).
% 0.33/5.22 thf(fact_6472_nat__of__num__mult, axiom, (! [X:num,Y3:num]: (((nat_of_num @ (((times_times @ num) @ X) @ Y3)) = (((times_times @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ Y3)))))).
% 0.33/5.22 thf(fact_6473_nat__of__num__inc, axiom, (! [X:num]: (((nat_of_num @ (inc @ X)) = (suc @ (nat_of_num @ X)))))).
% 0.33/5.22 thf(fact_6474_less__num__def, axiom, ((ord_less @ num) = (^ [M2:num,N3:num]: ((((ord_less @ nat) @ (nat_of_num @ M2)) @ (nat_of_num @ N3)))))).
% 0.33/5.22 thf(fact_6475_less__eq__num__def, axiom, ((ord_less_eq @ num) = (^ [M2:num,N3:num]: ((((ord_less_eq @ nat) @ (nat_of_num @ M2)) @ (nat_of_num @ N3)))))).
% 0.33/5.22 thf(fact_6476_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)))))).
% 0.33/5.22 thf(fact_6477_nat__of__num__sqr, axiom, (! [X:num]: (((nat_of_num @ (sqr @ X)) = (((times_times @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ X)))))).
% 0.33/5.22 thf(fact_6478_nat__of__num_Osimps_I1_J, axiom, ((nat_of_num @ one2) = (suc @ (zero_zero @ nat)))).
% 0.33/5.22 thf(fact_6479_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))))))).
% 0.33/5.22 thf(fact_6480_num__of__nat__inverse, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((nat_of_num @ (num_of_nat @ N)) = N))))).
% 0.33/5.22 thf(fact_6481_plus__num__def, axiom, ((plus_plus @ num) = (^ [M2:num,N3:num]: ((num_of_nat @ (((plus_plus @ nat) @ (nat_of_num @ M2)) @ (nat_of_num @ N3))))))).
% 0.33/5.22 thf(fact_6482_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))))))).
% 0.33/5.22 thf(fact_6483_is__num_Ocases, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A3:A]: ((((neg_numeral_is_num @ A) @ A3) => ((A3 != (one_one @ A)) => ((! [X3:A]: (((A3 = ((uminus_uminus @ A) @ X3)) => (~ (((neg_numeral_is_num @ A) @ X3)))))) => (~ ((! [X3:A,Y5:A]: (((A3 = (((plus_plus @ A) @ X3) @ Y5)) => (((neg_numeral_is_num @ A) @ X3) => (~ (((neg_numeral_is_num @ A) @ Y5)))))))))))))))))).
% 0.33/5.22 thf(fact_6484_is__num_Osimps, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_is_num @ A) = (^ [A5:A]: (((A5 = (one_one @ A)) | ((? [X2:A]: (((A5 = ((uminus_uminus @ A) @ X2)) & ((neg_numeral_is_num @ A) @ X2)))) | (? [X2:A,Y6:A]: (((A5 = (((plus_plus @ A) @ X2) @ Y6)) & (((neg_numeral_is_num @ A) @ X2) & ((neg_numeral_is_num @ A) @ Y6)))))))))))))).
% 0.33/5.22 thf(fact_6485_arg__max__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B2:nat]: (((P @ K) => ((! [Y5:A]: (((P @ Y5) => (((ord_less @ nat) @ (F @ Y5)) @ B2)))) => ((P @ ((((lattices_ord_arg_max @ A) @ nat) @ F) @ P)) & (! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ nat) @ (F @ Y4)) @ (F @ ((((lattices_ord_arg_max @ A) @ nat) @ F) @ P)))))))))))).
% 0.33/5.22 thf(fact_6486_arg__max__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,F:(A > nat),B2:nat]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (((ord_less @ nat) @ (F @ Y5)) @ B2)))) => (((ord_less_eq @ nat) @ (F @ X)) @ (F @ ((((lattices_ord_arg_max @ A) @ nat) @ F) @ P)))))))).
% 0.33/5.22 thf(fact_6487_arg__max__natI, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B2:nat]: (((P @ K) => ((! [Y5:A]: (((P @ Y5) => (((ord_less @ nat) @ (F @ Y5)) @ B2)))) => (P @ ((((lattices_ord_arg_max @ A) @ nat) @ F) @ P))))))).
% 0.33/5.22 thf(fact_6488_arg__max__equality, axiom, (! [A:$tType,C:$tType]: (((order @ A) => (! [P:(C > $o),K:C,F:(C > A)]: (((P @ K) => ((! [X3:C]: (((P @ X3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ K))))) => ((F @ ((((lattices_ord_arg_max @ C) @ A) @ F) @ P)) = (F @ K)))))))))).
% 0.33/5.22 thf(fact_6489_arg__maxI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [P:(A > $o),X:A,F:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (~ ((((ord_less @ B) @ (F @ X)) @ (F @ Y5))))))) => ((! [X3:A]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (~ ((((ord_less @ B) @ (F @ X3)) @ (F @ Y4))))))) => (Q @ X3))))) => (Q @ ((((lattices_ord_arg_max @ A) @ B) @ F) @ P))))))))))).
% 0.33/5.22 thf(fact_6490_arg__max__on__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattic1883929316492267755max_on @ B) @ A) = (^ [F2:(B > A),S7:(set @ B)]: (((((lattices_ord_arg_max @ B) @ A) @ F2) @ (^ [X2:B]: ((((member @ B) @ X2) @ S7))))))))))).
% 0.33/5.22 thf(fact_6491_arg__max__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattices_ord_arg_max @ B) @ A) = (^ [F2:(B > A),P4:(B > $o)]: (((fChoice @ B) @ ((((lattic501386751176901750rg_max @ B) @ A) @ F2) @ P4))))))))).
% 0.33/5.22 thf(fact_6492_is__arg__max__linorder, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (((lattic501386751176901750rg_max @ A) @ B) = (^ [F2:(A > B),P4:(A > $o),X2:A]: (((P4 @ X2) & (! [Y6:A]: (((P4 @ Y6) => (((ord_less_eq @ B) @ (F2 @ Y6)) @ (F2 @ X2))))))))))))).
% 0.33/5.22 thf(fact_6493_is__arg__max__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattic501386751176901750rg_max @ B) @ A) = (^ [F2:(B > A),P4:(B > $o),X2:B]: (((P4 @ X2) & (~ ((? [Y6:B]: (((P4 @ Y6) & (((ord_less @ A) @ (F2 @ X2)) @ (F2 @ Y6))))))))))))))).
% 0.33/5.22 thf(fact_6494_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)))))).
% 0.33/5.22 thf(fact_6495_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)))))).
% 0.33/5.22 thf(fact_6496_zero__natural_Orep__eq, axiom, ((code_nat_of_natural @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 0.33/5.22 thf(fact_6497_plus__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((plus_plus @ code_natural) @ X) @ Xa3)) = (((plus_plus @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6498_nat__of__natural__numeral, axiom, (! [K:num]: (((code_nat_of_natural @ ((numeral_numeral @ code_natural) @ K)) = ((numeral_numeral @ nat) @ K))))).
% 0.33/5.22 thf(fact_6499_minus__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((minus_minus @ code_natural) @ X) @ Xa3)) = (((minus_minus @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6500_times__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((times_times @ code_natural) @ X) @ Xa3)) = (((times_times @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6501_one__natural_Orep__eq, axiom, ((code_nat_of_natural @ (one_one @ code_natural)) = (one_one @ nat))).
% 0.33/5.22 thf(fact_6502_modulo__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((modulo_modulo @ code_natural) @ X) @ Xa3)) = (((modulo_modulo @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6503_divide__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((divide_divide @ code_natural) @ X) @ Xa3)) = (((divide_divide @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6504_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)))))).
% 0.33/5.22 thf(fact_6505_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)))))).
% 0.33/5.22 thf(fact_6506_division__segment__natural_Orep__eq, axiom, (! [X:code_natural]: (((code_nat_of_natural @ ((euclid7384307370059645450egment @ code_natural) @ X)) = ((euclid7384307370059645450egment @ nat) @ (code_nat_of_natural @ X)))))).
% 0.33/5.22 thf(fact_6507_euclidean__size__natural_Orep__eq, axiom, ((euclid6346220572633701492n_size @ code_natural) = (^ [X2:code_natural]: (((euclid6346220572633701492n_size @ nat) @ (code_nat_of_natural @ X2)))))).
% 0.33/5.22 thf(fact_6508_of__nat__of__natural, axiom, (! [N:code_natural]: ((((semiring_1_of_nat @ code_natural) @ (code_nat_of_natural @ N)) = N)))).
% 0.33/5.22 thf(fact_6509_nat__of__natural__of__nat, axiom, (! [N:nat]: (((code_nat_of_natural @ ((semiring_1_of_nat @ code_natural) @ N)) = N)))).
% 0.33/5.22 thf(fact_6510_nat__of__natural__induct, axiom, (! [Y3:nat,P:(nat > $o)]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => ((! [X3:code_natural]: ((P @ (code_nat_of_natural @ X3)))) => (P @ Y3)))))).
% 0.33/5.22 thf(fact_6511_nat__of__natural__cases, axiom, (! [Y3:nat]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => (~ ((! [X3:code_natural]: ((Y3 != (code_nat_of_natural @ X3)))))))))).
% 0.33/5.22 thf(fact_6512_nat__of__natural, axiom, (! [X:code_natural]: ((((member @ nat) @ (code_nat_of_natural @ X)) @ (top_top @ (set @ nat)))))).
% 0.33/5.22 thf(fact_6513_set__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se5668285175392031749et_bit @ code_natural) @ X) @ Xa3)) = (((bit_se5668285175392031749et_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6514_unset__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se2638667681897837118et_bit @ code_natural) @ X) @ Xa3)) = (((bit_se2638667681897837118et_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6515_flip__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se8732182000553998342ip_bit @ code_natural) @ X) @ Xa3)) = (((bit_se8732182000553998342ip_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6516_drop__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se4197421643247451524op_bit @ code_natural) @ X) @ Xa3)) = (((bit_se4197421643247451524op_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6517_bit__natural_Orep__eq, axiom, ((bit_se5641148757651400278ts_bit @ code_natural) = (^ [X2:code_natural]: (((bit_se5641148757651400278ts_bit @ nat) @ (code_nat_of_natural @ X2)))))).
% 0.33/5.22 thf(fact_6518_take__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se2584673776208193580ke_bit @ code_natural) @ X) @ Xa3)) = (((bit_se2584673776208193580ke_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6519_or__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se1065995026697491101ons_or @ code_natural) @ X) @ Xa3)) = (((bit_se1065995026697491101ons_or @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6520_xor__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se5824344971392196577ns_xor @ code_natural) @ X) @ Xa3)) = (((bit_se5824344971392196577ns_xor @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6521_and__natural_Orep__eq, axiom, (! [X:code_natural,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se5824344872417868541ns_and @ code_natural) @ X) @ Xa3)) = (((bit_se5824344872417868541ns_and @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6522_mask__natural_Orep__eq, axiom, (! [X:nat]: (((code_nat_of_natural @ ((bit_se2239418461657761734s_mask @ code_natural) @ X)) = ((bit_se2239418461657761734s_mask @ nat) @ X))))).
% 0.33/5.22 thf(fact_6523_push__bit__natural_Orep__eq, axiom, (! [X:nat,Xa3:code_natural]: (((code_nat_of_natural @ (((bit_se4730199178511100633sh_bit @ code_natural) @ X) @ Xa3)) = (((bit_se4730199178511100633sh_bit @ nat) @ X) @ (code_nat_of_natural @ Xa3)))))).
% 0.33/5.22 thf(fact_6524_natural__eq__iff, axiom, ((^ [Y:code_natural,Z2:code_natural]: ((Y = Z2))) = (^ [M2:code_natural,N3:code_natural]: (((code_nat_of_natural @ M2) = (code_nat_of_natural @ N3)))))).
% 0.33/5.22 thf(fact_6525_natural__eqI, axiom, (! [M:code_natural,N:code_natural]: ((((code_nat_of_natural @ M) = (code_nat_of_natural @ N)) => (M = N))))).
% 0.33/5.22 thf(fact_6526_nat__of__natural__inject, axiom, (! [X:code_natural,Y3:code_natural]: ((((code_nat_of_natural @ X) = (code_nat_of_natural @ Y3)) = (X = Y3))))).
% 0.33/5.22 thf(fact_6527_less__natural_Orep__eq, axiom, ((ord_less @ code_natural) = (^ [X2:code_natural,Xa4:code_natural]: ((((ord_less @ nat) @ (code_nat_of_natural @ X2)) @ (code_nat_of_natural @ Xa4)))))).
% 0.33/5.22 thf(fact_6528_less__eq__natural_Orep__eq, axiom, ((ord_less_eq @ code_natural) = (^ [X2:code_natural,Xa4:code_natural]: ((((ord_less_eq @ nat) @ (code_nat_of_natural @ X2)) @ (code_nat_of_natural @ Xa4)))))).
% 0.33/5.22 thf(fact_6529_natural__of__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_nat_of_natural @ (code_n4118661773612635043nteger @ X)) = (nat2 @ (code_int_of_integer @ X)))))).
% 0.33/5.22 thf(fact_6530_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)))))).
% 0.33/5.22 thf(fact_6531_Nat_Orep__eq, axiom, (! [X:code_integer]: (((code_nat_of_natural @ (code_Nat @ X)) = (nat2 @ (code_int_of_integer @ X)))))).
% 0.33/5.22 thf(fact_6532_Code__Numeral_Osize__nat__def, axiom, (code_size_nat = code_nat_of_natural)).
% 0.33/5.22 thf(fact_6533_natural__zero__minus__one, axiom, ((((minus_minus @ code_natural) @ (zero_zero @ code_natural)) @ (one_one @ code_natural)) = (zero_zero @ code_natural))).
% 0.33/5.22 thf(fact_6534_log_Osimps, axiom, (log = (^ [B3:code_natural,I5:code_natural]: (((((if @ code_natural) @ ((((ord_less_eq @ code_natural) @ B3) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ I5) @ B3))) @ (one_one @ code_natural)) @ (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ B3) @ (((divide_divide @ code_natural) @ I5) @ B3)))))))).
% 0.33/5.22 thf(fact_6535_log_Oelims, axiom, (! [X:code_natural,Xa3:code_natural,Y3:code_natural]: (((((log @ X) @ Xa3) = Y3) => ((((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa3) @ X)) => (Y3 = (one_one @ code_natural))) & ((~ (((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa3) @ X)))) => (Y3 = (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ X) @ (((divide_divide @ code_natural) @ Xa3) @ X)))))))))).
% 0.33/5.22 thf(fact_6536_minus__shift__def, axiom, (minus_shift = (^ [R5:code_natural,K3:code_natural,L3:code_natural]: (((((if @ code_natural) @ (((ord_less @ code_natural) @ K3) @ L3)) @ (((minus_minus @ code_natural) @ (((plus_plus @ code_natural) @ R5) @ K3)) @ L3)) @ (((minus_minus @ code_natural) @ K3) @ L3)))))).
% 0.33/5.23 thf(fact_6537_natural_Osimps_I4_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T)]: ((((((code_case_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 0.33/5.23 thf(fact_6538_range, axiom, (! [K:code_natural,S:((product_prod @ code_natural) @ code_natural)]: (((((ord_less @ code_natural) @ (zero_zero @ code_natural)) @ K) => (((ord_less @ code_natural) @ (((product_fst @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((range @ K) @ S))) @ K))))).
% 0.33/5.23 thf(fact_6539_select__weight__member, axiom, (! [A:$tType,Xs:(list @ ((product_prod @ code_natural) @ A)),S:((product_prod @ code_natural) @ code_natural)]: (((((ord_less @ code_natural) @ (zero_zero @ code_natural)) @ ((groups8242544230860333062m_list @ code_natural) @ ((((map @ ((product_prod @ code_natural) @ A)) @ code_natural) @ ((product_fst @ code_natural) @ A)) @ Xs))) => (((member @ A) @ (((product_fst @ A) @ ((product_prod @ code_natural) @ code_natural)) @ (((select_weight @ A) @ Xs) @ S))) @ ((set2 @ A) @ ((((map @ ((product_prod @ code_natural) @ A)) @ A) @ ((product_snd @ code_natural) @ A)) @ Xs))))))).
% 0.33/5.23 thf(fact_6540_select__weight__drop__zero, axiom, (! [A:$tType,Xs:(list @ ((product_prod @ code_natural) @ A))]: ((((select_weight @ A) @ (((filter @ ((product_prod @ code_natural) @ A)) @ ((((product_case_prod @ code_natural) @ A) @ $o) @ (^ [K3:code_natural,Uu:A]: ((((ord_less @ code_natural) @ (zero_zero @ code_natural)) @ K3))))) @ Xs)) = ((select_weight @ A) @ Xs))))).
% 0.33/5.23 thf(fact_6541_pick__member, axiom, (! [A:$tType,I:code_natural,Xs:(list @ ((product_prod @ code_natural) @ A))]: (((((ord_less @ code_natural) @ I) @ ((groups8242544230860333062m_list @ code_natural) @ ((((map @ ((product_prod @ code_natural) @ A)) @ code_natural) @ ((product_fst @ code_natural) @ A)) @ Xs))) => (((member @ A) @ (((pick @ A) @ Xs) @ I)) @ ((set2 @ A) @ ((((map @ ((product_prod @ code_natural) @ A)) @ A) @ ((product_snd @ code_natural) @ A)) @ Xs))))))).
% 0.33/5.23 thf(fact_6542_pick_Osimps, axiom, (! [A:$tType,I:code_natural,X:((product_prod @ code_natural) @ A),Xs:(list @ ((product_prod @ code_natural) @ A))]: ((((((ord_less @ code_natural) @ I) @ (((product_fst @ code_natural) @ A) @ X)) => ((((pick @ A) @ (((cons @ ((product_prod @ code_natural) @ A)) @ X) @ Xs)) @ I) = (((product_snd @ code_natural) @ A) @ X))) & ((~ ((((ord_less @ code_natural) @ I) @ (((product_fst @ code_natural) @ A) @ X)))) => ((((pick @ A) @ (((cons @ ((product_prod @ code_natural) @ A)) @ X) @ Xs)) @ I) = (((pick @ A) @ Xs) @ (((minus_minus @ code_natural) @ I) @ (((product_fst @ code_natural) @ A) @ X))))))))).
% 0.33/5.23 thf(fact_6543_pick__drop__zero, axiom, (! [A:$tType,Xs:(list @ ((product_prod @ code_natural) @ A))]: ((((pick @ A) @ (((filter @ ((product_prod @ code_natural) @ A)) @ ((((product_case_prod @ code_natural) @ A) @ $o) @ (^ [K3:code_natural,Uu:A]: ((((ord_less @ code_natural) @ (zero_zero @ code_natural)) @ K3))))) @ Xs)) = ((pick @ A) @ Xs))))).
% 0.33/5.23 thf(fact_6544_log_Opelims, axiom, (! [X:code_natural,Xa3:code_natural,Y3:code_natural]: (((((log @ X) @ Xa3) = Y3) => ((((accp @ ((product_prod @ code_natural) @ code_natural)) @ log_rel) @ ((((product_Pair @ code_natural) @ code_natural) @ X) @ Xa3)) => (~ ((((((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa3) @ X)) => (Y3 = (one_one @ code_natural))) & ((~ (((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa3) @ X)))) => (Y3 = (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ X) @ (((divide_divide @ code_natural) @ Xa3) @ X)))))) => (~ ((((accp @ ((product_prod @ code_natural) @ code_natural)) @ log_rel) @ ((((product_Pair @ code_natural) @ code_natural) @ X) @ Xa3)))))))))))).
% 0.33/5.23 thf(fact_6545_bit__natural__def, axiom, ((bit_se5641148757651400278ts_bit @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > $o)) @ (nat > $o)) @ code_nat_of_natural) @ (id @ (nat > $o))) @ (bit_se5641148757651400278ts_bit @ nat)))).
% 0.33/5.23 thf(fact_6546_pick__same, axiom, (! [A:$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)))))).
% 0.33/5.23 thf(fact_6547_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)))).
% 0.33/5.23 thf(fact_6548_natural__of__nat__of__natural__inverse, axiom, (! [N:code_natural]: (((code_natural_of_nat @ (code_nat_of_natural @ N)) = N)))).
% 0.33/5.23 thf(fact_6549_nat__of__natural__of__nat__inverse, axiom, (! [N:nat]: (((code_nat_of_natural @ (code_natural_of_nat @ N)) = N)))).
% 0.33/5.23 thf(fact_6550_nat__of__natural__inverse, axiom, (! [X:code_natural]: (((code_natural_of_nat @ (code_nat_of_natural @ X)) = X)))).
% 0.33/5.23 thf(fact_6551_natural__of__nat__of__nat, axiom, (code_natural_of_nat = (semiring_1_of_nat @ code_natural))).
% 0.33/5.23 thf(fact_6552_natural__of__nat__inverse, axiom, (! [Y3:nat]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => ((code_nat_of_natural @ (code_natural_of_nat @ Y3)) = Y3))))).
% 0.33/5.23 thf(fact_6553_natural__of__nat__cases, axiom, (! [X:code_natural]: ((~ ((! [Y5:nat]: (((X = (code_natural_of_nat @ Y5)) => (~ ((((member @ nat) @ Y5) @ (top_top @ (set @ nat))))))))))))).
% 0.33/5.23 thf(fact_6554_natural__of__nat__induct, axiom, (! [P:(code_natural > $o),X:code_natural]: (((! [Y5:nat]: (((((member @ nat) @ Y5) @ (top_top @ (set @ nat))) => (P @ (code_natural_of_nat @ Y5))))) => (P @ X))))).
% 0.33/5.23 thf(fact_6555_natural__of__nat__inject, axiom, (! [X:nat,Y3:nat]: (((((member @ nat) @ X) @ (top_top @ (set @ nat))) => ((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => (((code_natural_of_nat @ X) = (code_natural_of_nat @ Y3)) = (X = Y3))))))).
% 0.33/5.23 thf(fact_6556_minus__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((minus_minus @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((minus_minus @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6557_one__natural__def, axiom, ((one_one @ code_natural) = (code_natural_of_nat @ (one_one @ nat)))).
% 0.33/5.23 thf(fact_6558_divide__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((divide_divide @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((divide_divide @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6559_plus__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((plus_plus @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((plus_plus @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6560_zero__natural__def, axiom, ((zero_zero @ code_natural) = (code_natural_of_nat @ (zero_zero @ nat)))).
% 0.33/5.23 thf(fact_6561_less__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((ord_less @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (((ord_less @ nat) @ Xa3) @ X))))).
% 0.33/5.23 thf(fact_6562_less__eq__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((ord_less_eq @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (((ord_less_eq @ nat) @ Xa3) @ X))))).
% 0.33/5.23 thf(fact_6563_times__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((times_times @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((times_times @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6564_modulo__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((modulo_modulo @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((modulo_modulo @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6565_take__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se2584673776208193580ke_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se2584673776208193580ke_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6566_xor__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se5824344971392196577ns_xor @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5824344971392196577ns_xor @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6567_or__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se1065995026697491101ons_or @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se1065995026697491101ons_or @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6568_and__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se5824344872417868541ns_and @ code_natural) @ (code_natural_of_nat @ Xa3)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5824344872417868541ns_and @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6569_bit__natural_Oabs__eq, axiom, (! [X:nat]: ((((bit_se5641148757651400278ts_bit @ code_natural) @ (code_natural_of_nat @ X)) = ((bit_se5641148757651400278ts_bit @ nat) @ X))))).
% 0.33/5.23 thf(fact_6570_push__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se4730199178511100633sh_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se4730199178511100633sh_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6571_mask__natural_Oabs__eq, axiom, ((bit_se2239418461657761734s_mask @ code_natural) = (^ [X2:nat]: ((code_natural_of_nat @ ((bit_se2239418461657761734s_mask @ nat) @ X2)))))).
% 0.33/5.23 thf(fact_6572_drop__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se4197421643247451524op_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se4197421643247451524op_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6573_euclidean__size__natural_Oabs__eq, axiom, (! [X:nat]: ((((euclid6346220572633701492n_size @ code_natural) @ (code_natural_of_nat @ X)) = ((euclid6346220572633701492n_size @ nat) @ X))))).
% 0.33/5.23 thf(fact_6574_flip__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se8732182000553998342ip_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se8732182000553998342ip_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6575_unset__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se2638667681897837118et_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se2638667681897837118et_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6576_division__segment__natural_Oabs__eq, axiom, (! [X:nat]: ((((euclid7384307370059645450egment @ code_natural) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ ((euclid7384307370059645450egment @ nat) @ X)))))).
% 0.33/5.23 thf(fact_6577_set__bit__natural_Oabs__eq, axiom, (! [Xa3:nat,X:nat]: (((((bit_se5668285175392031749et_bit @ code_natural) @ Xa3) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5668285175392031749et_bit @ nat) @ Xa3) @ X)))))).
% 0.33/5.23 thf(fact_6578_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)))))).
% 0.33/5.23 thf(fact_6579_natural__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_n4118661773612635043nteger @ (code_integer_of_int @ X)) = (code_natural_of_nat @ (nat2 @ X)))))).
% 0.33/5.23 thf(fact_6580_Nat_Oabs__eq, axiom, (! [X:int]: (((code_Nat @ (code_integer_of_int @ X)) = (code_natural_of_nat @ (nat2 @ X)))))).
% 0.33/5.23 thf(fact_6581_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)))).
% 0.33/5.23 thf(fact_6582_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)))))))))).
% 0.33/5.23 thf(fact_6583_natural_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T))]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 0.33/5.23 thf(fact_6584_rec__natural__def, axiom, (! [T:$tType]: (((code_rec_natural @ T) = (^ [F12:T,F23:(code_natural > (T > T)),X2:code_natural]: (((the @ T) @ ((((code_rec_set_natural @ T) @ F12) @ F23) @ X2)))))))).
% 0.33/5.23 thf(fact_6585_Nat__def, axiom, (code_Nat = (((((((map_fun @ code_integer) @ int) @ nat) @ code_natural) @ code_int_of_integer) @ code_natural_of_nat) @ nat2))).
% 0.33/5.23 thf(fact_6586_natural__of__integer__def, axiom, (code_n4118661773612635043nteger = (((((((map_fun @ code_integer) @ int) @ nat) @ code_natural) @ code_int_of_integer) @ code_natural_of_nat) @ nat2))).
% 0.33/5.23 thf(fact_6587_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)))).
% 0.33/5.23 thf(fact_6588_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)))).
% 0.33/5.23 thf(fact_6589_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)))).
% 0.33/5.23 thf(fact_6590_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)))).
% 0.33/5.23 thf(fact_6591_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)))).
% 0.33/5.23 thf(fact_6592_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)))).
% 0.33/5.23 thf(fact_6593_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)))).
% 0.33/5.23 thf(fact_6594_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)))).
% 0.33/5.23 thf(fact_6595_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)))).
% 0.33/5.23 thf(fact_6596_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)))).
% 0.33/5.23 thf(fact_6597_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)))).
% 0.33/5.23 thf(fact_6598_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)))).
% 0.33/5.23 thf(fact_6599_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)))).
% 0.33/5.23 thf(fact_6600_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)))).
% 0.33/5.23 thf(fact_6601_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)))).
% 0.33/5.23 thf(fact_6602_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)))).
% 0.33/5.23 thf(fact_6603_Code__Numeral_OSuc__def, axiom, (code_Suc = (((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat) @ suc))).
% 0.33/5.23 thf(fact_6604_cr__natural__def, axiom, (code_cr_natural = (^ [X2:nat,Y6:code_natural]: ((X2 = (code_nat_of_natural @ Y6)))))).
% 0.33/5.23 thf(fact_6605_natural_Oinject, axiom, (! [Natural:code_natural,Natural2:code_natural]: ((((code_Suc @ Natural) = (code_Suc @ Natural2)) = (Natural = Natural2))))).
% 0.33/5.23 thf(fact_6606_natural_Osimps_I7_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T)),Natural:code_natural]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = ((F22 @ Natural) @ ((((code_rec_natural @ T) @ F1) @ F22) @ Natural)))))).
% 0.33/5.23 thf(fact_6607_natural_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T),Natural:code_natural]: ((((((code_case_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = (F22 @ Natural))))).
% 0.33/5.23 thf(fact_6608_Suc_Orep__eq, axiom, (! [X:code_natural]: (((code_nat_of_natural @ (code_Suc @ X)) = (suc @ (code_nat_of_natural @ X)))))).
% 0.33/5.23 thf(fact_6609_natural_Oexhaust, axiom, (! [Y3:code_natural]: (((Y3 != (zero_zero @ code_natural)) => (~ ((! [Natural3:code_natural]: ((Y3 != (code_Suc @ Natural3)))))))))).
% 0.33/5.23 thf(fact_6610_natural_Odistinct_I1_J, axiom, (! [Natural2:code_natural]: (((zero_zero @ code_natural) != (code_Suc @ Natural2))))).
% 0.33/5.23 thf(fact_6611_natural_Odistinct_I2_J, axiom, (! [Natural4:code_natural]: (((code_Suc @ Natural4) != (zero_zero @ code_natural))))).
% 0.33/5.23 thf(fact_6612_Suc_Oabs__eq, axiom, (! [X:nat]: (((code_Suc @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (suc @ X)))))).
% 0.33/5.23 thf(fact_6613_Suc__natural__minus__one, axiom, (! [N:code_natural]: (((((minus_minus @ code_natural) @ (code_Suc @ N)) @ (one_one @ code_natural)) = N)))).
% 0.33/5.23 thf(fact_6614_Nat_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ code_natural) @ code_pcr_integer) @ code_pcr_natural) @ nat2) @ code_Nat)).
% 0.33/5.23 thf(fact_6615_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))))))))).
% 0.33/5.23 thf(fact_6616_Suc_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural) @ suc) @ code_Suc)).
% 0.33/5.23 thf(fact_6617_natural_Opcr__cr__eq, axiom, (code_pcr_natural = code_cr_natural)).
% 0.33/5.23 thf(fact_6618_natural_Oid__abs__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ code_natural) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ code_pcr_natural) @ (^ [X2:nat]: (X2))) @ code_natural_of_nat)).
% 0.33/5.23 thf(fact_6619_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) @ $o) @ code_pcr_natural) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less @ nat)) @ (ord_less @ code_natural))).
% 0.33/5.23 thf(fact_6620_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))).
% 0.33/5.23 thf(fact_6621_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))).
% 0.33/5.23 thf(fact_6622_zero__natural_Otransfer, axiom, ((code_pcr_natural @ (zero_zero @ nat)) @ (zero_zero @ code_natural))).
% 0.33/5.23 thf(fact_6623_one__natural_Otransfer, axiom, ((code_pcr_natural @ (one_one @ nat)) @ (one_one @ code_natural))).
% 0.33/5.23 thf(fact_6624_natural_Obi__total, axiom, (((bi_total @ nat) @ code_natural) @ code_pcr_natural)).
% 0.33/5.23 thf(fact_6625_natural_Orep__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ nat) @ code_pcr_natural) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (^ [X2:nat]: (X2))) @ code_nat_of_natural)).
% 0.33/5.23 thf(fact_6626_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) @ $o) @ code_pcr_natural) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (ord_less_eq @ nat)) @ (ord_less_eq @ code_natural))).
% 0.33/5.23 thf(fact_6627_divide__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (divide_divide @ nat)) @ (divide_divide @ code_natural))).
% 0.33/5.23 thf(fact_6628_times__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (times_times @ nat)) @ (times_times @ code_natural))).
% 0.33/5.23 thf(fact_6629_modulo__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (modulo_modulo @ nat)) @ (modulo_modulo @ code_natural))).
% 0.33/5.23 thf(fact_6630_fold__atLeastAtMost__nat_Opinduct, axiom, (! [A:$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)))) => ((! [F3:(nat > (A > A)),A8:nat,B5:nat,Acc3: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))) @ F3) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A8) @ ((((product_Pair @ nat) @ A) @ B5) @ Acc3)))) => (((~ ((((ord_less @ nat) @ B5) @ A8))) => ((((P @ F3) @ (((plus_plus @ nat) @ A8) @ (one_one @ nat))) @ B5) @ ((F3 @ A8) @ Acc3))) => ((((P @ F3) @ A8) @ B5) @ Acc3))))) => ((((P @ A0) @ A1) @ A22) @ A32)))))).
% 0.33/5.23 thf(fact_6631_take__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se2584673776208193580ke_bit @ nat)) @ (bit_se2584673776208193580ke_bit @ code_natural))).
% 0.33/5.23 thf(fact_6632_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))).
% 0.33/5.23 thf(fact_6633_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))).
% 0.33/5.23 thf(fact_6634_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))).
% 0.33/5.23 thf(fact_6635_bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > $o)) @ (nat > $o)) @ code_pcr_natural) @ (^ [Y:(nat > $o),Z2:(nat > $o)]: ((Y = Z2)))) @ (bit_se5641148757651400278ts_bit @ nat)) @ (bit_se5641148757651400278ts_bit @ code_natural))).
% 0.33/5.23 thf(fact_6636_mask__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ code_natural) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ code_pcr_natural) @ (bit_se2239418461657761734s_mask @ nat)) @ (bit_se2239418461657761734s_mask @ code_natural))).
% 0.33/5.23 thf(fact_6637_push__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se4730199178511100633sh_bit @ nat)) @ (bit_se4730199178511100633sh_bit @ code_natural))).
% 0.33/5.23 thf(fact_6638_drop__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se4197421643247451524op_bit @ nat)) @ (bit_se4197421643247451524op_bit @ code_natural))).
% 0.33/5.23 thf(fact_6639_set__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se5668285175392031749et_bit @ nat)) @ (bit_se5668285175392031749et_bit @ code_natural))).
% 0.33/5.23 thf(fact_6640_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))).
% 0.33/5.23 thf(fact_6641_unset__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se2638667681897837118et_bit @ nat)) @ (bit_se2638667681897837118et_bit @ code_natural))).
% 0.33/5.23 thf(fact_6642_flip__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se8732182000553998342ip_bit @ nat)) @ (bit_se8732182000553998342ip_bit @ code_natural))).
% 0.33/5.23 thf(fact_6643_euclidean__size__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ nat) @ code_pcr_natural) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ (euclid6346220572633701492n_size @ nat)) @ (euclid6346220572633701492n_size @ code_natural))).
% 0.33/5.23 thf(fact_6644_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)).
% 0.33/5.23 thf(fact_6645_natural__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ code_natural) @ code_pcr_integer) @ code_pcr_natural) @ nat2) @ code_n4118661773612635043nteger)).
% 0.33/5.23 thf(fact_6646_fold__atLeastAtMost__nat_Opelims, axiom, (! [A:$tType,X:(nat > (A > A)),Xa3:nat,Xb3:nat,Xc:A,Y3:A]: ((((((((set_fo6178422350223883121st_nat @ A) @ X) @ Xa3) @ Xb3) @ Xc) = Y3) => ((((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)) @ Xa3) @ ((((product_Pair @ nat) @ A) @ Xb3) @ Xc)))) => (~ (((((((ord_less @ nat) @ Xb3) @ Xa3) => (Y3 = Xc)) & ((~ ((((ord_less @ nat) @ Xb3) @ Xa3))) => (Y3 = (((((set_fo6178422350223883121st_nat @ A) @ X) @ (((plus_plus @ nat) @ Xa3) @ (one_one @ nat))) @ Xb3) @ ((X @ Xa3) @ 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)) @ Xa3) @ ((((product_Pair @ nat) @ A) @ Xb3) @ Xc)))))))))))))).
% 0.33/5.23 thf(fact_6647_prod_Ocomm__monoid__list__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups4802862169904069756st_set @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.23 thf(fact_6648_prod_Oreindex__bij__betw__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S5:(set @ B),T5:(set @ C),H2:(B > C),S3:(set @ B),T6:(set @ C),G:(C > A)]: ((((finite_finite2 @ B) @ S5) => (((finite_finite2 @ C) @ T5) => ((((((bij_betw @ B) @ C) @ H2) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((! [A8:B]: (((((member @ B) @ A8) @ S5) => ((G @ (H2 @ A8)) = (one_one @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T5) => ((G @ B5) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X2:B]: ((G @ (H2 @ X2))))) @ S3) = ((((groups7121269368397514597t_prod @ C) @ A) @ G) @ T6))))))))))))).
% 0.33/5.23 thf(fact_6649_bij__betw__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N6:(set @ nat),A6:(set @ A)]: (((((((bij_betw @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6) @ A6) = (((((image @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6) = A6)))))))).
% 0.33/5.23 thf(fact_6650_bij__betw__Suc, axiom, (! [M7:(set @ nat),N6:(set @ nat)]: (((((((bij_betw @ nat) @ nat) @ suc) @ M7) @ N6) = (((((image @ nat) @ nat) @ suc) @ M7) = N6))))).
% 0.33/5.23 thf(fact_6651_bijection__def, axiom, (! [A:$tType]: (((hilbert_bijection @ A) = (^ [F2:(A > A)]: ((((((bij_betw @ A) @ A) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))))).
% 0.33/5.23 thf(fact_6652_bijection_Ointro, axiom, (! [A:$tType,F:(A > A)]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => ((hilbert_bijection @ A) @ F))))).
% 0.33/5.23 thf(fact_6653_bijection_Obij, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => (((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 0.33/5.23 thf(fact_6654_bij__betw__funpow, axiom, (! [A:$tType,F:(A > A),S3:(set @ A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ S3) @ S3) => (((((bij_betw @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ S3) @ S3))))).
% 0.33/5.23 thf(fact_6655_bij__fn, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => (((((bij_betw @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 0.33/5.23 thf(fact_6656_bij__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (((((bij_betw @ A) @ A) @ (uminus_uminus @ A)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 0.33/5.23 thf(fact_6657_bij__imp__bij__inv, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ (top_top @ (set @ B))) @ (top_top @ (set @ A))))))).
% 0.33/5.23 thf(fact_6658_bij__inv__eq__iff, axiom, (! [A:$tType,B:$tType,P5:(A > B),X:A,Y3:B]: (((((((bij_betw @ A) @ B) @ P5) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((X = (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ P5) @ Y3)) = ((P5 @ X) = Y3)))))).
% 0.33/5.23 thf(fact_6659_inv__inv__eq, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) = F))))).
% 0.33/5.23 thf(fact_6660_bij__betw__inv__into__right, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ A),A11:(set @ B),A4:B]: (((((((bij_betw @ A) @ B) @ F) @ A6) @ A11) => ((((member @ B) @ A4) @ A11) => ((F @ (((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ A4)) = A4)))))).
% 0.33/5.23 thf(fact_6661_bij__betw__inv__into__left, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),A11:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A6) @ A11) => ((((member @ A) @ A3) @ A6) => ((((((hilbert_inv_into @ A) @ B) @ A6) @ F) @ (F @ A3)) = A3)))))).
% 0.33/5.23 thf(fact_6662_inv__into__inv__into__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),A11:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A6) @ A11) => ((((member @ A) @ A3) @ A6) => ((((((hilbert_inv_into @ B) @ A) @ A11) @ ((((hilbert_inv_into @ A) @ B) @ A6) @ F)) @ A3) = (F @ A3))))))).
% 0.33/5.23 thf(fact_6663_bij__betw__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),B7:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A6) @ B7) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A6) @ F)) @ B7) @ A6))))).
% 0.33/5.23 thf(fact_6664_bij__betw__inv__into__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A),A11:(set @ B),B7:(set @ A),B13:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A6) @ A11) => ((((ord_less_eq @ (set @ A)) @ B7) @ A6) => ((((((image @ A) @ B) @ F) @ B7) = B13) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A6) @ F)) @ B13) @ B7))))))).
% 0.33/5.23 thf(fact_6665_bij__image__Compl__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A6:(set @ A)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A6)) = ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A6))))))).
% 0.33/5.23 thf(fact_6666_o__inv__distrib, axiom, (! [C:$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)))))))).
% 0.33/5.23 thf(fact_6667_inv__fn, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (((compow @ (A > A)) @ N) @ F)) = (((compow @ (A > A)) @ N) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F))))))).
% 0.33/5.23 thf(fact_6668_mono__inv, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F:(A > B)]: (((((order_mono @ A) @ B) @ F) => ((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((order_mono @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)))))))))).
% 0.33/5.23 thf(fact_6669_bijection_Obij__inv, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => (((((bij_betw @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 0.33/5.23 thf(fact_6670_bij__image__Collect__eq, axiom, (! [A:$tType,B:$tType,F:(A > B),P:(A > $o)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F) @ ((collect @ A) @ P)) = ((collect @ B) @ (^ [Y6:B]: ((P @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ Y6)))))))))).
% 0.33/5.23 thf(fact_6671_bij__int__decode, axiom, (((((bij_betw @ nat) @ int) @ nat_int_decode) @ (top_top @ (set @ nat))) @ (top_top @ (set @ int)))).
% 0.33/5.23 thf(fact_6672_bij__int__encode, axiom, (((((bij_betw @ int) @ nat) @ nat_int_encode) @ (top_top @ (set @ int))) @ (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6673_infinite__imp__bij__betw2, axiom, (! [A:$tType,A6:(set @ A),A3:A]: (((~ (((finite_finite2 @ A) @ A6))) => (? [H6:(A > A)]: ((((((bij_betw @ A) @ A) @ H6) @ A6) @ (((sup_sup @ (set @ A)) @ A6) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 0.33/5.23 thf(fact_6674_infinite__imp__bij__betw, axiom, (! [A:$tType,A6:(set @ A),A3:A]: (((~ (((finite_finite2 @ A) @ A6))) => (? [H6:(A > A)]: ((((((bij_betw @ A) @ A) @ H6) @ A6) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 0.33/5.23 thf(fact_6675_ex__bij__betw__finite__nat, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(A > nat)]: ((((((bij_betw @ A) @ nat) @ H6) @ M7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M7))))))))).
% 0.33/5.23 thf(fact_6676_ex__bij__betw__nat__finite, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M7))) @ M7))))))).
% 0.33/5.23 thf(fact_6677_mono__bij__Inf, axiom, (! [B:$tType,A:$tType]: ((((comple5582772986160207858norder @ A) & (comple5582772986160207858norder @ B)) => (! [F:(A > B),A6:(set @ A)]: (((((order_mono @ A) @ B) @ F) => ((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((F @ ((complete_Inf_Inf @ A) @ A6)) = ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F) @ A6))))))))))).
% 0.33/5.23 thf(fact_6678_bij__prod__decode, axiom, (((((bij_betw @ nat) @ ((product_prod @ nat) @ nat)) @ nat_prod_decode) @ (top_top @ (set @ nat))) @ (top_top @ (set @ ((product_prod @ nat) @ nat))))).
% 0.33/5.23 thf(fact_6679_inj__imp__bij__betw__inv, axiom, (! [B:$tType,A:$tType,F:(A > B),M7:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ ((((image @ A) @ B) @ F) @ M7)) @ M7))))).
% 0.33/5.23 thf(fact_6680_ex__bij__betw__nat__finite__1, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H6) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ ((finite_card @ A) @ M7))) @ M7))))))).
% 0.33/5.23 thf(fact_6681_bij__list__decode, axiom, (((((bij_betw @ nat) @ (list @ nat)) @ nat_list_decode) @ (top_top @ (set @ nat))) @ (top_top @ (set @ (list @ nat))))).
% 0.33/5.23 thf(fact_6682_bij__list__encode, axiom, (((((bij_betw @ (list @ nat)) @ nat) @ nat_list_encode) @ (top_top @ (set @ (list @ nat)))) @ (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6683_bij__prod__encode, axiom, (((((bij_betw @ ((product_prod @ nat) @ nat)) @ nat) @ nat_prod_encode) @ (top_top @ (set @ ((product_prod @ nat) @ nat)))) @ (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6684_bij__vimage__eq__inv__image, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((vimage @ A) @ B) @ F) @ A6) = ((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ A6)))))).
% 0.33/5.23 thf(fact_6685_fn__o__inv__fn__is__id, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => ((((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ (((compow @ (A > A)) @ N) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F))) = (^ [X2:A]: (X2))))))).
% 0.33/5.23 thf(fact_6686_inv__fn__o__fn__is__id, axiom, (! [A:$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)) = (^ [X2:A]: (X2))))))).
% 0.33/5.23 thf(fact_6687_sum_Oreindex__bij__betw__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S5:(set @ B),T5:(set @ C),H2:(B > C),S3:(set @ B),T6:(set @ C),G:(C > A)]: ((((finite_finite2 @ B) @ S5) => (((finite_finite2 @ C) @ T5) => ((((((bij_betw @ B) @ C) @ H2) @ (((minus_minus @ (set @ B)) @ S3) @ S5)) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((! [A8:B]: (((((member @ B) @ A8) @ S5) => ((G @ (H2 @ A8)) = (zero_zero @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T5) => ((G @ B5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X2:B]: ((G @ (H2 @ X2))))) @ S3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ T6))))))))))))).
% 0.33/5.23 thf(fact_6688_sum_Ocomm__monoid__list__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups4802862169904069756st_set @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.23 thf(fact_6689_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))))))).
% 0.33/5.23 thf(fact_6690_natLeq__on__wo__rel, axiom, (! [N:nat]: (((bNF_Wellorder_wo_rel @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))))).
% 0.33/5.23 thf(fact_6691_sum_H__def, axiom, (! [C:$tType,A:$tType]: (((comm_monoid_add @ A) => (((groups1027152243600224163dd_sum @ C) @ A) = ((((groups_comm_monoid_G @ A) @ C) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 0.33/5.23 thf(fact_6692_Heap_Osize__gen__o__map, axiom, (! [B:$tType,A:$tType,F:(B > nat),G:(A > B)]: (((((((comp @ (heap_Time_Heap @ B)) @ nat) @ (heap_Time_Heap @ A)) @ ((heap_Time_size_Heap @ B) @ F)) @ (((heap_Time_map_Heap @ A) @ B) @ G)) = ((heap_Time_size_Heap @ A) @ (((((comp @ B) @ nat) @ A) @ F) @ G)))))).
% 0.33/5.23 thf(fact_6693_max_Osemilattice__order__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((semilattice_order @ A) @ (ord_max @ A)) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_6694_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)))))))).
% 0.33/5.23 thf(fact_6695_semilattice__order__set_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less) => ((((semilattice_order @ A) @ F) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_6696_Heap_Omap__comp, axiom, (! [B:$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))))).
% 0.33/5.23 thf(fact_6697_semilattice__neutr__order_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less) => ((((semilattice_order @ A) @ F) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_6698_Heap_Omap__ident, axiom, (! [A:$tType,T2:(heap_Time_Heap @ A)]: ((((((heap_Time_map_Heap @ A) @ A) @ (^ [X2:A]: (X2))) @ T2) = T2)))).
% 0.33/5.23 thf(fact_6699_semilattice__order_Omono, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,C2:A,B2:A,D2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ C2) => (((Less_eq @ B2) @ D2) => ((Less_eq @ ((F @ A3) @ B2)) @ ((F @ C2) @ D2)))))))).
% 0.33/5.23 thf(fact_6700_semilattice__order_OorderE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) => (A3 = ((F @ A3) @ B2))))))).
% 0.33/5.23 thf(fact_6701_semilattice__order_OorderI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((A3 = ((F @ A3) @ B2)) => ((Less_eq @ A3) @ B2)))))).
% 0.33/5.23 thf(fact_6702_semilattice__order_Oabsorb1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) => (((F @ A3) @ B2) = A3)))))).
% 0.33/5.23 thf(fact_6703_semilattice__order_Oabsorb2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,A3:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ A3) => (((F @ A3) @ B2) = B2)))))).
% 0.33/5.23 thf(fact_6704_semilattice__order_Oabsorb3, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => (((F @ A3) @ B2) = A3)))))).
% 0.33/5.23 thf(fact_6705_semilattice__order_Oabsorb4, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,A3:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ B2) @ A3) => (((F @ A3) @ B2) = B2)))))).
% 0.33/5.23 thf(fact_6706_semilattice__order_OboundedE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ ((F @ B2) @ C2)) => (~ ((((Less_eq @ A3) @ B2) => (~ (((Less_eq @ A3) @ C2))))))))))).
% 0.33/5.23 thf(fact_6707_semilattice__order_OboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) => (((Less_eq @ A3) @ C2) => ((Less_eq @ A3) @ ((F @ B2) @ C2)))))))).
% 0.33/5.23 thf(fact_6708_semilattice__order_Oorder__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) = (A3 = ((F @ A3) @ B2))))))).
% 0.33/5.23 thf(fact_6709_semilattice__order_Ocobounded1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((Less_eq @ ((F @ A3) @ B2)) @ A3))))).
% 0.33/5.23 thf(fact_6710_semilattice__order_Ocobounded2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((Less_eq @ ((F @ A3) @ B2)) @ B2))))).
% 0.33/5.23 thf(fact_6711_semilattice__order_Oabsorb__iff1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) = (((F @ A3) @ B2) = A3)))))).
% 0.33/5.23 thf(fact_6712_semilattice__order_Oabsorb__iff2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,A3:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ A3) = (((F @ A3) @ B2) = B2)))))).
% 0.33/5.23 thf(fact_6713_semilattice__order_Obounded__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ ((F @ B2) @ C2)) = (((Less_eq @ A3) @ B2) & ((Less_eq @ A3) @ C2))))))).
% 0.33/5.23 thf(fact_6714_semilattice__order_OcoboundedI1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,C2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A3) @ C2) => ((Less_eq @ ((F @ A3) @ B2)) @ C2)))))).
% 0.33/5.23 thf(fact_6715_semilattice__order_OcoboundedI2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,C2:A,A3:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ C2) => ((Less_eq @ ((F @ A3) @ B2)) @ C2)))))).
% 0.33/5.23 thf(fact_6716_semilattice__order_Ostrict__boundedE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A3) @ ((F @ B2) @ C2)) => (~ ((((Less @ A3) @ B2) => (~ (((Less @ A3) @ C2))))))))))).
% 0.33/5.23 thf(fact_6717_semilattice__order_Ostrict__order__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A3) @ B2) = ((A3 = ((F @ A3) @ B2)) & (A3 != B2))))))).
% 0.33/5.23 thf(fact_6718_semilattice__order_Ostrict__coboundedI1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,C2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A3) @ C2) => ((Less @ ((F @ A3) @ B2)) @ C2)))))).
% 0.33/5.23 thf(fact_6719_semilattice__order_Ostrict__coboundedI2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,C2:A,A3:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ B2) @ C2) => ((Less @ ((F @ A3) @ B2)) @ C2)))))).
% 0.33/5.23 thf(fact_6720_Heap_Omap__id, axiom, (! [A:$tType,T2:(heap_Time_Heap @ A)]: ((((((heap_Time_map_Heap @ A) @ A) @ (id @ A)) @ T2) = T2)))).
% 0.33/5.23 thf(fact_6721_Heap_Omap__id0, axiom, (! [A:$tType]: (((((heap_Time_map_Heap @ A) @ A) @ (id @ A)) = (id @ (heap_Time_Heap @ A)))))).
% 0.33/5.23 thf(fact_6722_semilattice__order__set__def, axiom, (! [A:$tType]: (((lattic4895041142388067077er_set @ A) = (^ [F2:(A > (A > A)),Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((((((semilattice_order @ A) @ F2) @ Less_eq2) @ Less2) & ((lattic149705377957585745ce_set @ A) @ F2)))))))).
% 0.33/5.23 thf(fact_6723_semilattice__order__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((lattic149705377957585745ce_set @ A) @ F) => ((((lattic4895041142388067077er_set @ A) @ F) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_6724_semilattice__neutr__order__def, axiom, (! [A:$tType]: (((semila1105856199041335345_order @ A) = (^ [F2:(A > (A > A)),Z7:A,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((((semilattice_neutr @ A) @ F2) @ Z7) & ((((semilattice_order @ A) @ F2) @ Less_eq2) @ Less2)))))))).
% 0.33/5.23 thf(fact_6725_semilattice__neutr__order_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((semilattice_neutr @ A) @ F) @ Z3) => (((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((((semila1105856199041335345_order @ A) @ F) @ Z3) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_6726_inf_Osemilattice__order__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((((semilattice_order @ A) @ (inf_inf @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_6727_min_Osemilattice__order__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((semilattice_order @ A) @ (ord_min @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_6728_sup_Osemilattice__order__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((((semilattice_order @ A) @ (sup_sup @ A)) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_6729_lexordp__eq_Osimps, axiom, (! [A:$tType]: (((ord @ A) => ((ord_lexordp_eq @ A) = (^ [A12:(list @ A),A23:(list @ A)]: (((? [Ys3:(list @ A)]: (((A12 = (nil @ A)) & (A23 = Ys3)))) | ((? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((A12 = (((cons @ A) @ X2) @ Xs3)) & ((A23 = (((cons @ A) @ Y6) @ Ys3)) & (((ord_less @ A) @ X2) @ Y6))))) | (? [X2:A,Y6:A,Xs3:(list @ A),Ys3:(list @ A)]: (((A12 = (((cons @ A) @ X2) @ Xs3)) & ((A23 = (((cons @ A) @ Y6) @ Ys3)) & ((~ ((((ord_less @ A) @ X2) @ Y6))) & ((~ ((((ord_less @ A) @ Y6) @ X2))) & (((ord_lexordp_eq @ A) @ Xs3) @ Ys3)))))))))))))))).
% 0.33/5.23 thf(fact_6730_lexordp__eq__simps_I4_J, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Xs:(list @ A),Y3:A,Ys2:(list @ A)]: (((((ord_lexordp_eq @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2)) = ((((ord_less @ A) @ X) @ Y3) | ((~ ((((ord_less @ A) @ Y3) @ X))) & (((ord_lexordp_eq @ A) @ Xs) @ Ys2)))))))))).
% 0.33/5.23 thf(fact_6731_lexordp__eq_OCons, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less @ A) @ X) @ Y3) => (((ord_lexordp_eq @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2))))))))).
% 0.33/5.23 thf(fact_6732_lexordp__eq_OCons__eq, axiom, (! [A:$tType]: (((ord @ A) => (! [X:A,Y3:A,Xs:(list @ A),Ys2:(list @ A)]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((~ ((((ord_less @ A) @ Y3) @ X))) => ((((ord_lexordp_eq @ A) @ Xs) @ Ys2) => (((ord_lexordp_eq @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y3) @ Ys2))))))))))).
% 0.33/5.23 thf(fact_6733_lexordp__eq_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [A1:(list @ A),A22:(list @ A)]: (((((ord_lexordp_eq @ A) @ A1) @ A22) => ((A1 != (nil @ A)) => ((! [X3:A]: (((? [Xs2:(list @ A)]: ((A1 = (((cons @ A) @ X3) @ Xs2)))) => (! [Y5:A]: (((? [Ys4:(list @ A)]: ((A22 = (((cons @ A) @ Y5) @ Ys4)))) => (~ ((((ord_less @ A) @ X3) @ Y5))))))))) => (~ ((! [X3:A,Y5:A,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X3) @ Xs2)) => (! [Ys4:(list @ A)]: (((A22 = (((cons @ A) @ Y5) @ Ys4)) => ((~ ((((ord_less @ A) @ X3) @ Y5))) => ((~ ((((ord_less @ A) @ Y5) @ X3))) => (~ ((((ord_lexordp_eq @ A) @ Xs2) @ Ys4)))))))))))))))))))))).
% 0.33/5.23 thf(fact_6734_bdd__below__primitive__def, axiom, (! [A:$tType]: (((preorder @ A) => ((condit1013018076250108175_below @ A) = ((condit16957441358409770ng_bdd @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2))))))))).
% 0.33/5.23 thf(fact_6735_Quotient__natural, axiom, ((((((quotient @ nat) @ code_natural) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) @ code_natural_of_nat) @ code_nat_of_natural) @ code_cr_natural)).
% 0.33/5.23 thf(fact_6736_bdd__above__primitive__def, axiom, (! [A:$tType]: (((preorder @ A) => ((condit941137186595557371_above @ A) = ((condit16957441358409770ng_bdd @ A) @ (ord_less_eq @ A))))))).
% 0.33/5.23 thf(fact_6737_card_Ofolding__on__axioms, axiom, (! [A:$tType]: (((((finite_folding_on @ A) @ nat) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (suc)))))).
% 0.33/5.23 thf(fact_6738_bdd__below_Opreordering__bdd__axioms, axiom, (! [A:$tType]: (((preorder @ A) => (((condit622319405099724424ng_bdd @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_6739_bdd__above_Opreordering__bdd__axioms, axiom, (! [A:$tType]: (((preorder @ A) => (((condit622319405099724424ng_bdd @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_6740_Quotient__integer, axiom, ((((((quotient @ int) @ code_integer) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) @ code_integer_of_int) @ code_int_of_integer) @ code_cr_integer)).
% 0.33/5.23 thf(fact_6741_type__definition__integer, axiom, (((((type_definition @ code_integer) @ int) @ code_int_of_integer) @ code_integer_of_int) @ (top_top @ (set @ int)))).
% 0.33/5.23 thf(fact_6742_finite__subset__Union__chain, axiom, (! [A:$tType,A6:(set @ A),B12:(set @ (set @ A)),A16:(set @ (set @ A))]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ ((complete_Sup_Sup @ (set @ A)) @ B12)) => ((B12 != (bot_bot @ (set @ (set @ A)))) => (((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ B12) => (~ ((! [B9:(set @ A)]: (((((member @ (set @ A)) @ B9) @ B12) => (~ ((((ord_less_eq @ (set @ A)) @ A6) @ B9))))))))))))))).
% 0.33/5.23 thf(fact_6743_wait__def, axiom, (heap_Time_wait = (^ [N3:nat]: (((heap_Time_Heap2 @ product_unit) @ (^ [H4:(heap_ext @ product_unit)]: (((some @ ((product_prod @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ product_Unity) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ N3)))))))))).
% 0.33/5.23 thf(fact_6744_subset_Ochain__empty, axiom, (! [A:$tType,A6:(set @ (set @ A))]: (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ (bot_bot @ (set @ (set @ A))))))).
% 0.33/5.23 thf(fact_6745_subset__chain__def, axiom, (! [A:$tType,A16:(set @ (set @ A)),C7:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ C7) = ((((ord_less_eq @ (set @ (set @ A))) @ C7) @ A16) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C7) => (! [Y6:(set @ A)]: (((((member @ (set @ A)) @ Y6) @ C7) => ((((ord_less_eq @ (set @ A)) @ X2) @ Y6) | (((ord_less_eq @ (set @ A)) @ Y6) @ X2))))))))))))).
% 0.33/5.23 thf(fact_6746_subset__Zorn, axiom, (! [A:$tType,A6:(set @ (set @ A))]: (((! [C8:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C8) => (? [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ A6) & (! [Xa2:(set @ A)]: (((((member @ (set @ A)) @ Xa2) @ C8) => (((ord_less_eq @ (set @ A)) @ Xa2) @ X4)))))))))) => (? [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ A6) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A6) => ((((ord_less_eq @ (set @ A)) @ X3) @ Xa) => (Xa = X3)))))))))))).
% 0.33/5.23 thf(fact_6747_subset__chain__insert, axiom, (! [A:$tType,A16:(set @ (set @ A)),B7:(set @ A),B12:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ (((insert @ (set @ A)) @ B7) @ B12)) = ((((member @ (set @ A)) @ B7) @ A16) & ((! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ B12) => ((((ord_less_eq @ (set @ A)) @ X2) @ B7) | (((ord_less_eq @ (set @ A)) @ B7) @ X2))))) & ((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ B12))))))).
% 0.33/5.23 thf(fact_6748_subset__Zorn_H, axiom, (! [A:$tType,A6:(set @ (set @ A))]: (((! [C8:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C8) => (((member @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ C8)) @ A6)))) => (? [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ A6) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A6) => ((((ord_less_eq @ (set @ A)) @ X3) @ Xa) => (Xa = X3)))))))))))).
% 0.33/5.23 thf(fact_6749_subset_Ochain__extend, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A)),Z3:(set @ A)]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((((member @ (set @ A)) @ Z3) @ A6) => ((! [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ C6) => (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ X3) @ Z3)))) => ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ (((sup_sup @ (set @ (set @ A))) @ (((insert @ (set @ A)) @ Z3) @ (bot_bot @ (set @ (set @ A))))) @ C6)))))))).
% 0.33/5.23 thf(fact_6750_subset_Ochain__total, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A)),X:(set @ A),Y3:(set @ A)]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((((member @ (set @ A)) @ X) @ C6) => ((((member @ (set @ A)) @ Y3) @ C6) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ X) @ Y3) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ Y3) @ X)))))))).
% 0.33/5.23 thf(fact_6751_subset_OchainI, axiom, (! [A:$tType,C6:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ C6) @ A6) => ((! [X3:(set @ A),Y5:(set @ A)]: (((((member @ (set @ A)) @ X3) @ C6) => ((((member @ (set @ A)) @ Y5) @ C6) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ X3) @ Y5) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ Y5) @ X3)))))) => ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6)))))).
% 0.33/5.23 thf(fact_6752_subset_Ochain__def, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) = ((((ord_less_eq @ (set @ (set @ A))) @ C6) @ A6) & (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C6) => (! [Y6:(set @ A)]: (((((member @ (set @ A)) @ Y6) @ C6) => ((((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ X2) @ Y6) | (((((sup_sup @ ((set @ A) > ((set @ A) > $o))) @ (ord_less @ (set @ A))) @ (^ [Y:(set @ A),Z2:(set @ A)]: ((Y = Z2)))) @ Y6) @ X2))))))))))))).
% 0.33/5.23 thf(fact_6753_uminus__unit__def, axiom, ((uminus_uminus @ product_unit) = (^ [Uu2:product_unit]: (product_Unity)))).
% 0.33/5.23 thf(fact_6754_subset__Zorn__nonempty, axiom, (! [A:$tType,A16:(set @ (set @ A))]: (((A16 != (bot_bot @ (set @ (set @ A)))) => ((! [C9:(set @ (set @ A))]: (((C9 != (bot_bot @ (set @ (set @ A)))) => (((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ C9) => (((member @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ C9)) @ A16))))) => (? [X3:(set @ A)]: (((((member @ (set @ A)) @ X3) @ A16) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A16) => ((((ord_less_eq @ (set @ A)) @ X3) @ Xa) => (Xa = X3))))))))))))).
% 0.33/5.23 thf(fact_6755_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: (((wellorder @ A) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R3:B,F3:(A > B),G5:(A > B),X3:A]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X3) => ((F3 @ Y4) = (G5 @ Y4))))) => ((((P @ F3) @ X3) @ R3) = (((P @ G5) @ X3) @ R3))))) => ((! [X3:A,F3:(A > B)]: (((! [Y4:A]: (((((ord_less @ A) @ Y4) @ X3) => (((P @ F3) @ Y4) @ (F3 @ Y4))))) => (? [X_1:B]: ((((P @ F3) @ X3) @ X_1)))))) => (? [F3:(A > B)]: ((! [X4:A]: ((((P @ F3) @ X4) @ (F3 @ X4)))))))))))))).
% 0.33/5.23 thf(fact_6756_Union__in__chain, axiom, (! [A:$tType,B12:(set @ (set @ A)),A16:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ B12) => ((B12 != (bot_bot @ (set @ (set @ A)))) => (((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ B12) => (((member @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ B12)) @ B12))))))).
% 0.33/5.23 thf(fact_6757_Inter__in__chain, axiom, (! [A:$tType,B12:(set @ (set @ A)),A16:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ B12) => ((B12 != (bot_bot @ (set @ (set @ A)))) => (((((pred_chain @ (set @ A)) @ A16) @ (ord_less @ (set @ A))) @ B12) => (((member @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ B12)) @ B12))))))).
% 0.33/5.23 thf(fact_6758_chain__subset__alt__def, axiom, (! [A:$tType]: (((chain_subset @ A) = (((pred_chain @ (set @ A)) @ (top_top @ (set @ (set @ A)))) @ (ord_less @ (set @ A))))))).
% 0.33/5.23 thf(fact_6759_chains__alt__def, axiom, (! [A:$tType]: (((chains @ A) = (^ [A7:(set @ (set @ A))]: (((collect @ (set @ (set @ A))) @ (((pred_chain @ (set @ A)) @ A7) @ (ord_less @ (set @ A)))))))))).
% 0.33/5.23 thf(fact_6760_sup2CI, axiom, (! [A:$tType,B:$tType,B7:(A > (B > $o)),X:A,Y3:B,A6:(A > (B > $o))]: ((((~ (((B7 @ X) @ Y3))) => ((A6 @ X) @ Y3)) => (((((sup_sup @ (A > (B > $o))) @ A6) @ B7) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_6761_sup2I2, axiom, (! [A:$tType,B:$tType,B7:(A > (B > $o)),X:A,Y3:B,A6:(A > (B > $o))]: ((((B7 @ X) @ Y3) => (((((sup_sup @ (A > (B > $o))) @ A6) @ B7) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_6762_sup2I1, axiom, (! [A:$tType,B:$tType,A6:(A > (B > $o)),X:A,Y3:B,B7:(A > (B > $o))]: ((((A6 @ X) @ Y3) => (((((sup_sup @ (A > (B > $o))) @ A6) @ B7) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_6763_sup2E, axiom, (! [A:$tType,B:$tType,A6:(A > (B > $o)),B7:(A > (B > $o)),X:A,Y3:B]: (((((((sup_sup @ (A > (B > $o))) @ A6) @ B7) @ X) @ Y3) => ((~ (((A6 @ X) @ Y3))) => ((B7 @ X) @ Y3)))))).
% 0.33/5.23 thf(fact_6764_pred__on_Onot__maxchain__Some, axiom, (! [A:$tType,A6:(set @ A),P:(A > (A > $o)),C6:(set @ A)]: ((((((pred_chain @ A) @ A6) @ P) @ C6) => ((~ (((((pred_maxchain @ A) @ A6) @ P) @ C6))) => (((((pred_chain @ A) @ A6) @ P) @ ((fChoice @ (set @ A)) @ (^ [D7:(set @ A)]: ((((((pred_chain @ A) @ A6) @ P) @ D7) & (((ord_less @ (set @ A)) @ C6) @ D7)))))) & (((ord_less @ (set @ A)) @ C6) @ ((fChoice @ (set @ A)) @ (^ [D7:(set @ A)]: ((((((pred_chain @ A) @ A6) @ P) @ D7) & (((ord_less @ (set @ A)) @ C6) @ D7)))))))))))).
% 0.33/5.23 thf(fact_6765_subset__maxchain__max, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A)),X8:(set @ A)]: ((((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((((member @ (set @ A)) @ X8) @ A6) => ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ C6)) @ X8) => (((complete_Sup_Sup @ (set @ A)) @ C6) = X8))))))).
% 0.33/5.23 thf(fact_6766_subset_OHausdorff, axiom, (! [A:$tType,A6:(set @ (set @ A))]: ((? [X_12:(set @ (set @ A))]: (((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X_12)))))).
% 0.33/5.23 thf(fact_6767_subset_Omaxchain__def, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: ((((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) = (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) & (~ ((? [S7:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ S7) & (((ord_less @ (set @ (set @ A))) @ C6) @ S7))))))))))).
% 0.33/5.23 thf(fact_6768_subset_Omaxchain__imp__chain, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: ((((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6))))).
% 0.33/5.23 thf(fact_6769_subset_Onot__maxchain__Some, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((~ (((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6))) => (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((fChoice @ (set @ (set @ A))) @ (^ [D7:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ D7) & (((ord_less @ (set @ (set @ A))) @ C6) @ D7)))))) & (((ord_less @ (set @ (set @ A))) @ C6) @ ((fChoice @ (set @ (set @ A))) @ (^ [D7:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ D7) & (((ord_less @ (set @ (set @ A))) @ C6) @ D7)))))))))))).
% 0.33/5.23 thf(fact_6770_pred__on_Omaxchain__def, axiom, (! [A:$tType]: (((pred_maxchain @ A) = (^ [A7:(set @ A),P4:(A > (A > $o)),C10:(set @ A)]: ((((((pred_chain @ A) @ A7) @ P4) @ C10) & (~ ((? [S7:(set @ A)]: ((((((pred_chain @ A) @ A7) @ P4) @ S7) & (((ord_less @ (set @ A)) @ C10) @ S7))))))))))))).
% 0.33/5.23 thf(fact_6771_pred__on_Osuc__def, axiom, (! [A:$tType]: (((pred_suc @ A) = (^ [A7:(set @ A),P4:(A > (A > $o)),C10:(set @ A)]: (((((if @ (set @ A)) @ ((~ (((((pred_chain @ A) @ A7) @ P4) @ C10))) | ((((pred_maxchain @ A) @ A7) @ P4) @ C10))) @ C10) @ ((fChoice @ (set @ A)) @ (^ [D7:(set @ A)]: ((((((pred_chain @ A) @ A7) @ P4) @ D7) & (((ord_less @ (set @ A)) @ C10) @ D7)))))))))))).
% 0.33/5.23 thf(fact_6772_subset_Osuc__def, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: (((((~ (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6))) | ((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6)) => (((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) = C6)) & ((~ (((~ (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6))) | ((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6)))) => (((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) = ((fChoice @ (set @ (set @ A))) @ (^ [D7:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ D7) & (((ord_less @ (set @ (set @ A))) @ C6) @ D7))))))))))).
% 0.33/5.23 thf(fact_6773_subset_Onot__chain__suc, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ A))]: (((~ (((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8))) => (((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) = X8))))).
% 0.33/5.23 thf(fact_6774_subset_Omaxchain__suc, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ A))]: ((((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) => (((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) = X8))))).
% 0.33/5.23 thf(fact_6775_subset_Ochain__suc, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) => ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8)))))).
% 0.33/5.23 thf(fact_6776_subset_Osubset__suc, axiom, (! [A:$tType,X8:(set @ (set @ A)),Y7:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ X8) @ Y7) => (((ord_less_eq @ (set @ (set @ A))) @ X8) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Y7)))))).
% 0.33/5.23 thf(fact_6777_subset_Osuc__subset, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: ((((ord_less_eq @ (set @ (set @ A))) @ X8) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8))))).
% 0.33/5.23 thf(fact_6778_subset_Osuc__in__carrier, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ X8) @ A6) => (((ord_less_eq @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8)) @ A6))))).
% 0.33/5.23 thf(fact_6779_subset_Ochain__sucD, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) => ((((ord_less_eq @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8)) @ A6) & ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8))))))).
% 0.33/5.23 thf(fact_6780_subset_Osuc__not__equals, axiom, (! [A:$tType,A6:(set @ (set @ A)),C6:(set @ (set @ A))]: ((((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) => ((~ (((((pred_maxchain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6))) => (((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ C6) != C6)))))).
% 0.33/5.23 thf(fact_6781_alloc__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_alloc @ A) = (^ [Xs3:(list @ A),H4:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ ((array2 @ A) @ ((lim @ product_unit) @ H4))) @ ((((array_set @ A) @ ((array2 @ A) @ ((lim @ product_unit) @ H4))) @ Xs3) @ (((lim_update @ product_unit) @ (^ [Uu:nat]: ((((plus_plus @ nat) @ ((lim @ product_unit) @ H4)) @ (one_one @ nat))))) @ H4)))))))))).
% 0.33/5.23 thf(fact_6782_Code__Numeral_Osub__def, axiom, (code_sub = (((((((map_fun @ num) @ num) @ (num > int)) @ (num > code_integer)) @ (id @ num)) @ ((((((map_fun @ num) @ num) @ int) @ code_integer) @ (id @ num)) @ code_integer_of_int)) @ (^ [M2:num,N3:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N3))))))).
% 0.33/5.23 thf(fact_6783_present__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_present @ A) = (^ [H4:(heap_ext @ product_unit),A5:(array @ A)]: ((((ord_less @ nat) @ ((addr_of_array @ A) @ A5)) @ ((lim @ product_unit) @ H4))))))))).
% 0.33/5.23 thf(fact_6784_array_Osize_I2_J, axiom, (! [A:$tType,X:nat]: ((((size_size @ (array @ A)) @ ((array2 @ A) @ X)) = (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6785_array_Osize__gen, axiom, (! [A:$tType,Xa3:(A > nat),X:nat]: (((((size_array @ A) @ Xa3) @ ((array2 @ A) @ X)) = (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6786_noteq__def, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (((array_noteq @ A) @ B) = (^ [R5:(array @ A),S6:(array @ B)]: (((((typerep_typerep @ A) @ (type2 @ A)) != ((typerep_typerep @ B) @ (type2 @ B))) | (((addr_of_array @ A) @ R5) != ((addr_of_array @ B) @ S6)))))))))).
% 0.33/5.23 thf(fact_6787_get__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_get @ A) = (^ [H4:(heap_ext @ product_unit),A5:(array @ A)]: (((((map @ nat) @ A) @ (from_nat @ A)) @ ((((arrays @ product_unit) @ H4) @ ((typerep_typerep @ A) @ (type2 @ A))) @ ((addr_of_array @ A) @ A5)))))))))).
% 0.33/5.23 thf(fact_6788_Heap_Oempty__def, axiom, (empty = (((((heap_ext2 @ product_unit) @ (^ [Uu:typerep,Uv:nat]: ((nil @ nat)))) @ (^ [Uu:typerep,Uv:nat]: ((zero_zero @ nat)))) @ (zero_zero @ nat)) @ product_Unity))).
% 0.33/5.23 thf(fact_6789_coset__def, axiom, (! [A:$tType]: (((coset @ A) = (^ [Xs3:(list @ A)]: (((uminus_uminus @ (set @ A)) @ ((set2 @ A) @ Xs3)))))))).
% 0.33/5.23 thf(fact_6790_compl__coset, axiom, (! [A:$tType,Xs:(list @ A)]: ((((uminus_uminus @ (set @ A)) @ ((coset @ A) @ Xs)) = ((set2 @ A) @ Xs))))).
% 0.33/5.23 thf(fact_6791_Nitpick_Otranclp__unfold, axiom, (! [A:$tType]: (((transitive_tranclp @ A) = (^ [R5:(A > (A > $o)),A5:A,B3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B3)) @ ((transitive_trancl @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R5)))))))))).
% 0.33/5.23 thf(fact_6792_Field__natLeq__on, axiom, (! [N:nat]: ((((field2 @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6)))))))) = ((collect @ nat) @ (^ [X2:nat]: ((((ord_less @ nat) @ X2) @ N)))))))).
% 0.33/5.23 thf(fact_6793_less__nat__rel, axiom, ((ord_less @ nat) = ((transitive_tranclp @ nat) @ (^ [M2:nat,N3:nat]: ((N3 = (suc @ M2))))))).
% 0.33/5.23 thf(fact_6794_tranclp__power, axiom, (! [A:$tType]: (((transitive_tranclp @ A) = (^ [P4:(A > (A > $o)),X2:A,Y6:A]: ((? [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((((compow @ (A > (A > $o))) @ N3) @ P4) @ X2) @ Y6)))))))))).
% 0.33/5.23 thf(fact_6795_underS__Field3, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((field2 @ A) @ R) != (bot_bot @ (set @ A))) => (((ord_less @ (set @ A)) @ (((order_underS @ A) @ R) @ A3)) @ ((field2 @ A) @ R)))))).
% 0.33/5.23 thf(fact_6796_underS__Field2, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ A3) @ ((field2 @ A) @ R)) => (((ord_less @ (set @ A)) @ (((order_underS @ A) @ R) @ A3)) @ ((field2 @ A) @ R)))))).
% 0.33/5.23 thf(fact_6797_natLeq__on__Well__order, axiom, (! [N:nat]: ((((order_well_order_on @ nat) @ ((field2 @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))))).
% 0.33/5.23 thf(fact_6798_Quotient3__int, axiom, (((((quotient3 @ ((product_prod @ nat) @ nat)) @ int) @ intrel) @ abs_Integ) @ rep_Integ)).
% 0.33/5.23 thf(fact_6799_natLeq__on__well__order__on, axiom, (! [N:nat]: ((((order_well_order_on @ nat) @ ((collect @ nat) @ (^ [X2:nat]: ((((ord_less @ nat) @ X2) @ N))))) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X2:nat,Y6:nat]: (((((ord_less @ nat) @ X2) @ N) & ((((ord_less @ nat) @ Y6) @ N) & (((ord_less_eq @ nat) @ X2) @ Y6))))))))))).
% 0.33/5.23 thf(fact_6800_embedS__Field, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((((bNF_Wellorder_embedS @ A) @ B) @ R) @ R4) @ F) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((field2 @ A) @ R))) @ ((field2 @ B) @ R4))))))).
% 0.33/5.23 thf(fact_6801_list__all__length, axiom, (! [A:$tType]: (((list_all @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((! [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs3)) => (P4 @ (((nth @ A) @ Xs3) @ N3))))))))))).
% 0.33/5.23 thf(fact_6802_ofilter__subset__embedS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A6:(set @ A),B7:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A6) => ((((order_ofilter @ A) @ R) @ B7) => ((((ord_less @ (set @ A)) @ A6) @ B7) = (((((bNF_Wellorder_embedS @ A) @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A6) @ (^ [Uu:A]: (A6))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B7) @ (^ [Uu:A]: (B7))))) @ (id @ A))))))))).
% 0.33/5.23 thf(fact_6803_embedS__iff, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((((bNF_Wellorder_embed @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embedS @ A) @ B) @ R) @ R4) @ F) = (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((field2 @ A) @ R))) @ ((field2 @ B) @ R4)))))))).
% 0.33/5.23 thf(fact_6804_ofilterIncl__def, axiom, (! [A:$tType]: (((bNF_We413866401316099525erIncl @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [A7:(set @ A),B6:(set @ A)]: (((((order_ofilter @ A) @ R5) @ A7) & ((A7 != ((field2 @ A) @ R5)) & ((((order_ofilter @ A) @ R5) @ B6) & ((B6 != ((field2 @ A) @ R5)) & (((ord_less @ (set @ A)) @ A7) @ B6))))))))))))))).
% 0.33/5.23 thf(fact_6805_ofilter__subset__embedS__iso, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A6:(set @ A),B7:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A6) => ((((order_ofilter @ A) @ R) @ B7) => (((((ord_less @ (set @ A)) @ A6) @ B7) = (((((bNF_Wellorder_embedS @ A) @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A6) @ (^ [Uu:A]: (A6))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B7) @ (^ [Uu:A]: (B7))))) @ (id @ A))) & ((A6 = B7) = (((((bNF_Wellorder_iso @ A) @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A6) @ (^ [Uu:A]: (A6))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B7) @ (^ [Uu:A]: (B7))))) @ (id @ A)))))))))).
% 0.33/5.23 thf(fact_6806_ofilter__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A6:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A6) => ((((ord_less @ (set @ A)) @ A6) @ ((field2 @ A) @ R)) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A6) @ (^ [Uu:A]: (A6))))) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ A)))))))).
% 0.33/5.23 thf(fact_6807_ofilter__subset__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A6:(set @ A),B7:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A6) => ((((order_ofilter @ A) @ R) @ B7) => ((((ord_less @ (set @ A)) @ A6) @ B7) = (((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) @ A6) @ (^ [Uu:A]: (A6))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B7) @ (^ [Uu:A]: (B7)))))) @ ((bNF_We4044943003108391690rdLess @ A) @ A))))))))).
% 0.33/5.23 thf(fact_6808_bsqr__ofilter, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),D4:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ ((product_prod @ A) @ A)) @ ((bNF_Wellorder_bsqr @ A) @ R)) @ D4) => ((((ord_less @ (set @ ((product_prod @ A) @ A))) @ D4) @ ((((product_Sigma @ A) @ A) @ ((field2 @ A) @ R)) @ (^ [Uu:A]: (((field2 @ A) @ R))))) => ((~ ((? [A8:A]: ((((field2 @ A) @ R) = (((order_under @ A) @ R) @ A8)))))) => (? [A9:(set @ A)]: (((((order_ofilter @ A) @ R) @ A9) & ((((ord_less @ (set @ A)) @ A9) @ ((field2 @ A) @ R)) & (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ D4) @ ((((product_Sigma @ A) @ A) @ A9) @ (^ [Uu:A]: (A9))))))))))))))).
% 0.33/5.23 thf(fact_6809_internalize__ordLess, axiom, (! [A:$tType,B:$tType,R4:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) = (? [P6:(set @ ((product_prod @ B) @ B))]: (((((ord_less @ (set @ B)) @ ((field2 @ B) @ P6)) @ ((field2 @ B) @ R)) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ P6)) @ ((bNF_Wellorder_ordIso @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ P6) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ B)))))))))).
% 0.33/5.23 thf(fact_6810_of__int__code_I1_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ (neg @ K)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))))))))).
% 0.33/5.23 thf(fact_6811_type__definition__natural, axiom, (((((type_definition @ code_natural) @ nat) @ code_nat_of_natural) @ code_natural_of_nat) @ (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6812_less__int__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less @ int) @ (zero_zero @ int)) @ (neg @ L))))))).
% 0.33/5.23 thf(fact_6813_less__int__code_I7_J, axiom, (! [K:num]: ((((ord_less @ int) @ (neg @ K)) @ (zero_zero @ int))))).
% 0.33/5.23 thf(fact_6814_less__eq__int__code_I7_J, axiom, (! [K:num]: ((((ord_less_eq @ int) @ (neg @ K)) @ (zero_zero @ int))))).
% 0.33/5.23 thf(fact_6815_less__eq__int__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (neg @ L))))))).
% 0.33/5.23 thf(fact_6816_nat__code_I1_J, axiom, (! [K:num]: (((nat2 @ (neg @ K)) = (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6817_plus__int__code_I6_J, axiom, (! [M:num,N:num]: (((((plus_plus @ int) @ (neg @ M)) @ (neg @ N)) = (neg @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6818_less__eq__int__code_I9_J, axiom, (! [K:num,L:num]: (((((ord_less_eq @ int) @ (neg @ K)) @ (neg @ L)) = (((ord_less_eq @ num) @ L) @ K))))).
% 0.33/5.23 thf(fact_6819_less__int__code_I9_J, axiom, (! [K:num,L:num]: (((((ord_less @ int) @ (neg @ K)) @ (neg @ L)) = (((ord_less @ num) @ L) @ K))))).
% 0.33/5.23 thf(fact_6820_Int_Osub__code_I4_J, axiom, (! [N:num]: ((((sub @ one2) @ (bit0 @ N)) = (neg @ (bitM @ N)))))).
% 0.33/5.23 thf(fact_6821_Int_Osub__code_I5_J, axiom, (! [N:num]: ((((sub @ one2) @ (bit1 @ N)) = (neg @ (bit0 @ N)))))).
% 0.33/5.23 thf(fact_6822_Int_Osub__code_I1_J, axiom, (((sub @ one2) @ one2) = (zero_zero @ int))).
% 0.33/5.23 thf(fact_6823_Int_Osub__def, axiom, (sub = (^ [M2:num,N3:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N3)))))).
% 0.33/5.23 thf(fact_6824_minus__int__code_I6_J, axiom, (! [M:num,N:num]: (((((minus_minus @ int) @ (neg @ M)) @ (neg @ N)) = ((sub @ N) @ M))))).
% 0.33/5.23 thf(fact_6825_Int_Osub__code_I9_J, axiom, (! [M:num,N:num]: ((((sub @ (bit0 @ M)) @ (bit1 @ N)) = (((minus_minus @ int) @ (dup @ ((sub @ M) @ N))) @ (one_one @ int)))))).
% 0.33/5.23 thf(fact_6826_Int_Osub__code_I8_J, axiom, (! [M:num,N:num]: ((((sub @ (bit1 @ M)) @ (bit0 @ N)) = (((plus_plus @ int) @ (dup @ ((sub @ M) @ N))) @ (one_one @ int)))))).
% 0.33/5.23 thf(fact_6827_Int_Odup__code_I1_J, axiom, ((dup @ (zero_zero @ int)) = (zero_zero @ int))).
% 0.33/5.23 thf(fact_6828_Int_Odup__def, axiom, (dup = (^ [K3:int]: ((((plus_plus @ int) @ K3) @ K3))))).
% 0.33/5.23 thf(fact_6829_Int_Odup__code_I3_J, axiom, (! [N:num]: (((dup @ (neg @ N)) = (neg @ (bit0 @ N)))))).
% 0.33/5.23 thf(fact_6830_Int_Osub__code_I6_J, axiom, (! [M:num,N:num]: ((((sub @ (bit0 @ M)) @ (bit0 @ N)) = (dup @ ((sub @ M) @ N)))))).
% 0.33/5.23 thf(fact_6831_Int_Osub__code_I7_J, axiom, (! [M:num,N:num]: ((((sub @ (bit1 @ M)) @ (bit1 @ N)) = (dup @ ((sub @ M) @ N)))))).
% 0.33/5.23 thf(fact_6832_Int_Osub__code_I2_J, axiom, (! [M:num]: ((((sub @ (bit0 @ M)) @ one2) = (pos @ (bitM @ M)))))).
% 0.33/5.23 thf(fact_6833_Int_Osub__code_I3_J, axiom, (! [M:num]: ((((sub @ (bit1 @ M)) @ one2) = (pos @ (bit0 @ M)))))).
% 0.33/5.23 thf(fact_6834_less__int__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less @ int) @ (neg @ K)) @ (pos @ L))))).
% 0.33/5.23 thf(fact_6835_less__int__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less @ int) @ (pos @ K)) @ (neg @ L))))))).
% 0.33/5.23 thf(fact_6836_less__eq__int__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less_eq @ int) @ (pos @ K)) @ (neg @ L))))))).
% 0.33/5.23 thf(fact_6837_less__eq__int__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less_eq @ int) @ (neg @ K)) @ (pos @ L))))).
% 0.33/5.23 thf(fact_6838_Int_ONeg__def, axiom, (neg = (^ [N3:num]: (((uminus_uminus @ int) @ (pos @ N3)))))).
% 0.33/5.23 thf(fact_6839_uminus__int__code_I2_J, axiom, (! [M:num]: ((((uminus_uminus @ int) @ (pos @ M)) = (neg @ M))))).
% 0.33/5.23 thf(fact_6840_uminus__int__code_I3_J, axiom, (! [M:num]: ((((uminus_uminus @ int) @ (neg @ M)) = (pos @ M))))).
% 0.33/5.23 thf(fact_6841_nat__code_I3_J, axiom, (! [K:num]: (((nat2 @ (pos @ K)) = (nat_of_num @ K))))).
% 0.33/5.23 thf(fact_6842_times__int__code_I3_J, axiom, (! [M:num,N:num]: (((((times_times @ int) @ (pos @ M)) @ (pos @ N)) = (pos @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6843_Int_OPos__def, axiom, (pos = (numeral_numeral @ int))).
% 0.33/5.23 thf(fact_6844_less__eq__int__code_I2_J, axiom, (! [L:num]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (pos @ L))))).
% 0.33/5.23 thf(fact_6845_less__eq__int__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less_eq @ int) @ (pos @ K)) @ (zero_zero @ int))))))).
% 0.33/5.23 thf(fact_6846_of__int__code_I3_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ (pos @ K)) = ((numeral_numeral @ A) @ K)))))))).
% 0.33/5.23 thf(fact_6847_one__int__code, axiom, ((one_one @ int) = (pos @ one2))).
% 0.33/5.23 thf(fact_6848_plus__int__code_I3_J, axiom, (! [M:num,N:num]: (((((plus_plus @ int) @ (pos @ M)) @ (pos @ N)) = (pos @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6849_less__int__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less @ int) @ (pos @ K)) @ (zero_zero @ int))))))).
% 0.33/5.23 thf(fact_6850_less__int__code_I2_J, axiom, (! [L:num]: ((((ord_less @ int) @ (zero_zero @ int)) @ (pos @ L))))).
% 0.33/5.23 thf(fact_6851_less__eq__int__code_I5_J, axiom, (! [K:num,L:num]: (((((ord_less_eq @ int) @ (pos @ K)) @ (pos @ L)) = (((ord_less_eq @ num) @ K) @ L))))).
% 0.33/5.23 thf(fact_6852_less__int__code_I5_J, axiom, (! [K:num,L:num]: (((((ord_less @ int) @ (pos @ K)) @ (pos @ L)) = (((ord_less @ num) @ K) @ L))))).
% 0.33/5.23 thf(fact_6853_minus__int__code_I3_J, axiom, (! [M:num,N:num]: (((((minus_minus @ int) @ (pos @ M)) @ (pos @ N)) = ((sub @ M) @ N))))).
% 0.33/5.23 thf(fact_6854_Int_Odup__code_I2_J, axiom, (! [N:num]: (((dup @ (pos @ N)) = (pos @ (bit0 @ N)))))).
% 0.33/5.23 thf(fact_6855_minus__int__code_I4_J, axiom, (! [M:num,N:num]: (((((minus_minus @ int) @ (pos @ M)) @ (neg @ N)) = (pos @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6856_minus__int__code_I5_J, axiom, (! [M:num,N:num]: (((((minus_minus @ int) @ (neg @ M)) @ (pos @ N)) = (neg @ (((plus_plus @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6857_times__int__code_I4_J, axiom, (! [M:num,N:num]: (((((times_times @ int) @ (pos @ M)) @ (neg @ N)) = (neg @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6858_times__int__code_I5_J, axiom, (! [M:num,N:num]: (((((times_times @ int) @ (neg @ M)) @ (pos @ N)) = (neg @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6859_times__int__code_I6_J, axiom, (! [M:num,N:num]: (((((times_times @ int) @ (neg @ M)) @ (neg @ N)) = (pos @ (((times_times @ num) @ M) @ N)))))).
% 0.33/5.23 thf(fact_6860_plus__int__code_I5_J, axiom, (! [M:num,N:num]: (((((plus_plus @ int) @ (neg @ M)) @ (pos @ N)) = ((sub @ N) @ M))))).
% 0.33/5.23 thf(fact_6861_plus__int__code_I4_J, axiom, (! [M:num,N:num]: (((((plus_plus @ int) @ (pos @ M)) @ (neg @ N)) = ((sub @ M) @ N))))).
% 0.33/5.23 thf(fact_6862_prod__list_Ocomm__monoid__list__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups1828464146339083142d_list @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.23 thf(fact_6863_Least__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_Least @ A) = (^ [P4:(A > $o)]: (((the @ A) @ (^ [X2:A]: (((P4 @ X2) & (! [Y6:A]: (((P4 @ Y6) => (((ord_less_eq @ A) @ X2) @ Y6))))))))))))))).
% 0.33/5.23 thf(fact_6864_Ball__def__raw, axiom, (! [A:$tType]: (((ball @ A) = (^ [A7:(set @ A),P4:(A > $o)]: ((! [X2:A]: (((((member @ A) @ X2) @ A7) => (P4 @ X2)))))))))).
% 0.33/5.23 thf(fact_6865_sum__list_Ocomm__monoid__list__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups1828464146339083142d_list @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.23 thf(fact_6866_Greatest__def, axiom, (! [A:$tType]: (((order @ A) => ((order_Greatest @ A) = (^ [P4:(A > $o)]: (((the @ A) @ (^ [X2:A]: (((P4 @ X2) & (! [Y6:A]: (((P4 @ Y6) => (((ord_less_eq @ A) @ Y6) @ X2))))))))))))))).
% 0.33/5.23 thf(fact_6867_ord_OLeast__def, axiom, (! [A:$tType]: (((least @ A) = (^ [Less_eq2:(A > (A > $o)),P4:(A > $o)]: (((the @ A) @ (^ [X2:A]: (((P4 @ X2) & (! [Y6:A]: (((P4 @ Y6) => ((Less_eq2 @ X2) @ Y6)))))))))))))).
% 0.33/5.23 thf(fact_6868_ord_OLeast_Ocong, axiom, (! [A:$tType]: (((least @ A) = (least @ A))))).
% 0.33/5.23 thf(fact_6869_GreatestI__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))).
% 0.33/5.23 thf(fact_6870_Greatest__le__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ B2)))) => (((ord_less_eq @ nat) @ K) @ ((order_Greatest @ nat) @ P))))))).
% 0.33/5.23 thf(fact_6871_GreatestI__ex__nat, axiom, (! [P:(nat > $o),B2:nat]: (((? [X_1:nat]: ((P @ X_1))) => ((! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))).
% 0.33/5.23 thf(fact_6872_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X)))) => ((! [X3:A]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 0.33/5.23 thf(fact_6873_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X)))) => (((order_Greatest @ A) @ P) = X))))))))).
% 0.33/5.23 thf(fact_6874_sum__encode__def, axiom, (nat_sum_encode = (((((sum_case_sum @ nat) @ nat) @ nat) @ ((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (^ [B3:nat]: ((suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B3))))))).
% 0.33/5.23 thf(fact_6875_disjnt__ge__max, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y7:(set @ A),X8:(set @ A)]: ((((finite_finite2 @ A) @ Y7) => ((! [X3:A]: (((((member @ A) @ X3) @ X8) => (((ord_less @ A) @ ((lattic643756798349783984er_Max @ A) @ Y7)) @ X3)))) => (((disjnt @ A) @ X8) @ Y7))))))))).
% 0.33/5.23 thf(fact_6876_inj__sum__encode, axiom, (! [A6:(set @ ((sum_sum @ nat) @ nat))]: (((((inj_on @ ((sum_sum @ nat) @ nat)) @ nat) @ nat_sum_encode) @ A6)))).
% 0.33/5.23 thf(fact_6877_sum__encode__eq, axiom, (! [X:((sum_sum @ nat) @ nat),Y3:((sum_sum @ nat) @ nat)]: ((((nat_sum_encode @ X) = (nat_sum_encode @ Y3)) = (X = Y3))))).
% 0.33/5.23 thf(fact_6878_bij__sum__encode, axiom, (((((bij_betw @ ((sum_sum @ nat) @ nat)) @ nat) @ nat_sum_encode) @ (top_top @ (set @ ((sum_sum @ nat) @ nat)))) @ (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6879_surj__sum__encode, axiom, (((((image @ ((sum_sum @ nat) @ nat)) @ nat) @ nat_sum_encode) @ (top_top @ (set @ ((sum_sum @ nat) @ nat)))) = (top_top @ (set @ nat)))).
% 0.33/5.23 thf(fact_6880_int__encode__def, axiom, (nat_int_encode = (^ [I5:int]: ((nat_sum_encode @ ((((if @ ((sum_sum @ nat) @ nat)) @ (((ord_less_eq @ int) @ (zero_zero @ int)) @ I5)) @ (((sum_Inl @ nat) @ nat) @ (nat2 @ I5))) @ (((sum_Inr @ nat) @ nat) @ (nat2 @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ I5)) @ (one_one @ int)))))))))).
% 0.33/5.23 thf(fact_6881_ordering__top_Oaxioms_I2_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((ordering_top_axioms @ A) @ Less_eq) @ Top))))).
% 0.33/5.23 thf(fact_6882_prod_OPlus, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [A6:(set @ B),B7:(set @ C),G:(((sum_sum @ B) @ C) > A)]: ((((finite_finite2 @ B) @ A6) => (((finite_finite2 @ C) @ B7) => (((((groups7121269368397514597t_prod @ ((sum_sum @ B) @ C)) @ A) @ G) @ ((((sum_Plus @ B) @ C) @ A6) @ B7)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ ((sum_sum @ B) @ C)) @ A) @ B) @ G) @ ((sum_Inl @ B) @ C))) @ A6)) @ ((((groups7121269368397514597t_prod @ C) @ A) @ (((((comp @ ((sum_sum @ B) @ C)) @ A) @ C) @ G) @ ((sum_Inr @ C) @ B))) @ B7))))))))))).
% 0.33/5.23 thf(fact_6883_ordering__top__axioms__def, axiom, (! [A:$tType]: (((ordering_top_axioms @ A) = (^ [Less_eq2:(A > (A > $o)),Top2:A]: ((! [A5:A]: (((Less_eq2 @ A5) @ Top2))))))))).
% 0.33/5.23 thf(fact_6884_ordering__top__axioms_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Top:A]: (((! [A8:A]: (((Less_eq @ A8) @ Top))) => (((ordering_top_axioms @ A) @ Less_eq) @ Top))))).
% 0.33/5.23 thf(fact_6885_ndepth__Push__Node__aux, axiom, (! [A:$tType,I:nat,F:(nat > ((sum_sum @ A) @ nat)),K:nat]: (((((((case_nat @ ((sum_sum @ A) @ nat)) @ (((sum_Inr @ nat) @ A) @ (suc @ I))) @ F) @ K) = (((sum_Inr @ nat) @ A) @ (zero_zero @ nat))) => (((ord_less_eq @ nat) @ (suc @ ((ord_Least @ nat) @ (^ [X2:nat]: (((F @ X2) = (((sum_Inr @ nat) @ A) @ (zero_zero @ nat)))))))) @ K))))).
% 0.33/5.23 thf(fact_6886_le__sum__encode__Inl, axiom, (! [X:nat,Y3:nat]: (((((ord_less_eq @ nat) @ X) @ Y3) => (((ord_less_eq @ nat) @ X) @ (nat_sum_encode @ (((sum_Inl @ nat) @ nat) @ Y3))))))).
% 0.33/5.23 thf(fact_6887_le__sum__encode__Inr, axiom, (! [X:nat,Y3:nat]: (((((ord_less_eq @ nat) @ X) @ Y3) => (((ord_less_eq @ nat) @ X) @ (nat_sum_encode @ (((sum_Inr @ nat) @ nat) @ Y3))))))).
% 0.33/5.23 thf(fact_6888_sum_Osize_I3_J, axiom, (! [A:$tType,B:$tType,X13:A]: ((((size_size @ ((sum_sum @ A) @ B)) @ (((sum_Inl @ A) @ B) @ X13)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.23 thf(fact_6889_sum_Osize_I4_J, axiom, (! [B:$tType,A:$tType,X22:B]: ((((size_size @ ((sum_sum @ A) @ B)) @ (((sum_Inr @ B) @ A) @ X22)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.23 thf(fact_6890_sum_Osize__neq, axiom, (! [A:$tType,B:$tType,X:((sum_sum @ A) @ B)]: ((((size_size @ ((sum_sum @ A) @ B)) @ X) != (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6891_prod_Osize__neq, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B)]: ((((size_size @ ((product_prod @ A) @ B)) @ X) != (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6892_sum_Osize__gen_I1_J, axiom, (! [B:$tType,A:$tType,Xa3:(A > nat),X:(B > nat),X13:A]: (((((((basic_BNF_size_sum @ A) @ B) @ Xa3) @ X) @ (((sum_Inl @ A) @ B) @ X13)) = (((plus_plus @ nat) @ (Xa3 @ X13)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.23 thf(fact_6893_sum_Osize__gen_I2_J, axiom, (! [A:$tType,B:$tType,Xa3:(A > nat),X:(B > nat),X22:B]: (((((((basic_BNF_size_sum @ A) @ B) @ Xa3) @ X) @ (((sum_Inr @ B) @ A) @ X22)) = (((plus_plus @ nat) @ (X @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.23 thf(fact_6894_sum__decode__def, axiom, (nat_sum_decode = (^ [N3:nat]: (((((if @ ((sum_sum @ nat) @ nat)) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)) @ (((sum_Inl @ nat) @ nat) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((sum_Inr @ nat) @ nat) @ (((divide_divide @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))).
% 0.33/5.23 thf(fact_6895_sum__decode__inverse, axiom, (! [N:nat]: (((nat_sum_encode @ (nat_sum_decode @ N)) = N)))).
% 0.33/5.23 thf(fact_6896_sum__encode__inverse, axiom, (! [X:((sum_sum @ nat) @ nat)]: (((nat_sum_decode @ (nat_sum_encode @ X)) = X)))).
% 0.33/5.23 thf(fact_6897_sum__decode__eq, axiom, (! [X:nat,Y3:nat]: ((((nat_sum_decode @ X) = (nat_sum_decode @ Y3)) = (X = Y3))))).
% 0.33/5.23 thf(fact_6898_inj__sum__decode, axiom, (! [A6:(set @ nat)]: (((((inj_on @ nat) @ ((sum_sum @ nat) @ nat)) @ nat_sum_decode) @ A6)))).
% 0.33/5.23 thf(fact_6899_surj__sum__decode, axiom, (((((image @ nat) @ ((sum_sum @ nat) @ nat)) @ nat_sum_decode) @ (top_top @ (set @ nat))) = (top_top @ (set @ ((sum_sum @ nat) @ nat))))).
% 0.33/5.23 thf(fact_6900_bij__sum__decode, axiom, (((((bij_betw @ nat) @ ((sum_sum @ nat) @ nat)) @ nat_sum_decode) @ (top_top @ (set @ nat))) @ (top_top @ (set @ ((sum_sum @ nat) @ nat))))).
% 0.33/5.23 thf(fact_6901_int__decode__def, axiom, (nat_int_decode = (^ [N3:nat]: (((((((sum_case_sum @ nat) @ int) @ nat) @ (semiring_1_of_nat @ int)) @ (^ [B3:nat]: ((((minus_minus @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ B3))) @ (one_one @ int))))) @ (nat_sum_decode @ N3)))))).
% 0.33/5.23 thf(fact_6902_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)),X2:((sum_sum @ A) @ nat),K3:nat]: (((P6 = ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ F2) @ X2)) & ((F2 @ K3) = (((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))))))))))).
% 0.33/5.23 thf(fact_6903_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)) @ (^ [K3:nat]: ((((sum_Inr @ nat) @ A) @ (zero_zero @ nat))))) @ A3)) @ ((old_Node @ A) @ B))))).
% 0.33/5.23 thf(fact_6904_Push__neq__K0, axiom, (! [A:$tType,K:nat,F:(nat > ((sum_sum @ A) @ nat))]: (((((old_Push @ A) @ (((sum_Inr @ nat) @ A) @ (suc @ K))) @ F) != (^ [Z7:nat]: ((((sum_Inr @ nat) @ A) @ (zero_zero @ nat)))))))).
% 0.33/5.23 thf(fact_6905_ndepth__def, axiom, (! [B:$tType,A:$tType]: ((((old_ndepth @ A) @ B) = (^ [N3:((old_node @ A) @ B)]: ((((((product_case_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ nat) @ (^ [F2:(nat > ((sum_sum @ B) @ nat)),X2:((sum_sum @ A) @ nat)]: (((ord_Least @ nat) @ (^ [K3:nat]: (((F2 @ K3) = (((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))))))) @ (((old_Rep_Node @ A) @ B) @ N3)))))))).
% 0.33/5.23 thf(fact_6906_ndepth__K0, axiom, (! [A:$tType,B:$tType,X:((sum_sum @ A) @ nat)]: (((((old_ndepth @ A) @ B) @ (((old_Abs_Node @ B) @ A) @ ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ (^ [K3:nat]: ((((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))) @ X))) = (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_6907_ndepth__Push__Node, axiom, (! [B:$tType,A:$tType,I:nat,N:((old_node @ A) @ B)]: (((((old_ndepth @ A) @ B) @ ((((old_Push_Node @ B) @ A) @ (((sum_Inr @ nat) @ B) @ (suc @ I))) @ N)) = (suc @ (((old_ndepth @ A) @ B) @ N)))))).
% 0.33/5.23 thf(fact_6908_Atom__def, axiom, (! [B:$tType,A:$tType]: ((((old_Atom @ A) @ B) = (^ [X2:((sum_sum @ A) @ nat)]: ((((insert @ ((old_node @ A) @ B)) @ (((old_Abs_Node @ B) @ A) @ ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ (^ [K3:nat]: ((((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))) @ X2))) @ (bot_bot @ (set @ ((old_node @ A) @ B)))))))))).
% 0.33/5.23 thf(fact_6909_Scons__def, axiom, (! [B:$tType,A:$tType]: ((((old_Scons @ A) @ B) = (^ [M9:(set @ ((old_node @ A) @ B)),N7:(set @ ((old_node @ A) @ B))]: ((((sup_sup @ (set @ ((old_node @ A) @ B))) @ ((((image @ ((old_node @ A) @ B)) @ ((old_node @ A) @ B)) @ (((old_Push_Node @ B) @ A) @ (((sum_Inr @ nat) @ B) @ (one_one @ nat)))) @ M9)) @ ((((image @ ((old_node @ A) @ B)) @ ((old_node @ A) @ B)) @ (((old_Push_Node @ B) @ A) @ (((sum_Inr @ nat) @ B) @ (suc @ (one_one @ nat))))) @ N7)))))))).
% 0.33/5.23 thf(fact_6910_ntrunc__0, axiom, (! [B:$tType,A:$tType,M7:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (zero_zero @ nat)) @ M7) = (bot_bot @ (set @ ((old_node @ A) @ B))))))).
% 0.33/5.23 thf(fact_6911_ntrunc__Atom, axiom, (! [B:$tType,A:$tType,K:nat,A3:((sum_sum @ A) @ nat)]: ((((((old_ntrunc @ A) @ B) @ (suc @ K)) @ (((old_Atom @ A) @ B) @ A3)) = (((old_Atom @ A) @ B) @ A3))))).
% 0.33/5.23 thf(fact_6912_ntrunc__Scons, axiom, (! [B:$tType,A:$tType,K:nat,M7:(set @ ((old_node @ A) @ B)),N6:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (suc @ K)) @ ((((old_Scons @ A) @ B) @ M7) @ N6)) = ((((old_Scons @ A) @ B) @ ((((old_ntrunc @ A) @ B) @ K) @ M7)) @ ((((old_ntrunc @ A) @ B) @ K) @ N6)))))).
% 0.33/5.23 thf(fact_6913_ntrunc__def, axiom, (! [A:$tType,B:$tType]: ((((old_ntrunc @ A) @ B) = (^ [K3:nat,N7:(set @ ((old_node @ A) @ B))]: (((collect @ ((old_node @ A) @ B)) @ (^ [N3:((old_node @ A) @ B)]: (((((member @ ((old_node @ A) @ B)) @ N3) @ N7) & (((ord_less @ nat) @ (((old_ndepth @ A) @ B) @ N3)) @ K3))))))))))).
% 0.33/5.23 thf(fact_6914_ntrunc__one__In0, axiom, (! [B:$tType,A:$tType,M7:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (suc @ (zero_zero @ nat))) @ (((old_In0 @ A) @ B) @ M7)) = (bot_bot @ (set @ ((old_node @ A) @ B))))))).
% 0.33/5.23 thf(fact_6915_ntrunc__one__In1, axiom, (! [B:$tType,A:$tType,M7:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (suc @ (zero_zero @ nat))) @ (((old_In1 @ A) @ B) @ M7)) = (bot_bot @ (set @ ((old_node @ A) @ B))))))).
% 0.33/5.23 thf(fact_6916_ntrunc__In0, axiom, (! [B:$tType,A:$tType,K:nat,M7:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (suc @ (suc @ K))) @ (((old_In0 @ A) @ B) @ M7)) = (((old_In0 @ A) @ B) @ ((((old_ntrunc @ A) @ B) @ (suc @ K)) @ M7)))))).
% 0.33/5.23 thf(fact_6917_ntrunc__In1, axiom, (! [B:$tType,A:$tType,K:nat,M7:(set @ ((old_node @ A) @ B))]: ((((((old_ntrunc @ A) @ B) @ (suc @ (suc @ K))) @ (((old_In1 @ A) @ B) @ M7)) = (((old_In1 @ A) @ B) @ ((((old_ntrunc @ A) @ B) @ (suc @ K)) @ M7)))))).
% 0.33/5.23 thf(fact_6918_ntrunc__Leaf, axiom, (! [B:$tType,A:$tType,K:nat,A3:A]: ((((((old_ntrunc @ A) @ B) @ (suc @ K)) @ (((old_Leaf @ A) @ B) @ A3)) = (((old_Leaf @ A) @ B) @ A3))))).
% 0.33/5.23 thf(fact_6919_In1__def, axiom, (! [A:$tType,B:$tType]: ((((old_In1 @ A) @ B) = (((old_Scons @ A) @ B) @ (((old_Numb @ A) @ B) @ (one_one @ nat))))))).
% 0.33/5.23 thf(fact_6920_ntrunc__Numb, axiom, (! [A:$tType,B:$tType,K:nat,I:nat]: ((((((old_ntrunc @ A) @ B) @ (suc @ K)) @ (((old_Numb @ A) @ B) @ I)) = (((old_Numb @ A) @ B) @ I))))).
% 0.33/5.23 thf(fact_6921_In0__def, axiom, (! [A:$tType,B:$tType]: ((((old_In0 @ A) @ B) = (((old_Scons @ A) @ B) @ (((old_Numb @ A) @ B) @ (zero_zero @ nat))))))).
% 0.33/5.23 thf(fact_6922_ordering__top_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((((ordering_top_axioms @ A) @ Less_eq) @ Top) => ((((ordering_top @ A) @ Less_eq) @ Less) @ Top)))))).
% 0.33/5.23 thf(fact_6923_ordering__top__def, axiom, (! [A:$tType]: (((ordering_top @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),Top2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) & (((ordering_top_axioms @ A) @ Less_eq2) @ Top2)))))))).
% 0.33/5.23 thf(fact_6924_ordering__top_Oaxioms_I1_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((ordering @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_6925_ordering__strictI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((! [A8:A,B5:A]: ((((Less_eq @ A8) @ B5) = (((Less @ A8) @ B5) | (A8 = B5))))) => ((! [A8:A,B5:A]: ((((Less @ A8) @ B5) => (~ (((Less @ B5) @ A8)))))) => ((! [A8:A]: ((~ (((Less @ A8) @ A8))))) => ((! [A8:A,B5:A,C3:A]: ((((Less @ A8) @ B5) => (((Less @ B5) @ C3) => ((Less @ A8) @ C3))))) => (((ordering @ A) @ Less_eq) @ Less)))))))).
% 0.33/5.23 thf(fact_6926_ordering_Onot__eq__order__implies__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((A3 != B2) => (((Less_eq @ A3) @ B2) => ((Less @ A3) @ B2))))))).
% 0.33/5.23 thf(fact_6927_ordering_Ostrict__implies__not__eq, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => (A3 != B2)))))).
% 0.33/5.23 thf(fact_6928_ordering_Ostrict__iff__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) = (((Less_eq @ A3) @ B2) & (A3 != B2))))))).
% 0.33/5.23 thf(fact_6929_ordering_Oorder__iff__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) = (((Less @ A3) @ B2) | (A3 = B2))))))).
% 0.33/5.23 thf(fact_6930_ordering_Oantisym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) => (((Less_eq @ B2) @ A3) => (A3 = B2))))))).
% 0.33/5.23 thf(fact_6931_ordering_Oeq__iff, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((A3 = B2) = (((Less_eq @ A3) @ B2) & ((Less_eq @ B2) @ A3))))))).
% 0.33/5.23 thf(fact_6932_ordering__dualI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((ordering @ A) @ (^ [A5:A,B3:A]: (((Less_eq @ B3) @ A5)))) @ (^ [A5:A,B3:A]: (((Less @ B3) @ A5)))) => (((ordering @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_6933_order_Oordering__axioms, axiom, (! [A:$tType]: (((order @ A) => (((ordering @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_6934_dual__order_Oordering__axioms, axiom, (! [A:$tType]: (((order @ A) => (((ordering @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_6935_nth__item_Oelims, axiom, (! [A:$tType]: (((countable @ A) => (! [X:nat,Y3:(set @ ((old_node @ A) @ product_unit))]: (((((nth_item @ A) @ X) = Y3) => (((X = (zero_zero @ nat)) => (Y3 != (undefined @ (set @ ((old_node @ A) @ product_unit))))) => (~ ((! [N4:nat]: (((X = (suc @ N4)) => (Y3 != ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (^ [J3:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (nat_sum_decode @ I5))))) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J3))))) @ (^ [J3:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A5:nat,B3:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A5)) @ ((nth_item @ A) @ B3))))) @ (nat_prod_decode @ J3))))) @ (nat_sum_decode @ I5))))) @ (nat_sum_decode @ N4)))))))))))))))).
% 0.33/5.23 thf(fact_6936_group_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((((group_axioms @ A) @ F) @ Z3) @ Inverse))))).
% 0.33/5.23 thf(fact_6937_group__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A)]: (((! [A8:A]: ((((F @ Z3) @ A8) = A8))) => ((! [A8:A]: ((((F @ (Inverse @ A8)) @ A8) = Z3))) => ((((group_axioms @ A) @ F) @ Z3) @ Inverse)))))).
% 0.33/5.23 thf(fact_6938_group__axioms__def, axiom, (! [A:$tType]: (((group_axioms @ A) = (^ [F2:(A > (A > A)),Z7:A,Inverse2:(A > A)]: (((! [A5:A]: ((((F2 @ Z7) @ A5) = A5))) & (! [A5:A]: ((((F2 @ (Inverse2 @ A5)) @ A5) = Z7)))))))))).
% 0.33/5.23 thf(fact_6939_nth__item_Osimps_I1_J, axiom, (! [A:$tType]: (((countable @ A) => (((nth_item @ A) @ (zero_zero @ nat)) = (undefined @ (set @ ((old_node @ A) @ product_unit)))))))).
% 0.33/5.23 thf(fact_6940_nth__item_Osimps_I2_J, axiom, (! [A:$tType]: (((countable @ A) => (! [N:nat]: ((((nth_item @ A) @ (suc @ N)) = ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (^ [J3:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (nat_sum_decode @ I5))))) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J3))))) @ (^ [J3:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A5:nat,B3:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A5)) @ ((nth_item @ A) @ B3))))) @ (nat_prod_decode @ J3))))) @ (nat_sum_decode @ I5))))) @ (nat_sum_decode @ N))))))))).
% 0.33/5.23 thf(fact_6941_nth__item_Opelims, axiom, (! [A:$tType]: (((countable @ A) => (! [X:nat,Y3:(set @ ((old_node @ A) @ product_unit))]: (((((nth_item @ A) @ X) = Y3) => ((((accp @ nat) @ nth_item_rel) @ X) => (((X = (zero_zero @ nat)) => ((Y3 = (undefined @ (set @ ((old_node @ A) @ product_unit)))) => (~ ((((accp @ nat) @ nth_item_rel) @ (zero_zero @ nat)))))) => (~ ((! [N4:nat]: (((X = (suc @ N4)) => ((Y3 = ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (^ [J3:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (nat_sum_decode @ I5))))) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J3))))) @ (^ [J3:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A5:nat,B3:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A5)) @ ((nth_item @ A) @ B3))))) @ (nat_prod_decode @ J3))))) @ (nat_sum_decode @ I5))))) @ (nat_sum_decode @ N4))) => (~ ((((accp @ nat) @ nth_item_rel) @ (suc @ N4))))))))))))))))))).
% 0.33/5.23 thf(fact_6942_nth__item_Opsimps_I2_J, axiom, (! [A:$tType]: (((countable @ A) => (! [N:nat]: (((((accp @ nat) @ nth_item_rel) @ (suc @ N)) => (((nth_item @ A) @ (suc @ N)) = ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (^ [J3:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J3))))) @ (nat_sum_decode @ I5))))) @ (^ [I5:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J3:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J3))))) @ (^ [J3:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A5:nat,B3:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A5)) @ ((nth_item @ A) @ B3))))) @ (nat_prod_decode @ J3))))) @ (nat_sum_decode @ I5))))) @ (nat_sum_decode @ N)))))))))).
% 0.33/5.23 thf(fact_6943_nth__item_Opsimps_I1_J, axiom, (! [A:$tType]: (((countable @ A) => ((((accp @ nat) @ nth_item_rel) @ (zero_zero @ nat)) => (((nth_item @ A) @ (zero_zero @ nat)) = (undefined @ (set @ ((old_node @ A) @ product_unit))))))))).
% 0.33/5.23 thf(fact_6944_group_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A)]: ((((semigroup @ A) @ F) => (((((group_axioms @ A) @ F) @ Z3) @ Inverse) => ((((group @ A) @ F) @ Z3) @ Inverse)))))).
% 0.33/5.23 thf(fact_6945_group__def, axiom, (! [A:$tType]: (((group @ A) = (^ [F2:(A > (A > A)),Z7:A,Inverse2:(A > A)]: ((((semigroup @ A) @ F2) & ((((group_axioms @ A) @ F2) @ Z7) @ Inverse2)))))))).
% 0.33/5.23 thf(fact_6946_sup_Osemigroup__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((semigroup @ A) @ (sup_sup @ A)))))).
% 0.33/5.23 thf(fact_6947_semigroup_Oassoc, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A,C2:A]: ((((semigroup @ A) @ F) => (((F @ ((F @ A3) @ B2)) @ C2) = ((F @ A3) @ ((F @ B2) @ C2))))))).
% 0.33/5.23 thf(fact_6948_semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A8:A,B5:A,C3:A]: ((((F @ ((F @ A8) @ B5)) @ C3) = ((F @ A8) @ ((F @ B5) @ C3))))) => ((semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_6949_semigroup__def, axiom, (! [A:$tType]: (((semigroup @ A) = (^ [F2:(A > (A > A))]: ((! [A5:A,B3:A,C4:A]: ((((F2 @ ((F2 @ A5) @ B3)) @ C4) = ((F2 @ A5) @ ((F2 @ B3) @ C4))))))))))).
% 0.33/5.23 thf(fact_6950_group_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z3) @ Inverse) => ((semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_6951_monoid_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((monoid @ A) @ F) @ Z3) => ((semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_6952_min_Osemigroup__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((semigroup @ A) @ (ord_min @ A)))))).
% 0.33/5.23 thf(fact_6953_max_Osemigroup__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((semigroup @ A) @ (ord_max @ A)))))).
% 0.33/5.23 thf(fact_6954_add_Osemigroup__axioms, axiom, (! [A:$tType]: (((semigroup_add @ A) => ((semigroup @ A) @ (plus_plus @ A)))))).
% 0.33/5.23 thf(fact_6955_mult_Osemigroup__axioms, axiom, (! [A:$tType]: (((semigroup_mult @ A) => ((semigroup @ A) @ (times_times @ A)))))).
% 0.33/5.23 thf(fact_6956_inf_Osemigroup__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((semigroup @ A) @ (inf_inf @ A)))))).
% 0.33/5.23 thf(fact_6957_monoid_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: ((((semigroup @ A) @ F) => ((((monoid_axioms @ A) @ F) @ Z3) => (((monoid @ A) @ F) @ Z3)))))).
% 0.33/5.23 thf(fact_6958_monoid__def, axiom, (! [A:$tType]: (((monoid @ A) = (^ [F2:(A > (A > A)),Z7:A]: ((((semigroup @ A) @ F2) & (((monoid_axioms @ A) @ F2) @ Z7)))))))).
% 0.33/5.23 thf(fact_6959_monoid__axioms__def, axiom, (! [A:$tType]: (((monoid_axioms @ A) = (^ [F2:(A > (A > A)),Z7:A]: (((! [A5:A]: ((((F2 @ Z7) @ A5) = A5))) & (! [A5:A]: ((((F2 @ A5) @ Z7) = A5)))))))))).
% 0.33/5.23 thf(fact_6960_monoid__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((! [A8:A]: ((((F @ Z3) @ A8) = A8))) => ((! [A8:A]: ((((F @ A8) @ Z3) = A8))) => (((monoid_axioms @ A) @ F) @ Z3)))))).
% 0.33/5.23 thf(fact_6961_monoid_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((monoid @ A) @ F) @ Z3) => (((monoid_axioms @ A) @ F) @ Z3))))).
% 0.33/5.23 thf(fact_6962_nths__nths, axiom, (! [A:$tType,Xs:(list @ A),A6:(set @ nat),B7:(set @ nat)]: (((((nths @ A) @ (((nths @ A) @ Xs) @ A6)) @ B7) = (((nths @ A) @ Xs) @ ((collect @ nat) @ (^ [I5:nat]: (((((member @ nat) @ I5) @ A6) & (((member @ nat) @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I9:nat]: (((((member @ nat) @ I9) @ A6) & (((ord_less @ nat) @ I9) @ I5))))))) @ B7)))))))))).
% 0.33/5.23 thf(fact_6963_ordering_Oaxioms_I2_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((ordering @ A) @ Less_eq) @ Less) => (((ordering_axioms @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_6964_ordering__axioms_Ointro, axiom, (! [A:$tType,Less:(A > (A > $o)),Less_eq:(A > (A > $o))]: (((! [A8:A,B5:A]: ((((Less @ A8) @ B5) = (((Less_eq @ A8) @ B5) & (A8 != B5))))) => ((! [A8:A,B5:A]: ((((Less_eq @ A8) @ B5) => (((Less_eq @ B5) @ A8) => (A8 = B5))))) => (((ordering_axioms @ A) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_6965_ordering__axioms__def, axiom, (! [A:$tType]: (((ordering_axioms @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((! [A5:A,B3:A]: ((((Less2 @ A5) @ B3) = (((Less_eq2 @ A5) @ B3) & (A5 != B3))))) & (! [A5:A,B3:A]: ((((Less_eq2 @ A5) @ B3) => (((Less_eq2 @ B3) @ A5) => (A5 = B3)))))))))))).
% 0.33/5.23 thf(fact_6966_Bex__def__raw, axiom, (! [A:$tType]: (((bex @ A) = (^ [A7:(set @ A),P4:(A > $o)]: ((? [X2:A]: (((((member @ A) @ X2) @ A7) & (P4 @ X2)))))))))).
% 0.33/5.23 thf(fact_6967_ordering__def, axiom, (! [A:$tType]: (((ordering @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((((partial_preordering @ A) @ Less_eq2) & (((ordering_axioms @ A) @ Less_eq2) @ Less2)))))))).
% 0.33/5.23 thf(fact_6968_ordering_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((partial_preordering @ A) @ Less_eq) => ((((ordering_axioms @ A) @ Less_eq) @ Less) => (((ordering @ A) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_6969_dual__order_Opartial__preordering__axioms, axiom, (! [A:$tType]: (((preorder @ A) => ((partial_preordering @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_6970_order_Opartial__preordering__axioms, axiom, (! [A:$tType]: (((preorder @ A) => ((partial_preordering @ A) @ (ord_less_eq @ A)))))).
% 0.33/5.23 thf(fact_6971_partial__preordering__def, axiom, (! [A:$tType]: (((partial_preordering @ A) = (^ [Less_eq2:(A > (A > $o))]: (((! [A5:A]: (((Less_eq2 @ A5) @ A5))) & (! [A5:A,B3:A,C4:A]: ((((Less_eq2 @ A5) @ B3) => (((Less_eq2 @ B3) @ C4) => ((Less_eq2 @ A5) @ C4)))))))))))).
% 0.33/5.23 thf(fact_6972_partial__preordering_Otrans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),A3:A,B2:A,C2:A]: ((((partial_preordering @ A) @ Less_eq) => (((Less_eq @ A3) @ B2) => (((Less_eq @ B2) @ C2) => ((Less_eq @ A3) @ C2))))))).
% 0.33/5.23 thf(fact_6973_partial__preordering_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o))]: (((! [A8:A]: (((Less_eq @ A8) @ A8))) => ((! [A8:A,B5:A,C3:A]: ((((Less_eq @ A8) @ B5) => (((Less_eq @ B5) @ C3) => ((Less_eq @ A8) @ C3))))) => ((partial_preordering @ A) @ Less_eq)))))).
% 0.33/5.23 thf(fact_6974_partial__preordering_Orefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),A3:A]: ((((partial_preordering @ A) @ Less_eq) => ((Less_eq @ A3) @ A3))))).
% 0.33/5.23 thf(fact_6975_ordering_Oaxioms_I1_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((ordering @ A) @ Less_eq) @ Less) => ((partial_preordering @ A) @ Less_eq))))).
% 0.33/5.23 thf(fact_6976_Abs__int__cases, axiom, (! [X:int]: ((~ ((! [Y5:(set @ ((product_prod @ nat) @ nat))]: (((X = (abs_int @ Y5)) => (~ ((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y5) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))))))))))))).
% 0.33/5.23 thf(fact_6977_Abs__int__induct, axiom, (! [P:(int > $o),X:int]: (((! [Y5:(set @ ((product_prod @ nat) @ nat))]: (((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y5) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => (P @ (abs_int @ Y5))))) => (P @ X))))).
% 0.33/5.23 thf(fact_6978_Abs__int__inject, axiom, (! [X:(set @ ((product_prod @ nat) @ nat)),Y3:(set @ ((product_prod @ nat) @ nat))]: (((((member @ (set @ ((product_prod @ nat) @ nat))) @ X) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => ((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y3) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => (((abs_int @ X) = (abs_int @ Y3)) = (X = Y3))))))).
% 0.33/5.23 thf(fact_6979_Abs__Integ__def, axiom, (abs_Integ = ((((quot_abs @ ((product_prod @ nat) @ nat)) @ int) @ intrel) @ abs_int))).
% 0.33/5.23 thf(fact_6980_Abs__int__inverse, axiom, (! [Y3:(set @ ((product_prod @ nat) @ nat))]: (((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y3) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => ((rep_int @ (abs_int @ Y3)) = Y3))))).
% 0.33/5.23 thf(fact_6981_Rep__int__inject, axiom, (! [X:int,Y3:int]: ((((rep_int @ X) = (rep_int @ Y3)) = (X = Y3))))).
% 0.33/5.23 thf(fact_6982_Rep__int__inverse, axiom, (! [X:int]: (((abs_int @ (rep_int @ X)) = X)))).
% 0.33/5.23 thf(fact_6983_Rep__int, axiom, (! [X:int]: ((((member @ (set @ ((product_prod @ nat) @ nat))) @ (rep_int @ X)) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2))))))))))))).
% 0.33/5.23 thf(fact_6984_Rep__int__cases, axiom, (! [Y3:(set @ ((product_prod @ nat) @ nat))]: (((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y3) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => (~ ((! [X3:int]: ((Y3 != (rep_int @ X3)))))))))).
% 0.33/5.23 thf(fact_6985_Rep__int__induct, axiom, (! [Y3:(set @ ((product_prod @ nat) @ nat)),P:((set @ ((product_prod @ nat) @ nat)) > $o)]: (((((member @ (set @ ((product_prod @ nat) @ nat))) @ Y3) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2)))))))))) => ((! [X3:int]: ((P @ (rep_int @ X3)))) => (P @ Y3)))))).
% 0.33/5.23 thf(fact_6986_Rep__Integ__def, axiom, (rep_Integ = (((quot_rep @ int) @ ((product_prod @ nat) @ nat)) @ rep_int))).
% 0.33/5.23 thf(fact_6987_type__definition__int, axiom, (((((type_definition @ int) @ (set @ ((product_prod @ nat) @ nat))) @ rep_int) @ abs_int) @ ((collect @ (set @ ((product_prod @ nat) @ nat))) @ (^ [C4:(set @ ((product_prod @ nat) @ nat))]: ((? [X2:((product_prod @ nat) @ nat)]: ((((intrel @ X2) @ X2) & (C4 = ((collect @ ((product_prod @ nat) @ nat)) @ (intrel @ X2))))))))))).
% 0.33/5.23 thf(fact_6988_Domainp__cr__rat, axiom, ((((domainp @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) = (^ [X2:((product_prod @ int) @ int)]: (((((product_snd @ int) @ int) @ X2) != (zero_zero @ int)))))).
% 0.33/5.23 thf(fact_6989_semilattice__neutr_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: ((((semilattice @ A) @ F) => ((((comm_monoid @ A) @ F) @ Z3) => (((semilattice_neutr @ A) @ F) @ Z3)))))).
% 0.33/5.23 thf(fact_6990_inf_Osemilattice__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((semilattice @ A) @ (inf_inf @ A)))))).
% 0.33/5.23 thf(fact_6991_semilattice__order_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((semilattice @ A) @ F))))).
% 0.33/5.23 thf(fact_6992_max_Osemilattice__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((semilattice @ A) @ (ord_max @ A)))))).
% 0.33/5.23 thf(fact_6993_min_Osemilattice__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((semilattice @ A) @ (ord_min @ A)))))).
% 0.33/5.23 thf(fact_6994_semilattice_Oright__idem, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ ((F @ A3) @ B2)) @ B2) = ((F @ A3) @ B2)))))).
% 0.33/5.23 thf(fact_6995_semilattice_Oleft__idem, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ A3) @ ((F @ A3) @ B2)) = ((F @ A3) @ B2)))))).
% 0.33/5.23 thf(fact_6996_semilattice_Oidem, axiom, (! [A:$tType,F:(A > (A > A)),A3:A]: ((((semilattice @ A) @ F) => (((F @ A3) @ A3) = A3))))).
% 0.33/5.23 thf(fact_6997_semilattice__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((lattic149705377957585745ce_set @ A) @ F))))).
% 0.33/5.23 thf(fact_6998_semilattice__set_Oaxioms, axiom, (! [A:$tType,F:(A > (A > A))]: ((((lattic149705377957585745ce_set @ A) @ F) => ((semilattice @ A) @ F))))).
% 0.33/5.23 thf(fact_6999_semilattice__set__def, axiom, (! [A:$tType]: (((lattic149705377957585745ce_set @ A) = (semilattice @ A))))).
% 0.33/5.23 thf(fact_7000_semilattice__neutr_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((semilattice_neutr @ A) @ F) @ Z3) => ((semilattice @ A) @ F))))).
% 0.33/5.23 thf(fact_7001_sup_Osemilattice__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((semilattice @ A) @ (sup_sup @ A)))))).
% 0.33/5.23 thf(fact_7002_semilattice__neutr__def, axiom, (! [A:$tType]: (((semilattice_neutr @ A) = (^ [F2:(A > (A > A)),Z7:A]: ((((semilattice @ A) @ F2) & (((comm_monoid @ A) @ F2) @ Z7)))))))).
% 0.33/5.23 thf(fact_7003_semilattice__order_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((semilattice @ A) @ F) => (((((semila6385135966242565138axioms @ A) @ F) @ Less_eq) @ Less) => ((((semilattice_order @ A) @ F) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_7004_semilattice__order__def, axiom, (! [A:$tType]: (((semilattice_order @ A) = (^ [F2:(A > (A > A)),Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((((semilattice @ A) @ F2) & ((((semila6385135966242565138axioms @ A) @ F2) @ Less_eq2) @ Less2)))))))).
% 0.33/5.23 thf(fact_7005_semilattice__order__axioms_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),F:(A > (A > A)),Less:(A > (A > $o))]: (((! [A8:A,B5:A]: ((((Less_eq @ A8) @ B5) = (A8 = ((F @ A8) @ B5))))) => ((! [A8:A,B5:A]: ((((Less @ A8) @ B5) = ((A8 = ((F @ A8) @ B5)) & (A8 != B5))))) => ((((semila6385135966242565138axioms @ A) @ F) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_7006_semilattice__order__axioms__def, axiom, (! [A:$tType]: (((semila6385135966242565138axioms @ A) = (^ [F2:(A > (A > A)),Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((! [A5:A,B3:A]: ((((Less_eq2 @ A5) @ B3) = (A5 = ((F2 @ A5) @ B3))))) & (! [A5:A,B3:A]: ((((Less2 @ A5) @ B3) = ((A5 = ((F2 @ A5) @ B3)) & (A5 != B3)))))))))))).
% 0.33/5.23 thf(fact_7007_semilattice__order_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((((semila6385135966242565138axioms @ A) @ F) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_7008_prod__list_Omonoid__list__axioms, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((groups_monoid_list @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.23 thf(fact_7009_comm__monoid_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((comm_monoid @ A) @ F) @ Z3) => (((comm_monoid_axioms @ A) @ F) @ Z3))))).
% 0.33/5.23 thf(fact_7010_comm__monoid__axioms__def, axiom, (! [A:$tType]: (((comm_monoid_axioms @ A) = (^ [F2:(A > (A > A)),Z7:A]: ((! [A5:A]: ((((F2 @ A5) @ Z7) = A5))))))))).
% 0.33/5.23 thf(fact_7011_comm__monoid__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((! [A8:A]: ((((F @ A8) @ Z3) = A8))) => (((comm_monoid_axioms @ A) @ F) @ Z3))))).
% 0.33/5.23 thf(fact_7012_sum__list_Omonoid__list__axioms, axiom, (! [A:$tType]: (((monoid_add @ A) => (((groups_monoid_list @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.23 thf(fact_7013_comm__monoid__def, axiom, (! [A:$tType]: (((comm_monoid @ A) = (^ [F2:(A > (A > A)),Z7:A]: ((((abel_semigroup @ A) @ F2) & (((comm_monoid_axioms @ A) @ F2) @ Z7)))))))).
% 0.33/5.23 thf(fact_7014_comm__monoid_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: ((((abel_semigroup @ A) @ F) => ((((comm_monoid_axioms @ A) @ F) @ Z3) => (((comm_monoid @ A) @ F) @ Z3)))))).
% 0.33/5.23 thf(fact_7015_semilattice_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((abel_semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_7016_sup_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((abel_semigroup @ A) @ (sup_sup @ A)))))).
% 0.33/5.23 thf(fact_7017_comm__monoid_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A]: (((((comm_monoid @ A) @ F) @ Z3) => ((abel_semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_7018_abel__semigroup_Ocommute, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A]: ((((abel_semigroup @ A) @ F) => (((F @ A3) @ B2) = ((F @ B2) @ A3)))))).
% 0.33/5.23 thf(fact_7019_abel__semigroup_Oleft__commute, axiom, (! [A:$tType,F:(A > (A > A)),B2:A,A3:A,C2:A]: ((((abel_semigroup @ A) @ F) => (((F @ B2) @ ((F @ A3) @ C2)) = ((F @ A3) @ ((F @ B2) @ C2))))))).
% 0.33/5.23 thf(fact_7020_abstract__boolean__algebra_Oaxioms_I1_J, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((abel_semigroup @ A) @ Conj))))).
% 0.33/5.23 thf(fact_7021_abstract__boolean__algebra_Oaxioms_I2_J, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((abel_semigroup @ A) @ Disj))))).
% 0.33/5.23 thf(fact_7022_min_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((abel_semigroup @ A) @ (ord_min @ A)))))).
% 0.33/5.23 thf(fact_7023_add_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((abel_semigroup @ A) @ (plus_plus @ A)))))).
% 0.33/5.23 thf(fact_7024_max_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((abel_semigroup @ A) @ (ord_max @ A)))))).
% 0.33/5.23 thf(fact_7025_mult_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((abel_semigroup @ A) @ (times_times @ A)))))).
% 0.33/5.23 thf(fact_7026_inf_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((abel_semigroup @ A) @ (inf_inf @ A)))))).
% 0.33/5.23 thf(fact_7027_abel__semigroup_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((semigroup @ A) @ F))))).
% 0.33/5.23 thf(fact_7028_abel__semigroup__def, axiom, (! [A:$tType]: (((abel_semigroup @ A) = (^ [F2:(A > (A > A))]: ((((semigroup @ A) @ F2) & ((abel_s757365448890700780axioms @ A) @ F2)))))))).
% 0.33/5.23 thf(fact_7029_abel__semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semigroup @ A) @ F) => (((abel_s757365448890700780axioms @ A) @ F) => ((abel_semigroup @ A) @ F)))))).
% 0.33/5.23 thf(fact_7030_abel__semigroup__axioms__def, axiom, (! [A:$tType]: (((abel_s757365448890700780axioms @ A) = (^ [F2:(A > (A > A))]: ((! [A5:A,B3:A]: ((((F2 @ A5) @ B3) = ((F2 @ B3) @ A5)))))))))).
% 0.33/5.23 thf(fact_7031_abel__semigroup__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A8:A,B5:A]: ((((F @ A8) @ B5) = ((F @ B5) @ A8)))) => ((abel_s757365448890700780axioms @ A) @ F))))).
% 0.33/5.23 thf(fact_7032_abel__semigroup_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((abel_s757365448890700780axioms @ A) @ F))))).
% 0.33/5.23 thf(fact_7033_semilattice_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => (((semilattice_axioms @ A) @ F) => ((semilattice @ A) @ F)))))).
% 0.33/5.23 thf(fact_7034_semilattice__def, axiom, (! [A:$tType]: (((semilattice @ A) = (^ [F2:(A > (A > A))]: ((((abel_semigroup @ A) @ F2) & ((semilattice_axioms @ A) @ F2)))))))).
% 0.33/5.23 thf(fact_7035_semilattice__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A8:A]: ((((F @ A8) @ A8) = A8))) => ((semilattice_axioms @ A) @ F))))).
% 0.33/5.23 thf(fact_7036_semilattice__axioms__def, axiom, (! [A:$tType]: (((semilattice_axioms @ A) = (^ [F2:(A > (A > A))]: ((! [A5:A]: ((((F2 @ A5) @ A5) = A5))))))))).
% 0.33/5.23 thf(fact_7037_semilattice_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((semilattice_axioms @ A) @ F))))).
% 0.33/5.23 thf(fact_7038_abstract__boolean__algebra__def, axiom, (! [A:$tType]: (((boolea2506097494486148201lgebra @ A) = (^ [Conj2:(A > (A > A)),Disj2:(A > (A > A)),Compl2:(A > A),Zero2:A,One:A]: ((((abel_semigroup @ A) @ Conj2) & (((abel_semigroup @ A) @ Disj2) & ((((((boolea6902313364301356556axioms @ A) @ Conj2) @ Disj2) @ Compl2) @ Zero2) @ One))))))))).
% 0.33/5.23 thf(fact_7039_abstract__boolean__algebra_Ointro, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((abel_semigroup @ A) @ Conj) => (((abel_semigroup @ A) @ Disj) => (((((((boolea6902313364301356556axioms @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2))))))).
% 0.33/5.23 thf(fact_7040_abstract__boolean__algebra__axioms__def, axiom, (! [A:$tType]: (((boolea6902313364301356556axioms @ A) = (^ [Conj2:(A > (A > A)),Disj2:(A > (A > A)),Compl2:(A > A),Zero2:A,One:A]: (((! [X2:A,Y6:A,Z7:A]: ((((Conj2 @ X2) @ ((Disj2 @ Y6) @ Z7)) = ((Disj2 @ ((Conj2 @ X2) @ Y6)) @ ((Conj2 @ X2) @ Z7))))) & ((! [X2:A,Y6:A,Z7:A]: ((((Disj2 @ X2) @ ((Conj2 @ Y6) @ Z7)) = ((Conj2 @ ((Disj2 @ X2) @ Y6)) @ ((Disj2 @ X2) @ Z7))))) & ((! [X2:A]: ((((Conj2 @ X2) @ One) = X2))) & ((! [X2:A]: ((((Disj2 @ X2) @ Zero2) = X2))) & ((! [X2:A]: ((((Conj2 @ X2) @ (Compl2 @ X2)) = Zero2))) & (! [X2:A]: ((((Disj2 @ X2) @ (Compl2 @ X2)) = One)))))))))))))).
% 0.33/5.23 thf(fact_7041_abstract__boolean__algebra__axioms_Ointro, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),One2:A,Zero:A,Compl:(A > A)]: (((! [X3:A,Y5:A,Z4:A]: ((((Conj @ X3) @ ((Disj @ Y5) @ Z4)) = ((Disj @ ((Conj @ X3) @ Y5)) @ ((Conj @ X3) @ Z4))))) => ((! [X3:A,Y5:A,Z4:A]: ((((Disj @ X3) @ ((Conj @ Y5) @ Z4)) = ((Conj @ ((Disj @ X3) @ Y5)) @ ((Disj @ X3) @ Z4))))) => ((! [X3:A]: ((((Conj @ X3) @ One2) = X3))) => ((! [X3:A]: ((((Disj @ X3) @ Zero) = X3))) => ((! [X3:A]: ((((Conj @ X3) @ (Compl @ X3)) = Zero))) => ((! [X3:A]: ((((Disj @ X3) @ (Compl @ X3)) = One2))) => ((((((boolea6902313364301356556axioms @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2)))))))))).
% 0.33/5.23 thf(fact_7042_abstract__boolean__algebra_Oaxioms_I3_J, axiom, (! [A:$tType,Conj:(A > (A > A)),Disj:(A > (A > A)),Compl:(A > A),Zero:A,One2:A]: ((((((((boolea2506097494486148201lgebra @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2) => ((((((boolea6902313364301356556axioms @ A) @ Conj) @ Disj) @ Compl) @ Zero) @ One2))))).
% 0.33/5.23 thf(fact_7043_typerep_Osize__neq, axiom, (! [X:typerep]: ((((size_size @ typerep) @ X) != (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_7044_tuple__isomorphism_Osize__neq, axiom, (! [A:$tType,B:$tType,C:$tType,X:(((tuple_isomorphism @ A) @ B) @ C)]: ((((size_size @ (((tuple_isomorphism @ A) @ B) @ C)) @ X) != (zero_zero @ nat))))).
% 0.33/5.23 thf(fact_7045_tuple__isomorphism_Osize_I2_J, axiom, (! [B:$tType,C:$tType,A:$tType,X13:(A > ((product_prod @ B) @ C)),X22:(((product_prod @ B) @ C) > A)]: ((((size_size @ (((tuple_isomorphism @ A) @ B) @ C)) @ (((((tuple_1188178415141063261rphism @ A) @ B) @ C) @ X13) @ X22)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.23 thf(fact_7046_typerep_Osize_I2_J, axiom, (! [X13:literal,X22:(list @ typerep)]: ((((size_size @ typerep) @ ((typerep3 @ X13) @ X22)) = (((plus_plus @ nat) @ (((size_list @ typerep) @ (size_size @ typerep)) @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.23 thf(fact_7047_typerep_Osize__gen, axiom, (! [X13:literal,X22:(list @ typerep)]: (((size_typerep @ ((typerep3 @ X13) @ X22)) = (((plus_plus @ nat) @ (((size_list @ typerep) @ size_typerep) @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 0.33/5.23 thf(fact_7048_tuple__isomorphism_Osize__gen, axiom, (! [B:$tType,C:$tType,A:$tType,Xb3:(A > nat),Xa3:(B > nat),X:(C > nat),X13:(A > ((product_prod @ B) @ C)),X22:(((product_prod @ B) @ C) > A)]: (((((((((tuple_9181185373184732606rphism @ A) @ B) @ C) @ Xb3) @ Xa3) @ X) @ (((((tuple_1188178415141063261rphism @ A) @ B) @ C) @ X13) @ X22)) = (suc @ (zero_zero @ nat)))))).
% 0.33/5.23 thf(fact_7049_typerep__fun__box__def, axiom, (! [B:$tType,A:$tType]: ((((typerep2 @ A) & (typerep2 @ B)) => ((typerep_typerep @ ((fun_box @ A) @ B)) = (^ [T7:(itself @ ((fun_box @ A) @ B))]: (((typerep3 @ ((((((((literal2 @ $false) @ $true) @ $true) @ $true) @ $false) @ $false) @ $true) @ ((((((((literal2 @ $true) @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $false) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $false) @ $false) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $true) @ $true) @ $true) @ $false) @ $true) @ $false) @ ((((((((literal2 @ $false) @ $true) @ $true) @ $false) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $false) @ $true) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $true) @ $true) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $true) @ $true) @ $true) @ $false) @ $true) @ ((((((((literal2 @ $false) @ $true) @ $false) @ $false) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $true) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $false) @ $true) @ $true) @ $true) @ $true) @ (zero_zero @ literal))))))))))))))))) @ (((cons @ typerep) @ ((typerep_typerep @ A) @ (type2 @ A))) @ (((cons @ typerep) @ ((typerep_typerep @ B) @ (type2 @ B))) @ (nil @ typerep))))))))))).
% 0.33/5.23 thf(fact_7050_typerep__word__def, axiom, (! [A:$tType]: (((typerep2 @ A) => ((typerep_typerep @ (word @ A)) = (^ [T7:(itself @ (word @ A))]: (((typerep3 @ ((((((((literal2 @ $false) @ $true) @ $true) @ $true) @ $false) @ $false) @ $true) @ ((((((((literal2 @ $true) @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $false) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $false) @ $false) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $false) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $true) @ $true) @ $true) @ $false) @ $true) @ $false) @ ((((((((literal2 @ $true) @ $true) @ $true) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $true) @ $true) @ $true) @ $true) @ $false) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $true) @ $false) @ $false) @ $true) @ $true) @ $true) @ ((((((((literal2 @ $false) @ $false) @ $true) @ $false) @ $false) @ $true) @ $true) @ (zero_zero @ literal)))))))))))))) @ (((cons @ typerep) @ ((typerep_typerep @ A) @ (type2 @ A))) @ (nil @ typerep)))))))))).
% 0.33/5.23 thf(fact_7051_dual__order_Opreordering__axioms, axiom, (! [A:$tType]: (((preorder @ A) => (((preordering @ A) @ (^ [X2:A,Y6:A]: ((((ord_less_eq @ A) @ Y6) @ X2)))) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_7052_irreflp__greater, axiom, (! [A:$tType]: (((preorder @ A) => ((irreflp @ A) @ (^ [X2:A,Y6:A]: ((((ord_less @ A) @ Y6) @ X2)))))))).
% 0.33/5.23 thf(fact_7053_preordering_Oaxioms_I1_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((preordering @ A) @ Less_eq) @ Less) => ((partial_preordering @ A) @ Less_eq))))).
% 0.33/5.23 thf(fact_7054_irreflp__less, axiom, (! [A:$tType]: (((preorder @ A) => ((irreflp @ A) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_7055_preordering__dualI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((preordering @ A) @ (^ [A5:A,B3:A]: (((Less_eq @ B3) @ A5)))) @ (^ [A5:A,B3:A]: (((Less @ B3) @ A5)))) => (((preordering @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_7056_preordering_Oasym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => (~ (((Less @ B2) @ A3)))))))).
% 0.33/5.23 thf(fact_7057_preordering_Oirrefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A]: (((((preordering @ A) @ Less_eq) @ Less) => (~ (((Less @ A3) @ A3))))))).
% 0.33/5.23 thf(fact_7058_preordering_Ostrict__trans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => (((Less @ B2) @ C2) => ((Less @ A3) @ C2))))))).
% 0.33/5.23 thf(fact_7059_preordering_Ostrict__trans1, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A3) @ B2) => (((Less @ B2) @ C2) => ((Less @ A3) @ C2))))))).
% 0.33/5.23 thf(fact_7060_preordering_Ostrict__trans2, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A,C2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => (((Less_eq @ B2) @ C2) => ((Less @ A3) @ C2))))))).
% 0.33/5.23 thf(fact_7061_preordering_Ostrict__iff__not, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) = (((Less_eq @ A3) @ B2) & (~ (((Less_eq @ B2) @ A3))))))))).
% 0.33/5.23 thf(fact_7062_preordering_Ostrict__implies__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A3:A,B2:A]: (((((preordering @ A) @ Less_eq) @ Less) => (((Less @ A3) @ B2) => ((Less_eq @ A3) @ B2)))))).
% 0.33/5.23 thf(fact_7063_preordering__strictI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((! [A8:A,B5:A]: ((((Less_eq @ A8) @ B5) = (((Less @ A8) @ B5) | (A8 = B5))))) => ((! [A8:A,B5:A]: ((((Less @ A8) @ B5) => (~ (((Less @ B5) @ A8)))))) => ((! [A8:A]: ((~ (((Less @ A8) @ A8))))) => ((! [A8:A,B5:A,C3:A]: ((((Less @ A8) @ B5) => (((Less @ B5) @ C3) => ((Less @ A8) @ C3))))) => (((preordering @ A) @ Less_eq) @ Less)))))))).
% 0.33/5.23 thf(fact_7064_order_Opreordering__axioms, axiom, (! [A:$tType]: (((preorder @ A) => (((preordering @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 0.33/5.23 thf(fact_7065_preordering__def, axiom, (! [A:$tType]: (((preordering @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((((partial_preordering @ A) @ Less_eq2) & (((preordering_axioms @ A) @ Less_eq2) @ Less2)))))))).
% 0.33/5.23 thf(fact_7066_preordering_Ointro, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((partial_preordering @ A) @ Less_eq) => ((((preordering_axioms @ A) @ Less_eq) @ Less) => (((preordering @ A) @ Less_eq) @ Less)))))).
% 0.33/5.23 thf(fact_7067_preordering__axioms__def, axiom, (! [A:$tType]: (((preordering_axioms @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((! [A5:A,B3:A]: ((((Less2 @ A5) @ B3) = (((Less_eq2 @ A5) @ B3) & (~ (((Less_eq2 @ B3) @ A5))))))))))))).
% 0.33/5.23 thf(fact_7068_preordering__axioms_Ointro, axiom, (! [A:$tType,Less:(A > (A > $o)),Less_eq:(A > (A > $o))]: (((! [A8:A,B5:A]: ((((Less @ A8) @ B5) = (((Less_eq @ A8) @ B5) & (~ (((Less_eq @ B5) @ A8))))))) => (((preordering_axioms @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_7069_preordering_Oaxioms_I2_J, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((((preordering @ A) @ Less_eq) @ Less) => (((preordering_axioms @ A) @ Less_eq) @ Less))))).
% 0.33/5.23 thf(fact_7070_comm__monoid__set_Ozero__middle, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,P5:nat,K:nat,G:(nat > A),H2:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ ((((if @ A) @ (J3 = K)) @ Z3) @ (H2 @ (((minus_minus @ nat) @ J3) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ (H2 @ J3))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))).
% 0.33/5.23 thf(fact_7071_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)) = ((((image @ A) @ B) @ F) @ ((heap_Time_set_Heap @ A) @ V)))))).
% 0.33/5.23 thf(fact_7072_comm__monoid__set_OatLeastAtMost__shift__0, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M))))))))).
% 0.33/5.23 thf(fact_7073_comm__monoid__set_OatLeast__atMost__pred__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7074_comm__monoid__set_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N3:nat]: ((((minus_minus @ nat) @ N3) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7075_comm__monoid__set_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 0.33/5.23 thf(fact_7076_comm__monoid__set_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 0.33/5.23 thf(fact_7077_comm__monoid__set_OatLeastLessThan__shift__0, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)))))))).
% 0.33/5.23 thf(fact_7078_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))))))).
% 0.33/5.23 thf(fact_7079_comm__monoid__set_OatMost__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_atMost @ nat) @ N)) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))).
% 0.33/5.23 thf(fact_7080_comm__monoid__set_OlessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_lessThan @ nat) @ N)))))))).
% 0.33/5.23 thf(fact_7081_comm__monoid__set_OatMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ ((set_ord_atMost @ nat) @ N)))))))).
% 0.33/5.23 thf(fact_7082_sum_Ocomm__monoid__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups778175481326437816id_set @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 0.33/5.23 thf(fact_7083_comm__monoid__set_OatLeast0__atMost__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ (suc @ N)))))))).
% 0.33/5.23 thf(fact_7084_comm__monoid__set_OatLeast0__lessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ N))))))).
% 0.33/5.23 thf(fact_7085_comm__monoid__set_Onested__swap, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (A3 @ I5)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [J3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 0.33/5.23 thf(fact_7086_comm__monoid__set_OatLeast1__atMost__eq, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [K3:nat]: ((G @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7087_comm__monoid__set_Oin__pairs__0, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((F @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ ((set_ord_atMost @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7088_comm__monoid__set_Ocl__ivl__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,N:nat,M:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((ord_less @ nat) @ (suc @ N)) @ M) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = Z3)) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N)))))))))).
% 0.33/5.23 thf(fact_7089_comm__monoid__set_OatLeast__Suc__lessThan, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less @ nat) @ M) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N)) = ((F @ (G @ M)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ N))))))))).
% 0.33/5.23 thf(fact_7090_comm__monoid__set_Oop__ivl__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,N:nat,M:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((ord_less @ nat) @ N) @ M) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = Z3)) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))).
% 0.33/5.23 thf(fact_7091_comm__monoid__set_Ohead__if, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,N:nat,M:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((ord_less @ nat) @ N) @ M) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = Z3)) & ((~ ((((ord_less @ nat) @ N) @ M))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ (G @ N))))))))).
% 0.33/5.23 thf(fact_7092_comm__monoid__set_Oivl__cong, axiom, (! [A:$tType,B:$tType]: (((ord @ B) => (! [F:(A > (A > A)),Z3:A,A3:B,C2:B,B2:B,D2:B,G:(B > A),H2:(B > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((A3 = C2) => ((B2 = D2) => ((! [X3:B]: (((((ord_less_eq @ B) @ C2) @ X3) => ((((ord_less @ B) @ X3) @ D2) => ((G @ X3) = (H2 @ X3)))))) => (((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z3) @ H2) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D2))))))))))))).
% 0.33/5.23 thf(fact_7093_comm__monoid__set_Onested__swap_H, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (A3 @ I5)) @ ((set_ord_lessThan @ nat) @ I5))))) @ ((set_ord_atMost @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [J3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((A3 @ I5) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7094_comm__monoid__set_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7095_comm__monoid__set_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7096_comm__monoid__set_OlessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))).
% 0.33/5.23 thf(fact_7097_comm__monoid__set_OatMost__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_atMost @ nat) @ N))) @ (G @ (suc @ N)))))))).
% 0.33/5.23 thf(fact_7098_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat,M:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M))))))).
% 0.33/5.23 thf(fact_7099_comm__monoid__set_OatLeastLessThan__rev, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat,M:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ (suc @ I5)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M))))))).
% 0.33/5.23 thf(fact_7100_prod_Ocomm__monoid__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups778175481326437816id_set @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 0.33/5.23 thf(fact_7101_comm__monoid__set_Onat__diff__reindex, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I5)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_lessThan @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7102_comm__monoid__set_Onat__group, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [M2:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))).
% 0.33/5.23 thf(fact_7103_comm__monoid__set_Oub__add__nat, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A),P5:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ N) @ P5))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))).
% 0.33/5.23 thf(fact_7104_comm__monoid__set_Ohead, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((F @ (G @ M)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.23 thf(fact_7105_comm__monoid__set_OatLeast__Suc__atMost, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((F @ (G @ M)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N))))))))).
% 0.33/5.23 thf(fact_7106_comm__monoid__set_Onat__ivl__Suc_H, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ (suc @ N)) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N))) = ((F @ (G @ (suc @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.23 thf(fact_7107_comm__monoid__set_OatLeastLessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,A3:nat,B2:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ A3) @ B2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ (suc @ B2))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2))) @ (G @ B2)))))))).
% 0.33/5.23 thf(fact_7108_comm__monoid__set_OatLeastLessThan__concat, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,P5:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => (((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P5))))))))).
% 0.33/5.23 thf(fact_7109_comm__monoid__set_Olast__plus, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N)) = ((F @ (G @ N)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))))).
% 0.33/5.23 thf(fact_7110_comm__monoid__set_OSuc__reindex__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,M:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => ((((ord_less_eq @ nat) @ M) @ N) => (((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))) @ (G @ (suc @ N))) = ((F @ (G @ M)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: ((G @ (suc @ I5))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))))).
% 0.33/5.23 thf(fact_7111_Heap_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Xa3:(heap_Time_Heap @ A),F:(A > B),Fa:(A > B)]: (((! [Z4:A,Za:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ X)) => ((((member @ A) @ Za) @ ((heap_Time_set_Heap @ A) @ Xa3)) => (((F @ Z4) = (Fa @ Za)) => (Z4 = Za)))))) => ((((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ Fa) @ Xa3)) => (X = Xa3)))))).
% 0.33/5.23 thf(fact_7112_Heap_Omap__cong0, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((! [Z4:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ X)) => ((F @ Z4) = (G @ Z4))))) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ X)))))).
% 0.33/5.23 thf(fact_7113_Heap_Omap__cong, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z4:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((F @ Z4) = (G @ Z4))))) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ Ya))))))).
% 0.33/5.23 thf(fact_7114_comm__monoid__set_OIf__cases, axiom, (! [A:$tType,B:$tType,F:(A > (A > A)),Z3:A,A6:(set @ B),P:(B > $o),H2:(B > A),G:(B > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((finite_finite2 @ B) @ A6) => (((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z3) @ (^ [X2:B]: (((((if @ A) @ (P @ X2)) @ (H2 @ X2)) @ (G @ X2))))) @ A6) = ((F @ ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z3) @ H2) @ (((inf_inf @ (set @ B)) @ A6) @ ((collect @ B) @ P)))) @ ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z3) @ G) @ (((inf_inf @ (set @ B)) @ A6) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))).
% 0.33/5.23 thf(fact_7115_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7116_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7117_comm__monoid__mult__class_Oprod__def, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (((groups7121269368397514597t_prod @ B) @ A) = ((((groups_comm_monoid_F @ A) @ B) @ (times_times @ A)) @ (one_one @ A))))))).
% 0.33/5.23 thf(fact_7118_comm__monoid__set_Oin__pairs, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((F @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I5))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7119_comm__monoid__set_Otriangle__reindex, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ ((product_prod @ nat) @ nat)) @ F) @ Z3) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [K3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7120_comm__monoid__set_Otriangle__reindex__eq, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ ((product_prod @ nat) @ nat)) @ F) @ Z3) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I5:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I5) @ J3)) @ N)))))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [K3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (^ [I5:nat]: (((G @ I5) @ (((minus_minus @ nat) @ K3) @ I5))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_atMost @ nat) @ N))))))).
% 0.33/5.23 thf(fact_7121_comm__monoid__set_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(int > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ int) @ F) @ Z3) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7122_comm__monoid__set_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z3:A,G:(int > A),M:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z3) => (((((((groups_comm_monoid_F @ A) @ int) @ F) @ Z3) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z3) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N))))))).
% 0.33/5.23 thf(fact_7123_Heap_Oin__rel, axiom, (! [B:$tType,A:$tType]: ((((heap_Time_rel_Heap @ A) @ B) = (^ [R6:(A > (B > $o)),A5:(heap_Time_Heap @ A),B3:(heap_Time_Heap @ B)]: ((? [Z7:(heap_Time_Heap @ ((product_prod @ A) @ B))]: (((((member @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ Z7) @ ((collect @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (^ [X2:(heap_Time_Heap @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((heap_Time_set_Heap @ ((product_prod @ A) @ B)) @ X2)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) & ((((((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Z7) = A5) & (((((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Z7) = B3))))))))))).
% 0.33/5.23 thf(fact_7124_Heap_Opred__set, axiom, (! [A:$tType]: (((heap_Time_pred_Heap @ A) = (^ [P4:(A > $o),X2:(heap_Time_Heap @ A)]: ((! [Y6:A]: (((((member @ A) @ Y6) @ ((heap_Time_set_Heap @ A) @ X2)) => (P4 @ Y6)))))))))).
% 0.33/5.23 thf(fact_7125_Heap_ODomainp__rel, axiom, (! [B:$tType,A:$tType,R2:(A > (B > $o))]: (((((domainp @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ B) @ R2)) = ((heap_Time_pred_Heap @ A) @ (((domainp @ A) @ B) @ R2)))))).
% 0.33/5.23 thf(fact_7126_Heap_Opred__True, axiom, (! [A:$tType]: ((((heap_Time_pred_Heap @ A) @ (^ [Uu:A]: ($true))) = (^ [Uu:(heap_Time_Heap @ A)]: ($true)))))).
% 0.33/5.23 thf(fact_7127_Heap_Orel__eq, axiom, (! [A:$tType]: (((((heap_Time_rel_Heap @ A) @ A) @ (^ [Y:A,Z2:A]: ((Y = Z2)))) = (^ [Y:(heap_Time_Heap @ A),Z2:(heap_Time_Heap @ A)]: ((Y = Z2))))))).
% 0.33/5.23 thf(fact_7128_Heap_Orel__refl, axiom, (! [B:$tType,Ra:(B > (B > $o)),X:(heap_Time_Heap @ B)]: (((! [X3:B]: (((Ra @ X3) @ X3))) => (((((heap_Time_rel_Heap @ B) @ B) @ Ra) @ X) @ X))))).
% 0.33/5.23 thf(fact_7129_Heap_Orel__transp, axiom, (! [A:$tType,R2:(A > (A > $o))]: ((((transp @ A) @ R2) => ((transp @ (heap_Time_Heap @ A)) @ (((heap_Time_rel_Heap @ A) @ A) @ R2)))))).
% 0.33/5.23 thf(fact_7130_Heap_Opred__mono, axiom, (! [A:$tType,P:(A > $o),Pa:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ P) @ Pa) => (((ord_less_eq @ ((heap_Time_Heap @ A) > $o)) @ ((heap_Time_pred_Heap @ A) @ P)) @ ((heap_Time_pred_Heap @ A) @ Pa)))))).
% 0.33/5.23 thf(fact_7131_Heap_Orel__mono, axiom, (! [B:$tType,A:$tType,R2:(A > (B > $o)),Ra:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R2) @ Ra) => (((ord_less_eq @ ((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o))) @ (((heap_Time_rel_Heap @ A) @ B) @ R2)) @ (((heap_Time_rel_Heap @ A) @ B) @ Ra)))))).
% 0.33/5.23 thf(fact_7132_Heap_Obi__total__rel, axiom, (! [B:$tType,A:$tType,R2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ R2) => (((bi_total @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ B) @ R2)))))).
% 0.33/5.23 thf(fact_7133_Heap_Orel__transfer, axiom, (! [A:$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) @ $o) @ Sc) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))))) @ ((((((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) @ $o) @ (((heap_Time_rel_Heap @ B) @ D) @ Sc)) @ (^ [Y:$o,Z2:$o]: ((Y = Z2)))))) @ ((heap_Time_rel_Heap @ A) @ B)) @ ((heap_Time_rel_Heap @ C) @ D))))).
% 0.33/5.23 thf(fact_7134_Heap_Opred__transfer, axiom, (! [A:$tType,B:$tType,R2:(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) @ $o) @ R2) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ $o) @ $o) @ (((heap_Time_rel_Heap @ A) @ B) @ R2)) @ (^ [Y:$o,Z2:$o]: ((Y = Z2))))) @ (heap_Time_pred_Heap @ A)) @ (heap_Time_pred_Heap @ B))))).
% 0.33/5.23 thf(fact_7135_Heap_Omap__transfer, axiom, (! [A:$tType,B:$tType,F7:$tType,E4:$tType,Rb:(A > (E4 > $o)),Sd:(B > (F7 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (E4 > F7)) @ ((heap_Time_Heap @ A) > (heap_Time_Heap @ B))) @ ((heap_Time_Heap @ E4) > (heap_Time_Heap @ F7))) @ ((((((bNF_rel_fun @ A) @ E4) @ B) @ F7) @ Rb) @ Sd)) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ E4)) @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ F7)) @ (((heap_Time_rel_Heap @ A) @ E4) @ Rb)) @ (((heap_Time_rel_Heap @ B) @ F7) @ Sd))) @ ((heap_Time_map_Heap @ A) @ B)) @ ((heap_Time_map_Heap @ E4) @ F7))))).
% 0.33/5.23 thf(fact_7136_Heap_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((((heap_Time_pred_Heap @ A) @ (^ [Z7:A]: (((F @ Z7) = (G @ Z7))))) @ Ya) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ Ya))))))).
% 0.33/5.23 thf(fact_7137_Heap_Orel__map_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sb:(C > (B > $o)),I:(A > C),X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ B)]: (((((((heap_Time_rel_Heap @ C) @ B) @ Sb) @ ((((heap_Time_map_Heap @ A) @ C) @ I) @ X)) @ Y3) = (((((heap_Time_rel_Heap @ A) @ B) @ (^ [X2:A]: ((Sb @ (I @ X2))))) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_7138_Heap_Orel__map_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sa:(A > (C > $o)),X:(heap_Time_Heap @ A),G:(B > C),Y3:(heap_Time_Heap @ B)]: (((((((heap_Time_rel_Heap @ A) @ C) @ Sa) @ X) @ ((((heap_Time_map_Heap @ B) @ C) @ G) @ Y3)) = (((((heap_Time_rel_Heap @ A) @ B) @ (^ [X2:A,Y6:B]: (((Sa @ X2) @ (G @ Y6))))) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_7139_Heap_OQuotient, axiom, (! [B:$tType,A:$tType,R2:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T6:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ R2) @ Abs) @ Rep) @ T6) => ((((((quotient @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ A) @ R2)) @ (((heap_Time_map_Heap @ A) @ B) @ Abs)) @ (((heap_Time_map_Heap @ B) @ A) @ Rep)) @ (((heap_Time_rel_Heap @ A) @ B) @ T6)))))).
% 0.33/5.23 thf(fact_7140_Heap_Opred__map, axiom, (! [B:$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) @ $o) @ A) @ Q) @ F)) @ X))))).
% 0.33/5.23 thf(fact_7141_Heap_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X:(heap_Time_Heap @ A),Pa:(A > $o)]: (((((heap_Time_pred_Heap @ A) @ P) @ X) => ((! [Z4:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ X)) => ((P @ Z4) => (Pa @ Z4))))) => (((heap_Time_pred_Heap @ A) @ Pa) @ X)))))).
% 0.33/5.23 thf(fact_7142_Heap_Orel__refl__strong, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Ra:(A > (A > $o))]: (((! [Z4:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ X)) => ((Ra @ Z4) @ Z4)))) => (((((heap_Time_rel_Heap @ A) @ A) @ Ra) @ X) @ X))))).
% 0.33/5.23 thf(fact_7143_Heap_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ B),Ra:(A > (B > $o))]: (((((((heap_Time_rel_Heap @ A) @ B) @ R2) @ X) @ Y3) => ((! [Z4:A,Yb:B]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ X)) => ((((member @ B) @ Yb) @ ((heap_Time_set_Heap @ B) @ Y3)) => (((R2 @ Z4) @ Yb) => ((Ra @ Z4) @ Yb)))))) => (((((heap_Time_rel_Heap @ A) @ B) @ Ra) @ X) @ Y3)))))).
% 0.33/5.23 thf(fact_7144_Heap_Opred__cong, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),P:(A > $o),Pa:(A > $o)]: (((X = Ya) => ((! [Z4:A]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((P @ Z4) = (Pa @ Z4))))) => ((((heap_Time_pred_Heap @ A) @ P) @ X) = (((heap_Time_pred_Heap @ A) @ Pa) @ Ya))))))).
% 0.33/5.23 thf(fact_7145_Heap_Orel__cong, axiom, (! [A:$tType,B:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ B),Xa3:(heap_Time_Heap @ B),R2:(A > (B > $o)),Ra:(A > (B > $o))]: (((X = Ya) => ((Y3 = Xa3) => ((! [Z4:A,Yb:B]: (((((member @ A) @ Z4) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((((member @ B) @ Yb) @ ((heap_Time_set_Heap @ B) @ Xa3)) => (((R2 @ Z4) @ Yb) = ((Ra @ Z4) @ Yb)))))) => ((((((heap_Time_rel_Heap @ A) @ B) @ R2) @ X) @ Y3) = (((((heap_Time_rel_Heap @ A) @ B) @ Ra) @ Ya) @ Xa3)))))))).
% 0.33/5.23 thf(fact_7146_Fract__def, axiom, (fract = (((((((map_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > rat)) @ (id @ int)) @ ((((((map_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ rat) @ (id @ int)) @ abs_Rat)) @ (^ [A5:int,B3:int]: (((((if @ ((product_prod @ int) @ int)) @ (B3 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A5) @ B3))))))).
% 0.33/5.23 thf(fact_7147_subset_Osuc__Union__closed__suc, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A)),Y7:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((member @ (set @ (set @ A))) @ Y7) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Y7) = Y7) => (((ord_less_eq @ (set @ (set @ A))) @ X8) @ Y7))))))).
% 0.33/5.23 thf(fact_7148_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X3:A,Y5:B]: ((((P @ X3) @ Y5) => ((Q @ X3) @ Y5)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 0.33/5.23 thf(fact_7149_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 0.33/5.23 thf(fact_7150_of__int__eq__id, axiom, ((ring_1_of_int @ int) = (id @ int))).
% 0.33/5.23 thf(fact_7151_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 0.33/5.23 thf(fact_7152_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X:A,Y3:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X) @ Y3) => ((Q @ X) @ Y3)))))).
% 0.33/5.23 thf(fact_7153_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 0.33/5.23 thf(fact_7154_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:A,Y3:B,Q:(A > (B > $o))]: ((((P @ X) @ Y3) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X) @ Y3)))))).
% 0.33/5.23 thf(fact_7155_subset_Osuc__Union__closed_Ocases, axiom, (! [A:$tType,A3:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ A3) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((! [X10:(set @ (set @ A))]: (((A3 = ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X10)) => (~ ((((member @ (set @ (set @ A))) @ X10) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))))) => (~ ((! [X10:(set @ (set @ (set @ A)))]: (((A3 = ((complete_Sup_Sup @ (set @ (set @ A))) @ X10)) => (~ ((((member @ (set @ (set @ (set @ A)))) @ X10) @ ((pow2 @ (set @ (set @ A))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))))))))))))).
% 0.33/5.23 thf(fact_7156_subset_Osuc__Union__closed_Osimps, axiom, (! [A:$tType,A3:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ A3) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) = ((? [X7:(set @ (set @ A))]: (((A3 = ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X7)) & (((member @ (set @ (set @ A))) @ X7) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))) | (? [X7:(set @ (set @ (set @ A)))]: (((A3 = ((complete_Sup_Sup @ (set @ (set @ A))) @ X7)) & (((member @ (set @ (set @ (set @ A)))) @ X7) @ ((pow2 @ (set @ (set @ A))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))))))))).
% 0.33/5.23 thf(fact_7157_subset_Osuc__Union__closed_Osuc, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => (((member @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8)) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))).
% 0.33/5.23 thf(fact_7158_subset_Oeq__suc__Union, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) = X8) = (X8 = ((complete_Sup_Sup @ (set @ (set @ A))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))))).
% 0.33/5.23 thf(fact_7159_subset_Osuc__Union__closed__Union, axiom, (! [A:$tType,A6:(set @ (set @ A))]: ((((member @ (set @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))).
% 0.33/5.23 thf(fact_7160_subset_Osuc__Union__closed__in__carrier, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => (((ord_less_eq @ (set @ (set @ A))) @ X8) @ A6))))).
% 0.33/5.23 thf(fact_7161_subset_Osuc__Union__closed__total, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A)),Y7:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((member @ (set @ (set @ A))) @ Y7) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((ord_less_eq @ (set @ (set @ A))) @ X8) @ Y7) | (((ord_less_eq @ (set @ (set @ A))) @ Y7) @ X8))))))).
% 0.33/5.23 thf(fact_7162_subset_Osuc__Union__closed__induct, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A)),Q:((set @ (set @ A)) > $o)]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((! [X10:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X10) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((Q @ X10) => (Q @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X10)))))) => ((! [X10:(set @ (set @ (set @ A)))]: (((((ord_less_eq @ (set @ (set @ (set @ A)))) @ X10) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((! [X4:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X4) @ X10) => (Q @ X4)))) => (Q @ ((complete_Sup_Sup @ (set @ (set @ A))) @ X10)))))) => (Q @ X8))))))).
% 0.33/5.23 thf(fact_7163_subset_Osuc__Union__closed__cases, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((! [Y10:(set @ (set @ A))]: (((X8 = ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Y10)) => (~ ((((member @ (set @ (set @ A))) @ Y10) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))))) => (~ ((! [Y10:(set @ (set @ (set @ A)))]: (((X8 = ((complete_Sup_Sup @ (set @ (set @ A))) @ Y10)) => (~ ((((ord_less_eq @ (set @ (set @ (set @ A)))) @ Y10) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))))))))))).
% 0.33/5.23 thf(fact_7164_subset_Osuc__Union__closed_OUnion, axiom, (! [A:$tType,X8:(set @ (set @ (set @ A))),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ (set @ A)))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => (((member @ (set @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ X8)) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))).
% 0.33/5.23 thf(fact_7165_subset_Osuc__Union__closed_Ointros_I2_J, axiom, (! [A:$tType,Xa5:(set @ (set @ (set @ A))),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ (set @ A)))) @ Xa5) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => (((member @ (set @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ Xa5)) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))).
% 0.33/5.23 thf(fact_7166_subset_Osuc__Union__closed__chain, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((pred_chain @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8))))).
% 0.33/5.23 thf(fact_7167_subset_Osuc__Union__closed__empty, axiom, (! [A:$tType,A6:(set @ (set @ A))]: ((((member @ (set @ (set @ A))) @ (bot_bot @ (set @ (set @ A)))) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))))))).
% 0.33/5.23 thf(fact_7168_natural_Odomain, axiom, (! [P:(nat > $o)]: ((((((domainp @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) = P) => ((((domainp @ nat) @ code_natural) @ code_pcr_natural) = P))))).
% 0.33/5.23 thf(fact_7169_integer_Odomain, axiom, (! [P:(int > $o)]: ((((((domainp @ int) @ int) @ (^ [Y:int,Z2:int]: ((Y = Z2)))) = P) => ((((domainp @ int) @ code_integer) @ code_pcr_integer) = P))))).
% 0.33/5.23 thf(fact_7170_subset_Osuc__Union__closed__subsetD, axiom, (! [A:$tType,Y7:(set @ (set @ A)),X8:(set @ (set @ A)),A6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ Y7) @ X8) => ((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((member @ (set @ (set @ A))) @ Y7) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((X8 = Y7) | (((ord_less_eq @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Y7)) @ X8)))))))).
% 0.33/5.23 thf(fact_7171_subset_Osuc__Union__closed__total_H, axiom, (! [A:$tType,X8:(set @ (set @ A)),A6:(set @ (set @ A)),Y7:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X8) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((member @ (set @ (set @ A))) @ Y7) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((! [Z9:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ Z9) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) => ((((ord_less_eq @ (set @ (set @ A))) @ Z9) @ Y7) => ((Z9 = Y7) | (((ord_less_eq @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Z9)) @ Y7)))))) => ((((ord_less_eq @ (set @ (set @ A))) @ X8) @ Y7) | (((ord_less_eq @ (set @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ Y7)) @ X8)))))))).
% 0.33/5.23 thf(fact_7172_length__remdups__leq, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ ((remdups @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))).
% 0.33/5.23 thf(fact_7173_subset_Osuc__Union__closed__def, axiom, (! [A:$tType,A6:(set @ (set @ A))]: (((((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) = ((collect @ (set @ (set @ A))) @ (((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))).
% 0.33/5.23 thf(fact_7174_sorted__remdups, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((remdups @ A) @ Xs))))))))).
% 0.33/5.23 thf(fact_7175_subset_Osuc__Union__closedp__suc__Union__closed__eq, axiom, (! [A:$tType,A6:(set @ (set @ A))]: (((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) = (^ [X2:(set @ (set @ A))]: ((((member @ (set @ (set @ A))) @ X2) @ (((pred_s596693808085603175closed @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))))))))).
% 0.33/5.23 thf(fact_7176_subset_Osuc__Union__closedp_Osuc, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ A))]: ((((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8) => ((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X8)))))).
% 0.33/5.23 thf(fact_7177_subset_Osuc__Union__closedp_Osimps, axiom, (! [A:$tType,A6:(set @ (set @ A)),A3:(set @ (set @ A))]: ((((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ A3) = ((? [X7:(set @ (set @ A))]: (((A3 = ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X7)) & ((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X7)))) | (? [X7:(set @ (set @ (set @ A)))]: (((A3 = ((complete_Sup_Sup @ (set @ (set @ A))) @ X7)) & (((powp @ (set @ (set @ A))) @ (((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) @ X7))))))))).
% 0.33/5.23 thf(fact_7178_subset_Osuc__Union__closedp_Ointros_I2_J, axiom, (! [A:$tType,A6:(set @ (set @ A)),Xa5:(set @ (set @ (set @ A)))]: (((((powp @ (set @ (set @ A))) @ (((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) @ Xa5) => ((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ Xa5)))))).
% 0.33/5.23 thf(fact_7179_subset_Osuc__Union__closedp_OUnion, axiom, (! [A:$tType,A6:(set @ (set @ A)),X8:(set @ (set @ (set @ A)))]: (((((powp @ (set @ (set @ A))) @ (((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) @ X8) => ((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ X8)))))).
% 0.33/5.23 thf(fact_7180_subset_Osuc__Union__closedp_Ocases, axiom, (! [A:$tType,A6:(set @ (set @ A)),A3:(set @ (set @ A))]: ((((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ A3) => ((! [X10:(set @ (set @ A))]: (((A3 = ((((pred_suc @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X10)) => (~ (((((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A))) @ X10)))))) => (~ ((! [X10:(set @ (set @ (set @ A)))]: (((A3 = ((complete_Sup_Sup @ (set @ (set @ A))) @ X10)) => (~ ((((powp @ (set @ (set @ A))) @ (((pred_s7749564232668923593losedp @ (set @ A)) @ A6) @ (ord_less @ (set @ A)))) @ X10))))))))))))).
% 0.33/5.23 thf(fact_7181_int_Odomain, axiom, (! [DR1:(nat > $o),DR2:(nat > $o)]: ((((((domainp @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) = DR1) => (((((domainp @ nat) @ nat) @ (^ [Y:nat,Z2:nat]: ((Y = Z2)))) = DR2) => ((((domainp @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) = ((((basic_pred_prod @ nat) @ nat) @ DR1) @ DR2))))))).
% 0.33/5.23 thf(fact_7182_Heap_Ocase, axiom, (! [B:$tType,A:$tType,F:(((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > B),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((heap_Time_case_Heap @ A) @ B) @ F) @ ((heap_Time_Heap2 @ A) @ X)) = (F @ X))))).
% 0.33/5.23 thf(fact_7183_Heap_Ocase__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,H2:(B > C),F:(((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > B),Heap:(heap_Time_Heap @ A)]: (((H2 @ ((((heap_Time_case_Heap @ A) @ B) @ F) @ Heap)) = ((((heap_Time_case_Heap @ A) @ C) @ (^ [X2:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((H2 @ (F @ X2))))) @ Heap))))).
% 0.33/5.23 thf(fact_7184_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))))).
% 0.33/5.23 thf(fact_7185_enumerate__Suc__eq, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((enumerate @ A) @ (suc @ N)) @ Xs) = ((((map @ ((product_prod @ nat) @ A)) @ ((product_prod @ nat) @ A)) @ ((((product_apfst @ nat) @ nat) @ A) @ suc)) @ (((enumerate @ A) @ N) @ Xs)))))).
% 0.33/5.23 thf(fact_7186_Heap__ordE, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ A),H2:(heap_ext @ product_unit)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y3) => (((((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) @ Y3) @ H2))))))).
% 0.33/5.23 thf(fact_7187_Heap__ordI, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ A)]: (((! [H6:(heap_ext @ product_unit)]: ((((((heap_Time_execute @ A) @ X) @ H6) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) | ((((heap_Time_execute @ A) @ X) @ H6) = (((heap_Time_execute @ A) @ Y3) @ H6))))) => (((heap_Time_Heap_ord @ A) @ X) @ Y3))))).
% 0.33/5.23 thf(fact_7188_heap_Oleq__refl, axiom, (! [A:$tType,X:(heap_Time_Heap @ A)]: ((((heap_Time_Heap_ord @ A) @ X) @ X)))).
% 0.33/5.23 thf(fact_7189_heap_Oleq__trans, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ A),Z3:(heap_Time_Heap @ A)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y3) => ((((heap_Time_Heap_ord @ A) @ Y3) @ Z3) => (((heap_Time_Heap_ord @ A) @ X) @ Z3)))))).
% 0.33/5.23 thf(fact_7190_heap_Oleq__antisym, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y3:(heap_Time_Heap @ A)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y3) => ((((heap_Time_Heap_ord @ A) @ Y3) @ X) => (X = Y3)))))).
% 0.33/5.23 thf(fact_7191_heap_Olub__least, axiom, (! [A:$tType,A6:(set @ (heap_Time_Heap @ A)),Z3:(heap_Time_Heap @ A)]: (((((comple1602240252501008431_chain @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ A6) => ((! [X3:(heap_Time_Heap @ A)]: (((((member @ (heap_Time_Heap @ A)) @ X3) @ A6) => (((heap_Time_Heap_ord @ A) @ X3) @ Z3)))) => (((heap_Time_Heap_ord @ A) @ ((heap_Time_Heap_lub @ A) @ A6)) @ Z3)))))).
% 0.33/5.23 thf(fact_7192_heap_Olub__upper, axiom, (! [A:$tType,A6:(set @ (heap_Time_Heap @ A)),X:(heap_Time_Heap @ A)]: (((((comple1602240252501008431_chain @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ A6) => ((((member @ (heap_Time_Heap @ A)) @ X) @ A6) => (((heap_Time_Heap_ord @ A) @ X) @ ((heap_Time_Heap_lub @ A) @ A6))))))).
% 0.33/5.23 thf(fact_7193_heap_Oconst__mono, axiom, (! [A:$tType,B:$tType,Ord:(B > (B > $o)),C2:(heap_Time_Heap @ A)]: ((((((comple7038119648293358887notone @ B) @ (heap_Time_Heap @ A)) @ Ord) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:B]: (C2)))))).
% 0.33/5.23 thf(fact_7194_heap__interpretation, axiom, (! [A:$tType]: ((((partia7178651479351089652itions @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ (heap_Time_Heap_lub @ A))))).
% 0.33/5.23 thf(fact_7195_Heap__Time__Monad_Obind__mono, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,B7:((A > (heap_Time_Heap @ B)) > (heap_Time_Heap @ C)),C6:(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)) @ B7) => ((! [Y5: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)) @ (C6 @ Y5)))) => (((((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) @ (B7 @ F2)) @ (^ [Y6:C]: (((C6 @ Y6) @ F2)))))))))))).
% 0.33/5.23 thf(fact_7196_admissible__heap, axiom, (! [B:$tType,A:$tType,P:(A > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > (B > (nat > $o)))))]: (((((comple1908693960933563346ssible @ (A > (heap_Time_Heap @ B))) @ ((((partial_fun_lub @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_lub @ B))) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (^ [F2:(A > (heap_Time_Heap @ B))]: ((! [X2:A,H4:(heap_ext @ product_unit),H5:(heap_ext @ product_unit),R5:B,N3:nat]: ((((((((heap_Time_effect @ B) @ (F2 @ X2)) @ H4) @ H5) @ R5) @ N3) => (((((P @ X2) @ H4) @ H5) @ R5) @ N3)))))))))).
% 0.33/5.23 thf(fact_7197_heap_Ofixp__induct__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (heap_Time_Heap @ A))),F4:(C > C),C6:((B > (heap_Time_Heap @ A)) > C),F:C,P:((B > (heap_Time_Heap @ A)) > $o)]: (((! [X3:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((U4 @ (F4 @ (C6 @ F2))) @ X3)))))) => ((F = (C6 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (^ [F2:(B > (heap_Time_Heap @ A))]: ((U4 @ (F4 @ (C6 @ F2)))))))) => ((! [F3:(B > (heap_Time_Heap @ A))]: (((U4 @ (C6 @ F3)) = F3))) => (((((comple1908693960933563346ssible @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ P) => ((P @ (^ [Uu:B]: (((heap_Time_Heap2 @ A) @ (^ [X2:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))) => ((! [F3:C]: (((P @ (U4 @ F3)) => (P @ (U4 @ (F4 @ F3)))))) => (P @ (U4 @ F))))))))))).
% 0.33/5.23 thf(fact_7198_heap_Omono__body__fixp, axiom, (! [A:$tType,B:$tType,F4:((B > (heap_Time_Heap @ A)) > (B > (heap_Time_Heap @ A)))]: (((! [X3:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((F4 @ F2) @ X3)))))) => (((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ F4) = (F4 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ F4))))))).
% 0.33/5.23 thf(fact_7199_heap_Ofixp__rule__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (heap_Time_Heap @ A))),F4:(C > C),C6:((B > (heap_Time_Heap @ A)) > C),F:C]: (((! [X3: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 @ (F4 @ (C6 @ F2))) @ X3)))))) => ((F = (C6 @ ((((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 @ (F4 @ (C6 @ F2)))))))) => ((! [F3:C]: (((C6 @ (U4 @ F3)) = F3))) => (F = (F4 @ F)))))))).
% 0.33/5.23 thf(subcl_Heap_Oheap___HOL_Otype, axiom, (! [A:$tType]: (((heap @ A) => (type @ A))))).
% 0.33/5.23 thf(subcl_Heap_Oheap___Typerep_Otyperep, axiom, (! [A:$tType]: (((heap @ A) => (typerep2 @ A))))).
% 0.33/5.23 thf(subcl_Heap_Oheap___Countable_Ocountable, axiom, (! [A:$tType]: (((heap @ A) => (countable @ A))))).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice, axiom, (bounded_lattice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite, axiom, (finite_finite @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1, axiom, (! [A17:$tType,A18:$tType]: ((((finite_finite @ A17) & (finite_finite @ A18)) => (finite_finite @ ((product_prod @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Option_Ooption___Finite__Set_Ofinite_2, axiom, (! [A17:$tType]: (((finite_finite @ A17) => (finite_finite @ (option @ A17)))))).
% 0.33/5.23 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3, axiom, (! [A17:$tType,A18:$tType]: ((((finite_finite @ A17) & (finite_finite @ A18)) => (finite_finite @ ((sum_sum @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_String_Ochar___Finite__Set_Ofinite_4, axiom, (finite_finite @ char)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_5, axiom, (bounded_lattice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6, axiom, (finite_finite @ $o)).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Obounded__lattice_7, axiom, (! [A17:$tType]: ((bounded_lattice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Finite__Set_Ofinite_8, axiom, (! [A17:$tType]: (((finite_finite @ A17) => (finite_finite @ (set @ A17)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Obounded__lattice_9, axiom, (! [A17:$tType,A18:$tType]: (((bounded_lattice @ A18) => (bounded_lattice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Finite__Set_Ofinite_10, axiom, (! [A17:$tType,A18:$tType]: ((((finite_finite @ A17) & (finite_finite @ A18)) => (finite_finite @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A17:$tType,A18:$tType]: (((comple6319245703460814977attice @ A18) => (condit1219197933456340205attice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice, axiom, (! [A17:$tType,A18:$tType]: (((comple592849572758109894attice @ A18) => (comple592849572758109894attice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra, axiom, (! [A17:$tType,A18:$tType]: (((comple489889107523837845lgebra @ A18) => (comple489889107523837845lgebra @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A17:$tType,A18:$tType]: (((bounded_lattice @ A18) => (bounde4967611905675639751up_bot @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top, axiom, (! [A17:$tType,A18:$tType]: (((bounded_lattice @ A18) => (bounde4346867609351753570nf_top @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A17:$tType,A18:$tType]: (((comple6319245703460814977attice @ A18) => (comple6319245703460814977attice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra, axiom, (! [A17:$tType,A18:$tType]: (((boolea8198339166811842893lgebra @ A18) => (boolea8198339166811842893lgebra @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A17:$tType,A18:$tType]: (((bounded_lattice @ A18) => (bounded_lattice_top @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A17:$tType,A18:$tType]: (((bounded_lattice @ A18) => (bounded_lattice_bot @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Complete__Partial__Order_Occpo, axiom, (! [A17:$tType,A18:$tType]: (((comple6319245703460814977attice @ A18) => (comple9053668089753744459l_ccpo @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A17:$tType,A18:$tType]: (((semilattice_sup @ A18) => (semilattice_sup @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A17:$tType,A18:$tType]: (((semilattice_inf @ A18) => (semilattice_inf @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A17:$tType,A18:$tType]: (((distrib_lattice @ A18) => (distrib_lattice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A17:$tType,A18:$tType]: (((order_top @ A18) => (order_top @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A17:$tType,A18:$tType]: (((order_bot @ A18) => (order_bot @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Countable_Ocountable, axiom, (! [A17:$tType,A18:$tType]: ((((finite_finite @ A17) & (countable @ A18)) => (countable @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A17:$tType,A18:$tType]: (((preorder @ A18) => (preorder @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Lattices_Olattice, axiom, (! [A17:$tType,A18:$tType]: (((lattice @ A18) => (lattice @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Typerep_Otyperep, axiom, (! [A17:$tType,A18:$tType]: ((((typerep2 @ A17) & (typerep2 @ A18)) => (typerep2 @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Oorder, axiom, (! [A17:$tType,A18:$tType]: (((order @ A18) => (order @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Otop, axiom, (! [A17:$tType,A18:$tType]: (((top @ A18) => (top @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Oord, axiom, (! [A17:$tType,A18:$tType]: (((ord @ A18) => (ord @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Orderings_Obot, axiom, (! [A17:$tType,A18:$tType]: (((bot @ A18) => (bot @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Groups_Ouminus, axiom, (! [A17:$tType,A18:$tType]: (((uminus @ A18) => (uminus @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_fun___Groups_Ominus, axiom, (! [A17:$tType,A18:$tType]: (((minus @ A18) => (minus @ (A17 > A18)))))).
% 0.33/5.23 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit6923001295902523014norder @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11, axiom, (condit1219197933456340205attice @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, axiom, (bit_un5681908812861735899ations @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, (semiri1453513574482234551roduct @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat, axiom, (euclid5411537665997757685th_nat @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat, axiom, (euclid8789492081693882211th_nat @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere1937475149494474687imp_le @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring, axiom, (euclid3128863361964157862miring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel, axiom, (euclid4440199948858584721cancel @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative, axiom, (normal6328177297339901930cative @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1627219031080169319umeral @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel, axiom, (euclid8851590272496341667cancel @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel, axiom, (semiri6575147826004484403cancel @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict9044650504122735259up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere580206878836729694up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere2412721322843649153imp_le @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations, axiom, (bit_se359711467146920520ations @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict, axiom, (linord2810124833399127020strict @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add, axiom, (strict7427464778891057005id_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add, axiom, (ordere8940638589300402666id_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring, axiom, (euclid3725896446679973847miring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict, axiom, (linord715952674999750819strict @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add, axiom, (linord4140545234300271783up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations, axiom, (bit_ri3973907225187159222ations @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri2026040879449505780visors @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord181362715937106298miring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring, axiom, (euclid5891614535332579305n_ring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor, axiom, (semido2269285787275462019factor @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict, axiom, (linord8928482502909563296strict @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, (semiri3467727345109120633visors @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere6658533253407199908up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs, axiom, (ordere166539214618696060dd_abs @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize, axiom, (semiri6843258321239162965malize @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add, axiom, (ordere6911136660526730532id_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord5086331880401160121up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel2418104881723323429up_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors, axiom, (ring_15535105094025558882visors @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1802427076303600483id_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict, axiom, (linord4710134922213307826strict @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel, axiom, (comm_s4317794764714335236cancel @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits, axiom, (bit_semiring_bits @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oordered__comm__semiring, axiom, (ordere2520102378445227354miring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Onormalization__semidom, axiom, (normal8620421768224518004emidom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1, axiom, (linord6961819062388156250ring_1 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__semiring, axiom, (linordered_semiring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oordered__semiring__0, axiom, (ordered_semiring_0 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_12, axiom, (semilattice_sup @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_13, axiom, (semilattice_inf @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Lattices_Odistrib__lattice_14, axiom, (distrib_lattice @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult, axiom, (ab_semigroup_mult @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel, axiom, (semiring_1_cancel @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom, axiom, (algebraic_semidom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oordered__semiring, axiom, (ordered_semiring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oordered__ring__abs, axiom, (ordered_ring_abs @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Parity_Osemiring__parity, axiom, (semiring_parity @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__modulo, axiom, (semiring_modulo @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__ring, axiom, (linordered_ring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Osemigroup__mult, axiom, (semigroup_mult @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemidom__modulo, axiom, (semidom_modulo @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, (semidom_divide @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Num_Osemiring__numeral, axiom, (semiring_numeral @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ozero__less__one, axiom, (zero_less_one @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__semiring, axiom, (comm_semiring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Countable_Ocountable_15, axiom, (countable @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ozero__neq__one, axiom, (zero_neq_one @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oordered__ring, axiom, (ordered_ring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn, axiom, (idom_abs_sgn @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Parity_Oring__parity, axiom, (ring_parity @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Opreorder_16, axiom, (preorder @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oidom__modulo, axiom, (idom_modulo @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oidom__divide, axiom, (idom_divide @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, (monoid_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring__0, axiom, (semiring_0 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Lattices_Olattice_17, axiom, (lattice @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___GCD_Osemiring__gcd, axiom, (semiring_gcd @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___GCD_Osemiring__Gcd, axiom, (semiring_Gcd @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Typerep_Otyperep_18, axiom, (typerep2 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Omult__zero, axiom, (mult_zero @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Ocomm__ring, axiom, (comm_ring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Oorder_19, axiom, (order @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, (neg_numeral @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemiring, axiom, (semiring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Osemidom, axiom, (semidom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Orderings_Oord_20, axiom, (ord @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ouminus_21, axiom, (uminus @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oabs__if, axiom, (abs_if @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ominus_22, axiom, (minus @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___GCD_Oring__gcd, axiom, (ring_gcd @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Power_Opower, axiom, (power @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Num_Onumeral, axiom, (numeral @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oplus, axiom, (plus @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oring, axiom, (ring @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Oidom, axiom, (idom @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Rings_Odvd, axiom, (dvd @ int)).
% 0.33/5.23 thf(tcon_Int_Oint___Heap_Oheap, axiom, (heap @ int)).
% 0.33/5.23 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_23, axiom, (condit6923001295902523014norder @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24, axiom, (condit1219197933456340205attice @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_25, axiom, (bit_un5681908812861735899ations @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_26, axiom, (semiri1453513574482234551roduct @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_27, axiom, (euclid5411537665997757685th_nat @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28, axiom, (ordere1937475149494474687imp_le @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_29, axiom, (euclid3128863361964157862miring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_30, axiom, (euclid4440199948858584721cancel @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_31, axiom, (normal6328177297339901930cative @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_32, axiom, (unique1627219031080169319umeral @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_33, axiom, (semiri6575147826004484403cancel @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_34, axiom, (strict9044650504122735259up_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff, axiom, (ordere1170586879665033532d_diff @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_35, axiom, (ordere580206878836729694up_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_36, axiom, (ordere2412721322843649153imp_le @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_37, axiom, (bit_se359711467146920520ations @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_38, axiom, (linord2810124833399127020strict @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_39, axiom, (strict7427464778891057005id_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_40, axiom, (ordere8940638589300402666id_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni5634975068530333245id_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_41, axiom, (euclid3725896446679973847miring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_42, axiom, (linord4140545234300271783up_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43, axiom, (semiri2026040879449505780visors @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44, axiom, (linord181362715937106298miring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_45, axiom, (semido2269285787275462019factor @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46, axiom, (linord8928482502909563296strict @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47, axiom, (semiri3467727345109120633visors @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48, axiom, (ordere6658533253407199908up_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49, axiom, (semiri6843258321239162965malize @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50, axiom, (ordere6911136660526730532id_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51, axiom, (cancel2418104881723323429up_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52, axiom, (cancel1802427076303600483id_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53, axiom, (comm_s4317794764714335236cancel @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54, axiom, (bit_semiring_bits @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55, axiom, (ordere2520102378445227354miring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Onormalization__semidom_56, axiom, (normal8620421768224518004emidom @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_57, axiom, (cancel_semigroup_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Olinordered__semiring_58, axiom, (linordered_semiring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_59, axiom, (ordered_semiring_0 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_60, axiom, (linordered_semidom @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_61, axiom, (semilattice_sup @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_62, axiom, (semilattice_inf @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_63, axiom, (distrib_lattice @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64, axiom, (ab_semigroup_mult @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65, axiom, (semiring_1_cancel @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66, axiom, (algebraic_semidom @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67, axiom, (comm_monoid_mult @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_68, axiom, (ab_semigroup_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Oordered__semiring_69, axiom, (ordered_semiring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Parity_Osemiring__parity_70, axiom, (semiring_parity @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71, axiom, (comm_monoid_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_72, axiom, (semiring_modulo @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73, axiom, (comm_semiring_1 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74, axiom, (comm_semiring_0 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_75, axiom, (semigroup_mult @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_76, axiom, (semidom_modulo @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemidom__divide_77, axiom, (semidom_divide @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Num_Osemiring__numeral_78, axiom, (semiring_numeral @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Osemigroup__add_79, axiom, (semigroup_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ozero__less__one_80, axiom, (zero_less_one @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_81, axiom, (comm_semiring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Oorder__bot_82, axiom, (order_bot @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_83, axiom, (semiring_char_0 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Countable_Ocountable_84, axiom, (countable @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_85, axiom, (zero_neq_one @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Opreorder_86, axiom, (preorder @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Olinorder_87, axiom, (linorder @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Omonoid__mult_88, axiom, (monoid_mult @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Omonoid__add_89, axiom, (monoid_add @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__1_90, axiom, (semiring_1 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring__0_91, axiom, (semiring_0 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Ono__top_92, axiom, (no_top @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Lattices_Olattice_93, axiom, (lattice @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_94, axiom, (semiring_gcd @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_95, axiom, (semiring_Gcd @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Typerep_Otyperep_96, axiom, (typerep2 @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Omult__zero_97, axiom, (mult_zero @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Oorder_98, axiom, (order @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemiring_99, axiom, (semiring @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Osemidom_100, axiom, (semidom @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Oord_101, axiom, (ord @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Orderings_Obot_102, axiom, (bot @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ominus_103, axiom, (minus @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Power_Opower_104, axiom, (power @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Num_Onumeral_105, axiom, (numeral @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Ozero_106, axiom, (zero @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oplus_107, axiom, (plus @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Groups_Oone_108, axiom, (one @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Rings_Odvd_109, axiom, (dvd @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Heap_Oheap_110, axiom, (heap @ nat)).
% 0.33/5.23 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 0.33/5.23 thf(tcon_Num_Onum___Orderings_Opreorder_111, axiom, (preorder @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Orderings_Olinorder_112, axiom, (linorder @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Typerep_Otyperep_113, axiom, (typerep2 @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Orderings_Oorder_114, axiom, (order @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Orderings_Oord_115, axiom, (ord @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Groups_Oplus_116, axiom, (plus @ num)).
% 0.33/5.23 thf(tcon_Num_Onum___Nat_Osize_117, axiom, (size @ num)).
% 0.33/5.23 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_118, axiom, (semiri1453513574482234551roduct @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_119, axiom, (ordere1937475149494474687imp_le @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_120, axiom, (semiri6575147826004484403cancel @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_121, axiom, (strict9044650504122735259up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_122, axiom, (ordere580206878836729694up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_123, axiom, (ordere2412721322843649153imp_le @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_124, axiom, (linord2810124833399127020strict @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_125, axiom, (strict7427464778891057005id_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_126, axiom, (ordere8940638589300402666id_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field, axiom, (archim462609752435547400_field @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_127, axiom, (linord715952674999750819strict @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_128, axiom, (linord4140545234300271783up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_129, axiom, (semiri2026040879449505780visors @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_130, axiom, (linord181362715937106298miring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_131, axiom, (linord8928482502909563296strict @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_132, axiom, (semiri3467727345109120633visors @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_133, axiom, (ordere6658533253407199908up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_134, axiom, (ordere166539214618696060dd_abs @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling, axiom, (archim2362893244070406136eiling @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_135, axiom, (ordere6911136660526730532id_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_136, axiom, (linord5086331880401160121up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_137, axiom, (cancel2418104881723323429up_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_138, axiom, (ring_15535105094025558882visors @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_139, axiom, (cancel1802427076303600483id_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_140, axiom, (linord4710134922213307826strict @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_141, axiom, (comm_s4317794764714335236cancel @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_142, axiom, (ordere2520102378445227354miring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_143, axiom, (linord6961819062388156250ring_1 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_144, axiom, (ordered_ab_group_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_145, axiom, (cancel_semigroup_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__semiring_146, axiom, (linordered_semiring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_147, axiom, (ordered_semiring_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_148, axiom, (linordered_semidom @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Odense__linorder, axiom, (dense_linorder @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_149, axiom, (semilattice_sup @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_150, axiom, (semilattice_inf @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_151, axiom, (distrib_lattice @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_152, axiom, (ab_semigroup_mult @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_153, axiom, (semiring_1_cancel @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_154, axiom, (comm_monoid_mult @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_155, axiom, (ab_semigroup_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Olinordered__field, axiom, (linordered_field @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oordered__semiring_156, axiom, (ordered_semiring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_157, axiom, (ordered_ring_abs @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_158, axiom, (comm_monoid_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__ring_159, axiom, (linordered_ring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Olinordered__idom_160, axiom, (linordered_idom @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_161, axiom, (comm_semiring_1 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_162, axiom, (comm_semiring_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Odense__order, axiom, (dense_order @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_163, axiom, (semigroup_mult @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemidom__divide_164, axiom, (semidom_divide @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Num_Osemiring__numeral_165, axiom, (semiring_numeral @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Osemigroup__add_166, axiom, (semigroup_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn, axiom, (field_abs_sgn @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Odivision__ring, axiom, (division_ring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ozero__less__one_167, axiom, (zero_less_one @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_168, axiom, (comm_semiring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_169, axiom, (semiring_char_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oab__group__add_170, axiom, (ab_group_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Ofield__char__0, axiom, (field_char_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Countable_Ocountable_171, axiom, (countable @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_172, axiom, (zero_neq_one @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oordered__ring_173, axiom, (ordered_ring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_174, axiom, (idom_abs_sgn @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Opreorder_175, axiom, (preorder @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Olinorder_176, axiom, (linorder @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Omonoid__mult_177, axiom, (monoid_mult @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oidom__divide_178, axiom, (idom_divide @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_179, axiom, (comm_ring_1 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Omonoid__add_180, axiom, (monoid_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__1_181, axiom, (semiring_1 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring__0_182, axiom, (semiring_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Ono__top_183, axiom, (no_top @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Ono__bot_184, axiom, (no_bot @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Lattices_Olattice_185, axiom, (lattice @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ogroup__add_186, axiom, (group_add @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Typerep_Otyperep_187, axiom, (typerep2 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Omult__zero_188, axiom, (mult_zero @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Ocomm__ring_189, axiom, (comm_ring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Oorder_190, axiom, (order @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Num_Oneg__numeral_191, axiom, (neg_numeral @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Nat_Oring__char__0_192, axiom, (ring_char_0 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemiring_193, axiom, (semiring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Oinverse, axiom, (inverse @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Osemidom_194, axiom, (semidom @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Orderings_Oord_195, axiom, (ord @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ouminus_196, axiom, (uminus @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oring__1_197, axiom, (ring_1 @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oabs__if_198, axiom, (abs_if @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ominus_199, axiom, (minus @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Fields_Ofield, axiom, (field @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Power_Opower_200, axiom, (power @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Num_Onumeral_201, axiom, (numeral @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Ozero_202, axiom, (zero @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oplus_203, axiom, (plus @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oring_204, axiom, (ring @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Oidom_205, axiom, (idom @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Groups_Oone_206, axiom, (one @ rat)).
% 0.33/5.23 thf(tcon_Rat_Orat___Rings_Odvd_207, axiom, (dvd @ rat)).
% 0.33/5.23 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_208, axiom, (! [A17:$tType]: ((condit1219197933456340205attice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_209, axiom, (! [A17:$tType]: ((comple592849572758109894attice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_210, axiom, (! [A17:$tType]: ((comple489889107523837845lgebra @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_211, axiom, (! [A17:$tType]: ((bounde4967611905675639751up_bot @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_212, axiom, (! [A17:$tType]: ((bounde4346867609351753570nf_top @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_213, axiom, (! [A17:$tType]: ((comple6319245703460814977attice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_214, axiom, (! [A17:$tType]: ((boolea8198339166811842893lgebra @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_215, axiom, (! [A17:$tType]: ((bounded_lattice_top @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_216, axiom, (! [A17:$tType]: ((bounded_lattice_bot @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_217, axiom, (! [A17:$tType]: ((comple9053668089753744459l_ccpo @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_218, axiom, (! [A17:$tType]: ((semilattice_sup @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_219, axiom, (! [A17:$tType]: ((semilattice_inf @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_220, axiom, (! [A17:$tType]: ((distrib_lattice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Oorder__top_221, axiom, (! [A17:$tType]: ((order_top @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Oorder__bot_222, axiom, (! [A17:$tType]: ((order_bot @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Countable_Ocountable_223, axiom, (! [A17:$tType]: (((finite_finite @ A17) => (countable @ (set @ A17)))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Opreorder_224, axiom, (! [A17:$tType]: ((preorder @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Lattices_Olattice_225, axiom, (! [A17:$tType]: ((lattice @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Typerep_Otyperep_226, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (set @ A17)))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Oorder_227, axiom, (! [A17:$tType]: ((order @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Otop_228, axiom, (! [A17:$tType]: ((top @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Oord_229, axiom, (! [A17:$tType]: ((ord @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Orderings_Obot_230, axiom, (! [A17:$tType]: ((bot @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Groups_Ouminus_231, axiom, (! [A17:$tType]: ((uminus @ (set @ A17))))).
% 0.33/5.23 thf(tcon_Set_Oset___Groups_Ominus_232, axiom, (! [A17:$tType]: ((minus @ (set @ A17))))).
% 0.33/5.23 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233, axiom, (condit1219197933456340205attice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_234, axiom, (comple592849572758109894attice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_235, axiom, (comple489889107523837845lgebra @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_236, axiom, (bounde4967611905675639751up_bot @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_237, axiom, (bounde4346867609351753570nf_top @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_238, axiom, (comple6319245703460814977attice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_239, axiom, (boolea8198339166811842893lgebra @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_240, axiom, (bounded_lattice_top @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_241, axiom, (bounded_lattice_bot @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_242, axiom, (comple9053668089753744459l_ccpo @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_243, axiom, (semilattice_sup @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_244, axiom, (semilattice_inf @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_245, axiom, (distrib_lattice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Oorder__top_246, axiom, (order_top @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Oorder__bot_247, axiom, (order_bot @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Countable_Ocountable_248, axiom, (countable @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Opreorder_249, axiom, (preorder @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Olinorder_250, axiom, (linorder @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Lattices_Olattice_251, axiom, (lattice @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Typerep_Otyperep_252, axiom, (typerep2 @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Oorder_253, axiom, (order @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Otop_254, axiom, (top @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Oord_255, axiom, (ord @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Orderings_Obot_256, axiom, (bot @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Groups_Ouminus_257, axiom, (uminus @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Groups_Ominus_258, axiom, (minus @ $o)).
% 0.33/5.23 thf(tcon_HOL_Obool___Heap_Oheap_259, axiom, (heap @ $o)).
% 0.33/5.23 thf(tcon_List_Olist___Countable_Ocountable_260, axiom, (! [A17:$tType]: (((countable @ A17) => (countable @ (list @ A17)))))).
% 0.33/5.23 thf(tcon_List_Olist___Typerep_Otyperep_261, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (list @ A17)))))).
% 0.33/5.23 thf(tcon_List_Olist___Heap_Oheap_262, axiom, (! [A17:$tType]: (((heap @ A17) => (heap @ (list @ A17)))))).
% 0.33/5.23 thf(tcon_List_Olist___Nat_Osize_263, axiom, (! [A17:$tType]: ((size @ (list @ A17))))).
% 0.33/5.23 thf(tcon_Heap_Oarray___Countable_Ocountable_264, axiom, (! [A17:$tType]: ((countable @ (array @ A17))))).
% 0.33/5.23 thf(tcon_Heap_Oarray___Typerep_Otyperep_265, axiom, (! [A17:$tType]: ((typerep2 @ (array @ A17))))).
% 0.33/5.23 thf(tcon_Heap_Oarray___Heap_Oheap_266, axiom, (! [A17:$tType]: ((heap @ (array @ A17))))).
% 0.33/5.23 thf(tcon_Heap_Oarray___Nat_Osize_267, axiom, (! [A17:$tType]: ((size @ (array @ A17))))).
% 0.33/5.23 thf(tcon_String_Ochar___Countable_Ocountable_268, axiom, (countable @ char)).
% 0.33/5.23 thf(tcon_String_Ochar___Typerep_Otyperep_269, axiom, (typerep2 @ char)).
% 0.33/5.23 thf(tcon_String_Ochar___Heap_Oheap_270, axiom, (heap @ char)).
% 0.33/5.23 thf(tcon_String_Ochar___Nat_Osize_271, axiom, (size @ char)).
% 0.33/5.23 thf(tcon_Nitpick_Oword___Typerep_Otyperep_272, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (word @ A17)))))).
% 0.33/5.23 thf(tcon_Sum__Type_Osum___Countable_Ocountable_273, axiom, (! [A17:$tType,A18:$tType]: ((((countable @ A17) & (countable @ A18)) => (countable @ ((sum_sum @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_274, axiom, (! [A17:$tType,A18:$tType]: ((((typerep2 @ A17) & (typerep2 @ A18)) => (typerep2 @ ((sum_sum @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Sum__Type_Osum___Heap_Oheap_275, axiom, (! [A17:$tType,A18:$tType]: ((((heap @ A17) & (heap @ A18)) => (heap @ ((sum_sum @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Sum__Type_Osum___Nat_Osize_276, axiom, (! [A17:$tType,A18:$tType]: ((size @ ((sum_sum @ A17) @ A18))))).
% 0.33/5.23 thf(tcon_Option_Ooption___Countable_Ocountable_277, axiom, (! [A17:$tType]: (((countable @ A17) => (countable @ (option @ A17)))))).
% 0.33/5.23 thf(tcon_Option_Ooption___Typerep_Otyperep_278, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (option @ A17)))))).
% 0.33/5.23 thf(tcon_Option_Ooption___Heap_Oheap_279, axiom, (! [A17:$tType]: (((heap @ A17) => (heap @ (option @ A17)))))).
% 0.33/5.23 thf(tcon_Option_Ooption___Nat_Osize_280, axiom, (! [A17:$tType]: ((size @ (option @ A17))))).
% 0.33/5.23 thf(tcon_String_Oliteral___Groups_Osemigroup__add_281, axiom, (semigroup_add @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Countable_Ocountable_282, axiom, (countable @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Orderings_Opreorder_283, axiom, (preorder @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Orderings_Olinorder_284, axiom, (linorder @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Groups_Omonoid__add_285, axiom, (monoid_add @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Typerep_Otyperep_286, axiom, (typerep2 @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Orderings_Oorder_287, axiom, (order @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Orderings_Oord_288, axiom, (ord @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Groups_Ozero_289, axiom, (zero @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Groups_Oplus_290, axiom, (plus @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Heap_Oheap_291, axiom, (heap @ literal)).
% 0.33/5.23 thf(tcon_String_Oliteral___Nat_Osize_292, axiom, (size @ literal)).
% 0.33/5.23 thf(tcon_Nitpick_Ofun__box___Typerep_Otyperep_293, axiom, (! [A17:$tType,A18:$tType]: ((((typerep2 @ A17) & (typerep2 @ A18)) => (typerep2 @ ((fun_box @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Typerep_Otyperep___Countable_Ocountable_294, axiom, (countable @ typerep)).
% 0.33/5.23 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_295, axiom, (typerep2 @ typerep)).
% 0.33/5.23 thf(tcon_Typerep_Otyperep___Heap_Oheap_296, axiom, (heap @ typerep)).
% 0.33/5.23 thf(tcon_Typerep_Otyperep___Nat_Osize_297, axiom, (size @ typerep)).
% 0.33/5.23 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_298, axiom, (! [A17:$tType,A18:$tType]: ((((typerep2 @ A17) & (typerep2 @ A18)) => (typerep2 @ ((old_node @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Product__Type_Oprod___Countable_Ocountable_299, axiom, (! [A17:$tType,A18:$tType]: ((((countable @ A17) & (countable @ A18)) => (countable @ ((product_prod @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_300, axiom, (! [A17:$tType,A18:$tType]: ((((typerep2 @ A17) & (typerep2 @ A18)) => (typerep2 @ ((product_prod @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Product__Type_Oprod___Heap_Oheap_301, axiom, (! [A17:$tType,A18:$tType]: ((((heap @ A17) & (heap @ A18)) => (heap @ ((product_prod @ A17) @ A18)))))).
% 0.33/5.23 thf(tcon_Product__Type_Oprod___Nat_Osize_302, axiom, (! [A17:$tType,A18:$tType]: ((size @ ((product_prod @ A17) @ A18))))).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_303, axiom, (condit6923001295902523014norder @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_304, axiom, (condit1219197933456340205attice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_305, axiom, (comple592849572758109894attice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_306, axiom, (comple489889107523837845lgebra @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_307, axiom, (bounde4967611905675639751up_bot @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_308, axiom, (bounde4346867609351753570nf_top @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder, axiom, (comple5582772986160207858norder @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_309, axiom, (comple6319245703460814977attice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_310, axiom, (boolea8198339166811842893lgebra @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_311, axiom, (bounded_lattice_top @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_312, axiom, (bounded_lattice_bot @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_313, axiom, (comple9053668089753744459l_ccpo @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_314, axiom, (semilattice_sup @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_315, axiom, (semilattice_inf @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_316, axiom, (distrib_lattice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_317, axiom, (wellorder @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_318, axiom, (order_top @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_319, axiom, (order_bot @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Countable_Ocountable_320, axiom, (countable @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_321, axiom, (preorder @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_322, axiom, (linorder @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Lattices_Olattice_323, axiom, (lattice @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_324, axiom, (typerep2 @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Oorder_325, axiom, (order @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Otop_326, axiom, (top @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Oord_327, axiom, (ord @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Orderings_Obot_328, axiom, (bot @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Groups_Ouminus_329, axiom, (uminus @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Groups_Ominus_330, axiom, (minus @ product_unit)).
% 0.33/5.23 thf(tcon_Product__Type_Ounit___Heap_Oheap_331, axiom, (heap @ product_unit)).
% 0.33/5.23 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_332, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (heap_ext @ A17)))))).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_333, axiom, (bit_un5681908812861735899ations @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_334, axiom, (semiri1453513574482234551roduct @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_335, axiom, (euclid5411537665997757685th_nat @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_336, axiom, (euclid8789492081693882211th_nat @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_337, axiom, (ordere1937475149494474687imp_le @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_338, axiom, (euclid3128863361964157862miring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_339, axiom, (euclid4440199948858584721cancel @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_340, axiom, (unique1627219031080169319umeral @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_341, axiom, (euclid8851590272496341667cancel @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_342, axiom, (semiri6575147826004484403cancel @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_343, axiom, (strict9044650504122735259up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_344, axiom, (ordere580206878836729694up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_345, axiom, (ordere2412721322843649153imp_le @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_346, axiom, (bit_se359711467146920520ations @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_347, axiom, (linord2810124833399127020strict @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_348, axiom, (strict7427464778891057005id_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_349, axiom, (ordere8940638589300402666id_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_350, axiom, (euclid3725896446679973847miring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_351, axiom, (linord715952674999750819strict @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_352, axiom, (linord4140545234300271783up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_353, axiom, (bit_ri3973907225187159222ations @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_354, axiom, (semiri2026040879449505780visors @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_355, axiom, (linord181362715937106298miring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_356, axiom, (euclid5891614535332579305n_ring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_357, axiom, (linord8928482502909563296strict @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_358, axiom, (semiri3467727345109120633visors @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_359, axiom, (ordere6658533253407199908up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_360, axiom, (ordere166539214618696060dd_abs @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_361, axiom, (ordere6911136660526730532id_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_362, axiom, (linord5086331880401160121up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_363, axiom, (cancel2418104881723323429up_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_364, axiom, (ring_15535105094025558882visors @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_365, axiom, (cancel1802427076303600483id_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_366, axiom, (linord4710134922213307826strict @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_367, axiom, (comm_s4317794764714335236cancel @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_368, axiom, (bit_semiring_bits @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_369, axiom, (ordere2520102378445227354miring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_370, axiom, (linord6961819062388156250ring_1 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_371, axiom, (ordered_ab_group_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_372, axiom, (cancel_semigroup_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_373, axiom, (linordered_semiring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_374, axiom, (ordered_semiring_0 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_375, axiom, (linordered_semidom @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_376, axiom, (ab_semigroup_mult @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_377, axiom, (semiring_1_cancel @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_378, axiom, (algebraic_semidom @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_379, axiom, (comm_monoid_mult @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_380, axiom, (ab_semigroup_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_381, axiom, (ordered_semiring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_382, axiom, (ordered_ring_abs @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_383, axiom, (semiring_parity @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_384, axiom, (comm_monoid_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_385, axiom, (semiring_modulo @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_386, axiom, (linordered_ring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_387, axiom, (linordered_idom @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_388, axiom, (comm_semiring_1 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_389, axiom, (comm_semiring_0 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_390, axiom, (semigroup_mult @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_391, axiom, (semidom_modulo @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_392, axiom, (semidom_divide @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_393, axiom, (semiring_numeral @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_394, axiom, (semigroup_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_395, axiom, (zero_less_one @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_396, axiom, (comm_semiring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_397, axiom, (semiring_char_0 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_398, axiom, (ab_group_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_399, axiom, (zero_neq_one @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_400, axiom, (ordered_ring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_401, axiom, (idom_abs_sgn @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_402, axiom, (ring_parity @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_403, axiom, (preorder @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_404, axiom, (linorder @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_405, axiom, (monoid_mult @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_406, axiom, (idom_modulo @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_407, axiom, (idom_divide @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_408, axiom, (comm_ring_1 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_409, axiom, (monoid_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_410, axiom, (semiring_1 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_411, axiom, (semiring_0 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_412, axiom, (group_add @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_413, axiom, (typerep2 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414, axiom, (mult_zero @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415, axiom, (comm_ring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416, axiom, (order @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417, axiom, (neg_numeral @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418, axiom, (ring_char_0 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419, axiom, (semiring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_420, axiom, (semidom @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_421, axiom, (ord @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_422, axiom, (uminus @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_423, axiom, (ring_1 @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_424, axiom, (abs_if @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_425, axiom, (minus @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Power_Opower_426, axiom, (power @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_427, axiom, (numeral @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_428, axiom, (zero @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_429, axiom, (plus @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_430, axiom, (ring @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_431, axiom, (idom @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_432, axiom, (one @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_433, axiom, (dvd @ code_integer)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_434, axiom, (bit_un5681908812861735899ations @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_435, axiom, (euclid5411537665997757685th_nat @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_436, axiom, (ordere1937475149494474687imp_le @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_437, axiom, (euclid3128863361964157862miring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_438, axiom, (euclid4440199948858584721cancel @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_439, axiom, (semiri6575147826004484403cancel @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_440, axiom, (strict9044650504122735259up_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_441, axiom, (ordere580206878836729694up_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_442, axiom, (ordere2412721322843649153imp_le @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_443, axiom, (bit_se359711467146920520ations @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_444, axiom, (linord2810124833399127020strict @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_445, axiom, (strict7427464778891057005id_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_446, axiom, (ordere8940638589300402666id_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_447, axiom, (euclid3725896446679973847miring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_448, axiom, (linord4140545234300271783up_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_449, axiom, (semiri2026040879449505780visors @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_450, axiom, (linord181362715937106298miring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_451, axiom, (linord8928482502909563296strict @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_452, axiom, (semiri3467727345109120633visors @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_453, axiom, (ordere6658533253407199908up_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_454, axiom, (ordere6911136660526730532id_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_455, axiom, (cancel2418104881723323429up_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_456, axiom, (cancel1802427076303600483id_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_457, axiom, (comm_s4317794764714335236cancel @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_458, axiom, (bit_semiring_bits @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_459, axiom, (ordere2520102378445227354miring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_460, axiom, (cancel_semigroup_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_461, axiom, (linordered_semiring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_462, axiom, (ordered_semiring_0 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_463, axiom, (linordered_semidom @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_464, axiom, (ab_semigroup_mult @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_465, axiom, (semiring_1_cancel @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_466, axiom, (algebraic_semidom @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_467, axiom, (comm_monoid_mult @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_468, axiom, (comm_monoid_diff @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_469, axiom, (ab_semigroup_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_470, axiom, (ordered_semiring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_471, axiom, (semiring_parity @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_472, axiom, (comm_monoid_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_473, axiom, (semiring_modulo @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_474, axiom, (comm_semiring_1 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_475, axiom, (comm_semiring_0 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_476, axiom, (semigroup_mult @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_477, axiom, (semidom_modulo @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_478, axiom, (semidom_divide @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_479, axiom, (semiring_numeral @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_480, axiom, (semigroup_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_481, axiom, (zero_less_one @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_482, axiom, (comm_semiring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_483, axiom, (semiring_char_0 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_484, axiom, (zero_neq_one @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_485, axiom, (preorder @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_486, axiom, (linorder @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_487, axiom, (monoid_mult @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_488, axiom, (monoid_add @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_489, axiom, (semiring_1 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_490, axiom, (semiring_0 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_491, axiom, (typerep2 @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_492, axiom, (mult_zero @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_493, axiom, (order @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_494, axiom, (semiring @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_495, axiom, (semidom @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_496, axiom, (ord @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_497, axiom, (minus @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Power_Opower_498, axiom, (power @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_499, axiom, (numeral @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_500, axiom, (zero @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_501, axiom, (plus @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Groups_Oone_502, axiom, (one @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_503, axiom, (dvd @ code_natural)).
% 0.33/5.23 thf(tcon_Code__Numeral_Onatural___Nat_Osize_504, axiom, (size @ code_natural)).
% 0.33/5.23 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_505, axiom, (! [A17:$tType]: (((typerep2 @ A17) => (typerep2 @ (heap_Time_Heap @ A17)))))).
% 0.33/5.23 thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_506, axiom, (! [A17:$tType]: ((size @ (heap_Time_Heap @ A17))))).
% 0.33/5.23 thf(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_507, axiom, (! [A17:$tType,A18:$tType,A19:$tType]: ((((typerep2 @ A17) & ((typerep2 @ A18) & (typerep2 @ A19))) => (typerep2 @ (((tuple_isomorphism @ A17) @ A18) @ A19)))))).
% 0.33/5.23 thf(tcon_Record_Otuple__isomorphism___Nat_Osize_508, axiom, (! [A17:$tType,A18:$tType,A19:$tType]: ((size @ (((tuple_isomorphism @ A17) @ A18) @ A19))))).
% 0.33/5.23 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 0.33/5.23 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y3:A]: ((((((if @ A) @ $false) @ X) @ Y3) = Y3)))).
% 0.33/5.23 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y3:A]: ((((((if @ A) @ $true) @ X) @ Y3) = X)))).
% 0.33/5.23 thf(help_fChoice_1_1_T, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((fChoice @ A) @ P)) = (? [X7:A]: ((P @ X7))))))).
% 0.33/5.23 thf(tfree_0, hypothesis, (heap @ a)).
% 0.33/5.23 thf(conj_0, hypothesis, $true).
% 0.33/5.23 thf(conj_1, conjecture, thesis).
% 0.33/5.23 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2DTF31297.p
% 0.33/5.23 ------------------------
% 12.17/7.86 ---- Cleaned THF ---
% 12.17/7.86 thf(ty_t_Record_Otuple__isomorphism,type,
% 12.17/7.86 tuple_isomorphism: $tType > $tType > $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Heap__Time__Monad_OHeap,type,
% 12.17/7.86 heap_Time_Heap: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Code__Numeral_Onatural,type,
% 12.17/7.86 code_natural: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Code__Numeral_Ointeger,type,
% 12.17/7.86 code_integer: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Heap_Oheap_Oheap__ext,type,
% 12.17/7.86 heap_ext: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Product__Type_Ounit,type,
% 12.17/7.86 product_unit: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Product__Type_Oprod,type,
% 12.17/7.86 product_prod: $tType > $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Old__Datatype_Onode,type,
% 12.17/7.86 old_node: $tType > $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Nitpick_Ofun__box,type,
% 12.17/7.86 fun_box: $tType > $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Typerep_Otyperep,type,
% 12.17/7.86 typerep: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_String_Oliteral,type,
% 12.17/7.86 literal: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Sum__Type_Osum,type,
% 12.17/7.86 sum_sum: $tType > $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Option_Ooption,type,
% 12.17/7.86 option: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Nitpick_Oword,type,
% 12.17/7.86 word: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_String_Ochar,type,
% 12.17/7.86 char: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Heap_Oarray,type,
% 12.17/7.86 array: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_List_Olist,type,
% 12.17/7.86 list: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Set_Oset,type,
% 12.17/7.86 set: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Rat_Orat,type,
% 12.17/7.86 rat: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Num_Onum,type,
% 12.17/7.86 num: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Nat_Onat,type,
% 12.17/7.86 nat: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_Int_Oint,type,
% 12.17/7.86 int: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_t_itself,type,
% 12.17/7.86 itself: $tType > $tType ).
% 12.17/7.86
% 12.17/7.86 thf(ty_tf_a,type,
% 12.17/7.86 a: $tType ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Finite__Set_Ofinite,type,
% 12.17/7.86 finite_finite:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Obounded__lattice,type,
% 12.17/7.86 bounded_lattice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_HOL_Otype,type,
% 12.17/7.86 type:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Nat_Osize,type,
% 12.17/7.86 size:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Heap_Oheap,type,
% 12.17/7.86 heap:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Odvd,type,
% 12.17/7.86 dvd:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oone,type,
% 12.17/7.86 one:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oidom,type,
% 12.17/7.86 idom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oring,type,
% 12.17/7.86 ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oplus,type,
% 12.17/7.86 plus:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ozero,type,
% 12.17/7.86 zero:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Num_Onumeral,type,
% 12.17/7.86 numeral:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Power_Opower,type,
% 12.17/7.86 power:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Ofield,type,
% 12.17/7.86 field:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_GCD_Oring__gcd,type,
% 12.17/7.86 ring_gcd:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ominus,type,
% 12.17/7.86 minus:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oabs__if,type,
% 12.17/7.86 abs_if:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oring__1,type,
% 12.17/7.86 ring_1:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ouminus,type,
% 12.17/7.86 uminus:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Obot,type,
% 12.17/7.86 bot:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Oord,type,
% 12.17/7.86 ord:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Otop,type,
% 12.17/7.86 top:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemidom,type,
% 12.17/7.86 semidom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Oinverse,type,
% 12.17/7.86 inverse:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring,type,
% 12.17/7.86 semiring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Nat_Oring__char__0,type,
% 12.17/7.86 ring_char_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Num_Oneg__numeral,type,
% 12.17/7.86 neg_numeral:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Oorder,type,
% 12.17/7.86 order:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__ring,type,
% 12.17/7.86 comm_ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Omult__zero,type,
% 12.17/7.86 mult_zero:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Typerep_Otyperep,type,
% 12.17/7.86 typerep2:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_GCD_Osemiring__Gcd,type,
% 12.17/7.86 semiring_Gcd:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_GCD_Osemiring__gcd,type,
% 12.17/7.86 semiring_gcd:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ogroup__add,type,
% 12.17/7.86 group_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Olattice,type,
% 12.17/7.86 lattice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Ono__bot,type,
% 12.17/7.86 no_bot:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Ono__top,type,
% 12.17/7.86 no_top:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__0,type,
% 12.17/7.86 semiring_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__1,type,
% 12.17/7.86 semiring_1:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Omonoid__add,type,
% 12.17/7.86 monoid_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 12.17/7.86 comm_ring_1:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oidom__divide,type,
% 12.17/7.86 idom_divide:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oidom__modulo,type,
% 12.17/7.86 idom_modulo:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Omonoid__mult,type,
% 12.17/7.86 monoid_mult:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Olinorder,type,
% 12.17/7.86 linorder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Opreorder,type,
% 12.17/7.86 preorder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Parity_Oring__parity,type,
% 12.17/7.86 ring_parity:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 12.17/7.86 idom_abs_sgn:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oordered__ring,type,
% 12.17/7.86 ordered_ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ozero__neq__one,type,
% 12.17/7.86 zero_neq_one:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Countable_Ocountable,type,
% 12.17/7.86 countable:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Ofield__char__0,type,
% 12.17/7.86 field_char_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oab__group__add,type,
% 12.17/7.86 ab_group_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Nat_Osemiring__char__0,type,
% 12.17/7.86 semiring_char_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Oorder__bot,type,
% 12.17/7.86 order_bot:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Oorder__top,type,
% 12.17/7.86 order_top:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Owellorder,type,
% 12.17/7.86 wellorder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__semiring,type,
% 12.17/7.86 comm_semiring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ozero__less__one,type,
% 12.17/7.86 zero_less_one:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Odivision__ring,type,
% 12.17/7.86 division_ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Ofield__abs__sgn,type,
% 12.17/7.86 field_abs_sgn:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Osemigroup__add,type,
% 12.17/7.86 semigroup_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Num_Osemiring__numeral,type,
% 12.17/7.86 semiring_numeral:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemidom__divide,type,
% 12.17/7.86 semidom_divide:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemidom__modulo,type,
% 12.17/7.86 semidom_modulo:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Osemigroup__mult,type,
% 12.17/7.86 semigroup_mult:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Odense__order,type,
% 12.17/7.86 dense_order:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 12.17/7.86 comm_semiring_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 12.17/7.86 comm_semiring_1:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__idom,type,
% 12.17/7.86 linordered_idom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__ring,type,
% 12.17/7.86 linordered_ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__modulo,type,
% 12.17/7.86 semiring_modulo:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 12.17/7.86 comm_monoid_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Parity_Osemiring__parity,type,
% 12.17/7.86 semiring_parity:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oordered__ring__abs,type,
% 12.17/7.86 ordered_ring_abs:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oordered__semiring,type,
% 12.17/7.86 ordered_semiring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Fields_Olinordered__field,type,
% 12.17/7.86 linordered_field:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 12.17/7.86 ab_semigroup_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 12.17/7.86 comm_monoid_diff:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 12.17/7.86 comm_monoid_mult:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 12.17/7.86 algebraic_semidom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 12.17/7.86 semiring_1_cancel:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 12.17/7.86 ab_semigroup_mult:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 12.17/7.86 distrib_lattice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 12.17/7.86 semilattice_inf:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 12.17/7.86 semilattice_sup:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Orderings_Odense__linorder,type,
% 12.17/7.86 dense_linorder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__semidom,type,
% 12.17/7.86 linordered_semidom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 12.17/7.86 ordered_semiring_0:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__semiring,type,
% 12.17/7.86 linordered_semiring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 12.17/7.86 comple9053668089753744459l_ccpo:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
% 12.17/7.86 finite8700451911770168679attice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 12.17/7.86 cancel_semigroup_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 12.17/7.86 ordered_ab_group_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__semiring__1,type,
% 12.17/7.86 linord6961819062388156250ring_1:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Onormalization__semidom,type,
% 12.17/7.86 normal8620421768224518004emidom:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oordered__comm__semiring,type,
% 12.17/7.86 ordere2520102378445227354miring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Bit__Operations_Osemiring__bits,type,
% 12.17/7.86 bit_semiring_bits:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 12.17/7.86 bounded_lattice_bot:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 12.17/7.86 bounded_lattice_top:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 12.17/7.86 comm_s4317794764714335236cancel:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 12.17/7.86 linord4710134922213307826strict:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 12.17/7.86 cancel1802427076303600483id_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 12.17/7.86 ring_15535105094025558882visors:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 12.17/7.86 cancel2418104881723323429up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 12.17/7.86 linord5086331880401160121up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 12.17/7.86 ordere6911136660526730532id_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 12.17/7.86 archim2362893244070406136eiling:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
% 12.17/7.86 semiri6843258321239162965malize:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 12.17/7.86 ordere166539214618696060dd_abs:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 12.17/7.86 ordere6658533253407199908up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 12.17/7.86 semiri3467727345109120633visors:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
% 12.17/7.86 boolea8198339166811842893lgebra:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__semiring__strict,type,
% 12.17/7.86 linord8928482502909563296strict:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
% 12.17/7.86 semido2269285787275462019factor:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
% 12.17/7.86 euclid5891614535332579305n_ring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 12.17/7.86 linord181362715937106298miring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 12.17/7.86 semiri2026040879449505780visors:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
% 12.17/7.86 bit_ri3973907225187159222ations:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 12.17/7.86 comple6319245703460814977attice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
% 12.17/7.86 linord4140545234300271783up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
% 12.17/7.86 linord715952674999750819strict:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 12.17/7.86 archim462609752435547400_field:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 12.17/7.86 comple5582772986160207858norder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 12.17/7.86 bounde4346867609351753570nf_top:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 12.17/7.86 bounde4967611905675639751up_bot:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
% 12.17/7.86 euclid3725896446679973847miring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 12.17/7.86 canoni5634975068530333245id_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 12.17/7.86 ordere8940638589300402666id_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 12.17/7.86 strict7427464778891057005id_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
% 12.17/7.86 linord2810124833399127020strict:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
% 12.17/7.86 bit_se359711467146920520ations:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 12.17/7.86 ordere2412721322843649153imp_le:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 12.17/7.86 ordere580206878836729694up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 12.17/7.86 ordere1170586879665033532d_diff:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 12.17/7.86 strict9044650504122735259up_add:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 12.17/7.86 semiri6575147826004484403cancel:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
% 12.17/7.86 euclid8851590272496341667cancel:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 12.17/7.86 unique1627219031080169319umeral:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
% 12.17/7.86 comple489889107523837845lgebra:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
% 12.17/7.86 comple592849572758109894attice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
% 12.17/7.86 normal6328177297339901930cative:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
% 12.17/7.86 euclid4440199948858584721cancel:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
% 12.17/7.86 euclid3128863361964157862miring:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 12.17/7.86 ordere1937475149494474687imp_le:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 12.17/7.86 condit5016429287641298734tinuum:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
% 12.17/7.86 euclid8789492081693882211th_nat:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
% 12.17/7.86 euclid5411537665997757685th_nat:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 12.17/7.86 semiri1453513574482234551roduct:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
% 12.17/7.86 bit_un5681908812861735899ations:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 12.17/7.86 condit1219197933456340205attice:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 12.17/7.86 condit6923001295902523014norder:
% 12.17/7.86 !>[A: $tType] : $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Archimedean__Field_Oceiling,type,
% 12.17/7.86 archimedean_ceiling:
% 12.17/7.86 !>[A: $tType] : ( A > int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
% 12.17/7.86 archim6421214686448440834_floor:
% 12.17/7.86 !>[A: $tType] : ( A > int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Archimedean__Field_Ofrac,type,
% 12.17/7.86 archimedean_frac:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Archimedean__Field_Oround,type,
% 12.17/7.86 archimedean_round:
% 12.17/7.86 !>[A: $tType] : ( A > int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oalloc,type,
% 12.17/7.86 array_alloc:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Ofreeze,type,
% 12.17/7.86 array_freeze:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oget,type,
% 12.17/7.86 array_get:
% 12.17/7.86 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Olen,type,
% 12.17/7.86 array_len:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Olength,type,
% 12.17/7.86 array_length:
% 12.17/7.86 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Omake,type,
% 12.17/7.86 array_make:
% 12.17/7.86 !>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Omap__entry,type,
% 12.17/7.86 array_map_entry:
% 12.17/7.86 !>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Onew,type,
% 12.17/7.86 array_new:
% 12.17/7.86 !>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Onoteq,type,
% 12.17/7.86 array_noteq:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Onth,type,
% 12.17/7.86 array_nth:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oof__list,type,
% 12.17/7.86 array_of_list:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Opresent,type,
% 12.17/7.86 array_present:
% 12.17/7.86 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oset,type,
% 12.17/7.86 array_set:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oswap,type,
% 12.17/7.86 array_swap:
% 12.17/7.86 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oupd,type,
% 12.17/7.86 array_upd:
% 12.17/7.86 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Array__Time_Oupdate,type,
% 12.17/7.86 array_update:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 12.17/7.86 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
% 12.17/7.86 bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 12.17/7.86 bNF_Ca3754400796208372196lChain:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Def_Orel__fun,type,
% 12.17/7.86 bNF_rel_fun:
% 12.17/7.86 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
% 12.17/7.86 bNF_Wellorder_bsqr:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
% 12.17/7.86 bNF_We413866401316099525erIncl:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
% 12.17/7.86 bNF_Wellorder_ordIso:
% 12.17/7.86 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
% 12.17/7.86 bNF_We4044943003108391690rdLess:
% 12.17/7.86 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
% 12.17/7.86 bNF_Wellorder_embed:
% 12.17/7.86 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
% 12.17/7.86 bNF_Wellorder_embedS:
% 12.17/7.86 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
% 12.17/7.86 bNF_Wellorder_iso:
% 12.17/7.86 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
% 12.17/7.86 bNF_Wellorder_wo_rel:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 12.17/7.86 basic_BNF_size_prod:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
% 12.17/7.86 basic_BNF_size_sum:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Basic__BNFs_Opred__prod,type,
% 12.17/7.86 basic_pred_prod:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Binomial_Obinomial,type,
% 12.17/7.86 binomial: nat > nat > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Binomial_Ogbinomial,type,
% 12.17/7.86 gbinomial:
% 12.17/7.86 !>[A: $tType] : ( A > nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 12.17/7.86 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 12.17/7.86 bit_and_not_num: num > num > ( option @ num ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 12.17/7.86 bit_concat_bit: nat > int > int > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 12.17/7.86 bit_or_not_num_neg: num > num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
% 12.17/7.86 bit_ri4277139882892585799ns_not:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
% 12.17/7.86 bit_ri4674362597316999326ke_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
% 12.17/7.86 bit_se5824344872417868541ns_and:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
% 12.17/7.86 bit_se4197421643247451524op_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
% 12.17/7.86 bit_se8732182000553998342ip_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
% 12.17/7.86 bit_se2239418461657761734s_mask:
% 12.17/7.86 !>[A: $tType] : ( nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
% 12.17/7.86 bit_se1065995026697491101ons_or:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
% 12.17/7.86 bit_se4730199178511100633sh_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
% 12.17/7.86 bit_se5668285175392031749et_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
% 12.17/7.86 bit_se2584673776208193580ke_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
% 12.17/7.86 bit_se2638667681897837118et_bit:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
% 12.17/7.86 bit_se5824344971392196577ns_xor:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
% 12.17/7.86 bit_se5641148757651400278ts_bit:
% 12.17/7.86 !>[A: $tType] : ( A > nat > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
% 12.17/7.86 bit_se6407376104438227557le_bit:
% 12.17/7.86 !>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 12.17/7.86 bit_take_bit_num: nat > num > ( option @ num ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 12.17/7.86 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 12.17/7.86 bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
% 12.17/7.86 boolea2506097494486148201lgebra:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__axioms,type,
% 12.17/7.86 boolea6902313364301356556axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
% 12.17/7.86 boolea3799213064322606851m_diff:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
% 12.17/7.86 boolea5476839437570043046axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_ONat,type,
% 12.17/7.86 code_Nat: code_integer > code_natural ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_ONeg,type,
% 12.17/7.86 code_Neg: num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_OPos,type,
% 12.17/7.86 code_Pos: num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_OSuc,type,
% 12.17/7.86 code_Suc: code_natural > code_natural ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 12.17/7.86 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ocr__integer,type,
% 12.17/7.86 code_cr_integer: int > code_integer > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ocr__natural,type,
% 12.17/7.86 code_cr_natural: nat > code_natural > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 12.17/7.86 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 12.17/7.86 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Odup,type,
% 12.17/7.86 code_dup: code_integer > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 12.17/7.86 code_int_of_integer: code_integer > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 12.17/7.86 code_integer_of_int: int > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 12.17/7.86 code_integer_of_nat: nat > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
% 12.17/7.86 code_i5400310926305786745atural: code_natural > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 12.17/7.86 code_integer_of_num: num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 12.17/7.86 code_nat_of_integer: code_integer > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 12.17/7.86 code_case_natural:
% 12.17/7.86 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
% 12.17/7.86 code_nat_of_natural: code_natural > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
% 12.17/7.86 code_natural_of_nat: nat > code_natural ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 12.17/7.86 code_rec_natural:
% 12.17/7.86 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
% 12.17/7.86 code_rec_set_natural:
% 12.17/7.86 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onatural__of__integer,type,
% 12.17/7.86 code_n4118661773612635043nteger: code_integer > code_natural ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onegative,type,
% 12.17/7.86 code_negative: num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 12.17/7.86 code_num_of_integer: code_integer > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Opcr__integer,type,
% 12.17/7.86 code_pcr_integer: int > code_integer > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Opcr__natural,type,
% 12.17/7.86 code_pcr_natural: nat > code_natural > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Opositive,type,
% 12.17/7.86 code_positive: num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Osize__nat,type,
% 12.17/7.86 code_size_nat: code_natural > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Code__Numeral_Osub,type,
% 12.17/7.86 code_sub: num > num > code_integer ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 12.17/7.86 complete_Inf_Inf:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 12.17/7.86 complete_Sup_Sup:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
% 12.17/7.86 comple1908693960933563346ssible:
% 12.17/7.86 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
% 12.17/7.86 comple187402453842119260l_fixp:
% 12.17/7.86 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 12.17/7.86 comple115746919287870866o_fixp:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
% 12.17/7.86 comple6359979572994053840erates:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
% 12.17/7.86 comple7512665784863727008ratesp:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 12.17/7.86 comple1602240252501008431_chain:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 12.17/7.86 comple7038119648293358887notone:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
% 12.17/7.86 condit941137186595557371_above:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
% 12.17/7.86 condit1013018076250108175_below:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
% 12.17/7.86 condit622319405099724424ng_bdd:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
% 12.17/7.86 condit16957441358409770ng_bdd:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Countable_Ofrom__nat,type,
% 12.17/7.86 from_nat:
% 12.17/7.86 !>[A: $tType] : ( nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Countable_Onth__item,type,
% 12.17/7.86 nth_item:
% 12.17/7.86 !>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Countable_Onth__item__rel,type,
% 12.17/7.86 nth_item_rel: nat > nat > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Oadjust__div,type,
% 12.17/7.86 adjust_div: ( product_prod @ int @ int ) > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Oadjust__mod,type,
% 12.17/7.86 adjust_mod: int > int > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Odivmod__nat,type,
% 12.17/7.86 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Oeucl__rel__int,type,
% 12.17/7.86 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 12.17/7.86 unique5940410009612947441es_aux:
% 12.17/7.86 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
% 12.17/7.86 unique8689654367752047608divmod:
% 12.17/7.86 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
% 12.17/7.86 unique1321980374590559556d_step:
% 12.17/7.86 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
% 12.17/7.86 euclid6346220572633701492n_size:
% 12.17/7.86 !>[A: $tType] : ( A > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
% 12.17/7.86 euclid7384307370059645450egment:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
% 12.17/7.86 comm_s3205402744901411588hammer:
% 12.17/7.86 !>[A: $tType] : ( A > nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
% 12.17/7.86 semiring_char_0_fact:
% 12.17/7.86 !>[A: $tType] : ( nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 12.17/7.86 inverse_inverse:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Finite__Set_Ocard,type,
% 12.17/7.86 finite_card:
% 12.17/7.86 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Finite__Set_Ofinite,type,
% 12.17/7.86 finite_finite2:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Finite__Set_Ofold,type,
% 12.17/7.86 finite_fold:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Finite__Set_Ofolding__on,type,
% 12.17/7.86 finite_folding_on:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Finite__Set_Ofolding__on_OF,type,
% 12.17/7.86 finite_folding_F:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Obij__betw,type,
% 12.17/7.86 bij_betw:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Ocomp,type,
% 12.17/7.86 comp:
% 12.17/7.86 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Ofun__upd,type,
% 12.17/7.86 fun_upd:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Oid,type,
% 12.17/7.86 id:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Oinj__on,type,
% 12.17/7.86 inj_on:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun_Omap__fun,type,
% 12.17/7.86 map_fun:
% 12.17/7.86 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Ois__measure,type,
% 12.17/7.86 fun_is_measure:
% 12.17/7.86 !>[A: $tType] : ( ( A > nat ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Omax__strict,type,
% 12.17/7.86 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Omax__weak,type,
% 12.17/7.86 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Omin__strict,type,
% 12.17/7.86 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Omin__weak,type,
% 12.17/7.86 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Opair__leq,type,
% 12.17/7.86 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Opair__less,type,
% 12.17/7.86 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 12.17/7.86 fun_reduction_pair:
% 12.17/7.86 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 12.17/7.86 fun_rp_inv_image:
% 12.17/7.86 !>[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 ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_OGcd__class_OGcd,type,
% 12.17/7.86 gcd_Gcd:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_OGcd__class_OLcm,type,
% 12.17/7.86 gcd_Lcm:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Obezw,type,
% 12.17/7.86 bezw: nat > nat > ( product_prod @ int @ int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Obezw__rel,type,
% 12.17/7.86 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Obounded__quasi__semilattice,type,
% 12.17/7.86 bounde8507323023520639062attice:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
% 12.17/7.86 bounde6485984586167503788ce_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
% 12.17/7.86 bounde2362111253966948842tice_F:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Ogcd__class_Ogcd,type,
% 12.17/7.86 gcd_gcd:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Ogcd__class_Olcm,type,
% 12.17/7.86 gcd_lcm:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 12.17/7.86 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
% 12.17/7.86 semiri4206861660011772517g_char:
% 12.17/7.86 !>[A: $tType] : ( ( itself @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
% 12.17/7.86 semiring_gcd_Gcd_fin:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
% 12.17/7.86 semiring_gcd_Lcm_fin:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Oabel__semigroup,type,
% 12.17/7.86 abel_semigroup:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 12.17/7.86 abel_s757365448890700780axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 12.17/7.86 abs_abs:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ocomm__monoid,type,
% 12.17/7.86 comm_monoid:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ocomm__monoid__axioms,type,
% 12.17/7.86 comm_monoid_axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ogroup,type,
% 12.17/7.86 group:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ogroup__axioms,type,
% 12.17/7.86 group_axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 12.17/7.86 minus_minus:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Omonoid,type,
% 12.17/7.86 monoid:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Omonoid__axioms,type,
% 12.17/7.86 monoid_axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Oone__class_Oone,type,
% 12.17/7.86 one_one:
% 12.17/7.86 !>[A: $tType] : A ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 12.17/7.86 plus_plus:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Osemigroup,type,
% 12.17/7.86 semigroup:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 12.17/7.86 sgn_sgn:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 12.17/7.86 times_times:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 12.17/7.86 uminus_uminus:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 12.17/7.86 zero_zero:
% 12.17/7.86 !>[A: $tType] : A ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 12.17/7.86 groups7311177749621191930dd_sum:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
% 12.17/7.86 groups1027152243600224163dd_sum:
% 12.17/7.86 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
% 12.17/7.86 groups7121269368397514597t_prod:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
% 12.17/7.86 groups1962203154675924110t_prod:
% 12.17/7.86 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
% 12.17/7.86 groups778175481326437816id_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
% 12.17/7.86 groups_comm_monoid_F:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
% 12.17/7.86 groups_comm_monoid_G:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Ocomm__monoid__list,type,
% 12.17/7.86 groups1828464146339083142d_list:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
% 12.17/7.86 groups4802862169904069756st_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
% 12.17/7.86 groups4207007520872428315er_sum:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 12.17/7.86 groups8242544230860333062m_list:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Omonoid__list,type,
% 12.17/7.86 groups_monoid_list:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Omonoid__list_OF,type,
% 12.17/7.86 groups_monoid_F:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
% 12.17/7.86 groups5270119922927024881d_list:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_HOL_ONO__MATCH,type,
% 12.17/7.86 nO_MATCH:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_HOL_OThe,type,
% 12.17/7.86 the:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_HOL_OUniq,type,
% 12.17/7.86 uniq:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_HOL_Oundefined,type,
% 12.17/7.86 undefined:
% 12.17/7.86 !>[A: $tType] : A ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oaddr__of__array,type,
% 12.17/7.86 addr_of_array:
% 12.17/7.86 !>[A: $tType] : ( ( array @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oarray_OArray,type,
% 12.17/7.86 array2:
% 12.17/7.86 !>[A: $tType] : ( nat > ( array @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oarray_Osize__array,type,
% 12.17/7.86 size_array:
% 12.17/7.86 !>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oempty,type,
% 12.17/7.86 empty: heap_ext @ product_unit ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oheap_Oarrays,type,
% 12.17/7.86 arrays:
% 12.17/7.86 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oheap_Oheap__ext,type,
% 12.17/7.86 heap_ext2:
% 12.17/7.86 !>[Z: $tType] : ( ( typerep > nat > ( list @ nat ) ) > ( typerep > nat > nat ) > nat > Z > ( heap_ext @ Z ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oheap_Olim,type,
% 12.17/7.86 lim:
% 12.17/7.86 !>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap_Oheap_Olim__update,type,
% 12.17/7.86 lim_update:
% 12.17/7.86 !>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
% 12.17/7.86 heap_Time_Heap2:
% 12.17/7.86 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
% 12.17/7.86 heap_Time_case_Heap:
% 12.17/7.86 !>[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 ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
% 12.17/7.86 heap_Time_map_Heap:
% 12.17/7.86 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
% 12.17/7.86 heap_Time_pred_Heap:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
% 12.17/7.86 heap_Time_rec_Heap:
% 12.17/7.86 !>[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 ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
% 12.17/7.86 heap_Time_rel_Heap:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
% 12.17/7.86 heap_Time_set_Heap:
% 12.17/7.86 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
% 12.17/7.86 heap_Time_size_Heap:
% 12.17/7.86 !>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
% 12.17/7.86 heap_Time_Heap_lub:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
% 12.17/7.86 heap_Time_Heap_ord:
% 12.17/7.86 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oassert,type,
% 12.17/7.86 heap_Time_assert:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Obind,type,
% 12.17/7.86 heap_Time_bind:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oeffect,type,
% 12.17/7.86 heap_Time_effect:
% 12.17/7.86 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oexecute,type,
% 12.17/7.86 heap_Time_execute:
% 12.17/7.86 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oguard,type,
% 12.17/7.86 heap_Time_guard:
% 12.17/7.86 !>[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 ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oheap,type,
% 12.17/7.86 heap_Time_heap:
% 12.17/7.86 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Olift,type,
% 12.17/7.86 heap_Time_lift:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oraise,type,
% 12.17/7.86 heap_Time_raise:
% 12.17/7.86 !>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oreturn,type,
% 12.17/7.86 heap_Time_return:
% 12.17/7.86 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Osuccess,type,
% 12.17/7.86 heap_Time_success:
% 12.17/7.86 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Otap,type,
% 12.17/7.86 heap_Time_tap:
% 12.17/7.86 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
% 12.17/7.86 heap_Time_timeFrame:
% 12.17/7.86 !>[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 ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
% 12.17/7.86 heap_T5500966940807335491me_rel:
% 12.17/7.86 !>[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 ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Oureturn,type,
% 12.17/7.86 heap_Time_ureturn:
% 12.17/7.86 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Heap__Time__Monad_Owait,type,
% 12.17/7.86 heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Hilbert__Choice_Obijection,type,
% 12.17/7.86 hilbert_bijection:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 12.17/7.86 hilbert_inv_into:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_If,type,
% 12.17/7.86 if:
% 12.17/7.86 !>[A: $tType] : ( $o > A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
% 12.17/7.86 complete_lattice_gfp:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
% 12.17/7.86 complete_lattice_lfp:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_OAbs__Integ,type,
% 12.17/7.86 abs_Integ: ( product_prod @ nat @ nat ) > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_ONeg,type,
% 12.17/7.86 neg: num > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_OPos,type,
% 12.17/7.86 pos: num > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_ORep__Integ,type,
% 12.17/7.86 rep_Integ: int > ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Ocr__int,type,
% 12.17/7.86 cr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Odup,type,
% 12.17/7.86 dup: int > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oint_OAbs__int,type,
% 12.17/7.86 abs_int: ( set @ ( product_prod @ nat @ nat ) ) > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oint_ORep__int,type,
% 12.17/7.86 rep_int: int > ( set @ ( product_prod @ nat @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oint__ge__less__than,type,
% 12.17/7.86 int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oint__ge__less__than2,type,
% 12.17/7.86 int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Ointrel,type,
% 12.17/7.86 intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Onat,type,
% 12.17/7.86 nat2: int > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Opcr__int,type,
% 12.17/7.86 pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Opower__int,type,
% 12.17/7.86 power_int:
% 12.17/7.86 !>[A: $tType] : ( A > int > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oring__1__class_OInts,type,
% 12.17/7.86 ring_1_Ints:
% 12.17/7.86 !>[A: $tType] : ( set @ A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 12.17/7.86 ring_1_of_int:
% 12.17/7.86 !>[A: $tType] : ( int > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Int_Osub,type,
% 12.17/7.86 sub: num > num > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 12.17/7.86 inf_inf:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice,type,
% 12.17/7.86 semilattice:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice__axioms,type,
% 12.17/7.86 semilattice_axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 12.17/7.86 semilattice_neutr:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice__neutr__order,type,
% 12.17/7.86 semila1105856199041335345_order:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice__order,type,
% 12.17/7.86 semilattice_order:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osemilattice__order__axioms,type,
% 12.17/7.86 semila6385135966242565138axioms:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices_Osup__class_Osup,type,
% 12.17/7.86 sup_sup:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Olinorder_OMax,type,
% 12.17/7.86 lattices_Max:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Olinorder_OMin,type,
% 12.17/7.86 lattices_Min:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 12.17/7.86 lattic643756798349783984er_Max:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 12.17/7.86 lattic643756798350308766er_Min:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
% 12.17/7.86 lattices_ord_arg_max:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
% 12.17/7.86 lattic1883929316492267755max_on:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
% 12.17/7.86 lattices_ord_arg_min:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 12.17/7.86 lattic7623131987881927897min_on:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
% 12.17/7.86 lattic501386751176901750rg_max:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
% 12.17/7.86 lattic501386751177426532rg_min:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 12.17/7.86 lattic7752659483105999362nf_fin:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
% 12.17/7.86 lattic5652469242046573047tr_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
% 12.17/7.86 lattic5214292709420241887eutr_F:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__order__neutr__set,type,
% 12.17/7.86 lattic3600114342068043075tr_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 12.17/7.86 lattic4895041142388067077er_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 12.17/7.86 lattic149705377957585745ce_set:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
% 12.17/7.86 lattic1715443433743089157tice_F:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 12.17/7.86 lattic5882676163264333800up_fin:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Lifting_OQuotient,type,
% 12.17/7.86 quotient:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oappend,type,
% 12.17/7.86 append:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oarg__min__list,type,
% 12.17/7.86 arg_min_list:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oarg__min__list__rel,type,
% 12.17/7.86 arg_min_list_rel:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Obutlast,type,
% 12.17/7.86 butlast:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oconcat,type,
% 12.17/7.86 concat:
% 12.17/7.86 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ocoset,type,
% 12.17/7.86 coset:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ocount__list,type,
% 12.17/7.86 count_list:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Odistinct,type,
% 12.17/7.86 distinct:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Odrop,type,
% 12.17/7.86 drop:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_OdropWhile,type,
% 12.17/7.86 dropWhile:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oenumerate,type,
% 12.17/7.86 enumerate:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ofilter,type,
% 12.17/7.86 filter:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ofind,type,
% 12.17/7.86 find:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ofold,type,
% 12.17/7.86 fold:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ofolding__insort__key,type,
% 12.17/7.86 folding_insort_key:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ofoldr,type,
% 12.17/7.86 foldr:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ogen__length,type,
% 12.17/7.86 gen_length:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olast,type,
% 12.17/7.86 last:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olenlex,type,
% 12.17/7.86 lenlex:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olex,type,
% 12.17/7.86 lex:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olexord,type,
% 12.17/7.86 lexord:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
% 12.17/7.86 linord329482645794927042rt_key:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olinorder__class_Oinsort__key,type,
% 12.17/7.86 linorder_insort_key:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olinorder__class_Osort__key,type,
% 12.17/7.86 linorder_sort_key:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 12.17/7.86 linord4507533701916653071of_set:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_OCons,type,
% 12.17/7.86 cons:
% 12.17/7.86 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_ONil,type,
% 12.17/7.86 nil:
% 12.17/7.86 !>[A: $tType] : ( list @ A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Ohd,type,
% 12.17/7.86 hd:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Olist__all,type,
% 12.17/7.86 list_all:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Olist__all2,type,
% 12.17/7.86 list_all2:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Omap,type,
% 12.17/7.86 map:
% 12.17/7.86 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Oset,type,
% 12.17/7.86 set2:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Osize__list,type,
% 12.17/7.86 size_list:
% 12.17/7.86 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist_Otl,type,
% 12.17/7.86 tl:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist__ex,type,
% 12.17/7.86 list_ex:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olist__update,type,
% 12.17/7.86 list_update:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olistrel,type,
% 12.17/7.86 listrel:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Olistrel1,type,
% 12.17/7.86 listrel1:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Omeasures,type,
% 12.17/7.86 measures:
% 12.17/7.86 !>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_On__lists,type,
% 12.17/7.86 n_lists:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Onth,type,
% 12.17/7.86 nth:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Onths,type,
% 12.17/7.86 nths:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oord__class_Olexordp,type,
% 12.17/7.86 ord_lexordp:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oord__class_Olexordp__eq,type,
% 12.17/7.86 ord_lexordp_eq:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oproduct,type,
% 12.17/7.86 product:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oproduct__lists,type,
% 12.17/7.86 product_lists:
% 12.17/7.86 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oremdups,type,
% 12.17/7.86 remdups:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oremdups__adj,type,
% 12.17/7.86 remdups_adj:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oremove1,type,
% 12.17/7.86 remove1:
% 12.17/7.86 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_OremoveAll,type,
% 12.17/7.86 removeAll:
% 12.17/7.86 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oreplicate,type,
% 12.17/7.86 replicate:
% 12.17/7.86 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Orev,type,
% 12.17/7.86 rev:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Orotate,type,
% 12.17/7.86 rotate:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Orotate1,type,
% 12.17/7.86 rotate1:
% 12.17/7.86 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Osorted__wrt,type,
% 12.17/7.86 sorted_wrt:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Otake,type,
% 12.17/7.86 take:
% 12.17/7.86 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_OtakeWhile,type,
% 12.17/7.86 takeWhile:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Otranspose,type,
% 12.17/7.86 transpose:
% 12.17/7.86 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oupt,type,
% 12.17/7.86 upt: nat > nat > ( list @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oupto,type,
% 12.17/7.86 upto: int > int > ( list @ int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oupto__aux,type,
% 12.17/7.86 upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Oupto__rel,type,
% 12.17/7.86 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_List_Ozip,type,
% 12.17/7.86 zip:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Map_Omap__of,type,
% 12.17/7.86 map_of:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Map_Omap__upds,type,
% 12.17/7.86 map_upds:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Map_Orestrict__map,type,
% 12.17/7.86 restrict_map:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_OSuc,type,
% 12.17/7.86 suc: nat > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Ocompow,type,
% 12.17/7.86 compow:
% 12.17/7.86 !>[A: $tType] : ( nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Ofunpow,type,
% 12.17/7.86 funpow:
% 12.17/7.86 !>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 12.17/7.86 case_nat:
% 12.17/7.86 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Onat_Opred,type,
% 12.17/7.86 pred: nat > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 12.17/7.86 rec_nat:
% 12.17/7.86 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
% 12.17/7.86 rec_set_nat:
% 12.17/7.86 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Osemiring__1__class_ONats,type,
% 12.17/7.86 semiring_1_Nats:
% 12.17/7.86 !>[A: $tType] : ( set @ A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 12.17/7.86 semiring_1_of_nat:
% 12.17/7.86 !>[A: $tType] : ( nat > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 12.17/7.86 semiri8178284476397505188at_aux:
% 12.17/7.86 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat_Osize__class_Osize,type,
% 12.17/7.86 size_size:
% 12.17/7.86 !>[A: $tType] : ( A > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oint__decode,type,
% 12.17/7.86 nat_int_decode: nat > int ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oint__encode,type,
% 12.17/7.86 nat_int_encode: int > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 12.17/7.86 nat_list_decode: nat > ( list @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 12.17/7.86 nat_list_decode_rel: nat > nat > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 12.17/7.86 nat_list_encode: ( list @ nat ) > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 12.17/7.86 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 12.17/7.86 nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 12.17/7.86 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 12.17/7.86 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 12.17/7.86 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 12.17/7.86 nat_set_decode: nat > ( set @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 12.17/7.86 nat_set_encode: ( set @ nat ) > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 12.17/7.86 nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Osum__encode,type,
% 12.17/7.86 nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Nat__Bijection_Otriangle,type,
% 12.17/7.86 nat_triangle: nat > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_OBitM,type,
% 12.17/7.86 bitM: num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oinc,type,
% 12.17/7.86 inc: num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onat__of__num,type,
% 12.17/7.86 nat_of_num: num > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 12.17/7.86 neg_numeral_dbl:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
% 12.17/7.86 neg_numeral_dbl_dec:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 12.17/7.86 neg_numeral_dbl_inc:
% 12.17/7.86 !>[A: $tType] : ( A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
% 12.17/7.86 neg_numeral_is_num:
% 12.17/7.86 !>[A: $tType] : ( A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 12.17/7.86 neg_numeral_sub:
% 12.17/7.86 !>[A: $tType] : ( num > num > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_OBit0,type,
% 12.17/7.86 bit0: num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_OBit1,type,
% 12.17/7.86 bit1: num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_OOne,type,
% 12.17/7.86 one2: num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_Ocase__num,type,
% 12.17/7.86 case_num:
% 12.17/7.86 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_Orec__num,type,
% 12.17/7.86 rec_num:
% 12.17/7.86 !>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum_Osize__num,type,
% 12.17/7.86 size_num: num > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onum__of__nat,type,
% 12.17/7.86 num_of_nat: nat > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 12.17/7.86 numeral_numeral:
% 12.17/7.86 !>[A: $tType] : ( num > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Opow,type,
% 12.17/7.86 pow: num > num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Opred__numeral,type,
% 12.17/7.86 pred_numeral: num > nat ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Oring__1__class_Oiszero,type,
% 12.17/7.86 ring_1_iszero:
% 12.17/7.86 !>[A: $tType] : ( A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Num_Osqr,type,
% 12.17/7.86 sqr: num > num ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OAtom,type,
% 12.17/7.86 old_Atom:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OIn0,type,
% 12.17/7.86 old_In0:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OIn1,type,
% 12.17/7.86 old_In1:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OLeaf,type,
% 12.17/7.86 old_Leaf:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_ONode,type,
% 12.17/7.86 old_Node:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_ONumb,type,
% 12.17/7.86 old_Numb:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OPush,type,
% 12.17/7.86 old_Push:
% 12.17/7.86 !>[B: $tType] : ( ( sum_sum @ B @ nat ) > ( nat > ( sum_sum @ B @ nat ) ) > nat > ( sum_sum @ B @ nat ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OPush__Node,type,
% 12.17/7.86 old_Push_Node:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_OScons,type,
% 12.17/7.86 old_Scons:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_Ondepth,type,
% 12.17/7.86 old_ndepth:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
% 12.17/7.86 old_Abs_Node:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_Onode_ORep__Node,type,
% 12.17/7.86 old_Rep_Node:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Old__Datatype_Ontrunc,type,
% 12.17/7.86 old_ntrunc:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Option_Ooption_ONone,type,
% 12.17/7.86 none:
% 12.17/7.86 !>[A: $tType] : ( option @ A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Option_Ooption_OSome,type,
% 12.17/7.86 some:
% 12.17/7.86 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Option_Ooption_Ocase__option,type,
% 12.17/7.86 case_option:
% 12.17/7.86 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Option_Ooption_Osize__option,type,
% 12.17/7.86 size_option:
% 12.17/7.86 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Option_Ooption_Othe,type,
% 12.17/7.86 the2:
% 12.17/7.86 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Order__Relation_Oofilter,type,
% 12.17/7.86 order_ofilter:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Order__Relation_Ounder,type,
% 12.17/7.86 order_under:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Order__Relation_OunderS,type,
% 12.17/7.86 order_underS:
% 12.17/7.86 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Order__Relation_Owell__order__on,type,
% 12.17/7.86 order_well_order_on:
% 12.17/7.86 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Obot__class_Obot,type,
% 12.17/7.86 bot_bot:
% 12.17/7.86 !>[A: $tType] : A ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord_OLeast,type,
% 12.17/7.86 least:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord_Omax,type,
% 12.17/7.86 max:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord_Omin,type,
% 12.17/7.86 min:
% 12.17/7.86 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 12.17/7.86 ord_Least:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord__class_Oless,type,
% 12.17/7.86 ord_less:
% 12.17/7.86 !>[A: $tType] : ( A > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 12.17/7.86 ord_less_eq:
% 12.17/7.86 !>[A: $tType] : ( A > A > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord__class_Omax,type,
% 12.17/7.86 ord_max:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oord__class_Omin,type,
% 12.17/7.86 ord_min:
% 12.17/7.86 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 12.17/7.86 order_Greatest:
% 12.17/7.86 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 12.17/7.86 order_antimono:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 12.17/7.86
% 12.17/7.86 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 12.17/7.86 order_mono:
% 12.17/7.86 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Oordering,type,
% 12.17/7.87 ordering:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Oordering__axioms,type,
% 12.17/7.87 ordering_axioms:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Oordering__top,type,
% 12.17/7.87 ordering_top:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Oordering__top__axioms,type,
% 12.17/7.87 ordering_top_axioms:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Opartial__preordering,type,
% 12.17/7.87 partial_preordering:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Opreordering,type,
% 12.17/7.87 preordering:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Opreordering__axioms,type,
% 12.17/7.87 preordering_axioms:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Orderings_Otop__class_Otop,type,
% 12.17/7.87 top_top:
% 12.17/7.87 !>[A: $tType] : A ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Partial__Function_Ofun__lub,type,
% 12.17/7.87 partial_fun_lub:
% 12.17/7.87 !>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Partial__Function_Ofun__ord,type,
% 12.17/7.87 partial_fun_ord:
% 12.17/7.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Partial__Function_Opartial__function__definitions,type,
% 12.17/7.87 partia7178651479351089652itions:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Power_Opower_Opower,type,
% 12.17/7.87 power2:
% 12.17/7.87 !>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Power_Opower__class_Opower,type,
% 12.17/7.87 power_power:
% 12.17/7.87 !>[A: $tType] : ( A > nat > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_OPair,type,
% 12.17/7.87 product_Pair:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_OSigma,type,
% 12.17/7.87 product_Sigma:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_OUnity,type,
% 12.17/7.87 product_Unity: product_unit ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_Oapfst,type,
% 12.17/7.87 product_apfst:
% 12.17/7.87 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_Oapsnd,type,
% 12.17/7.87 product_apsnd:
% 12.17/7.87 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 12.17/7.87 product_case_prod:
% 12.17/7.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 12.17/7.87 product_fst:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 12.17/7.87 product_snd:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Pure_Otype,type,
% 12.17/7.87 type2:
% 12.17/7.87 !>[A: $tType] : ( itself @ A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Quotient_OQuotient3,type,
% 12.17/7.87 quotient3:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Quotient_Oquot__type_Oabs,type,
% 12.17/7.87 quot_abs:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > A > B ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Quotient_Oquot__type_Orep,type,
% 12.17/7.87 quot_rep:
% 12.17/7.87 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > B > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Olog,type,
% 12.17/7.87 log: code_natural > code_natural > code_natural ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Olog__rel,type,
% 12.17/7.87 log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Ominus__shift,type,
% 12.17/7.87 minus_shift: code_natural > code_natural > code_natural > code_natural ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Opick,type,
% 12.17/7.87 pick:
% 12.17/7.87 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Orange,type,
% 12.17/7.87 range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Random_Oselect__weight,type,
% 12.17/7.87 select_weight:
% 12.17/7.87 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_OAbs__Rat,type,
% 12.17/7.87 abs_Rat: ( product_prod @ int @ int ) > rat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_OFract,type,
% 12.17/7.87 fract: int > int > rat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_OFrct,type,
% 12.17/7.87 frct: ( product_prod @ int @ int ) > rat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_ORep__Rat,type,
% 12.17/7.87 rep_Rat: rat > ( product_prod @ int @ int ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
% 12.17/7.87 field_char_0_Rats:
% 12.17/7.87 !>[A: $tType] : ( set @ A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
% 12.17/7.87 field_char_0_of_rat:
% 12.17/7.87 !>[A: $tType] : ( rat > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Onormalize,type,
% 12.17/7.87 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Oof__int,type,
% 12.17/7.87 of_int: int > rat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Opcr__rat,type,
% 12.17/7.87 pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Opositive,type,
% 12.17/7.87 positive: rat > $o ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Oquotient__of,type,
% 12.17/7.87 quotient_of: rat > ( product_prod @ int @ int ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rat_Oratrel,type,
% 12.17/7.87 ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
% 12.17/7.87 tuple_1188178415141063261rphism:
% 12.17/7.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( ( product_prod @ B @ C ) > A ) > ( tuple_isomorphism @ A @ B @ C ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
% 12.17/7.87 tuple_9181185373184732606rphism:
% 12.17/7.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( tuple_isomorphism @ A @ B @ C ) > nat ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_ODomainp,type,
% 12.17/7.87 domainp:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_OField,type,
% 12.17/7.87 field2:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_OId,type,
% 12.17/7.87 id2:
% 12.17/7.87 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_OPowp,type,
% 12.17/7.87 powp:
% 12.17/7.87 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_Oinv__image,type,
% 12.17/7.87 inv_image:
% 12.17/7.87 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_Oirreflp,type,
% 12.17/7.87 irreflp:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_Orelcomp,type,
% 12.17/7.87 relcomp:
% 12.17/7.87 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Relation_Otransp,type,
% 12.17/7.87 transp:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
% 12.17/7.87 algebr8660921524188924756oprime:
% 12.17/7.87 !>[A: $tType] : ( A > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 12.17/7.87 divide_divide:
% 12.17/7.87 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 12.17/7.87 dvd_dvd:
% 12.17/7.87 !>[A: $tType] : ( A > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Omodulo__class_Omodulo,type,
% 12.17/7.87 modulo_modulo:
% 12.17/7.87 !>[A: $tType] : ( A > A > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
% 12.17/7.87 normal6383669964737779283malize:
% 12.17/7.87 !>[A: $tType] : ( A > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
% 12.17/7.87 unit_f5069060285200089521factor:
% 12.17/7.87 !>[A: $tType] : ( A > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 12.17/7.87 zero_neq_one_of_bool:
% 12.17/7.87 !>[A: $tType] : ( $o > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_OBall,type,
% 12.17/7.87 ball:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_OBex,type,
% 12.17/7.87 bex:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_OCollect,type,
% 12.17/7.87 collect:
% 12.17/7.87 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_OPow,type,
% 12.17/7.87 pow2:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_Odisjnt,type,
% 12.17/7.87 disjnt:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_Oimage,type,
% 12.17/7.87 image:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_Oinsert,type,
% 12.17/7.87 insert:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_Ois__singleton,type,
% 12.17/7.87 is_singleton:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set_Ovimage,type,
% 12.17/7.87 vimage:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
% 12.17/7.87 set_fo6178422350223883121st_nat:
% 12.17/7.87 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
% 12.17/7.87 set_fo1817059534552279752at_rel:
% 12.17/7.87 !>[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 ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 12.17/7.87 set_ord_atLeast:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 12.17/7.87 set_or1337092689740270186AtMost:
% 12.17/7.87 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 12.17/7.87 set_or7035219750837199246ssThan:
% 12.17/7.87 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 12.17/7.87 set_ord_atMost:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 12.17/7.87 set_ord_greaterThan:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 12.17/7.87 set_or3652927894154168847AtMost:
% 12.17/7.87 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 12.17/7.87 set_or5935395276787703475ssThan:
% 12.17/7.87 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 12.17/7.87 set_ord_lessThan:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_String_OLiteral,type,
% 12.17/7.87 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_String_Ochar_OChar,type,
% 12.17/7.87 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_String_Ochar_Osize__char,type,
% 12.17/7.87 size_char: char > nat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
% 12.17/7.87 comm_s6883823935334413003f_char:
% 12.17/7.87 !>[A: $tType] : ( char > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
% 12.17/7.87 unique5772411509450598832har_of:
% 12.17/7.87 !>[A: $tType] : ( A > char ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Sum__Type_OInl,type,
% 12.17/7.87 sum_Inl:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Sum__Type_OInr,type,
% 12.17/7.87 sum_Inr:
% 12.17/7.87 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Sum__Type_OPlus,type,
% 12.17/7.87 sum_Plus:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 12.17/7.87 sum_case_sum:
% 12.17/7.87 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Transfer_Obi__total,type,
% 12.17/7.87 bi_total:
% 12.17/7.87 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Transitive__Closure_Ontrancl,type,
% 12.17/7.87 transitive_ntrancl:
% 12.17/7.87 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 12.17/7.87 transitive_rtrancl:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Transitive__Closure_Otrancl,type,
% 12.17/7.87 transitive_trancl:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Transitive__Closure_Otranclp,type,
% 12.17/7.87 transitive_tranclp:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Typedef_Otype__definition,type,
% 12.17/7.87 type_definition:
% 12.17/7.87 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Typerep_Otyperep_OTyperep,type,
% 12.17/7.87 typerep3: literal > ( list @ typerep ) > typerep ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
% 12.17/7.87 size_typerep: typerep > nat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
% 12.17/7.87 typerep_typerep:
% 12.17/7.87 !>[A: $tType] : ( ( itself @ A ) > typerep ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Oaccp,type,
% 12.17/7.87 accp:
% 12.17/7.87 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Ofinite__psubset,type,
% 12.17/7.87 finite_psubset:
% 12.17/7.87 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Oless__than,type,
% 12.17/7.87 less_than: set @ ( product_prod @ nat @ nat ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Olex__prod,type,
% 12.17/7.87 lex_prod:
% 12.17/7.87 !>[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 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Omax__ext,type,
% 12.17/7.87 max_ext:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Omeasure,type,
% 12.17/7.87 measure:
% 12.17/7.87 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Omin__ext,type,
% 12.17/7.87 min_ext:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Omlex__prod,type,
% 12.17/7.87 mlex_prod:
% 12.17/7.87 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Opred__nat,type,
% 12.17/7.87 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Wellfounded_Owf,type,
% 12.17/7.87 wf:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Ochain__subset,type,
% 12.17/7.87 chain_subset:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Ochains,type,
% 12.17/7.87 chains:
% 12.17/7.87 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 12.17/7.87 pred_chain:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Opred__on_Omaxchain,type,
% 12.17/7.87 pred_maxchain:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Opred__on_Osuc,type,
% 12.17/7.87 pred_suc:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
% 12.17/7.87 pred_s596693808085603175closed:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ ( set @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
% 12.17/7.87 pred_s7749564232668923593losedp:
% 12.17/7.87 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_fChoice,type,
% 12.17/7.87 fChoice:
% 12.17/7.87 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_c_member,type,
% 12.17/7.87 member:
% 12.17/7.87 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_a,type,
% 12.17/7.87 a2: array @ a ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_h,type,
% 12.17/7.87 h: heap_ext @ product_unit ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_h_H,type,
% 12.17/7.87 h2: heap_ext @ product_unit ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_n,type,
% 12.17/7.87 n: nat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_r,type,
% 12.17/7.87 r: nat ).
% 12.17/7.87
% 12.17/7.87 thf(sy_v_thesis,type,
% 12.17/7.87 thesis: $o ).
% 12.17/7.87
% 12.17/7.87 thf(fact_0_assms,axiom,
% 12.17/7.87 heap_Time_effect @ nat @ ( array_len @ a @ a2 ) @ h @ h2 @ r @ n ).
% 12.17/7.87
% 12.17/7.87 thf(fact_1_that,axiom,
% 12.17/7.87 ( ( r
% 12.17/7.87 = ( array_length @ a @ h2 @ a2 ) )
% 12.17/7.87 => ( ( h2 = h )
% 12.17/7.87 => ( ( n
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 => thesis ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_2_effect__lengthI,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( heap @ A )
% 12.17/7.87 => ! [H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
% 12.17/7.87 ( ( H = H2 )
% 12.17/7.87 => ( ( R
% 12.17/7.87 = ( array_length @ A @ H2 @ A3 ) )
% 12.17/7.87 => ( ( N
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 => ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_3_one__natural_Orsp,axiom,
% 12.17/7.87 ( ( one_one @ nat )
% 12.17/7.87 = ( one_one @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_4_one__reorient,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( one @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ( ( ( one_one @ A )
% 12.17/7.87 = X )
% 12.17/7.87 = ( X
% 12.17/7.87 = ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_5_length__update,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ( heap @ A )
% 12.17/7.87 & ( heap @ B ) )
% 12.17/7.87 => ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
% 12.17/7.87 ( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
% 12.17/7.87 = ( array_length @ A @ H2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_6_euclidean__size__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
% 12.17/7.87 = ( one_one @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_7_fact__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_8_of__nat__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_1 @ A )
% 12.17/7.87 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_9_of__nat__1__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( ( one_one @ A )
% 12.17/7.87 = ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_10_of__nat__eq__1__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( ( semiring_1_of_nat @ A @ N )
% 12.17/7.87 = ( one_one @ A ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_11_dbl__dec__simps_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( neg_numeral @ A )
% 12.17/7.87 => ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_12_round__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.87 => ( ( archimedean_round @ A @ ( one_one @ A ) )
% 12.17/7.87 = ( one_one @ int ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_13_of__nat__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( semiring_1_of_nat @ A @ M )
% 12.17/7.87 = ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_14_of__nat__fact,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
% 12.17/7.87 = ( semiring_char_0_fact @ A @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_15_euclidean__size__of__nat,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_16_round__of__nat,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_17_one__integer_Orsp,axiom,
% 12.17/7.87 ( ( one_one @ int )
% 12.17/7.87 = ( one_one @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_18_update__swap__neqIndex,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( heap @ A )
% 12.17/7.87 => ! [I: nat,I2: nat,A3: array @ A,V: A,V2: A,H2: heap_ext @ product_unit] :
% 12.17/7.87 ( ( I != I2 )
% 12.17/7.87 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I2 @ V2 @ H2 ) )
% 12.17/7.87 = ( array_update @ A @ A3 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_19_int__ops_I2_J,axiom,
% 12.17/7.87 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 12.17/7.87 = ( one_one @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_20_effect__deterministic_I3_J,axiom,
% 12.17/7.87 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 12.17/7.87 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 12.17/7.87 => ( N = N2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_21_effect__deterministic_I2_J,axiom,
% 12.17/7.87 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 12.17/7.87 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 12.17/7.87 => ( H = H3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_22_effect__deterministic_I1_J,axiom,
% 12.17/7.87 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
% 12.17/7.87 => ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
% 12.17/7.87 => ( A3 = B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_23_effect__ifE,axiom,
% 12.17/7.87 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N )
% 12.17/7.87 => ( ( C2
% 12.17/7.87 => ~ ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
% 12.17/7.87 => ~ ( ~ C2
% 12.17/7.87 => ~ ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_24_effect__ifI,axiom,
% 12.17/7.87 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,E: heap_Time_Heap @ A] :
% 12.17/7.87 ( ( C2
% 12.17/7.87 => ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
% 12.17/7.87 => ( ( ~ C2
% 12.17/7.87 => ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) )
% 12.17/7.87 => ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_25_division__segment__of__nat,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_26_present__update,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ( heap @ A )
% 12.17/7.87 & ( heap @ B ) )
% 12.17/7.87 => ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
% 12.17/7.87 ( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
% 12.17/7.87 = ( array_present @ A @ H2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_27_fact__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_28_effect__tapI,axiom,
% 12.17/7.87 ! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
% 12.17/7.87 ( ( H = H2 )
% 12.17/7.87 => ( ( R
% 12.17/7.87 = ( F @ H2 ) )
% 12.17/7.87 => ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ ( one_one @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_29_effect__tapE,axiom,
% 12.17/7.87 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ N )
% 12.17/7.87 => ~ ( ( H = H2 )
% 12.17/7.87 => ( ( R
% 12.17/7.87 = ( F @ H2 ) )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( one_one @ nat ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_30_update__swap__neq,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ( heap @ A )
% 12.17/7.87 & ( heap @ B ) )
% 12.17/7.87 => ! [A3: array @ A,A4: array @ B,I: nat,V: A,I2: nat,V2: B,H2: heap_ext @ product_unit] :
% 12.17/7.87 ( ( array_noteq @ A @ B @ A3 @ A4 )
% 12.17/7.87 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A4 @ I2 @ V2 @ H2 ) )
% 12.17/7.87 = ( array_update @ B @ A4 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_31_unequal,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( heap @ A )
% 12.17/7.87 => ! [A3: array @ A,A4: array @ A] :
% 12.17/7.87 ( ( A3 != A4 )
% 12.17/7.87 = ( array_noteq @ A @ A @ A3 @ A4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_32_division__segment__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_33_round__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.87 => ( ( archimedean_round @ A @ ( zero_zero @ A ) )
% 12.17/7.87 = ( zero_zero @ int ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_34_of__nat__eq__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [M: nat] :
% 12.17/7.87 ( ( ( semiring_1_of_nat @ A @ M )
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 = ( M
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_35_of__nat__0__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( ( zero_zero @ A )
% 12.17/7.87 = ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( ( zero_zero @ nat )
% 12.17/7.87 = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_36_of__nat__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_1 @ A )
% 12.17/7.87 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_37_euclidean__size__eq__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.87 => ! [B2: A] :
% 12.17/7.87 ( ( ( euclid6346220572633701492n_size @ A @ B2 )
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 = ( B2
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_38_size__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.87 => ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_39_nat__int__comparison_I1_J,axiom,
% 12.17/7.87 ( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.17/7.87 = ( ^ [A5: nat,B3: nat] :
% 12.17/7.87 ( ( semiring_1_of_nat @ int @ A5 )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_40_zero__natural_Orsp,axiom,
% 12.17/7.87 ( ( zero_zero @ nat )
% 12.17/7.87 = ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_41_division__segment__not__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( euclid7384307370059645450egment @ A @ A3 )
% 12.17/7.87 != ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_42_mem__Collect__eq,axiom,
% 12.17/7.87 ! [A: $tType,A3: A,P: A > $o] :
% 12.17/7.87 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 12.17/7.87 = ( P @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_43_Collect__mem__eq,axiom,
% 12.17/7.87 ! [A: $tType,A6: set @ A] :
% 12.17/7.87 ( ( collect @ A
% 12.17/7.87 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 12.17/7.87 = A6 ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_44_Collect__cong,axiom,
% 12.17/7.87 ! [A: $tType,P: A > $o,Q: A > $o] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ( P @ X3 )
% 12.17/7.87 = ( Q @ X3 ) )
% 12.17/7.87 => ( ( collect @ A @ P )
% 12.17/7.87 = ( collect @ A @ Q ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_45_ext,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ( F @ X3 )
% 12.17/7.87 = ( G @ X3 ) )
% 12.17/7.87 => ( F = G ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_46_int__if,axiom,
% 12.17/7.87 ! [P: $o,A3: nat,B2: nat] :
% 12.17/7.87 ( ( P
% 12.17/7.87 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ A3 ) ) )
% 12.17/7.87 & ( ~ P
% 12.17/7.87 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_47_zero__reorient,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( zero @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ( ( ( zero_zero @ A )
% 12.17/7.87 = X )
% 12.17/7.87 = ( X
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_48_noteq__sym,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ( heap @ A )
% 12.17/7.87 & ( heap @ B ) )
% 12.17/7.87 => ! [A3: array @ A,B2: array @ B] :
% 12.17/7.87 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 12.17/7.87 => ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_49_noteq__irrefl,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( heap @ A )
% 12.17/7.87 => ! [R: array @ A] :
% 12.17/7.87 ~ ( array_noteq @ A @ A @ R @ R ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_50_fact__nonzero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ( semiring_char_0 @ A )
% 12.17/7.87 & ( semiri3467727345109120633visors @ A ) )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( semiring_char_0_fact @ A @ N )
% 12.17/7.87 != ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_51_division__segment__nat__def,axiom,
% 12.17/7.87 ( ( euclid7384307370059645450egment @ nat )
% 12.17/7.87 = ( ^ [N3: nat] : ( one_one @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_52_division__segment__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( euclid7384307370059645450egment @ A @ A3 )
% 12.17/7.87 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.17/7.87 => ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 12.17/7.87 = ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.17/7.87 => ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_53_zero__neq__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( zero_neq_one @ A )
% 12.17/7.87 => ( ( zero_zero @ A )
% 12.17/7.87 != ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_54_dbl__inc__simps_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( neg_numeral @ A )
% 12.17/7.87 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_55_effect__ureturnI,axiom,
% 12.17/7.87 ! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
% 12.17/7.87 ( ( H2 = H )
% 12.17/7.87 => ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ X @ ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_56_effect__ureturnE,axiom,
% 12.17/7.87 ! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ R @ N )
% 12.17/7.87 => ~ ( ( R = X )
% 12.17/7.87 => ( ( H = H2 )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_57_effect__assertE,axiom,
% 12.17/7.87 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N )
% 12.17/7.87 => ~ ( ( P @ X )
% 12.17/7.87 => ( ( R = X )
% 12.17/7.87 => ( ( H = H2 )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( one_one @ nat ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_58_effect__assertI,axiom,
% 12.17/7.87 ! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( P @ X )
% 12.17/7.87 => ( ( H = H2 )
% 12.17/7.87 => ( ( R = X )
% 12.17/7.87 => ( ( N
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 => ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_59_dbl__dec__simps_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( neg_numeral @ A )
% 12.17/7.87 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
% 12.17/7.87 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_60_division__segment__euclidean__size,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_61_euclidean__size__greater__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.87 => ! [B2: A] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.17/7.87 = ( B2
% 12.17/7.87 != ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_62_fact__Suc__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.87 = ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_63_of__nat__le__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 12.17/7.87 = ( M
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_64_effect__returnE,axiom,
% 12.17/7.87 ! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.17/7.87 ( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ R @ N )
% 12.17/7.87 => ~ ( ( R = X )
% 12.17/7.87 => ( ( H = H2 )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( one_one @ nat ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_65_verit__minus__simplify_I4_J,axiom,
% 12.17/7.87 ! [B: $tType] :
% 12.17/7.87 ( ( group_add @ B )
% 12.17/7.87 => ! [B2: B] :
% 12.17/7.87 ( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
% 12.17/7.87 = B2 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_66_neg__equal__iff__equal,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( uminus_uminus @ A @ A3 )
% 12.17/7.87 = ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( A3 = B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_67_add_Oinverse__inverse,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_68_Suc__le__mono,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_69_old_Onat_Oinject,axiom,
% 12.17/7.87 ! [Nat: nat,Nat2: nat] :
% 12.17/7.87 ( ( ( suc @ Nat )
% 12.17/7.87 = ( suc @ Nat2 ) )
% 12.17/7.87 = ( Nat = Nat2 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_70_nat_Oinject,axiom,
% 12.17/7.87 ! [X22: nat,Y2: nat] :
% 12.17/7.87 ( ( ( suc @ X22 )
% 12.17/7.87 = ( suc @ Y2 ) )
% 12.17/7.87 = ( X22 = Y2 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_71_mult__cancel2,axiom,
% 12.17/7.87 ! [M: nat,K: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ M @ K )
% 12.17/7.87 = ( times_times @ nat @ N @ K ) )
% 12.17/7.87 = ( ( M = N )
% 12.17/7.87 | ( K
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_72_mult__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ K @ M )
% 12.17/7.87 = ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ( M = N )
% 12.17/7.87 | ( K
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_73_mult__0__right,axiom,
% 12.17/7.87 ! [M: nat] :
% 12.17/7.87 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_74_mult__is__0,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ M @ N )
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 | ( N
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_75_le0,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_76_bot__nat__0_Oextremum,axiom,
% 12.17/7.87 ! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_77_nat__mult__eq__1__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ M @ N )
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_78_nat__1__eq__mult__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( one_one @ nat )
% 12.17/7.87 = ( times_times @ nat @ M @ N ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_79_le__zero__eq,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [N: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_80_not__gr__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [N: A] :
% 12.17/7.87 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_81_mult__zero__left,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( mult_zero @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 = ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_82_mult__zero__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( mult_zero @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 = ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_83_mult__eq__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri3467727345109120633visors @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 = ( ( A3
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( B2
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_84_mult__cancel__left,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri6575147826004484403cancel @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ C2 @ A3 )
% 12.17/7.87 = ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_85_mult__cancel__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri6575147826004484403cancel @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.87 = ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_86_neg__le__iff__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_87_neg__equal__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ( uminus_uminus @ A @ A3 )
% 12.17/7.87 = A3 )
% 12.17/7.87 = ( A3
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_88_equal__neg__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 = ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( A3
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_89_neg__equal__0__iff__equal,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ( uminus_uminus @ A @ A3 )
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 = ( A3
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_90_neg__0__equal__iff__equal,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ( zero_zero @ A )
% 12.17/7.87 = ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ( zero_zero @ A )
% 12.17/7.87 = A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_91_add_Oinverse__neutral,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 12.17/7.87 = ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_92_mult__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( monoid_mult @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_93_mult_Oright__neutral,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( monoid_mult @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_94_neg__less__iff__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_95_mult__minus__left,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.87 = ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_96_minus__mult__minus,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( times_times @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_97_mult__minus__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_98_of__nat__le__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_99_one__le__mult__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
% 12.17/7.87 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 12.17/7.87 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_100_one__eq__mult__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( suc @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( times_times @ nat @ M @ N ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_101_mult__eq__1__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ M @ N )
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_102_of__nat__mult,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_1 @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
% 12.17/7.87 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_103_Suc__less__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_104_Suc__mono,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_105_lessI,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_106_mult__le__cancel2,axiom,
% 12.17/7.87 ! [M: nat,K: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 12.17/7.87 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_107_nat__0__less__mult__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 12.17/7.87 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.87 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_108_mult__less__cancel2,axiom,
% 12.17/7.87 ! [M: nat,K: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 12.17/7.87 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 & ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_109_less__nat__zero__code,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_110_neq0__conv,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( N
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_111_bot__nat__0_Onot__eq__extremum,axiom,
% 12.17/7.87 ! [A3: nat] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_112_neg__less__eq__nonneg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 12.17/7.87 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_113_less__eq__neg__nonpos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_114_neg__le__0__iff__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.87 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_115_neg__0__le__iff__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_116_mult__cancel__right2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_15535105094025558882visors @ A )
% 12.17/7.87 => ! [A3: A,C2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.87 = C2 )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( A3
% 12.17/7.87 = ( one_one @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_117_mult__cancel__right1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_15535105094025558882visors @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( C2
% 12.17/7.87 = ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( B2
% 12.17/7.87 = ( one_one @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_118_mult__cancel__left2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_15535105094025558882visors @ A )
% 12.17/7.87 => ! [C2: A,A3: A] :
% 12.17/7.87 ( ( ( times_times @ A @ C2 @ A3 )
% 12.17/7.87 = C2 )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( A3
% 12.17/7.87 = ( one_one @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_119_mult__cancel__left1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_15535105094025558882visors @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( C2
% 12.17/7.87 = ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( C2
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( B2
% 12.17/7.87 = ( one_one @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_120_neg__less__0__iff__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.87 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_121_neg__0__less__iff__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_122_neg__less__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 12.17/7.87 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_123_less__neg__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.87 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_124_mult__minus1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_1 @ A )
% 12.17/7.87 => ! [Z3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z3 )
% 12.17/7.87 = ( uminus_uminus @ A @ Z3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_125_mult__minus1__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_1 @ A )
% 12.17/7.87 => ! [Z3: A] :
% 12.17/7.87 ( ( times_times @ A @ Z3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.87 = ( uminus_uminus @ A @ Z3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_126_of__nat__less__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_127_zero__less__Suc,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_128_less__Suc0,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_129_less__one,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_130_dbl__inc__simps_I4_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( neg_numeral @ A )
% 12.17/7.87 => ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.87 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_131_fact__Suc,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( semiring_char_0_fact @ A @ ( suc @ N ) )
% 12.17/7.87 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_132_of__nat__0__less__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_133_zero__integer_Orsp,axiom,
% 12.17/7.87 ( ( zero_zero @ int )
% 12.17/7.87 = ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_134_one__less__mult,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.17/7.87 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 12.17/7.87 => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_135_n__less__m__mult__n,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 12.17/7.87 => ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_136_n__less__n__mult__m,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 12.17/7.87 => ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_137_ex__least__nat__less,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ( P @ N )
% 12.17/7.87 => ( ~ ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 => ? [K2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ K2 @ N )
% 12.17/7.87 & ! [I3: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I3 @ K2 )
% 12.17/7.87 => ~ ( P @ I3 ) )
% 12.17/7.87 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_138_mult__le__cancel__iff1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_139_mult__le__cancel__iff2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_140_division__segment__int__def,axiom,
% 12.17/7.87 ( ( euclid7384307370059645450egment @ int )
% 12.17/7.87 = ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_141_verit__la__disequality,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( A3 = B2 )
% 12.17/7.87 | ~ ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 | ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_142_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord2810124833399127020strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_143_mult__less__cancel__right__disj,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 & ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.87 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_144_mult__strict__right__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_145_mult__strict__right__mono__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_146_mult__less__cancel__left__disj,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 & ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.87 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_147_mult__strict__left__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_148_mult__less__le__imp__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_149_mult__le__less__imp__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_150_mult__strict__left__mono__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_151_mult__less__cancel__left__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_152_mult__less__cancel__left__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_153_mult__right__le__imp__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_154_zero__less__mult__pos2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_155_mult__left__le__imp__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_156_mult__le__cancel__left__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_157_mult__le__cancel__left__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_158_zero__less__mult__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_159_mult__less__cancel__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_160_mult__strict__mono_H,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_161_mult__right__less__imp__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semiring @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_162_mult__less__cancel__left,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_163_mult__strict__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_164_mult__left__less__imp__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semiring @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_165_mult__le__cancel__right,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_166_mult__le__cancel__left,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_167_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordere2520102378445227354miring @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_168_zero__less__mult__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 12.17/7.87 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_169_mult__pos__neg2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_170_mult__pos__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_171_mult__pos__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_172_mult__neg__pos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord8928482502909563296strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_173_zero__le__mult__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 12.17/7.87 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_174_mult__nonneg__nonpos2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring_0 @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_175_mult__less__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 12.17/7.87 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_176_mult__nonpos__nonneg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring_0 @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_177_mult__nonneg__nonpos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring_0 @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_178_mult__nonneg__nonneg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring_0 @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_179_split__mult__neg__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring_0 @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 12.17/7.87 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_180_mult__le__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 12.17/7.87 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_181_not__square__less__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_ring @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_182_mult__neg__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord4710134922213307826strict @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_183_mult__right__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_184_mult__right__mono__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ring @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_185_mult__left__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_186_mult__nonpos__nonpos,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_187_mult__left__mono__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ring @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_188_split__mult__pos__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 12.17/7.87 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.87 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_189_zero__le__square,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_ring @ A )
% 12.17/7.87 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_190_mult__mono_H,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_191_mult__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_semiring @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_192_verit__negate__coefficient_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.87 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_193_verit__negate__coefficient_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( A3 = B2 )
% 12.17/7.87 => ( ( uminus_uminus @ A @ A3 )
% 12.17/7.87 = ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_194_verit__comp__simplify1_I1_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_195_verit__comp__simplify1_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_196_verit__comp__simplify1_I3_J,axiom,
% 12.17/7.87 ! [B: $tType] :
% 12.17/7.87 ( ( linorder @ B )
% 12.17/7.87 => ! [B4: B,A4: B] :
% 12.17/7.87 ( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
% 12.17/7.87 = ( ord_less @ B @ A4 @ B4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_197_square__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( idom @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ A3 )
% 12.17/7.87 = ( times_times @ A @ B2 @ B2 ) )
% 12.17/7.87 = ( ( A3 = B2 )
% 12.17/7.87 | ( A3
% 12.17/7.87 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_198_minus__mult__commute,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.87 = ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_199_linorder__neqE__linordered__idom,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( X != Y3 )
% 12.17/7.87 => ( ~ ( ord_less @ A @ X @ Y3 )
% 12.17/7.87 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_200_infinite__descent__measure,axiom,
% 12.17/7.87 ! [A: $tType,P: A > $o,V3: A > nat,X: A] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ~ ( P @ X3 )
% 12.17/7.87 => ? [Y4: A] :
% 12.17/7.87 ( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
% 12.17/7.87 & ~ ( P @ Y4 ) ) )
% 12.17/7.87 => ( P @ X ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_201_transitive__stepwise__le,axiom,
% 12.17/7.87 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 12.17/7.87 => ( ! [X3: nat,Y5: nat,Z4: nat] :
% 12.17/7.87 ( ( R2 @ X3 @ Y5 )
% 12.17/7.87 => ( ( R2 @ Y5 @ Z4 )
% 12.17/7.87 => ( R2 @ X3 @ Z4 ) ) )
% 12.17/7.87 => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
% 12.17/7.87 => ( R2 @ M @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_202_less__mono__imp__le__mono,axiom,
% 12.17/7.87 ! [F: nat > nat,I: nat,J: nat] :
% 12.17/7.87 ( ! [I4: nat,J2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I4 @ J2 )
% 12.17/7.87 => ( ord_less @ nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_203_Suc__mult__less__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_204_not__less__less__Suc__eq,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ~ ( ord_less @ nat @ N @ M )
% 12.17/7.87 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 12.17/7.87 = ( N = M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_205_nat__induct__at__least,axiom,
% 12.17/7.87 ! [M: nat,N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ( P @ M )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N4 )
% 12.17/7.87 => ( ( P @ N4 )
% 12.17/7.87 => ( P @ ( suc @ N4 ) ) ) )
% 12.17/7.87 => ( P @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_206_measure__induct__rule,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( wellorder @ B )
% 12.17/7.87 => ! [F: A > B,P: A > $o,A3: A] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ! [Y4: A] :
% 12.17/7.87 ( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
% 12.17/7.87 => ( P @ Y4 ) )
% 12.17/7.87 => ( P @ X3 ) )
% 12.17/7.87 => ( P @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_207_le__neq__implies__less,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ( M != N )
% 12.17/7.87 => ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_208_Suc__mult__le__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_209_strict__inc__induct,axiom,
% 12.17/7.87 ! [I: nat,J: nat,P: nat > $o] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J )
% 12.17/7.87 => ( ! [I4: nat] :
% 12.17/7.87 ( ( J
% 12.17/7.87 = ( suc @ I4 ) )
% 12.17/7.87 => ( P @ I4 ) )
% 12.17/7.87 => ( ! [I4: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I4 @ J )
% 12.17/7.87 => ( ( P @ ( suc @ I4 ) )
% 12.17/7.87 => ( P @ I4 ) ) )
% 12.17/7.87 => ( P @ I ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_210_linorder__neqE__nat,axiom,
% 12.17/7.87 ! [X: nat,Y3: nat] :
% 12.17/7.87 ( ( X != Y3 )
% 12.17/7.87 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 12.17/7.87 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_211_less__or__eq__imp__le,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 | ( M = N ) )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_212_le__eq__less__or__eq,axiom,
% 12.17/7.87 ( ( ord_less_eq @ nat )
% 12.17/7.87 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M2 @ N3 )
% 12.17/7.87 | ( M2 = N3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_213_infinite__descent,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( ~ ( P @ N4 )
% 12.17/7.87 => ? [M3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M3 @ N4 )
% 12.17/7.87 & ~ ( P @ M3 ) ) )
% 12.17/7.87 => ( P @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_214_Suc__mult__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ ( suc @ K ) @ M )
% 12.17/7.87 = ( times_times @ nat @ ( suc @ K ) @ N ) )
% 12.17/7.87 = ( M = N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_215_nat__less__induct,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( ! [M3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M3 @ N4 )
% 12.17/7.87 => ( P @ M3 ) )
% 12.17/7.87 => ( P @ N4 ) )
% 12.17/7.87 => ( P @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_216_less__irrefl__nat,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ N @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_217_less__imp__le__nat,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_218_less__Suc__induct,axiom,
% 12.17/7.87 ! [I: nat,J: nat,P: nat > nat > $o] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J )
% 12.17/7.87 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
% 12.17/7.87 => ( ! [I4: nat,J2: nat,K2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I4 @ J2 )
% 12.17/7.87 => ( ( ord_less @ nat @ J2 @ K2 )
% 12.17/7.87 => ( ( P @ I4 @ J2 )
% 12.17/7.87 => ( ( P @ J2 @ K2 )
% 12.17/7.87 => ( P @ I4 @ K2 ) ) ) ) )
% 12.17/7.87 => ( P @ I @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_219_le__imp__less__Suc,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_220_full__nat__induct,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( ! [M3: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N4 )
% 12.17/7.87 => ( P @ M3 ) )
% 12.17/7.87 => ( P @ N4 ) )
% 12.17/7.87 => ( P @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_221_not__less__eq__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_222_measure__induct,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( wellorder @ B )
% 12.17/7.87 => ! [F: A > B,P: A > $o,A3: A] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ! [Y4: A] :
% 12.17/7.87 ( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
% 12.17/7.87 => ( P @ Y4 ) )
% 12.17/7.87 => ( P @ X3 ) )
% 12.17/7.87 => ( P @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_223_less__trans__Suc,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J )
% 12.17/7.87 => ( ( ord_less @ nat @ J @ K )
% 12.17/7.87 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_224_less__not__refl3,axiom,
% 12.17/7.87 ! [S: nat,T2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ S @ T2 )
% 12.17/7.87 => ( S != T2 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_225_less__not__refl2,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ N @ M )
% 12.17/7.87 => ( M != N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_226_less__eq__Suc__le,axiom,
% 12.17/7.87 ( ( ord_less @ nat )
% 12.17/7.87 = ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_227_less__Suc__eq__le,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_228_le__less__Suc__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 12.17/7.87 = ( N = M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_229_Suc__n__not__le__n,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_230_less__not__refl,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ N @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_231_Suc__less__SucD,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.17/7.87 => ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_232_less__antisym,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ~ ( ord_less @ nat @ N @ M )
% 12.17/7.87 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 12.17/7.87 => ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_233_Suc__less__eq2,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.17/7.87 = ( ? [M4: nat] :
% 12.17/7.87 ( ( M
% 12.17/7.87 = ( suc @ M4 ) )
% 12.17/7.87 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_234_Suc__le__lessD,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 12.17/7.87 => ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_235_All__less__Suc,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ! [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 12.17/7.87 => ( P @ I5 ) ) )
% 12.17/7.87 = ( ( P @ N )
% 12.17/7.87 & ! [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ N )
% 12.17/7.87 => ( P @ I5 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_236_not__less__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_237_nat__neq__iff,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( M != N )
% 12.17/7.87 = ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 | ( ord_less @ nat @ N @ M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_238_nat__less__le,axiom,
% 12.17/7.87 ( ( ord_less @ nat )
% 12.17/7.87 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M2 @ N3 )
% 12.17/7.87 & ( M2 != N3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_239_n__not__Suc__n,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( N
% 12.17/7.87 != ( suc @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_240_less__Suc__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 = ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 | ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_241_Ex__less__Suc,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ? [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 12.17/7.87 & ( P @ I5 ) ) )
% 12.17/7.87 = ( ( P @ N )
% 12.17/7.87 | ? [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ N )
% 12.17/7.87 & ( P @ I5 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_242_inc__induct,axiom,
% 12.17/7.87 ! [I: nat,J: nat,P: nat > $o] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ( P @ J )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ N4 )
% 12.17/7.87 => ( ( ord_less @ nat @ N4 @ J )
% 12.17/7.87 => ( ( P @ ( suc @ N4 ) )
% 12.17/7.87 => ( P @ N4 ) ) ) )
% 12.17/7.87 => ( P @ I ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_243_dec__induct,axiom,
% 12.17/7.87 ! [I: nat,J: nat,P: nat > $o] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ( P @ I )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ N4 )
% 12.17/7.87 => ( ( ord_less @ nat @ N4 @ J )
% 12.17/7.87 => ( ( P @ N4 )
% 12.17/7.87 => ( P @ ( suc @ N4 ) ) ) ) )
% 12.17/7.87 => ( P @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_244_Suc__inject,axiom,
% 12.17/7.87 ! [X: nat,Y3: nat] :
% 12.17/7.87 ( ( ( suc @ X )
% 12.17/7.87 = ( suc @ Y3 ) )
% 12.17/7.87 => ( X = Y3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_245_less__SucI,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_246_less__SucE,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 => ( ~ ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_247_le__Suc__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 = ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 | ( M
% 12.17/7.87 = ( suc @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_248_Suc__lessI,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ( ( suc @ M )
% 12.17/7.87 != N )
% 12.17/7.87 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_249_Suc__lessE,axiom,
% 12.17/7.87 ! [I: nat,K: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ I ) @ K )
% 12.17/7.87 => ~ ! [J2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J2 )
% 12.17/7.87 => ( K
% 12.17/7.87 != ( suc @ J2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_250_Suc__lessD,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 12.17/7.87 => ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_251_Suc__le__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 12.17/7.87 = ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_252_Suc__le__D,axiom,
% 12.17/7.87 ! [N: nat,M5: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 12.17/7.87 => ? [M6: nat] :
% 12.17/7.87 ( M5
% 12.17/7.87 = ( suc @ M6 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_253_le__SucI,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_254_le__SucE,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( M
% 12.17/7.87 = ( suc @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_255_Suc__leI,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_256_Suc__leD,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_257_Nat_OlessE,axiom,
% 12.17/7.87 ! [I: nat,K: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ K )
% 12.17/7.87 => ( ( K
% 12.17/7.87 != ( suc @ I ) )
% 12.17/7.87 => ~ ! [J2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J2 )
% 12.17/7.87 => ( K
% 12.17/7.87 != ( suc @ J2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_258_square__eq__1__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_15535105094025558882visors @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ( ( ( times_times @ A @ X @ X )
% 12.17/7.87 = ( one_one @ A ) )
% 12.17/7.87 = ( ( X
% 12.17/7.87 = ( one_one @ A ) )
% 12.17/7.87 | ( X
% 12.17/7.87 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_259_minus__less__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.87 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_260_less__minus__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_261_mult_Oleft__commute,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ab_semigroup_mult @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
% 12.17/7.87 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_262_le__imp__neg__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_263_minus__le__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.87 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_264_le__minus__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ordered_ab_group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_265_mult_Ocommute,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ab_semigroup_mult @ A )
% 12.17/7.87 => ( ( times_times @ A )
% 12.17/7.87 = ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_266_less__1__mult,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [M: A,N: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( one_one @ A ) @ M )
% 12.17/7.87 => ( ( ord_less @ A @ ( one_one @ A ) @ N )
% 12.17/7.87 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_267_minus__equation__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( uminus_uminus @ A @ A3 )
% 12.17/7.87 = B2 )
% 12.17/7.87 = ( ( uminus_uminus @ A @ B2 )
% 12.17/7.87 = A3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_268_equation__minus__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( group_add @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 = ( uminus_uminus @ A @ B2 ) )
% 12.17/7.87 = ( B2
% 12.17/7.87 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_269_lift__Suc__mono__less__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [F: nat > A,N: nat,M: nat] :
% 12.17/7.87 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.17/7.87 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 12.17/7.87 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_270_mult_Oassoc,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semigroup_mult @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.87 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_271_lift__Suc__antimono__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [F: nat > A,N: nat,N2: nat] :
% 12.17/7.87 ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ N @ N2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_272_lift__Suc__mono__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [F: nat > A,N: nat,N2: nat] :
% 12.17/7.87 ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.17/7.87 => ( ( ord_less @ nat @ N @ N2 )
% 12.17/7.87 => ( ord_less @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_273_lift__Suc__mono__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [F: nat > A,N: nat,N2: nat] :
% 12.17/7.87 ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ N @ N2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_274_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ab_semigroup_mult @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.87 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_275_fold__if__return,axiom,
% 12.17/7.87 ! [A: $tType,B2: $o,C2: A,D2: A] :
% 12.17/7.87 ( ( B2
% 12.17/7.87 => ( ( heap_Time_return @ A @ C2 )
% 12.17/7.87 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) )
% 12.17/7.87 & ( ~ B2
% 12.17/7.87 => ( ( heap_Time_return @ A @ D2 )
% 12.17/7.87 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_276_mult__less__cancel__right2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_277_mult__less__cancel__right1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_278_mult__less__cancel__left2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,A3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_279_mult__less__cancel__left1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 12.17/7.87 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_280_mult__le__cancel__right2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_281_mult__le__cancel__right1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_282_mult__le__cancel__left2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_283_mult__le__cancel__left1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [C2: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 12.17/7.87 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_284_mult__less__iff1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 12.17/7.87 => ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
% 12.17/7.87 = ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_285_less__minus__one__simps_I4_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_286_less__minus__one__simps_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_287_le__minus__one__simps_I4_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_288_le__minus__one__simps_I2_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_289_less__Suc__eq__0__disj,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 | ? [J3: nat] :
% 12.17/7.87 ( ( M
% 12.17/7.87 = ( suc @ J3 ) )
% 12.17/7.87 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_290_gr0__implies__Suc,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 => ? [M6: nat] :
% 12.17/7.87 ( N
% 12.17/7.87 = ( suc @ M6 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_291_All__less__Suc2,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ! [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 12.17/7.87 => ( P @ I5 ) ) )
% 12.17/7.87 = ( ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 & ! [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ N )
% 12.17/7.87 => ( P @ ( suc @ I5 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_292_gr0__conv__Suc,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 = ( ? [M2: nat] :
% 12.17/7.87 ( N
% 12.17/7.87 = ( suc @ M2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_293_Ex__less__Suc2,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ? [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ ( suc @ N ) )
% 12.17/7.87 & ( P @ I5 ) ) )
% 12.17/7.87 = ( ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 | ? [I5: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I5 @ N )
% 12.17/7.87 & ( P @ ( suc @ I5 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_294_mult__less__mono2,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J )
% 12.17/7.87 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_295_mult__less__mono1,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I @ J )
% 12.17/7.87 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_296_ex__least__nat__le,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ( P @ N )
% 12.17/7.87 => ( ~ ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 => ? [K2: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ K2 @ N )
% 12.17/7.87 & ! [I3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ I3 @ K2 )
% 12.17/7.87 => ~ ( P @ I3 ) )
% 12.17/7.87 & ( P @ K2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_297_less__imp__of__nat__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_298_of__nat__less__imp__less,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.87 => ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_299_of__nat__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [I: nat,J: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_300_int__ops_I7_J,axiom,
% 12.17/7.87 ! [A3: nat,B2: nat] :
% 12.17/7.87 ( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B2 ) )
% 12.17/7.87 = ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_301_nat__int__comparison_I3_J,axiom,
% 12.17/7.87 ( ( ord_less_eq @ nat )
% 12.17/7.87 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_302_mult__not__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( mult_zero @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( A3
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 & ( B2
% 12.17/7.87 != ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_303_divisors__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri3467727345109120633visors @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 => ( ( A3
% 12.17/7.87 = ( zero_zero @ A ) )
% 12.17/7.87 | ( B2
% 12.17/7.87 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_304_no__zero__divisors,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri3467727345109120633visors @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( B2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.87 != ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_305_mult__left__cancel,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri6575147826004484403cancel @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( C2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ( times_times @ A @ C2 @ A3 )
% 12.17/7.87 = ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.87 = ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_306_mult__right__cancel,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiri6575147826004484403cancel @ A )
% 12.17/7.87 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.87 ( ( C2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.87 = ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.87 = ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_307_fact__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_308_euclidean__size__mult,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.87 = ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_309_round__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_310_mult__left__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [C2: A,A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_311_mult__le__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_312_mult__right__le__one__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y3 ) @ X ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_313_mult__left__le__one__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.87 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ Y3 @ X ) @ X ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_314_less__minus__one__simps_I1_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_315_less__minus__one__simps_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_316_le__minus__one__simps_I1_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_317_le__minus__one__simps_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_idom @ A )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_318_ex__less__of__nat__mult,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim462609752435547400_field @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.87 => ? [N4: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_319_nat__induct__non__zero,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 => ( ( P @ ( one_one @ nat ) )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.17/7.87 => ( ( P @ N4 )
% 12.17/7.87 => ( P @ ( suc @ N4 ) ) ) )
% 12.17/7.87 => ( P @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_320_size__mult__mono_H,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( B2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_321_size__mult__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( B2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_322_fact__less__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.87 => ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_323_fact__ge__Suc__0__nat,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_324_one__neq__neg__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_char_0 @ A )
% 12.17/7.87 => ( ( one_one @ A )
% 12.17/7.87 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_325_zero__less__iff__neq__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [N: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 12.17/7.87 = ( N
% 12.17/7.87 != ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_326_gr__implies__not__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [M: A,N: A] :
% 12.17/7.87 ( ( ord_less @ A @ M @ N )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( zero_zero @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_327_not__less__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [N: A] :
% 12.17/7.87 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_328_gr__zeroI,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [N: A] :
% 12.17/7.87 ( ( N
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_329_less__numeral__extra_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_330_mult_Ocomm__neutral,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( comm_monoid_mult @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_331_comm__monoid__mult__class_Omult__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( comm_monoid_mult @ A )
% 12.17/7.87 => ! [A3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 12.17/7.87 = A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_332_zero__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.87 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_333_le__numeral__extra_I3_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_334_less__numeral__extra_I4_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_335_exists__least__lemma,axiom,
% 12.17/7.87 ! [P: nat > $o] :
% 12.17/7.87 ( ~ ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 => ( ? [X_1: nat] : ( P @ X_1 )
% 12.17/7.87 => ? [N4: nat] :
% 12.17/7.87 ( ~ ( P @ N4 )
% 12.17/7.87 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_336_not0__implies__Suc,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( N
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 => ? [M6: nat] :
% 12.17/7.87 ( N
% 12.17/7.87 = ( suc @ M6 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_337_Zero__not__Suc,axiom,
% 12.17/7.87 ! [M: nat] :
% 12.17/7.87 ( ( zero_zero @ nat )
% 12.17/7.87 != ( suc @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_338_Zero__neq__Suc,axiom,
% 12.17/7.87 ! [M: nat] :
% 12.17/7.87 ( ( zero_zero @ nat )
% 12.17/7.87 != ( suc @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_339_Suc__neq__Zero,axiom,
% 12.17/7.87 ! [M: nat] :
% 12.17/7.87 ( ( suc @ M )
% 12.17/7.87 != ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_340_zero__induct,axiom,
% 12.17/7.87 ! [P: nat > $o,K: nat] :
% 12.17/7.87 ( ( P @ K )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( P @ ( suc @ N4 ) )
% 12.17/7.87 => ( P @ N4 ) )
% 12.17/7.87 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_341_diff__induct,axiom,
% 12.17/7.87 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.17/7.87 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 12.17/7.87 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 12.17/7.87 => ( ! [X3: nat,Y5: nat] :
% 12.17/7.87 ( ( P @ X3 @ Y5 )
% 12.17/7.87 => ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
% 12.17/7.87 => ( P @ M @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_342_nat__induct,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( P @ N4 )
% 12.17/7.87 => ( P @ ( suc @ N4 ) ) )
% 12.17/7.87 => ( P @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_343_old_Onat_Oexhaust,axiom,
% 12.17/7.87 ! [Y3: nat] :
% 12.17/7.87 ( ( Y3
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 => ~ ! [Nat3: nat] :
% 12.17/7.87 ( Y3
% 12.17/7.87 != ( suc @ Nat3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_344_nat_OdiscI,axiom,
% 12.17/7.87 ! [Nat: nat,X22: nat] :
% 12.17/7.87 ( ( Nat
% 12.17/7.87 = ( suc @ X22 ) )
% 12.17/7.87 => ( Nat
% 12.17/7.87 != ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_345_old_Onat_Odistinct_I1_J,axiom,
% 12.17/7.87 ! [Nat2: nat] :
% 12.17/7.87 ( ( zero_zero @ nat )
% 12.17/7.87 != ( suc @ Nat2 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_346_old_Onat_Odistinct_I2_J,axiom,
% 12.17/7.87 ! [Nat4: nat] :
% 12.17/7.87 ( ( suc @ Nat4 )
% 12.17/7.87 != ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_347_nat_Odistinct_I1_J,axiom,
% 12.17/7.87 ! [X22: nat] :
% 12.17/7.87 ( ( zero_zero @ nat )
% 12.17/7.87 != ( suc @ X22 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_348_le__numeral__extra_I4_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_349_mult__0,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.87 = ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_350_mult__of__nat__commute,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_1 @ A )
% 12.17/7.87 => ! [X: nat,Y3: A] :
% 12.17/7.87 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y3 )
% 12.17/7.87 = ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_351_infinite__descent0__measure,axiom,
% 12.17/7.87 ! [A: $tType,V3: A > nat,P: A > $o,X: A] :
% 12.17/7.87 ( ! [X3: A] :
% 12.17/7.87 ( ( ( V3 @ X3 )
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 => ( P @ X3 ) )
% 12.17/7.87 => ( ! [X3: A] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X3 ) )
% 12.17/7.87 => ( ~ ( P @ X3 )
% 12.17/7.87 => ? [Y4: A] :
% 12.17/7.87 ( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
% 12.17/7.87 & ~ ( P @ Y4 ) ) ) )
% 12.17/7.87 => ( P @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_352_infinite__descent0,axiom,
% 12.17/7.87 ! [P: nat > $o,N: nat] :
% 12.17/7.87 ( ( P @ ( zero_zero @ nat ) )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.17/7.87 => ( ~ ( P @ N4 )
% 12.17/7.87 => ? [M3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M3 @ N4 )
% 12.17/7.87 & ~ ( P @ M3 ) ) ) )
% 12.17/7.87 => ( P @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_353_gr__implies__not0,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( N
% 12.17/7.87 != ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_354_less__zeroE,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_355_not__less0,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_356_not__gr0,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_357_gr0I,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( N
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_358_bot__nat__0_Oextremum__strict,axiom,
% 12.17/7.87 ! [A3: nat] :
% 12.17/7.87 ~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_359_reals__Archimedean2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim462609752435547400_field @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ? [N4: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_360_le__0__eq,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( N
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_361_bot__nat__0_Oextremum__uniqueI,axiom,
% 12.17/7.87 ! [A3: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 12.17/7.87 => ( A3
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_362_bot__nat__0_Oextremum__unique,axiom,
% 12.17/7.87 ! [A3: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( A3
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_363_less__eq__nat_Osimps_I1_J,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_364_real__arch__simple,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( archim462609752435547400_field @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ? [N4: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_365_nat__mult__1__right,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( times_times @ nat @ N @ ( one_one @ nat ) )
% 12.17/7.87 = N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_366_nat__mult__1,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ( ( times_times @ nat @ ( one_one @ nat ) @ N )
% 12.17/7.87 = N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_367_not__one__less__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_368_zero__less__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( zero_less_one @ A )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_369_not__one__le__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_370_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_371_zero__less__one__class_Ozero__le__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( zero_less_one @ A )
% 12.17/7.87 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_372_fact__mono__nat,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_373_fact__ge__self,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_374_zero__neq__neg__one,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ring_char_0 @ A )
% 12.17/7.87 => ( ( zero_zero @ A )
% 12.17/7.87 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_375_less__numeral__extra_I1_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_376_of__nat__neq__0,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( semiring_char_0 @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 12.17/7.87 != ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_377_of__nat__less__0__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [M: nat] :
% 12.17/7.87 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_378_of__nat__0__le__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linord181362715937106298miring @ A )
% 12.17/7.87 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_379_One__nat__def,axiom,
% 12.17/7.87 ( ( one_one @ nat )
% 12.17/7.87 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_380_mult__eq__self__implies__10,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( M
% 12.17/7.87 = ( times_times @ nat @ M @ N ) )
% 12.17/7.87 => ( ( N
% 12.17/7.87 = ( one_one @ nat ) )
% 12.17/7.87 | ( M
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_381_fact__gt__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_382_fact__not__neg,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [N: nat] :
% 12.17/7.87 ~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_383_fact__ge__zero,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_384_division__segment__mult,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( B2
% 12.17/7.87 != ( zero_zero @ A ) )
% 12.17/7.87 => ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.87 = ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_385_fact__ge__1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_semidom @ A )
% 12.17/7.87 => ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_386_int__ops_I1_J,axiom,
% 12.17/7.87 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 12.17/7.87 = ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_387_fact__less__mono__nat,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.87 => ( ( ord_less @ nat @ M @ N )
% 12.17/7.87 => ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_388_effect__returnI,axiom,
% 12.17/7.87 ! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
% 12.17/7.87 ( ( H2 = H )
% 12.17/7.87 => ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ X @ ( one_one @ nat ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_389_negative__eq__positive,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ M ) )
% 12.17/7.87 = ( ( N
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 & ( M
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_390_nat__mult__less__cancel__disj,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 & ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_391_nat__mult__le__cancel__disj,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_392_compl__less__compl__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 12.17/7.87 = ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_393_compl__le__compl__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 12.17/7.87 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_394_field__le__mult__one__interval,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_field @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ! [Z4: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
% 12.17/7.87 => ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
% 12.17/7.87 => ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X ) @ Y3 ) ) )
% 12.17/7.87 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_395_int__cases3,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( K
% 12.17/7.87 != ( zero_zero @ int ) )
% 12.17/7.87 => ( ! [N4: nat] :
% 12.17/7.87 ( ( K
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( ( K
% 12.17/7.87 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
% 12.17/7.87 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_396_not__zle__0__negative,axiom,
% 12.17/7.87 ! [N: nat] :
% 12.17/7.87 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_397_int__cases4,axiom,
% 12.17/7.87 ! [M: int] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( M
% 12.17/7.87 != ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.17/7.87 => ( M
% 12.17/7.87 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_398_negative__zle,axiom,
% 12.17/7.87 ! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_399_lift__collapse,axiom,
% 12.17/7.87 ! [A: $tType,B: $tType] :
% 12.17/7.87 ( ( heap_Time_lift @ B @ A )
% 12.17/7.87 = ( ^ [F2: B > A,X2: B] : ( heap_Time_return @ A @ ( F2 @ X2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_400_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [X: A] :
% 12.17/7.87 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 12.17/7.87 = X ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_401_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ( uminus_uminus @ A @ X )
% 12.17/7.87 = ( uminus_uminus @ A @ Y3 ) )
% 12.17/7.87 = ( X = Y3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_402_negative__zless,axiom,
% 12.17/7.87 ! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_403_verit__la__generic,axiom,
% 12.17/7.87 ! [A3: int,X: int] :
% 12.17/7.87 ( ( ord_less_eq @ int @ A3 @ X )
% 12.17/7.87 | ( A3 = X )
% 12.17/7.87 | ( ord_less_eq @ int @ X @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_404_Nat_Oex__has__greatest__nat,axiom,
% 12.17/7.87 ! [P: nat > $o,K: nat,B2: nat] :
% 12.17/7.87 ( ( P @ K )
% 12.17/7.87 => ( ! [Y5: nat] :
% 12.17/7.87 ( ( P @ Y5 )
% 12.17/7.87 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 12.17/7.87 => ? [X3: nat] :
% 12.17/7.87 ( ( P @ X3 )
% 12.17/7.87 & ! [Y4: nat] :
% 12.17/7.87 ( ( P @ Y4 )
% 12.17/7.87 => ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_405_nat__le__linear,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_406_mult__le__mono2,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_407_mult__le__mono1,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_408_mult__le__mono,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ K @ L )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_409_le__antisym,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.87 => ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_410_le__square,axiom,
% 12.17/7.87 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_411_eq__imp__le,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( M = N )
% 12.17/7.87 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_412_le__trans,axiom,
% 12.17/7.87 ! [I: nat,J: nat,K: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ J @ K )
% 12.17/7.87 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_413_le__refl,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_414_le__cube,axiom,
% 12.17/7.87 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_415_less__int__code_I1_J,axiom,
% 12.17/7.87 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_416_zmult__zless__mono2,axiom,
% 12.17/7.87 ! [I: int,J: int,K: int] :
% 12.17/7.87 ( ( ord_less @ int @ I @ J )
% 12.17/7.87 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.87 => ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_417_not__int__zless__negative,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_418_int__one__le__iff__zero__less,axiom,
% 12.17/7.87 ! [Z3: int] :
% 12.17/7.87 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 )
% 12.17/7.87 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_419_pos__zmult__eq__1__iff,axiom,
% 12.17/7.87 ! [M: int,N: int] :
% 12.17/7.87 ( ( ord_less @ int @ ( zero_zero @ int ) @ M )
% 12.17/7.87 => ( ( ( times_times @ int @ M @ N )
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 = ( ( M
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( one_one @ int ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_420_zmult__zless__mono2__lemma,axiom,
% 12.17/7.87 ! [I: int,J: int,K: nat] :
% 12.17/7.87 ( ( ord_less @ int @ I @ J )
% 12.17/7.87 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_421_negative__zless__0,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_422_negD,axiom,
% 12.17/7.87 ! [X: int] :
% 12.17/7.87 ( ( ord_less @ int @ X @ ( zero_zero @ int ) )
% 12.17/7.87 => ? [N4: nat] :
% 12.17/7.87 ( X
% 12.17/7.87 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_423_nat__int__comparison_I2_J,axiom,
% 12.17/7.87 ( ( ord_less @ nat )
% 12.17/7.87 = ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_424_pos__int__cases,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( ( K
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_425_zero__less__imp__eq__int,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.87 => ? [N4: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.17/7.87 & ( K
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_426_neg__int__cases,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( ( K
% 12.17/7.87 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
% 12.17/7.87 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_427_linordered__field__no__ub,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_field @ A )
% 12.17/7.87 => ! [X4: A] :
% 12.17/7.87 ? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_428_linordered__field__no__lb,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linordered_field @ A )
% 12.17/7.87 => ! [X4: A] :
% 12.17/7.87 ? [Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_429_nat__mult__eq__cancel__disj,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ( times_times @ nat @ K @ M )
% 12.17/7.87 = ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ( K
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 | ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_430_times__int__code_I2_J,axiom,
% 12.17/7.87 ! [L: int] :
% 12.17/7.87 ( ( times_times @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.87 = ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_431_times__int__code_I1_J,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( times_times @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.87 = ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_432_uminus__int__code_I1_J,axiom,
% 12.17/7.87 ( ( uminus_uminus @ int @ ( zero_zero @ int ) )
% 12.17/7.87 = ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_433_less__eq__int__code_I1_J,axiom,
% 12.17/7.87 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_434_zmult__eq__1__iff,axiom,
% 12.17/7.87 ! [M: int,N: int] :
% 12.17/7.87 ( ( ( times_times @ int @ M @ N )
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 = ( ( ( M
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( one_one @ int ) ) )
% 12.17/7.87 | ( ( M
% 12.17/7.87 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.17/7.87 & ( N
% 12.17/7.87 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_435_pos__zmult__eq__1__iff__lemma,axiom,
% 12.17/7.87 ! [M: int,N: int] :
% 12.17/7.87 ( ( ( times_times @ int @ M @ N )
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 => ( ( M
% 12.17/7.87 = ( one_one @ int ) )
% 12.17/7.87 | ( M
% 12.17/7.87 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_436_int__int__eq,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ( semiring_1_of_nat @ int @ M )
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N ) )
% 12.17/7.87 = ( M = N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_437_int__cases2,axiom,
% 12.17/7.87 ! [Z3: int] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( Z3
% 12.17/7.87 != ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( Z3
% 12.17/7.87 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_438_zle__int,axiom,
% 12.17/7.87 ! [M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_439_compl__le__swap2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [Y3: A,X: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 12.17/7.87 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_440_compl__le__swap1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [Y3: A,X: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 12.17/7.87 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_441_compl__mono,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_442_compl__less__swap2,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [Y3: A,X: A] :
% 12.17/7.87 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 12.17/7.87 => ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_443_compl__less__swap1,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( boolea8198339166811842893lgebra @ A )
% 12.17/7.87 => ! [Y3: A,X: A] :
% 12.17/7.87 ( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 12.17/7.87 => ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_444_nat__mult__eq__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ( ( times_times @ nat @ K @ M )
% 12.17/7.87 = ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( M = N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_445_nat__mult__le__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_446_nat__mult__less__cancel1,axiom,
% 12.17/7.87 ! [K: nat,M: nat,N: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.87 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.87 = ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_447_int__of__nat__induct,axiom,
% 12.17/7.87 ! [P: int > $o,Z3: int] :
% 12.17/7.87 ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ( ! [N4: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) )
% 12.17/7.87 => ( P @ Z3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_448_int__cases,axiom,
% 12.17/7.87 ! [Z3: int] :
% 12.17/7.87 ( ! [N4: nat] :
% 12.17/7.87 ( Z3
% 12.17/7.87 != ( semiring_1_of_nat @ int @ N4 ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( Z3
% 12.17/7.87 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_449_int__zle__neg,axiom,
% 12.17/7.87 ! [N: nat,M: nat] :
% 12.17/7.87 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
% 12.17/7.87 = ( ( N
% 12.17/7.87 = ( zero_zero @ nat ) )
% 12.17/7.87 & ( M
% 12.17/7.87 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_450_zero__le__imp__eq__int,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.87 => ? [N4: nat] :
% 12.17/7.87 ( K
% 12.17/7.87 = ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_451_nonpos__int__cases,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( K
% 12.17/7.87 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_452_nonneg__int__cases,axiom,
% 12.17/7.87 ! [K: int] :
% 12.17/7.87 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( K
% 12.17/7.87 != ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_453_negative__zle__0,axiom,
% 12.17/7.87 ! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_454_uminus__apply,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( uminus @ B )
% 12.17/7.87 => ( ( uminus_uminus @ ( A > B ) )
% 12.17/7.87 = ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_455_dual__order_Orefl,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_456_order__refl,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_457_imp__le__cong,axiom,
% 12.17/7.87 ! [X: int,X5: int,P: $o,P2: $o] :
% 12.17/7.87 ( ( X = X5 )
% 12.17/7.87 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 12.17/7.87 => ( P = P2 ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.87 => P )
% 12.17/7.87 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 12.17/7.87 => P2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_458_conj__le__cong,axiom,
% 12.17/7.87 ! [X: int,X5: int,P: $o,P2: $o] :
% 12.17/7.87 ( ( X = X5 )
% 12.17/7.87 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 12.17/7.87 => ( P = P2 ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.87 & P )
% 12.17/7.87 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 12.17/7.87 & P2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_459_Lattices__Big_Oex__has__greatest__nat,axiom,
% 12.17/7.87 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 12.17/7.87 ( ( P @ K )
% 12.17/7.87 => ( ! [Y5: A] :
% 12.17/7.87 ( ( P @ Y5 )
% 12.17/7.87 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 12.17/7.87 => ? [X3: A] :
% 12.17/7.87 ( ( P @ X3 )
% 12.17/7.87 & ! [Y4: A] :
% 12.17/7.87 ( ( P @ Y4 )
% 12.17/7.87 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_460_nat__descend__induct,axiom,
% 12.17/7.87 ! [N: nat,P: nat > $o,M: nat] :
% 12.17/7.87 ( ! [K2: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ N @ K2 )
% 12.17/7.87 => ( P @ K2 ) )
% 12.17/7.87 => ( ! [K2: nat] :
% 12.17/7.87 ( ( ord_less_eq @ nat @ K2 @ N )
% 12.17/7.87 => ( ! [I3: nat] :
% 12.17/7.87 ( ( ord_less @ nat @ K2 @ I3 )
% 12.17/7.87 => ( P @ I3 ) )
% 12.17/7.87 => ( P @ K2 ) ) )
% 12.17/7.87 => ( P @ M ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_461_dependent__nat__choice,axiom,
% 12.17/7.87 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 12.17/7.87 ( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
% 12.17/7.87 => ( ! [X3: A,N4: nat] :
% 12.17/7.87 ( ( P @ N4 @ X3 )
% 12.17/7.87 => ? [Y4: A] :
% 12.17/7.87 ( ( P @ ( suc @ N4 ) @ Y4 )
% 12.17/7.87 & ( Q @ N4 @ X3 @ Y4 ) ) )
% 12.17/7.87 => ? [F3: nat > A] :
% 12.17/7.87 ! [N5: nat] :
% 12.17/7.87 ( ( P @ N5 @ ( F3 @ N5 ) )
% 12.17/7.87 & ( Q @ N5 @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_462_list__decode_Ocases,axiom,
% 12.17/7.87 ! [X: nat] :
% 12.17/7.87 ( ( X
% 12.17/7.87 != ( zero_zero @ nat ) )
% 12.17/7.87 => ~ ! [N4: nat] :
% 12.17/7.87 ( X
% 12.17/7.87 != ( suc @ N4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_463_order__le__imp__less__or__eq,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.87 | ( X = Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_464_nle__le,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.87 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 & ( B2 != A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_465_le__cases3,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.87 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 12.17/7.87 => ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 12.17/7.87 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 12.17/7.87 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_466_order__class_Oorder__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.17/7.87 = ( ^ [X2: A,Y6: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 12.17/7.87 & ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_467_ord__eq__le__trans,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ord @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( A3 = B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_468_ord__le__eq__trans,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( ord @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( B2 = C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_469_order__antisym,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.87 => ( X = Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_470_order_Otrans,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_471_order__trans,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 12.17/7.87 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_472_linorder__wlog,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [P: A > A > $o,A3: A,B2: A] :
% 12.17/7.87 ( ! [A8: A,B5: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A8 @ B5 )
% 12.17/7.87 => ( P @ A8 @ B5 ) )
% 12.17/7.87 => ( ! [A8: A,B5: A] :
% 12.17/7.87 ( ( P @ B5 @ A8 )
% 12.17/7.87 => ( P @ A8 @ B5 ) )
% 12.17/7.87 => ( P @ A3 @ B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_473_dual__order_Oeq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.17/7.87 = ( ^ [A5: A,B3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ B3 @ A5 )
% 12.17/7.87 & ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_474_dual__order_Oantisym,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [B2: A,A3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_475_dual__order_Otrans,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 12.17/7.87 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_476_antisym,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [A3: A,B2: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.87 => ( A3 = B2 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_477_le__funD,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ord @ B )
% 12.17/7.87 => ! [F: A > B,G: A > B,X: A] :
% 12.17/7.87 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 12.17/7.87 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_478_le__funE,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ord @ B )
% 12.17/7.87 => ! [F: A > B,G: A > B,X: A] :
% 12.17/7.87 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 12.17/7.87 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_479_le__funI,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ord @ B )
% 12.17/7.87 => ! [F: A > B,G: A > B] :
% 12.17/7.87 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.17/7.87 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_480_le__fun__def,axiom,
% 12.17/7.87 ! [B: $tType,A: $tType] :
% 12.17/7.87 ( ( ord @ B )
% 12.17/7.87 => ( ( ord_less_eq @ ( A > B ) )
% 12.17/7.87 = ( ^ [F2: A > B,G2: A > B] :
% 12.17/7.87 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_481_Orderings_Oorder__eq__iff,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.17/7.87 = ( ^ [A5: A,B3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A5 @ B3 )
% 12.17/7.87 & ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_482_order__subst1,axiom,
% 12.17/7.87 ! [A: $tType,B: $tType] :
% 12.17/7.87 ( ( ( order @ B )
% 12.17/7.87 & ( order @ A ) )
% 12.17/7.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 12.17/7.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 12.17/7.87 => ( ! [X3: B,Y5: B] :
% 12.17/7.87 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_483_order__subst2,axiom,
% 12.17/7.87 ! [A: $tType,C: $tType] :
% 12.17/7.87 ( ( ( order @ C )
% 12.17/7.87 & ( order @ A ) )
% 12.17/7.87 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 12.17/7.87 => ( ! [X3: A,Y5: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.17/7.87 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.87 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_484_order__eq__refl,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( preorder @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( X = Y3 )
% 12.17/7.87 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_485_linorder__linear,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_486_ord__eq__le__subst,axiom,
% 12.17/7.87 ! [A: $tType,B: $tType] :
% 12.17/7.87 ( ( ( ord @ B )
% 12.17/7.87 & ( ord @ A ) )
% 12.17/7.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.87 ( ( A3
% 12.17/7.87 = ( F @ B2 ) )
% 12.17/7.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 12.17/7.87 => ( ! [X3: B,Y5: B] :
% 12.17/7.87 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 12.17/7.87 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.87 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_487_ord__le__eq__subst,axiom,
% 12.17/7.87 ! [A: $tType,B: $tType] :
% 12.17/7.87 ( ( ( ord @ B )
% 12.17/7.87 & ( ord @ A ) )
% 12.17/7.87 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 12.17/7.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.87 => ( ( ( F @ B2 )
% 12.17/7.87 = C2 )
% 12.17/7.87 => ( ! [X3: A,Y5: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.17/7.87 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.87 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_488_linorder__le__cases,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [X: A,Y3: A] :
% 12.17/7.87 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_489_order__antisym__conv,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( order @ A )
% 12.17/7.87 => ! [Y3: A,X: A] :
% 12.17/7.87 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.87 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.87 = ( X = Y3 ) ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_490_minf_I11_J,axiom,
% 12.17/7.87 ! [C: $tType,D: $tType] :
% 12.17/7.87 ( ( ord @ C )
% 12.17/7.87 => ! [F4: D] :
% 12.17/7.87 ? [Z4: C] :
% 12.17/7.87 ! [X4: C] :
% 12.17/7.87 ( ( ord_less @ C @ X4 @ Z4 )
% 12.17/7.87 => ( F4 = F4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_491_minf_I7_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.87 ( ( linorder @ A )
% 12.17/7.87 => ! [T2: A] :
% 12.17/7.87 ? [Z4: A] :
% 12.17/7.87 ! [X4: A] :
% 12.17/7.87 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.87 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 12.17/7.87
% 12.17/7.87 thf(fact_492_minf_I5_J,axiom,
% 12.17/7.87 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_493_minf_I4_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( X4 != T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_494_minf_I3_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( X4 != T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_495_minf_I2_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 12.17/7.88 ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Z5 )
% 12.17/7.88 => ( ( P @ X3 )
% 12.17/7.88 = ( P2 @ X3 ) ) )
% 12.17/7.88 => ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Z5 )
% 12.17/7.88 => ( ( Q @ X3 )
% 12.17/7.88 = ( Q2 @ X3 ) ) )
% 12.17/7.88 => ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( ( ( P @ X4 )
% 12.17/7.88 | ( Q @ X4 ) )
% 12.17/7.88 = ( ( P2 @ X4 )
% 12.17/7.88 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_496_minf_I1_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 12.17/7.88 ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Z5 )
% 12.17/7.88 => ( ( P @ X3 )
% 12.17/7.88 = ( P2 @ X3 ) ) )
% 12.17/7.88 => ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Z5 )
% 12.17/7.88 => ( ( Q @ X3 )
% 12.17/7.88 = ( Q2 @ X3 ) ) )
% 12.17/7.88 => ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( ( ( P @ X4 )
% 12.17/7.88 & ( Q @ X4 ) )
% 12.17/7.88 = ( ( P2 @ X4 )
% 12.17/7.88 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_497_pinf_I11_J,axiom,
% 12.17/7.88 ! [C: $tType,D: $tType] :
% 12.17/7.88 ( ( ord @ C )
% 12.17/7.88 => ! [F4: D] :
% 12.17/7.88 ? [Z4: C] :
% 12.17/7.88 ! [X4: C] :
% 12.17/7.88 ( ( ord_less @ C @ Z4 @ X4 )
% 12.17/7.88 => ( F4 = F4 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_498_pinf_I7_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_499_pinf_I5_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_500_pinf_I4_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( X4 != T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_501_pinf_I3_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( X4 != T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_502_pinf_I2_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 12.17/7.88 ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z5 @ X3 )
% 12.17/7.88 => ( ( P @ X3 )
% 12.17/7.88 = ( P2 @ X3 ) ) )
% 12.17/7.88 => ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z5 @ X3 )
% 12.17/7.88 => ( ( Q @ X3 )
% 12.17/7.88 = ( Q2 @ X3 ) ) )
% 12.17/7.88 => ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( ( ( P @ X4 )
% 12.17/7.88 | ( Q @ X4 ) )
% 12.17/7.88 = ( ( P2 @ X4 )
% 12.17/7.88 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_503_pinf_I1_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 12.17/7.88 ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z5 @ X3 )
% 12.17/7.88 => ( ( P @ X3 )
% 12.17/7.88 = ( P2 @ X3 ) ) )
% 12.17/7.88 => ( ? [Z5: A] :
% 12.17/7.88 ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z5 @ X3 )
% 12.17/7.88 => ( ( Q @ X3 )
% 12.17/7.88 = ( Q2 @ X3 ) ) )
% 12.17/7.88 => ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( ( ( P @ X4 )
% 12.17/7.88 & ( Q @ X4 ) )
% 12.17/7.88 = ( ( P2 @ X4 )
% 12.17/7.88 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_504_lt__ex,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( no_bot @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_505_gt__ex,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( no_top @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_506_dense,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dense_order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ? [Z4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Z4 )
% 12.17/7.88 & ( ord_less @ A @ Z4 @ Y3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_507_less__imp__neq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( X != Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_508_order_Oasym,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_509_ord__eq__less__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ord @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( A3 = B2 )
% 12.17/7.88 => ( ( ord_less @ A @ B2 @ C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_510_ord__less__eq__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ord @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( B2 = C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_511_less__induct,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( wellorder @ A )
% 12.17/7.88 => ! [P: A > $o,A3: A] :
% 12.17/7.88 ( ! [X3: A] :
% 12.17/7.88 ( ! [Y4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Y4 @ X3 )
% 12.17/7.88 => ( P @ Y4 ) )
% 12.17/7.88 => ( P @ X3 ) )
% 12.17/7.88 => ( P @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_512_antisym__conv3,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [Y3: A,X: A] :
% 12.17/7.88 ( ~ ( ord_less @ A @ Y3 @ X )
% 12.17/7.88 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 12.17/7.88 = ( X = Y3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_513_linorder__cases,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ~ ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( X != Y3 )
% 12.17/7.88 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_514_dual__order_Oasym,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_515_dual__order_Oirrefl,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_516_exists__least__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( wellorder @ A )
% 12.17/7.88 => ( ( ^ [P3: A > $o] :
% 12.17/7.88 ? [X6: A] : ( P3 @ X6 ) )
% 12.17/7.88 = ( ^ [P4: A > $o] :
% 12.17/7.88 ? [N3: A] :
% 12.17/7.88 ( ( P4 @ N3 )
% 12.17/7.88 & ! [M2: A] :
% 12.17/7.88 ( ( ord_less @ A @ M2 @ N3 )
% 12.17/7.88 => ~ ( P4 @ M2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_517_linorder__less__wlog,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [P: A > A > $o,A3: A,B2: A] :
% 12.17/7.88 ( ! [A8: A,B5: A] :
% 12.17/7.88 ( ( ord_less @ A @ A8 @ B5 )
% 12.17/7.88 => ( P @ A8 @ B5 ) )
% 12.17/7.88 => ( ! [A8: A] : ( P @ A8 @ A8 )
% 12.17/7.88 => ( ! [A8: A,B5: A] :
% 12.17/7.88 ( ( P @ B5 @ A8 )
% 12.17/7.88 => ( P @ A8 @ B5 ) )
% 12.17/7.88 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_518_order_Ostrict__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ A @ B2 @ C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_519_not__less__iff__gr__or__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 12.17/7.88 = ( ( ord_less @ A @ Y3 @ X )
% 12.17/7.88 | ( X = Y3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_520_dual__order_Ostrict__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( ord_less @ A @ C2 @ B2 )
% 12.17/7.88 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_521_order_Ostrict__implies__not__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( A3 != B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_522_dual__order_Ostrict__implies__not__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( A3 != B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_523_linorder__neqE,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( X != Y3 )
% 12.17/7.88 => ( ~ ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_524_order__less__asym,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_525_linorder__neq__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( X != Y3 )
% 12.17/7.88 = ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 | ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_526_order__less__asym_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_527_order__less__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ Z3 )
% 12.17/7.88 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_528_ord__eq__less__subst,axiom,
% 12.17/7.88 ! [A: $tType,B: $tType] :
% 12.17/7.88 ( ( ( ord @ B )
% 12.17/7.88 & ( ord @ A ) )
% 12.17/7.88 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 = ( F @ B2 ) )
% 12.17/7.88 => ( ( ord_less @ B @ B2 @ C2 )
% 12.17/7.88 => ( ! [X3: B,Y5: B] :
% 12.17/7.88 ( ( ord_less @ B @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_529_ord__less__eq__subst,axiom,
% 12.17/7.88 ! [A: $tType,B: $tType] :
% 12.17/7.88 ( ( ( ord @ B )
% 12.17/7.88 & ( ord @ A ) )
% 12.17/7.88 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ( F @ B2 )
% 12.17/7.88 = C2 )
% 12.17/7.88 => ( ! [X3: A,Y5: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_530_order__less__irrefl,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ~ ( ord_less @ A @ X @ X ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_531_order__less__subst1,axiom,
% 12.17/7.88 ! [A: $tType,B: $tType] :
% 12.17/7.88 ( ( ( order @ B )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 12.17/7.88 => ( ( ord_less @ B @ B2 @ C2 )
% 12.17/7.88 => ( ! [X3: B,Y5: B] :
% 12.17/7.88 ( ( ord_less @ B @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_532_order__less__subst2,axiom,
% 12.17/7.88 ! [A: $tType,C: $tType] :
% 12.17/7.88 ( ( ( order @ C )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 12.17/7.88 => ( ! [X3: A,Y5: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_533_order__less__not__sym,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_534_order__less__imp__triv,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A,P: $o] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ X )
% 12.17/7.88 => P ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_535_linorder__less__linear,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 | ( X = Y3 )
% 12.17/7.88 | ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_536_order__less__imp__not__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( X != Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_537_order__less__imp__not__eq2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( Y3 != X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_538_order__less__imp__not__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_539_fun__Compl__def,axiom,
% 12.17/7.88 ! [B: $tType,A: $tType] :
% 12.17/7.88 ( ( uminus @ B )
% 12.17/7.88 => ( ( uminus_uminus @ ( A > B ) )
% 12.17/7.88 = ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_540_ex__has__least__nat,axiom,
% 12.17/7.88 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 12.17/7.88 ( ( P @ K )
% 12.17/7.88 => ? [X3: A] :
% 12.17/7.88 ( ( P @ X3 )
% 12.17/7.88 & ! [Y4: A] :
% 12.17/7.88 ( ( P @ Y4 )
% 12.17/7.88 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_541_minf_I8_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_542_minf_I6_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ X4 @ Z4 )
% 12.17/7.88 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_543_pinf_I8_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_544_pinf_I6_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [T2: A] :
% 12.17/7.88 ? [Z4: A] :
% 12.17/7.88 ! [X4: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z4 @ X4 )
% 12.17/7.88 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_545_leD,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [Y3: A,X: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.88 => ~ ( ord_less @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_546_leI,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ~ ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_547_nless__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ~ ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 | ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_548_antisym__conv1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ~ ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 = ( X = Y3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_549_antisym__conv2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 12.17/7.88 = ( X = Y3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_550_dense__ge,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dense_linorder @ A )
% 12.17/7.88 => ! [Z3: A,Y3: A] :
% 12.17/7.88 ( ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z3 @ X3 )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ X3 ) )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_551_dense__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dense_linorder @ A )
% 12.17/7.88 => ! [Y3: A,Z3: A] :
% 12.17/7.88 ( ! [X3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_552_less__le__not__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [X2: A,Y6: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 12.17/7.88 & ~ ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_553_not__le__imp__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [Y3: A,X: A] :
% 12.17/7.88 ( ~ ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.88 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_554_order_Oorder__iff__strict,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less_eq @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( ord_less @ A @ A5 @ B3 )
% 12.17/7.88 | ( A5 = B3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_555_order_Ostrict__iff__order,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A5 @ B3 )
% 12.17/7.88 & ( A5 != B3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_556_order_Ostrict__trans1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ A @ B2 @ C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_557_order_Ostrict__trans2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_558_order_Ostrict__iff__not,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A5 @ B3 )
% 12.17/7.88 & ~ ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_559_dense__ge__bounded,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dense_linorder @ A )
% 12.17/7.88 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z3 @ X )
% 12.17/7.88 => ( ! [W: A] :
% 12.17/7.88 ( ( ord_less @ A @ Z3 @ W )
% 12.17/7.88 => ( ( ord_less @ A @ W @ X )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ W ) ) )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_560_dense__le__bounded,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dense_linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ! [W: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ W )
% 12.17/7.88 => ( ( ord_less @ A @ W @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 12.17/7.88 => ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_561_dual__order_Oorder__iff__strict,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less_eq @ A )
% 12.17/7.88 = ( ^ [B3: A,A5: A] :
% 12.17/7.88 ( ( ord_less @ A @ B3 @ A5 )
% 12.17/7.88 | ( A5 = B3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_562_dual__order_Ostrict__iff__order,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [B3: A,A5: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ B3 @ A5 )
% 12.17/7.88 & ( A5 != B3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_563_dual__order_Ostrict__trans1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( ord_less @ A @ C2 @ B2 )
% 12.17/7.88 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_564_dual__order_Ostrict__trans2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 12.17/7.88 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_565_dual__order_Ostrict__iff__not,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [B3: A,A5: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ B3 @ A5 )
% 12.17/7.88 & ~ ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_566_order_Ostrict__implies__order,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_567_dual__order_Ostrict__implies__order,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_568_order__le__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less_eq @ A )
% 12.17/7.88 = ( ^ [X2: A,Y6: A] :
% 12.17/7.88 ( ( ord_less @ A @ X2 @ Y6 )
% 12.17/7.88 | ( X2 = Y6 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_569_order__less__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [X2: A,Y6: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 12.17/7.88 & ( X2 != Y6 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_570_linorder__not__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ~ ( ord_less_eq @ A @ X @ Y3 ) )
% 12.17/7.88 = ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_571_linorder__not__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_572_order__less__imp__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_573_order__le__neq__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( A3 != B2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_574_order__neq__le__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( order @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3 != B2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_575_order__le__less__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ Z3 )
% 12.17/7.88 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_576_order__less__le__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( preorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ Y3 @ Z3 )
% 12.17/7.88 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_577_order__le__less__subst1,axiom,
% 12.17/7.88 ! [A: $tType,B: $tType] :
% 12.17/7.88 ( ( ( order @ B )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 12.17/7.88 => ( ( ord_less @ B @ B2 @ C2 )
% 12.17/7.88 => ( ! [X3: B,Y5: B] :
% 12.17/7.88 ( ( ord_less @ B @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_578_order__le__less__subst2,axiom,
% 12.17/7.88 ! [A: $tType,C: $tType] :
% 12.17/7.88 ( ( ( order @ C )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 12.17/7.88 => ( ! [X3: A,Y5: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_579_order__less__le__subst1,axiom,
% 12.17/7.88 ! [A: $tType,B: $tType] :
% 12.17/7.88 ( ( ( order @ B )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 12.17/7.88 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 12.17/7.88 => ( ! [X3: B,Y5: B] :
% 12.17/7.88 ( ( ord_less_eq @ B @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_580_order__less__le__subst2,axiom,
% 12.17/7.88 ! [A: $tType,C: $tType] :
% 12.17/7.88 ( ( ( order @ C )
% 12.17/7.88 & ( order @ A ) )
% 12.17/7.88 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 12.17/7.88 => ( ! [X3: A,Y5: A] :
% 12.17/7.88 ( ( ord_less @ A @ X3 @ Y5 )
% 12.17/7.88 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.17/7.88 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_581_linorder__le__less__linear,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linorder @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 | ( ord_less @ A @ Y3 @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_582_complete__interval,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( condit6923001295902523014norder @ A )
% 12.17/7.88 => ! [A3: A,B2: A,P: A > $o] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( P @ A3 )
% 12.17/7.88 => ( ~ ( P @ B2 )
% 12.17/7.88 => ? [C3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ C3 )
% 12.17/7.88 & ( ord_less_eq @ A @ C3 @ B2 )
% 12.17/7.88 & ! [X4: A] :
% 12.17/7.88 ( ( ( ord_less_eq @ A @ A3 @ X4 )
% 12.17/7.88 & ( ord_less @ A @ X4 @ C3 ) )
% 12.17/7.88 => ( P @ X4 ) )
% 12.17/7.88 & ! [D3: A] :
% 12.17/7.88 ( ! [X3: A] :
% 12.17/7.88 ( ( ( ord_less_eq @ A @ A3 @ X3 )
% 12.17/7.88 & ( ord_less @ A @ X3 @ D3 ) )
% 12.17/7.88 => ( P @ X3 ) )
% 12.17/7.88 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_583_fact__reduce,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_char_0 @ A )
% 12.17/7.88 => ! [N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( semiring_char_0_fact @ A @ N )
% 12.17/7.88 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_584_fact__num__eq__if,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_char_0 @ A )
% 12.17/7.88 => ( ( semiring_char_0_fact @ A )
% 12.17/7.88 = ( ^ [M2: nat] :
% 12.17/7.88 ( if @ A
% 12.17/7.88 @ ( M2
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 @ ( one_one @ A )
% 12.17/7.88 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_585_zero__le__ceiling,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 12.17/7.88 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_586_ceiling__less__zero,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 12.17/7.88 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_587_one__less__nat__eq,axiom,
% 12.17/7.88 ! [Z3: int] :
% 12.17/7.88 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z3 ) )
% 12.17/7.88 = ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_588_split__div_H,axiom,
% 12.17/7.88 ! [P: nat > $o,M: nat,N: nat] :
% 12.17/7.88 ( ( P @ ( divide_divide @ nat @ M @ N ) )
% 12.17/7.88 = ( ( ( N
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 & ( P @ ( zero_zero @ nat ) ) )
% 12.17/7.88 | ? [Q3: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
% 12.17/7.88 & ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
% 12.17/7.88 & ( P @ Q3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_589_frac__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ( archimedean_frac @ A @ X )
% 12.17/7.88 = X )
% 12.17/7.88 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_590_euclidean__size__times__nonunit,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_591_sgn__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_592_Suc__times__mod__eq,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
% 12.17/7.88 = ( one_one @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_593_minus__apply,axiom,
% 12.17/7.88 ! [B: $tType,A: $tType] :
% 12.17/7.88 ( ( minus @ B )
% 12.17/7.88 => ( ( minus_minus @ ( A > B ) )
% 12.17/7.88 = ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_594_mod__mod__trivial,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_595_nat__dvd__1__iff__1,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 12.17/7.88 = ( M
% 12.17/7.88 = ( one_one @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_596_int__dvd__int__iff,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 12.17/7.88 = ( dvd_dvd @ nat @ M @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_597_sgn__sgn,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.17/7.88 = ( sgn_sgn @ A @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_598_frac__frac,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
% 12.17/7.88 = ( archimedean_frac @ A @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_599_diff__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_600_diff__0__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 = A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_601_zero__diff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_diff @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_602_diff__zero,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 = A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_603_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_604_dvd__0__left__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_semiring_1 @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_605_dvd__0__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_semiring_1 @ A )
% 12.17/7.88 => ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_606_division__ring__divide__zero,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_607_divide__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ A3 @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 | ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_608_divide__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ C2 @ A3 )
% 12.17/7.88 = ( divide_divide @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 | ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_609_divide__eq__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( ( A3
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 | ( B2
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_610_div__by__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_611_div__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_612_times__divide__eq__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_613_divide__divide__eq__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_614_divide__divide__eq__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_615_times__divide__eq__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.88 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_616_minus__diff__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_617_div__by__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 = A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_618_mod__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ A3 @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_619_mod__by__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 = A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_620_mod__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_621_minus__dvd__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_ring_1 @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y3 )
% 12.17/7.88 = ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_622_dvd__minus__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_ring_1 @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y3 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_623_div__minus__minus,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_624_dvd__1__left,axiom,
% 12.17/7.88 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_625_dvd__1__iff__1,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.88 = ( M
% 12.17/7.88 = ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_626_div__dvd__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( divide_divide @ A @ C2 @ A3 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_627_minus__mod__self2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_628_mod__minus__minus,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_629_diff__Suc__Suc,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.17/7.88 = ( minus_minus @ nat @ M @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_630_Suc__diff__diff,axiom,
% 12.17/7.88 ! [M: nat,N: nat,K: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 12.17/7.88 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_631_diff__0__eq__0,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 = ( zero_zero @ nat ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_632_diff__self__eq__0,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ M @ M )
% 12.17/7.88 = ( zero_zero @ nat ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_633_nat__mult__dvd__cancel__disj,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.88 = ( ( K
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 | ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_634_diff__diff__cancel,axiom,
% 12.17/7.88 ! [I: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ I @ N )
% 12.17/7.88 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 12.17/7.88 = I ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_635_sgn__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_636_sgn__mult__dvd__iff,axiom,
% 12.17/7.88 ! [R: int,L: int,K: int] :
% 12.17/7.88 ( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ L ) @ K )
% 12.17/7.88 = ( ( dvd_dvd @ int @ L @ K )
% 12.17/7.88 & ( ( R
% 12.17/7.88 = ( zero_zero @ int ) )
% 12.17/7.88 => ( K
% 12.17/7.88 = ( zero_zero @ int ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_637_mult__sgn__dvd__iff,axiom,
% 12.17/7.88 ! [L: int,R: int,K: int] :
% 12.17/7.88 ( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R ) ) @ K )
% 12.17/7.88 = ( ( dvd_dvd @ int @ L @ K )
% 12.17/7.88 & ( ( R
% 12.17/7.88 = ( zero_zero @ int ) )
% 12.17/7.88 => ( K
% 12.17/7.88 = ( zero_zero @ int ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_638_dvd__sgn__mult__iff,axiom,
% 12.17/7.88 ! [L: int,R: int,K: int] :
% 12.17/7.88 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ K ) )
% 12.17/7.88 = ( ( dvd_dvd @ int @ L @ K )
% 12.17/7.88 | ( R
% 12.17/7.88 = ( zero_zero @ int ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_639_dvd__mult__sgn__iff,axiom,
% 12.17/7.88 ! [L: int,K: int,R: int] :
% 12.17/7.88 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R ) ) )
% 12.17/7.88 = ( ( dvd_dvd @ int @ L @ K )
% 12.17/7.88 | ( R
% 12.17/7.88 = ( zero_zero @ int ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_640_sgn__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ( ( sgn_sgn @ A @ ( one_one @ A ) )
% 12.17/7.88 = ( one_one @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_641_sgn__minus,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.88 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_642_sgn__divide,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field_abs_sgn @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_643_mod__less,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ M @ N )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ M @ N )
% 12.17/7.88 = M ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_644_nat__int,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 12.17/7.88 = N ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_645_diff__ge__0__iff__ge,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_646_diff__gt__0__iff__gt,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_647_diff__numeral__special_I9_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( neg_numeral @ A )
% 12.17/7.88 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_648_dvd__mult__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 | ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_649_dvd__mult__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 | ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_650_dvd__times__left__cancel__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_651_dvd__times__right__cancel__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_652_mult__divide__mult__cancel__left__if,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( zero_zero @ A ) ) )
% 12.17/7.88 & ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_653_nonzero__mult__divide__mult__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_654_nonzero__mult__divide__mult__cancel__left2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_655_nonzero__mult__divide__mult__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_656_nonzero__mult__divide__mult__cancel__right2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_657_div__mult__mult1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_658_div__mult__mult2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_659_div__mult__mult1__if,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( zero_zero @ A ) ) )
% 12.17/7.88 & ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_660_nonzero__mult__div__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
% 12.17/7.88 = B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_661_nonzero__mult__div__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.88 = A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_662_diff__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_663_verit__minus__simplify_I3_J,axiom,
% 12.17/7.88 ! [B: $tType] :
% 12.17/7.88 ( ( group_add @ B )
% 12.17/7.88 => ! [B2: B] :
% 12.17/7.88 ( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
% 12.17/7.88 = ( uminus_uminus @ B @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_664_divide__eq__1__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( one_one @ A ) )
% 12.17/7.88 = ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 & ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_665_one__eq__divide__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( one_one @ A )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 & ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_666_divide__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ A3 )
% 12.17/7.88 = ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_667_divide__self__if,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ( A3
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) )
% 12.17/7.88 & ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ A3 )
% 12.17/7.88 = ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_668_divide__eq__eq__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ B2 @ A3 )
% 12.17/7.88 = ( one_one @ A ) )
% 12.17/7.88 = ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 & ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_669_eq__divide__eq__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ( one_one @ A )
% 12.17/7.88 = ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 & ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_670_one__divide__eq__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_671_zero__eq__1__divide__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ( zero_zero @ A )
% 12.17/7.88 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_672_div__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ A3 )
% 12.17/7.88 = ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_673_unit__prod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_674_mod__mult__self1__is__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_675_mod__mult__self2__is__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_676_mod__by__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_677_divide__minus1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.88 = ( uminus_uminus @ A @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_678_div__minus1__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.88 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_679_dvd__mult__div__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_680_dvd__div__mult__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 12.17/7.88 = B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_681_unit__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_682_unit__div__1__unit,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_683_unit__div__1__div__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 12.17/7.88 = A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_684_div__diff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_modulo @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_685_dvd__imp__mod__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( modulo_modulo @ A @ B2 @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_686_mod__div__trivial,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_687_minus__mod__self1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_688_sgn__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.88 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_689_sgn__greater,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
% 12.17/7.88 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_690_zero__less__diff,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 12.17/7.88 = ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_691_diff__is__0__eq,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ( minus_minus @ nat @ M @ N )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_692_diff__is__0__eq_H,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.88 => ( ( minus_minus @ nat @ M @ N )
% 12.17/7.88 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_693_divide__sgn,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
% 12.17/7.88 = ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_694_diff__Suc__1,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
% 12.17/7.88 = N ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_695_mod__by__Suc__0,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.88 = ( zero_zero @ nat ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_696_div__by__Suc__0,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.88 = M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_697_div__less,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ M @ N )
% 12.17/7.88 => ( ( divide_divide @ nat @ M @ N )
% 12.17/7.88 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_698_ceiling__zero,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 12.17/7.88 = ( zero_zero @ int ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_699_nat__mult__div__cancel__disj,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( ( K
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.88 = ( zero_zero @ nat ) ) )
% 12.17/7.88 & ( ( K
% 12.17/7.88 != ( zero_zero @ nat ) )
% 12.17/7.88 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.88 = ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_700_ceiling__one,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 12.17/7.88 = ( one_one @ int ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_701_ceiling__of__nat,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [N: nat] :
% 12.17/7.88 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.88 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_702_divide__le__0__1__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.88 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_703_zero__le__divide__1__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_704_divide__less__0__1__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.88 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_705_divide__less__eq__1__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.88 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_706_divide__less__eq__1__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.88 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_707_less__divide__eq__1__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_708_less__divide__eq__1__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_709_zero__less__divide__1__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 12.17/7.88 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_710_nonzero__divide__mult__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_711_nonzero__divide__mult__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_712_diff__numeral__special_I12_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( neg_numeral @ A )
% 12.17/7.88 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_713_unit__mult__div__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 12.17/7.88 = ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_714_unit__div__mult__self,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 12.17/7.88 = B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_715_mod__minus1__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_716_sgn__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 = ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_717_Suc__pred,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.17/7.88 = N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_718_nat__1,axiom,
% 12.17/7.88 ( ( nat2 @ ( one_one @ int ) )
% 12.17/7.88 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_719_nat__le__0,axiom,
% 12.17/7.88 ! [Z3: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 12.17/7.88 => ( ( nat2 @ Z3 )
% 12.17/7.88 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_720_nat__0__iff,axiom,
% 12.17/7.88 ! [I: int] :
% 12.17/7.88 ( ( ( nat2 @ I )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_721_zless__nat__conj,axiom,
% 12.17/7.88 ! [W2: int,Z3: int] :
% 12.17/7.88 ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 12.17/7.88 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 & ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_722_div__mult__self1__is__m,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
% 12.17/7.88 = M ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_723_div__mult__self__is__m,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
% 12.17/7.88 = M ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_724_ceiling__diff__one,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 12.17/7.88 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_725_int__nat__eq,axiom,
% 12.17/7.88 ! [Z3: int] :
% 12.17/7.88 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 12.17/7.88 = Z3 ) )
% 12.17/7.88 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 12.17/7.88 = ( zero_zero @ int ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_726_nat__zminus__int,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.88 = ( zero_zero @ nat ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_727_divide__le__eq__1__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.88 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_728_divide__le__eq__1__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.88 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_729_le__divide__eq__1__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_730_le__divide__eq__1__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_731_Suc__diff__1,axiom,
% 12.17/7.88 ! [N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 12.17/7.88 = N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_732_ceiling__le__zero,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 12.17/7.88 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_733_zero__less__ceiling,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 12.17/7.88 = ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_734_zero__less__nat__eq,axiom,
% 12.17/7.88 ! [Z3: int] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z3 ) )
% 12.17/7.88 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_735_ceiling__less__one,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 12.17/7.88 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_736_one__le__ceiling,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 12.17/7.88 = ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_737_ceiling__le__one,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 12.17/7.88 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_738_one__less__ceiling,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] :
% 12.17/7.88 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 12.17/7.88 = ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_739_less__fun__def,axiom,
% 12.17/7.88 ! [B: $tType,A: $tType] :
% 12.17/7.88 ( ( ord @ B )
% 12.17/7.88 => ( ( ord_less @ ( A > B ) )
% 12.17/7.88 = ( ^ [F2: A > B,G2: A > B] :
% 12.17/7.88 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 12.17/7.88 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_740_fun__diff__def,axiom,
% 12.17/7.88 ! [B: $tType,A: $tType] :
% 12.17/7.88 ( ( minus @ B )
% 12.17/7.88 => ( ( minus_minus @ ( A > B ) )
% 12.17/7.88 = ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_741_mod__eq__0__iff__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_742_dvd__eq__mod__eq__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ( ( dvd_dvd @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ B3 @ A5 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_743_mod__0__imp__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_744_mod__eq__self__iff__div__eq__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.88 = A3 )
% 12.17/7.88 = ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_745_dvd__div__eq__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_746_dvd__diff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_ring_1 @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ X @ Y3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ X @ Z3 )
% 12.17/7.88 => ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_747_dvd__refl,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_748_dvd__trans,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ C2 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_749_dvd__mod__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_750_dvd__minus__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_751_dvd__div__eq__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( ( divide_divide @ A @ A3 @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( A3 = B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_752_dvd__mod__imp__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_modulo @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_753_dvd__div__eq__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semidom_divide @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ A3 @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( A3 = B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_754_div__div__div__same,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [D2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ D2 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_755_diff__eq__diff__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.88 ( ( ( minus_minus @ A @ A3 @ B2 )
% 12.17/7.88 = ( minus_minus @ A @ C2 @ D2 ) )
% 12.17/7.88 => ( ( A3 = B2 )
% 12.17/7.88 = ( C2 = D2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_756_diff__right__commute,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 12.17/7.88 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_757_dvd__diff__commute,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid5891614535332579305n_ring @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_758_mod__diff__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_759_dvd__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [K: A,M: A,N: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ K @ M )
% 12.17/7.88 => ( ( dvd_dvd @ A @ K @ N )
% 12.17/7.88 => ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_760_mod__diff__cong,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 12.17/7.88 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 12.17/7.88 => ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_761_mod__eq__dvd__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_762_mod__diff__left__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_763_mod__diff__right__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_764_mod__mod__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_765_diff__commute,axiom,
% 12.17/7.88 ! [I: nat,J: nat,K: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 12.17/7.88 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_766_dvd__diff__nat,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ K @ M )
% 12.17/7.88 => ( ( dvd_dvd @ nat @ K @ N )
% 12.17/7.88 => ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_767_field__char__0__class_Oof__nat__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field_char_0 @ A )
% 12.17/7.88 => ! [M: nat,N: nat] :
% 12.17/7.88 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_768_diff__divide__distrib,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_769_modulo__nat__def,axiom,
% 12.17/7.88 ( ( modulo_modulo @ nat )
% 12.17/7.88 = ( ^ [M2: nat,N3: nat] : ( minus_minus @ nat @ M2 @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_770_minus__mult__div__eq__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 12.17/7.88 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_771_minus__mod__eq__mult__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_772_minus__mod__eq__div__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_773_minus__div__mult__eq__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_modulo @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 12.17/7.88 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_774_div__mult__div__if__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,D2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ D2 @ C2 )
% 12.17/7.88 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_775_dvd__mult__imp__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_776_dvd__div__mult2__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_777_div__div__eq__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_778_div__mult__swap,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_779_dvd__div__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_780_unit__div__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 12.17/7.88 = ( divide_divide @ A @ C2 @ A3 ) )
% 12.17/7.88 = ( B2 = C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_781_div__unit__dvd__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_782_dvd__div__unit__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_783_of__nat__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.88 => ! [M: nat,N: nat] :
% 12.17/7.88 ( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
% 12.17/7.88 = ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_784_dvd__neg__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_divide @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.88 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_785_dvd__div__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_divide @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_786_mod__if,axiom,
% 12.17/7.88 ( ( modulo_modulo @ nat )
% 12.17/7.88 = ( ^ [M2: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N3 ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_787_le__mod__geq,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ M @ N )
% 12.17/7.88 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_788_dvd__minus__self,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
% 12.17/7.88 = ( ( ord_less @ nat @ N @ M )
% 12.17/7.88 | ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_789_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.88 => ! [M: nat,N: nat] :
% 12.17/7.88 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 12.17/7.88 = ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_790_of__nat__dvd__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.88 => ! [M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.88 = ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_791_less__eq__dvd__minus,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.88 => ( ( dvd_dvd @ nat @ M @ N )
% 12.17/7.88 = ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_792_dvd__diffD1,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 12.17/7.88 => ( ( dvd_dvd @ nat @ K @ M )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_793_dvd__diffD,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 12.17/7.88 => ( ( dvd_dvd @ nat @ K @ N )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_794_mod__nat__eqI,axiom,
% 12.17/7.88 ! [R: nat,N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ R @ N )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ R @ M )
% 12.17/7.88 => ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R ) )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ M @ N )
% 12.17/7.88 = R ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_795_unit__imp__mod__eq__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_796_of__nat__ceiling,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_797_add__divide__eq__if__simps_I4_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( Z3
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 12.17/7.88 = A3 ) )
% 12.17/7.88 & ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_798_diff__frac__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 12.17/7.88 ( ( Y3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_799_diff__divide__eq__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.88 ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_800_divide__diff__eq__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.88 ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_801_dvd__div__eq__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 12.17/7.88 = C2 )
% 12.17/7.88 = ( B2
% 12.17/7.88 = ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_802_div__dvd__iff__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_803_dvd__div__iff__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_804_dvd__div__div__eq__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ D2 )
% 12.17/7.88 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 12.17/7.88 = ( divide_divide @ A @ D2 @ C2 ) )
% 12.17/7.88 = ( ( times_times @ A @ B2 @ C2 )
% 12.17/7.88 = ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_805_unit__div__eq__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_806_unit__eq__div1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = C2 )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_807_unit__eq__div2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( A3
% 12.17/7.88 = ( divide_divide @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.88 = C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_808_div__mult__unit2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_809_unit__div__commute,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_810_unit__div__mult__swap,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_811_is__unit__div__mult2__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_812_div__less__mono,axiom,
% 12.17/7.88 ! [A6: nat,B7: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ A6 @ B7 )
% 12.17/7.88 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( ( modulo_modulo @ nat @ A6 @ N )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 => ( ( ( modulo_modulo @ nat @ B7 @ N )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 => ( ord_less @ nat @ ( divide_divide @ nat @ A6 @ N ) @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_813_mod__greater__zero__iff__not__dvd,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
% 12.17/7.88 = ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_814_of__nat__diff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( semiring_1_cancel @ A )
% 12.17/7.88 => ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 12.17/7.88 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_815_division__segment__mod,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ~ ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_816_mod__mult__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_817_mod__mult__cong,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 12.17/7.88 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 12.17/7.88 => ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_818_mod__mult__mult2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_819_mult__mod__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_820_mod__mult__left__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_821_mod__mult__right__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_822_diff__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,D2: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_823_diff__left__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_824_diff__right__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_825_diff__eq__diff__less__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.88 ( ( ( minus_minus @ A @ A3 @ B2 )
% 12.17/7.88 = ( minus_minus @ A @ C2 @ D2 ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_826_mod__minus__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_827_mod__minus__cong,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A,A4: A] :
% 12.17/7.88 ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ A4 @ B2 ) )
% 12.17/7.88 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.88 = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A4 ) @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_828_mod__minus__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_829_eq__iff__diff__eq__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( group_add @ A )
% 12.17/7.88 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ A5 @ B3 )
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_830_diff__strict__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,D2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ A @ D2 @ C2 )
% 12.17/7.88 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_831_diff__eq__diff__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.88 ( ( ( minus_minus @ A @ A3 @ B2 )
% 12.17/7.88 = ( minus_minus @ A @ C2 @ D2 ) )
% 12.17/7.88 => ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_832_diff__strict__left__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_833_diff__strict__right__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_834_right__diff__distrib_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_835_left__diff__distrib_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.88 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.88 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.88 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_836_right__diff__distrib,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_837_left__diff__distrib,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_838_inf__period_I1_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ( comm_ring @ A )
% 12.17/7.88 & ( dvd @ A ) )
% 12.17/7.88 => ! [P: A > $o,D4: A,Q: A > $o] :
% 12.17/7.88 ( ! [X3: A,K2: A] :
% 12.17/7.88 ( ( P @ X3 )
% 12.17/7.88 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 12.17/7.88 => ( ! [X3: A,K2: A] :
% 12.17/7.88 ( ( Q @ X3 )
% 12.17/7.88 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 12.17/7.88 => ! [X4: A,K4: A] :
% 12.17/7.88 ( ( ( P @ X4 )
% 12.17/7.88 & ( Q @ X4 ) )
% 12.17/7.88 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
% 12.17/7.88 & ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_839_inf__period_I2_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ( comm_ring @ A )
% 12.17/7.88 & ( dvd @ A ) )
% 12.17/7.88 => ! [P: A > $o,D4: A,Q: A > $o] :
% 12.17/7.88 ( ! [X3: A,K2: A] :
% 12.17/7.88 ( ( P @ X3 )
% 12.17/7.88 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 12.17/7.88 => ( ! [X3: A,K2: A] :
% 12.17/7.88 ( ( Q @ X3 )
% 12.17/7.88 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
% 12.17/7.88 => ! [X4: A,K4: A] :
% 12.17/7.88 ( ( ( P @ X4 )
% 12.17/7.88 | ( Q @ X4 ) )
% 12.17/7.88 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
% 12.17/7.88 | ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_840_dvd__field__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ( ( dvd_dvd @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] :
% 12.17/7.88 ( ( A5
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( B3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_841_dvd__0__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_semiring_1 @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 => ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_842_mod__Suc__eq,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
% 12.17/7.88 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_843_mod__Suc__Suc__eq,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
% 12.17/7.88 = ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_844_minus__diff__commute,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ab_group_add @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
% 12.17/7.88 = ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_845_dvd__triv__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_846_dvd__mult__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 => ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_847_mult__dvd__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ C2 @ D2 )
% 12.17/7.88 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_848_dvd__triv__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_849_dvd__mult__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_850_dvd__mult2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_851_dvd__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_852_dvd__def,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dvd @ A )
% 12.17/7.88 => ( ( dvd_dvd @ A )
% 12.17/7.88 = ( ^ [B3: A,A5: A] :
% 12.17/7.88 ? [K3: A] :
% 12.17/7.88 ( A5
% 12.17/7.88 = ( times_times @ A @ B3 @ K3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_853_dvdI,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dvd @ A )
% 12.17/7.88 => ! [A3: A,B2: A,K: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 = ( times_times @ A @ B2 @ K ) )
% 12.17/7.88 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_854_dvdE,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( dvd @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ~ ! [K2: A] :
% 12.17/7.88 ( A3
% 12.17/7.88 != ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_855_one__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( comm_monoid_mult @ A )
% 12.17/7.88 => ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_856_unit__imp__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_857_dvd__unit__imp__unit,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_858_divide__divide__eq__left_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_859_divide__divide__times__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A,W2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_860_times__divide__times__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [X: A,Y3: A,Z3: A,W2: A] :
% 12.17/7.88 ( ( times_times @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ W2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_861_mod__less__eq__dividend,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_862_minus__divide__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_863_minus__divide__divide,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_864_minus__divide__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_865_div__minus__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_866_zero__induct__lemma,axiom,
% 12.17/7.88 ! [P: nat > $o,K: nat,I: nat] :
% 12.17/7.88 ( ( P @ K )
% 12.17/7.88 => ( ! [N4: nat] :
% 12.17/7.88 ( ( P @ ( suc @ N4 ) )
% 12.17/7.88 => ( P @ N4 ) )
% 12.17/7.88 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_867_minus__nat_Odiff__0,axiom,
% 12.17/7.88 ! [M: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 12.17/7.88 = M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_868_diffs0__imp__equal,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ( minus_minus @ nat @ M @ N )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 => ( ( ( minus_minus @ nat @ N @ M )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 => ( M = N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_869_sgn__eq__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_870_sgn__0__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 = ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_871_diff__less__mono2,axiom,
% 12.17/7.88 ! [M: nat,N: nat,L: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ M @ N )
% 12.17/7.88 => ( ( ord_less @ nat @ M @ L )
% 12.17/7.88 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_872_less__imp__diff__less,axiom,
% 12.17/7.88 ! [J: nat,K: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ J @ K )
% 12.17/7.88 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_873_div__le__dividend,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_874_div__le__mono,axiom,
% 12.17/7.88 ! [M: nat,N: nat,K: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.88 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_875_diff__le__mono2,axiom,
% 12.17/7.88 ! [M: nat,N: nat,L: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.88 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_876_le__diff__iff_H,axiom,
% 12.17/7.88 ! [A3: nat,C2: nat,B2: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ A3 @ C2 )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 12.17/7.88 = ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_877_diff__le__self,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_878_diff__le__mono,axiom,
% 12.17/7.88 ! [M: nat,N: nat,L: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.88 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_879_Nat_Odiff__diff__eq,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ K @ M )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.88 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 12.17/7.88 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_880_le__diff__iff,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ K @ M )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 12.17/7.88 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_881_eq__diff__iff,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ K @ M )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.88 => ( ( ( minus_minus @ nat @ M @ K )
% 12.17/7.88 = ( minus_minus @ nat @ N @ K ) )
% 12.17/7.88 = ( M = N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_882_sgn__mult,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_883_div__mult2__eq,axiom,
% 12.17/7.88 ! [M: nat,N: nat,Q4: nat] :
% 12.17/7.88 ( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
% 12.17/7.88 = ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_884_diff__mult__distrib2,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 12.17/7.88 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_885_diff__mult__distrib,axiom,
% 12.17/7.88 ! [M: nat,N: nat,K: nat] :
% 12.17/7.88 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 12.17/7.88 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_886_uminus__dvd__conv_I2_J,axiom,
% 12.17/7.88 ( ( dvd_dvd @ int )
% 12.17/7.88 = ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_887_uminus__dvd__conv_I1_J,axiom,
% 12.17/7.88 ( ( dvd_dvd @ int )
% 12.17/7.88 = ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_888_nat__dvd__iff,axiom,
% 12.17/7.88 ! [Z3: int,M: nat] :
% 12.17/7.88 ( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M )
% 12.17/7.88 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M ) ) )
% 12.17/7.88 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( M
% 12.17/7.88 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_889_frac__le__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 12.17/7.88 ( ( Y3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 12.17/7.88 = ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_890_frac__less__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 12.17/7.88 ( ( Y3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 12.17/7.88 = ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_891_add__divide__eq__if__simps_I6_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( Z3
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 12.17/7.88 = ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.88 & ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_892_add__divide__eq__if__simps_I5_J,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.88 ( ( ( Z3
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 12.17/7.88 = ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.88 & ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_893_minus__divide__diff__eq__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.88 ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
% 12.17/7.88 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_894_is__unitE,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ~ ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ! [B5: A] :
% 12.17/7.88 ( ( B5
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 12.17/7.88 = B5 )
% 12.17/7.88 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
% 12.17/7.88 = A3 )
% 12.17/7.88 => ( ( ( times_times @ A @ A3 @ B5 )
% 12.17/7.88 = ( one_one @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ C2 @ A3 )
% 12.17/7.88 != ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_895_is__unit__div__mult__cancel__left,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_896_is__unit__div__mult__cancel__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
% 12.17/7.88 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_897_div__if,axiom,
% 12.17/7.88 ( ( divide_divide @ nat )
% 12.17/7.88 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.88 ( if @ nat
% 12.17/7.88 @ ( ( ord_less @ nat @ M2 @ N3 )
% 12.17/7.88 | ( N3
% 12.17/7.88 = ( zero_zero @ nat ) ) )
% 12.17/7.88 @ ( zero_zero @ nat )
% 12.17/7.88 @ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_898_le__div__geq,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( ( divide_divide @ nat @ M @ N )
% 12.17/7.88 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_899_nat__zero__as__int,axiom,
% 12.17/7.88 ( ( zero_zero @ nat )
% 12.17/7.88 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_900_le__iff__diff__le__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ( ( ord_less_eq @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_901_less__iff__diff__less__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( ordered_ab_group_add @ A )
% 12.17/7.88 => ( ( ord_less @ A )
% 12.17/7.88 = ( ^ [A5: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_902_int__sgnE,axiom,
% 12.17/7.88 ! [K: int] :
% 12.17/7.88 ~ ! [N4: nat,L2: int] :
% 12.17/7.88 ( K
% 12.17/7.88 != ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_903_ex__nat,axiom,
% 12.17/7.88 ( ( ^ [P3: nat > $o] :
% 12.17/7.88 ? [X6: nat] : ( P3 @ X6 ) )
% 12.17/7.88 = ( ^ [P4: nat > $o] :
% 12.17/7.88 ? [X2: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
% 12.17/7.88 & ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_904_all__nat,axiom,
% 12.17/7.88 ( ( ^ [P3: nat > $o] :
% 12.17/7.88 ! [X6: nat] : ( P3 @ X6 ) )
% 12.17/7.88 = ( ^ [P4: nat > $o] :
% 12.17/7.88 ! [X2: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
% 12.17/7.88 => ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_905_eq__nat__nat__iff,axiom,
% 12.17/7.88 ! [Z3: int,Z6: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 12.17/7.88 => ( ( ( nat2 @ Z3 )
% 12.17/7.88 = ( nat2 @ Z6 ) )
% 12.17/7.88 = ( Z3 = Z6 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_906_divide__le__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.17/7.88 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 12.17/7.88 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_907_divide__right__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_908_zero__le__divide__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 12.17/7.88 | ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_909_divide__nonneg__nonneg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_910_divide__nonneg__nonpos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_911_divide__nonpos__nonneg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_912_divide__nonpos__nonpos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_913_divide__right__mono__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_914_nat__mono,axiom,
% 12.17/7.88 ! [X: int,Y3: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ X @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_915_divide__neg__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_916_divide__neg__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_917_divide__pos__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_918_divide__pos__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_919_divide__less__0__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.17/7.88 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 12.17/7.88 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_920_divide__less__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.88 => ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.88 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less @ A @ B2 @ A3 ) )
% 12.17/7.88 & ( C2
% 12.17/7.88 != ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_921_zero__less__divide__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.88 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 12.17/7.88 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.88 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_922_divide__strict__right__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_923_divide__strict__right__mono__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.88 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_924_not__is__unit__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_925_frac__eq__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( field @ A )
% 12.17/7.88 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 12.17/7.88 ( ( Y3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( Z3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ X @ Y3 )
% 12.17/7.88 = ( divide_divide @ A @ W2 @ Z3 ) )
% 12.17/7.88 = ( ( times_times @ A @ X @ Z3 )
% 12.17/7.88 = ( times_times @ A @ W2 @ Y3 ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_926_divide__eq__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.88 ( ( ( divide_divide @ A @ B2 @ C2 )
% 12.17/7.88 = A3 )
% 12.17/7.88 = ( ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( B2
% 12.17/7.88 = ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.88 & ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_927_eq__divide__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.88 = B2 ) )
% 12.17/7.88 & ( ( C2
% 12.17/7.88 = ( zero_zero @ A ) )
% 12.17/7.88 => ( A3
% 12.17/7.88 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_928_divide__eq__imp,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( B2
% 12.17/7.88 = ( times_times @ A @ A3 @ C2 ) )
% 12.17/7.88 => ( ( divide_divide @ A @ B2 @ C2 )
% 12.17/7.88 = A3 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_929_eq__divide__imp,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.88 = B2 )
% 12.17/7.88 => ( A3
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_930_nonzero__divide__eq__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ B2 @ C2 )
% 12.17/7.88 = A3 )
% 12.17/7.88 = ( B2
% 12.17/7.88 = ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_931_nonzero__eq__divide__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.88 ( ( C2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( A3
% 12.17/7.88 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.88 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_932_right__inverse__eq,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.88 = ( one_one @ A ) )
% 12.17/7.88 = ( A3 = B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_933_mod__Suc,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
% 12.17/7.88 = N )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
% 12.17/7.88 = ( zero_zero @ nat ) ) )
% 12.17/7.88 & ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
% 12.17/7.88 != N )
% 12.17/7.88 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
% 12.17/7.88 = ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_934_is__unit__mult__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 12.17/7.88 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_935_dvd__mult__unit__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_936_mult__unit__dvd__iff,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_937_dvd__mult__unit__iff_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_938_mult__unit__dvd__iff_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.88 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_939_unit__mult__left__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( times_times @ A @ A3 @ B2 )
% 12.17/7.88 = ( times_times @ A @ A3 @ C2 ) )
% 12.17/7.88 = ( B2 = C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_940_unit__mult__right__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( algebraic_semidom @ A )
% 12.17/7.88 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( ( times_times @ A @ B2 @ A3 )
% 12.17/7.88 = ( times_times @ A @ C2 @ A3 ) )
% 12.17/7.88 = ( B2 = C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_941_nonzero__minus__divide__right,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_942_nonzero__minus__divide__divide,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( division_ring @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( B2
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.88 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_943_mod__induct,axiom,
% 12.17/7.88 ! [P: nat > $o,N: nat,P5: nat,M: nat] :
% 12.17/7.88 ( ( P @ N )
% 12.17/7.88 => ( ( ord_less @ nat @ N @ P5 )
% 12.17/7.88 => ( ( ord_less @ nat @ M @ P5 )
% 12.17/7.88 => ( ! [N4: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ N4 @ P5 )
% 12.17/7.88 => ( ( P @ N4 )
% 12.17/7.88 => ( P @ ( modulo_modulo @ nat @ ( suc @ N4 ) @ P5 ) ) ) )
% 12.17/7.88 => ( P @ M ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_944_nat__one__as__int,axiom,
% 12.17/7.88 ( ( one_one @ nat )
% 12.17/7.88 = ( nat2 @ ( one_one @ int ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_945_mod__less__divisor,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_946_mod__Suc__le__divisor,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_947_ceiling__mono,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [Y3: A,X: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.17/7.88 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y3 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_948_ceiling__less__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) )
% 12.17/7.88 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_949_div__mult2__eq_H,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.88 => ! [A3: A,M: nat,N: nat] :
% 12.17/7.88 ( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 12.17/7.88 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_950_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ( divide_divide @ nat @ M @ N )
% 12.17/7.88 = ( zero_zero @ nat ) )
% 12.17/7.88 = ( ( ord_less @ nat @ M @ N )
% 12.17/7.88 | ( N
% 12.17/7.88 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_951_Suc__diff__Suc,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ N @ M )
% 12.17/7.88 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
% 12.17/7.88 = ( minus_minus @ nat @ M @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_952_diff__less__Suc,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_953_nat__dvd__not__less,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.88 => ( ( ord_less @ nat @ M @ N )
% 12.17/7.88 => ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_954_Suc__div__le__mono,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_955_diff__less,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.88 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.88 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_956_Suc__diff__le,axiom,
% 12.17/7.88 ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 12.17/7.88 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_957_less__diff__iff,axiom,
% 12.17/7.88 ! [K: nat,M: nat,N: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ K @ M )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.88 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 12.17/7.88 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_958_diff__less__mono,axiom,
% 12.17/7.88 ! [A3: nat,B2: nat,C2: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ A3 @ B2 )
% 12.17/7.88 => ( ( ord_less_eq @ nat @ C2 @ A3 )
% 12.17/7.88 => ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_959_zdvd__antisym__nonneg,axiom,
% 12.17/7.88 ! [M: int,N: int] :
% 12.17/7.88 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
% 12.17/7.88 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.17/7.88 => ( ( dvd_dvd @ int @ M @ N )
% 12.17/7.88 => ( ( dvd_dvd @ int @ N @ M )
% 12.17/7.88 => ( M = N ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_960_sgn__not__eq__imp,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_idom @ A )
% 12.17/7.88 => ! [B2: A,A3: A] :
% 12.17/7.88 ( ( ( sgn_sgn @ A @ B2 )
% 12.17/7.88 != ( sgn_sgn @ A @ A3 ) )
% 12.17/7.88 => ( ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( sgn_sgn @ A @ B2 )
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( sgn_sgn @ A @ A3 )
% 12.17/7.88 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_961_zdvd__not__zless,axiom,
% 12.17/7.88 ! [M: int,N: int] :
% 12.17/7.88 ( ( ord_less @ int @ ( zero_zero @ int ) @ M )
% 12.17/7.88 => ( ( ord_less @ int @ M @ N )
% 12.17/7.88 => ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_962_less__mult__imp__div__less,axiom,
% 12.17/7.88 ! [M: nat,I: nat,N: nat] :
% 12.17/7.88 ( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
% 12.17/7.88 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_963_times__div__less__eq__dividend,axiom,
% 12.17/7.88 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_964_div__times__less__eq__dividend,axiom,
% 12.17/7.88 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_965_zdvd__mult__cancel,axiom,
% 12.17/7.88 ! [K: int,M: int,N: int] :
% 12.17/7.88 ( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) )
% 12.17/7.88 => ( ( K
% 12.17/7.88 != ( zero_zero @ int ) )
% 12.17/7.88 => ( dvd_dvd @ int @ M @ N ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_966_zdvd__mono,axiom,
% 12.17/7.88 ! [K: int,M: int,T2: int] :
% 12.17/7.88 ( ( K
% 12.17/7.88 != ( zero_zero @ int ) )
% 12.17/7.88 => ( ( dvd_dvd @ int @ M @ T2 )
% 12.17/7.88 = ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_967_diff__Suc__eq__diff__pred,axiom,
% 12.17/7.88 ! [M: nat,N: nat] :
% 12.17/7.88 ( ( minus_minus @ nat @ M @ ( suc @ N ) )
% 12.17/7.88 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_968_sgn__minus__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( idom_abs_sgn @ A )
% 12.17/7.88 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.88 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_969_dvd__euclidean__size__eq__imp__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.88 => ! [A3: A,B2: A] :
% 12.17/7.88 ( ( A3
% 12.17/7.88 != ( zero_zero @ A ) )
% 12.17/7.88 => ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 12.17/7.88 = ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.17/7.88 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.88 => ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_970_euclidean__size__unit,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.88 => ! [A3: A] :
% 12.17/7.88 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.88 => ( ( euclid6346220572633701492n_size @ A @ A3 )
% 12.17/7.88 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_971_fact__dvd,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_semidom @ A )
% 12.17/7.88 => ! [N: nat,M: nat] :
% 12.17/7.88 ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.88 => ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_972_is__unit__division__segment,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.88 => ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_973_division__segment__eq__sgn,axiom,
% 12.17/7.88 ! [K: int] :
% 12.17/7.88 ( ( K
% 12.17/7.88 != ( zero_zero @ int ) )
% 12.17/7.88 => ( ( euclid7384307370059645450egment @ int @ K )
% 12.17/7.88 = ( sgn_sgn @ int @ K ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_974_ceiling__ge__round,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_975_frac__ge__0,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_976_frac__lt__1,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.88 => ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_977_nat__mono__iff,axiom,
% 12.17/7.88 ! [Z3: int,W2: int] :
% 12.17/7.88 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.88 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 12.17/7.88 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_978_frac__le,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [Y3: A,X: A,W2: A,Z3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ W2 @ Z3 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_979_frac__less,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A,W2: A,Z3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 12.17/7.88 => ( ( ord_less_eq @ A @ W2 @ Z3 )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_980_frac__less2,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A,W2: A,Z3: A] :
% 12.17/7.88 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
% 12.17/7.88 => ( ( ord_less @ A @ W2 @ Z3 )
% 12.17/7.88 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_981_divide__le__cancel,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.88 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.88 => ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.88 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_982_divide__nonneg__neg,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.88 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_983_divide__nonneg__pos,axiom,
% 12.17/7.88 ! [A: $tType] :
% 12.17/7.88 ( ( linordered_field @ A )
% 12.17/7.88 => ! [X: A,Y3: A] :
% 12.17/7.88 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.88 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.88 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.88
% 12.17/7.88 thf(fact_984_divide__nonpos__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_985_divide__nonpos__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_986_divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_987_less__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_988_neg__divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_989_neg__less__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_990_pos__divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_991_pos__less__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_992_mult__imp__div__pos__less,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [Y3: A,X: A,Z3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
% 12.17/7.89 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_993_mult__imp__less__div__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [Y3: A,Z3: A,X: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
% 12.17/7.89 => ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_994_divide__strict__left__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ B2 @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.89 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_995_divide__strict__left__mono__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.89 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_996_unit__dvdE,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( algebraic_semidom @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.89 => ~ ( ( A3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ! [C3: A] :
% 12.17/7.89 ( B2
% 12.17/7.89 != ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_997_divide__less__eq__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 & ( ord_less @ A @ B2 @ A3 ) )
% 12.17/7.89 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 & ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.89 | ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_998_less__divide__eq__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 & ( ord_less @ A @ A3 @ B2 ) )
% 12.17/7.89 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_999_zless__nat__eq__int__zless,axiom,
% 12.17/7.89 ! [M: nat,Z3: int] :
% 12.17/7.89 ( ( ord_less @ nat @ M @ ( nat2 @ Z3 ) )
% 12.17/7.89 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1000_int__eq__iff,axiom,
% 12.17/7.89 ! [M: nat,Z3: int] :
% 12.17/7.89 ( ( ( semiring_1_of_nat @ int @ M )
% 12.17/7.89 = Z3 )
% 12.17/7.89 = ( ( M
% 12.17/7.89 = ( nat2 @ Z3 ) )
% 12.17/7.89 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1001_nat__0__le,axiom,
% 12.17/7.89 ! [Z3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.89 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 12.17/7.89 = Z3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1002_nat__le__iff,axiom,
% 12.17/7.89 ! [X: int,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
% 12.17/7.89 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1003_eq__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 = ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ( ( C2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( times_times @ A @ A3 @ C2 )
% 12.17/7.89 = ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.89 & ( ( C2
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1004_minus__divide__eq__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = A3 )
% 12.17/7.89 = ( ( ( C2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( uminus_uminus @ A @ B2 )
% 12.17/7.89 = ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ( C2
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1005_nonzero__neg__divide__eq__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.17/7.89 = C2 )
% 12.17/7.89 = ( ( uminus_uminus @ A @ A3 )
% 12.17/7.89 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1006_nonzero__neg__divide__eq__eq2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( C2
% 12.17/7.89 = ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 12.17/7.89 = ( ( times_times @ A @ C2 @ B2 )
% 12.17/7.89 = ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1007_divide__eq__minus__1__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.89 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.89 = ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 & ( A3
% 12.17/7.89 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1008_mod__le__divisor,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1009_fact__mod,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ( linordered_semidom @ A )
% 12.17/7.89 & ( semidom_modulo @ A ) )
% 12.17/7.89 => ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.89 => ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1010_mod__size__less,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1011_diff__Suc__less,axiom,
% 12.17/7.89 ! [N: nat,I: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1012_div__greater__zero__iff,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 12.17/7.89 = ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.89 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1013_div__le__mono2,axiom,
% 12.17/7.89 ! [M: nat,N: nat,K: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1014_sgn__1__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ( sgn_sgn @ A @ A3 )
% 12.17/7.89 = ( one_one @ A ) )
% 12.17/7.89 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1015_dvd__imp__le,axiom,
% 12.17/7.89 ! [K: nat,N: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ K @ N )
% 12.17/7.89 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1016_nat__mult__div__cancel1,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.89 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.89 = ( divide_divide @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1017_div__less__iff__less__mult,axiom,
% 12.17/7.89 ! [Q4: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
% 12.17/7.89 => ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N )
% 12.17/7.89 = ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q4 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1018_nat__mult__dvd__cancel1,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.89 = ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1019_dvd__mult__cancel,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 12.17/7.89 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.89 => ( dvd_dvd @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1020_unit__iff__euclidean__size,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.89 = ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 12.17/7.89 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
% 12.17/7.89 & ( A3
% 12.17/7.89 != ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1021_div__less__dividend,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 12.17/7.89 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1022_div__eq__dividend__iff,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 => ( ( ( divide_divide @ nat @ M @ N )
% 12.17/7.89 = M )
% 12.17/7.89 = ( N
% 12.17/7.89 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1023_euclidean__size__times__unit,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.89 => ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1024_zdvd__imp__le,axiom,
% 12.17/7.89 ! [Z3: int,N: int] :
% 12.17/7.89 ( ( dvd_dvd @ int @ Z3 @ N )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.17/7.89 => ( ord_less_eq @ int @ Z3 @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1025_dvd__proper__imp__size__less,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.89 => ( ~ ( dvd_dvd @ A @ B2 @ A3 )
% 12.17/7.89 => ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1026_dvd__imp__size__le,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3725896446679973847miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1027_ex__gt__or__lt,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( condit5016429287641298734tinuum @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ? [B5: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B5 )
% 12.17/7.89 | ( ord_less @ A @ B5 @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1028_of__nat__euclidean__size,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) )
% 12.17/7.89 = ( divide_divide @ A @ A3 @ ( euclid7384307370059645450egment @ A @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1029_dvd__fact,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.89 => ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1030_zsgn__def,axiom,
% 12.17/7.89 ( ( sgn_sgn @ int )
% 12.17/7.89 = ( ^ [I5: int] :
% 12.17/7.89 ( if @ int
% 12.17/7.89 @ ( I5
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 @ ( zero_zero @ int )
% 12.17/7.89 @ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I5 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1031_nat__less__eq__zless,axiom,
% 12.17/7.89 ! [W2: int,Z3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 12.17/7.89 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1032_nat__le__eq__zle,axiom,
% 12.17/7.89 ! [W2: int,Z3: int] :
% 12.17/7.89 ( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
% 12.17/7.89 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1033_nat__eq__iff,axiom,
% 12.17/7.89 ! [W2: int,M: nat] :
% 12.17/7.89 ( ( ( nat2 @ W2 )
% 12.17/7.89 = M )
% 12.17/7.89 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( W2
% 12.17/7.89 = ( semiring_1_of_nat @ int @ M ) ) )
% 12.17/7.89 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( M
% 12.17/7.89 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1034_nat__eq__iff2,axiom,
% 12.17/7.89 ! [M: nat,W2: int] :
% 12.17/7.89 ( ( M
% 12.17/7.89 = ( nat2 @ W2 ) )
% 12.17/7.89 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( W2
% 12.17/7.89 = ( semiring_1_of_nat @ int @ M ) ) )
% 12.17/7.89 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( M
% 12.17/7.89 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1035_divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1036_le__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1037_divide__left__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1038_neg__divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1039_neg__le__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1040_pos__divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1041_pos__le__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1042_mult__imp__div__pos__le,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [Y3: A,X: A,Z3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1043_mult__imp__le__div__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [Y3: A,Z3: A,X: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
% 12.17/7.89 => ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1044_divide__left__mono__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1045_split__nat,axiom,
% 12.17/7.89 ! [P: nat > $o,I: int] :
% 12.17/7.89 ( ( P @ ( nat2 @ I ) )
% 12.17/7.89 = ( ! [N3: nat] :
% 12.17/7.89 ( ( I
% 12.17/7.89 = ( semiring_1_of_nat @ int @ N3 ) )
% 12.17/7.89 => ( P @ N3 ) )
% 12.17/7.89 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
% 12.17/7.89 => ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1046_divide__le__eq__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 & ( ord_less_eq @ A @ B2 @ A3 ) )
% 12.17/7.89 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.89 | ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1047_le__divide__eq__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 12.17/7.89 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1048_le__nat__iff,axiom,
% 12.17/7.89 ! [K: int,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
% 12.17/7.89 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1049_nat__mult__distrib,axiom,
% 12.17/7.89 ! [Z3: int,Z6: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.89 => ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
% 12.17/7.89 = ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1050_less__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1051_minus__divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1052_neg__less__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1053_neg__minus__divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1054_pos__less__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1055_pos__minus__divide__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1056_div__nat__eqI,axiom,
% 12.17/7.89 ! [N: nat,Q4: nat,M: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M )
% 12.17/7.89 => ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
% 12.17/7.89 => ( ( divide_divide @ nat @ M @ N )
% 12.17/7.89 = Q4 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1057_less__eq__div__iff__mult__less__eq,axiom,
% 12.17/7.89 ! [Q4: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q4 ) )
% 12.17/7.89 = ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1058_sgn__if,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ( ( sgn_sgn @ A )
% 12.17/7.89 = ( ^ [X2: A] :
% 12.17/7.89 ( if @ A
% 12.17/7.89 @ ( X2
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 @ ( zero_zero @ A )
% 12.17/7.89 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1059_sgn__1__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ( sgn_sgn @ A @ A3 )
% 12.17/7.89 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.89 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1060_Suc__pred_H,axiom,
% 12.17/7.89 ! [N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( N
% 12.17/7.89 = ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1061_Suc__diff__eq__diff__pred,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 12.17/7.89 = ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1062_dvd__mult__cancel1,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
% 12.17/7.89 = ( N
% 12.17/7.89 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1063_dvd__mult__cancel2,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
% 12.17/7.89 = ( N
% 12.17/7.89 = ( one_one @ nat ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1064_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( euclid7384307370059645450egment @ A @ A3 )
% 12.17/7.89 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.17/7.89 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.17/7.89 | ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1065_nat__less__iff,axiom,
% 12.17/7.89 ! [W2: int,M: nat] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
% 12.17/7.89 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
% 12.17/7.89 = ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1066_le__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1067_minus__divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1068_neg__le__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1069_neg__minus__divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1070_pos__le__minus__divide__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1071_pos__minus__divide__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
% 12.17/7.89 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1072_nat__mult__distrib__neg,axiom,
% 12.17/7.89 ! [Z3: int,Z6: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
% 12.17/7.89 = ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1073_nat__approx__posE,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.89 => ! [E: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 12.17/7.89 => ~ ! [N4: nat] :
% 12.17/7.89 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ E ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1074_mult__ceiling__le,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1075_bits__mod__div__trivial,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1076_bits__mod__by__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1077_choose__dvd,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [K: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.89 => ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1078_div__geq,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ~ ( ord_less @ nat @ M @ N )
% 12.17/7.89 => ( ( divide_divide @ nat @ M @ N )
% 12.17/7.89 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1079_bits__mod__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1080_bits__div__by__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1081_bits__div__by__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1082_bits__div__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( bit_semiring_bits @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1083_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.89 = A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1084_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1085_mod__neg__neg__trivial,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ L @ K )
% 12.17/7.89 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.89 = K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1086_mod__pos__pos__trivial,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 => ( ( ord_less @ int @ K @ L )
% 12.17/7.89 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.89 = K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1087_div__neg__neg__trivial,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ L @ K )
% 12.17/7.89 => ( ( divide_divide @ int @ K @ L )
% 12.17/7.89 = ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1088_div__pos__pos__trivial,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 => ( ( ord_less @ int @ K @ L )
% 12.17/7.89 => ( ( divide_divide @ int @ K @ L )
% 12.17/7.89 = ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1089_zle__diff1__eq,axiom,
% 12.17/7.89 ! [W2: int,Z3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
% 12.17/7.89 = ( ord_less @ int @ W2 @ Z3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1090_dvd__antisym,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ M @ N )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ N @ M )
% 12.17/7.89 => ( M = N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1091_zdvd__zdiffD,axiom,
% 12.17/7.89 ! [K: int,M: int,N: int] :
% 12.17/7.89 ( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
% 12.17/7.89 => ( ( dvd_dvd @ int @ K @ N )
% 12.17/7.89 => ( dvd_dvd @ int @ K @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1092_zmod__zminus1__eq__if,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 12.17/7.89 = ( zero_zero @ int ) ) )
% 12.17/7.89 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 12.17/7.89 = ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1093_zmod__zminus2__eq__if,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 12.17/7.89 = ( zero_zero @ int ) ) )
% 12.17/7.89 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 12.17/7.89 = ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1094_zdiv__zminus1__eq__if,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 12.17/7.89 = ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
% 12.17/7.89 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
% 12.17/7.89 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1095_zdiv__zminus2__eq__if,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 12.17/7.89 = ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
% 12.17/7.89 & ( ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
% 12.17/7.89 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1096_zdiv__mono__strict,axiom,
% 12.17/7.89 ! [A6: int,B7: int,N: int] :
% 12.17/7.89 ( ( ord_less @ int @ A6 @ B7 )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.17/7.89 => ( ( ( modulo_modulo @ int @ A6 @ N )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ( modulo_modulo @ int @ B7 @ N )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ( ord_less @ int @ ( divide_divide @ int @ A6 @ N ) @ ( divide_divide @ int @ B7 @ N ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1097_mod__pos__geq,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 => ( ( ord_less_eq @ int @ L @ K )
% 12.17/7.89 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.89 = ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1098_sgn__mod,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( L
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ~ ( dvd_dvd @ int @ L @ K )
% 12.17/7.89 => ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
% 12.17/7.89 = ( sgn_sgn @ int @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1099_minus__int__code_I1_J,axiom,
% 12.17/7.89 ! [K: int] :
% 12.17/7.89 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.89 = K ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1100_int__distrib_I3_J,axiom,
% 12.17/7.89 ! [Z1: int,Z22: int,W2: int] :
% 12.17/7.89 ( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
% 12.17/7.89 = ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1101_int__distrib_I4_J,axiom,
% 12.17/7.89 ! [W2: int,Z1: int,Z22: int] :
% 12.17/7.89 ( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
% 12.17/7.89 = ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1102_int__diff__cases,axiom,
% 12.17/7.89 ! [Z3: int] :
% 12.17/7.89 ~ ! [M6: nat,N4: nat] :
% 12.17/7.89 ( Z3
% 12.17/7.89 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M6 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1103_zmod__le__nonneg__dividend,axiom,
% 12.17/7.89 ! [M: int,K: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
% 12.17/7.89 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1104_zmod__eq__0__iff,axiom,
% 12.17/7.89 ! [M: int,D2: int] :
% 12.17/7.89 ( ( ( modulo_modulo @ int @ M @ D2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 = ( ? [Q3: int] :
% 12.17/7.89 ( M
% 12.17/7.89 = ( times_times @ int @ D2 @ Q3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1105_zmod__eq__0D,axiom,
% 12.17/7.89 ! [M: int,D2: int] :
% 12.17/7.89 ( ( ( modulo_modulo @ int @ M @ D2 )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 => ? [Q5: int] :
% 12.17/7.89 ( M
% 12.17/7.89 = ( times_times @ int @ D2 @ Q5 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1106_zmod__zminus2__not__zero,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.89 != ( zero_zero @ int ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1107_zmod__zminus1__not__zero,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
% 12.17/7.89 != ( zero_zero @ int ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.89 != ( zero_zero @ int ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1108_div__neg__pos__less0,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1109_neg__imp__zdiv__neg__iff,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 12.17/7.89 = ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1110_pos__imp__zdiv__neg__iff,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 12.17/7.89 = ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1111_zmod__int,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B2 ) )
% 12.17/7.89 = ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1112_zdiv__int,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B2 ) )
% 12.17/7.89 = ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1113_minus__mod__int__eq,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
% 12.17/7.89 = ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1114_zmod__minus1,axiom,
% 12.17/7.89 ! [B2: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
% 12.17/7.89 = ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1115_neg__mod__conj,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
% 12.17/7.89 & ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1116_pos__mod__conj,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
% 12.17/7.89 & ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1117_zmod__trivial__iff,axiom,
% 12.17/7.89 ! [I: int,K: int] :
% 12.17/7.89 ( ( ( modulo_modulo @ int @ I @ K )
% 12.17/7.89 = I )
% 12.17/7.89 = ( ( K
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 12.17/7.89 & ( ord_less @ int @ I @ K ) )
% 12.17/7.89 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 12.17/7.89 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1118_mod__int__pos__iff,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
% 12.17/7.89 = ( ( dvd_dvd @ int @ L @ K )
% 12.17/7.89 | ( ( L
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
% 12.17/7.89 | ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1119_neg__mod__bound,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1120_Euclidean__Division_Opos__mod__bound,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 => ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1121_zdiv__mono1,axiom,
% 12.17/7.89 ! [A3: int,A4: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ A3 @ A4 )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A4 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1122_zdiv__mono2,axiom,
% 12.17/7.89 ! [A3: int,B4: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B4 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1123_zdiv__eq__0__iff,axiom,
% 12.17/7.89 ! [I: int,K: int] :
% 12.17/7.89 ( ( ( divide_divide @ int @ I @ K )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 = ( ( K
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 12.17/7.89 & ( ord_less @ int @ I @ K ) )
% 12.17/7.89 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 12.17/7.89 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1124_zdiv__mono1__neg,axiom,
% 12.17/7.89 ! [A3: int,A4: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ A3 @ A4 )
% 12.17/7.89 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A4 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1125_zdiv__mono2__neg,axiom,
% 12.17/7.89 ! [A3: int,B4: int,B2: int] :
% 12.17/7.89 ( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B4 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1126_div__int__pos__iff,axiom,
% 12.17/7.89 ! [K: int,L: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
% 12.17/7.89 = ( ( K
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( L
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
% 12.17/7.89 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.89 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1127_div__positive__int,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ L @ K )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1128_div__nonneg__neg__le0,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1129_div__nonpos__pos__le0,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1130_pos__imp__zdiv__pos__iff,axiom,
% 12.17/7.89 ! [K: int,I: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
% 12.17/7.89 = ( ord_less_eq @ int @ K @ I ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1131_neg__imp__zdiv__nonneg__iff,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 12.17/7.89 = ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1132_pos__imp__zdiv__nonneg__iff,axiom,
% 12.17/7.89 ! [B2: int,A3: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 12.17/7.89 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1133_nonneg1__imp__zdiv__pos__iff,axiom,
% 12.17/7.89 ! [A3: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
% 12.17/7.89 = ( ( ord_less_eq @ int @ B2 @ A3 )
% 12.17/7.89 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1134_int__div__less__self,axiom,
% 12.17/7.89 ! [X: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.89 => ( ( ord_less @ int @ ( one_one @ int ) @ K )
% 12.17/7.89 => ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1135_minus__int__code_I2_J,axiom,
% 12.17/7.89 ! [L: int] :
% 12.17/7.89 ( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 = ( uminus_uminus @ int @ L ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1136_int__le__induct,axiom,
% 12.17/7.89 ! [I: int,K: int,P: int > $o] :
% 12.17/7.89 ( ( ord_less_eq @ int @ I @ K )
% 12.17/7.89 => ( ( P @ K )
% 12.17/7.89 => ( ! [I4: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ I4 @ K )
% 12.17/7.89 => ( ( P @ I4 )
% 12.17/7.89 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.89 => ( P @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1137_int__less__induct,axiom,
% 12.17/7.89 ! [I: int,K: int,P: int > $o] :
% 12.17/7.89 ( ( ord_less @ int @ I @ K )
% 12.17/7.89 => ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
% 12.17/7.89 => ( ! [I4: int] :
% 12.17/7.89 ( ( ord_less @ int @ I4 @ K )
% 12.17/7.89 => ( ( P @ I4 )
% 12.17/7.89 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.89 => ( P @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1138_div__eq__minus1,axiom,
% 12.17/7.89 ! [B2: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.17/7.89 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
% 12.17/7.89 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1139_neg__mod__sign,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 12.17/7.89 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1140_Euclidean__Division_Opos__mod__sign,axiom,
% 12.17/7.89 ! [L: int,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1141_plusinfinity,axiom,
% 12.17/7.89 ! [D2: int,P2: int > $o,P: int > $o] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.17/7.89 => ( ! [X3: int,K2: int] :
% 12.17/7.89 ( ( P2 @ X3 )
% 12.17/7.89 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 12.17/7.89 => ( ? [Z5: int] :
% 12.17/7.89 ! [X3: int] :
% 12.17/7.89 ( ( ord_less @ int @ Z5 @ X3 )
% 12.17/7.89 => ( ( P @ X3 )
% 12.17/7.89 = ( P2 @ X3 ) ) )
% 12.17/7.89 => ( ? [X_1: int] : ( P2 @ X_1 )
% 12.17/7.89 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1142_minusinfinity,axiom,
% 12.17/7.89 ! [D2: int,P1: int > $o,P: int > $o] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.17/7.89 => ( ! [X3: int,K2: int] :
% 12.17/7.89 ( ( P1 @ X3 )
% 12.17/7.89 = ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 12.17/7.89 => ( ? [Z5: int] :
% 12.17/7.89 ! [X3: int] :
% 12.17/7.89 ( ( ord_less @ int @ X3 @ Z5 )
% 12.17/7.89 => ( ( P @ X3 )
% 12.17/7.89 = ( P1 @ X3 ) ) )
% 12.17/7.89 => ( ? [X_1: int] : ( P1 @ X_1 )
% 12.17/7.89 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1143_zdiv__zmult2__eq,axiom,
% 12.17/7.89 ! [C2: int,A3: int,B2: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 12.17/7.89 => ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 12.17/7.89 = ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1144_int__minus,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
% 12.17/7.89 = ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1145_nat__mod__distrib,axiom,
% 12.17/7.89 ! [X: int,Y3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.17/7.89 => ( ( nat2 @ ( modulo_modulo @ int @ X @ Y3 ) )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1146_decr__mult__lemma,axiom,
% 12.17/7.89 ! [D2: int,P: int > $o,K: int] :
% 12.17/7.89 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.17/7.89 => ( ! [X3: int] :
% 12.17/7.89 ( ( P @ X3 )
% 12.17/7.89 => ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.89 => ! [X4: int] :
% 12.17/7.89 ( ( P @ X4 )
% 12.17/7.89 => ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1147_int__ops_I6_J,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
% 12.17/7.89 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
% 12.17/7.89 = ( zero_zero @ int ) ) )
% 12.17/7.89 & ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
% 12.17/7.89 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
% 12.17/7.89 = ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1148_nat__diff__distrib,axiom,
% 12.17/7.89 ! [Z6: int,Z3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ Z6 @ Z3 )
% 12.17/7.89 => ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) )
% 12.17/7.89 = ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1149_nat__diff__distrib_H,axiom,
% 12.17/7.89 ! [X: int,Y3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.17/7.89 => ( ( nat2 @ ( minus_minus @ int @ X @ Y3 ) )
% 12.17/7.89 = ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1150_verit__less__mono__div__int2,axiom,
% 12.17/7.89 ! [A6: int,B7: int,N: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ A6 @ B7 )
% 12.17/7.89 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
% 12.17/7.89 => ( ord_less_eq @ int @ ( divide_divide @ int @ B7 @ N ) @ ( divide_divide @ int @ A6 @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1151_nat__div__distrib_H,axiom,
% 12.17/7.89 ! [Y3: int,X: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.17/7.89 => ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
% 12.17/7.89 = ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1152_nat__div__distrib,axiom,
% 12.17/7.89 ! [X: int,Y3: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.17/7.89 => ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
% 12.17/7.89 = ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1153_zdiff__int__split,axiom,
% 12.17/7.89 ! [P: int > $o,X: nat,Y3: nat] :
% 12.17/7.89 ( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y3 ) ) )
% 12.17/7.89 = ( ( ( ord_less_eq @ nat @ Y3 @ X )
% 12.17/7.89 => ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) )
% 12.17/7.89 & ( ( ord_less @ nat @ X @ Y3 )
% 12.17/7.89 => ( P @ ( zero_zero @ int ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1154_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1155_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1156_mod__eq__0D,axiom,
% 12.17/7.89 ! [M: nat,D2: nat] :
% 12.17/7.89 ( ( ( modulo_modulo @ nat @ M @ D2 )
% 12.17/7.89 = ( zero_zero @ nat ) )
% 12.17/7.89 => ? [Q5: nat] :
% 12.17/7.89 ( M
% 12.17/7.89 = ( times_times @ nat @ D2 @ Q5 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1157_mod__geq,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ~ ( ord_less @ nat @ M @ N )
% 12.17/7.89 => ( ( modulo_modulo @ nat @ M @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1158_mod__eq__dvd__iff__nat,axiom,
% 12.17/7.89 ! [N: nat,M: nat,Q4: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.89 => ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 12.17/7.89 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 12.17/7.89 = ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1159_div__positive,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1160_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1161_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1162_bezout1__nat,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ? [D6: nat,X3: nat,Y5: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 12.17/7.89 & ( dvd_dvd @ nat @ D6 @ B2 )
% 12.17/7.89 & ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
% 12.17/7.89 = D6 )
% 12.17/7.89 | ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
% 12.17/7.89 = D6 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1163_gcd__nat__induct,axiom,
% 12.17/7.89 ! [P: nat > nat > $o,M: nat,N: nat] :
% 12.17/7.89 ( ! [M6: nat] : ( P @ M6 @ ( zero_zero @ nat ) )
% 12.17/7.89 => ( ! [M6: nat,N4: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.17/7.89 => ( ( P @ N4 @ ( modulo_modulo @ nat @ M6 @ N4 ) )
% 12.17/7.89 => ( P @ M6 @ N4 ) ) )
% 12.17/7.89 => ( P @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1164_dvd__pos__nat,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ M @ N )
% 12.17/7.89 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1165_gbinomial__absorption_H,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [K: nat,A3: A] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.89 => ( ( gbinomial @ A @ A3 @ K )
% 12.17/7.89 = ( 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 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1166_modulo__int__unfold,axiom,
% 12.17/7.89 ! [L: int,K: int,N: nat,M: nat] :
% 12.17/7.89 ( ( ( ( ( sgn_sgn @ int @ L )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( ( sgn_sgn @ int @ K )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.89 = ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
% 12.17/7.89 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( ( sgn_sgn @ int @ K )
% 12.17/7.89 = ( zero_zero @ int ) )
% 12.17/7.89 | ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) )
% 12.17/7.89 => ( ( ( ( sgn_sgn @ int @ K )
% 12.17/7.89 = ( sgn_sgn @ int @ L ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.89 = ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
% 12.17/7.89 & ( ( ( sgn_sgn @ int @ K )
% 12.17/7.89 != ( sgn_sgn @ int @ L ) )
% 12.17/7.89 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.89 = ( times_times @ int @ ( sgn_sgn @ int @ L )
% 12.17/7.89 @ ( minus_minus @ int
% 12.17/7.89 @ ( semiring_1_of_nat @ int
% 12.17/7.89 @ ( times_times @ nat @ N
% 12.17/7.89 @ ( zero_neq_one_of_bool @ nat
% 12.17/7.89 @ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
% 12.17/7.89 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1167_divmod__cases,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( A3
% 12.17/7.89 != ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
% 12.17/7.89 => ( ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ! [Q5: A,R3: A] :
% 12.17/7.89 ( ( ( euclid7384307370059645450egment @ A @ R3 )
% 12.17/7.89 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.17/7.89 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.17/7.89 => ( ( R3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 12.17/7.89 = Q5 )
% 12.17/7.89 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.17/7.89 = R3 )
% 12.17/7.89 => ( A3
% 12.17/7.89 != ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R3 ) ) ) ) ) ) ) )
% 12.17/7.89 => ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1168_gcd__nat_Oextremum,axiom,
% 12.17/7.89 ! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1169_add__left__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel_semigroup_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ C2 ) )
% 12.17/7.89 = ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1170_add__right__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel_semigroup_add @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ B2 @ A3 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ A3 ) )
% 12.17/7.89 = ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1171_add__Suc__right,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 12.17/7.89 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1172_add__is__0,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ( plus_plus @ nat @ M @ N )
% 12.17/7.89 = ( zero_zero @ nat ) )
% 12.17/7.89 = ( ( M
% 12.17/7.89 = ( zero_zero @ nat ) )
% 12.17/7.89 & ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1173_Nat_Oadd__0__right,axiom,
% 12.17/7.89 ! [M: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 12.17/7.89 = M ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1174_nat__add__left__cancel__less,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 12.17/7.89 = ( ord_less @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1175_nat__add__left__cancel__le,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 12.17/7.89 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1176_diff__diff__left,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 12.17/7.89 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1177_add__le__cancel__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1178_add__le__cancel__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1179_double__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ A3 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1180_add_Oright__neutral,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( monoid_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1181_double__zero__sym,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ( zero_zero @ A )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ A3 ) )
% 12.17/7.89 = ( A3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1182_add__cancel__left__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ B2 @ A3 )
% 12.17/7.89 = A3 )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1183_add__cancel__left__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = A3 )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1184_add__cancel__right__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 = ( plus_plus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1185_add__cancel__right__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 = ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1186_add__eq__0__iff__both__eq__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ X @ Y3 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( ( X
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 & ( Y3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1187_zero__eq__add__iff__both__eq__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ( zero_zero @ A )
% 12.17/7.89 = ( plus_plus @ A @ X @ Y3 ) )
% 12.17/7.89 = ( ( X
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 & ( Y3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1188_add__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( monoid_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1189_add__less__cancel__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1190_add__less__cancel__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1191_add__diff__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1192_diff__add__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1193_add__diff__cancel__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1194_add__diff__cancel__left_H,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
% 12.17/7.89 = B2 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1195_add__diff__cancel__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1196_add__diff__cancel__right_H,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1197_add__minus__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
% 12.17/7.89 = B2 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1198_minus__add__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = B2 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1199_minus__add__distrib,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1200_dvd__add__triv__left__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1201_dvd__add__triv__right__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1202_of__nat__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring_1 @ A )
% 12.17/7.89 => ! [M: nat,N: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
% 12.17/7.89 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1203_mod__add__self1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1204_mod__add__self2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1205_add__gr__0,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 12.17/7.89 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.17/7.89 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1206_mult__Suc__right,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( times_times @ nat @ M @ ( suc @ N ) )
% 12.17/7.89 = ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1207_Nat_Odiff__diff__right,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 12.17/7.89 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1208_Nat_Oadd__diff__assoc2,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 12.17/7.89 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1209_Nat_Oadd__diff__assoc,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 12.17/7.89 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1210_of__bool__less__eq__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord181362715937106298miring @ A )
% 12.17/7.89 => ! [P: $o,Q: $o] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
% 12.17/7.89 = ( P
% 12.17/7.89 => Q ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1211_of__bool__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( ( zero_neq_one_of_bool @ A @ P )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ~ P ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1212_of__bool__eq_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ( ( zero_neq_one_of_bool @ A @ $false )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1213_of__bool__less__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord181362715937106298miring @ A )
% 12.17/7.89 => ! [P: $o,Q: $o] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
% 12.17/7.89 = ( ~ P
% 12.17/7.89 & Q ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1214_of__bool__eq_I2_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ( ( zero_neq_one_of_bool @ A @ $true )
% 12.17/7.89 = ( one_one @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1215_of__bool__eq__1__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( ( zero_neq_one_of_bool @ A @ P )
% 12.17/7.89 = ( one_one @ A ) )
% 12.17/7.89 = P ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1216_gbinomial__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ( semiring_char_0 @ A )
% 12.17/7.89 & ( semidom_divide @ A ) )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1217_add__le__same__cancel1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1218_add__le__same__cancel2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1219_le__add__same__cancel1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1220_le__add__same__cancel2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1221_double__add__le__zero__iff__single__add__le__zero,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1222_zero__le__double__add__iff__zero__le__single__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1223_add__less__same__cancel1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.89 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1224_add__less__same__cancel2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1225_less__add__same__cancel1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1226_less__add__same__cancel2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1937475149494474687imp_le @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1227_double__add__less__zero__iff__single__add__less__zero,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 12.17/7.89 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1228_zero__less__double__add__iff__zero__less__single__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord5086331880401160121up_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 12.17/7.89 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1229_le__add__diff__inverse2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.89 => ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.89 = A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1230_le__add__diff__inverse,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.89 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.17/7.89 = A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1231_diff__add__zero,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_monoid_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1232_add_Oright__inverse,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1233_ab__left__minus,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1234_dvd__add__times__triv__right__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1235_dvd__add__times__triv__left__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1236_diff__minus__eq__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1237_uminus__add__conv__diff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.17/7.89 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1238_div__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( algebraic_semidom @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.17/7.89 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1239_mod__mult__self1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1240_mod__mult__self2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1241_mod__mult__self3,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1242_mod__mult__self4,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1243_zero__less__of__bool__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
% 12.17/7.89 = P ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1244_diff__Suc__diff__eq1,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
% 12.17/7.89 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1245_diff__Suc__diff__eq2,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
% 12.17/7.89 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1246_of__bool__less__one__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
% 12.17/7.89 = ~ P ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1247_of__bool__not__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ring_1 @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( zero_neq_one_of_bool @ A @ ~ P )
% 12.17/7.89 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1248_gbinomial__0_I2_J,axiom,
% 12.17/7.89 ! [B: $tType] :
% 12.17/7.89 ( ( ( semiring_char_0 @ B )
% 12.17/7.89 & ( semidom_divide @ B ) )
% 12.17/7.89 => ! [K: nat] :
% 12.17/7.89 ( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
% 12.17/7.89 = ( zero_zero @ B ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1249_zle__add1__eq__le,axiom,
% 12.17/7.89 ! [W2: int,Z3: int] :
% 12.17/7.89 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
% 12.17/7.89 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1250_gbinomial__0_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ( semiring_char_0 @ A )
% 12.17/7.89 & ( semidom_divide @ A ) )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
% 12.17/7.89 = ( one_one @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1251_Suc__mod__mult__self1,axiom,
% 12.17/7.89 ! [M: nat,K: nat,N: nat] :
% 12.17/7.89 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1252_Suc__mod__mult__self2,axiom,
% 12.17/7.89 ! [M: nat,N: nat,K: nat] :
% 12.17/7.89 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1253_Suc__mod__mult__self3,axiom,
% 12.17/7.89 ! [K: nat,N: nat,M: nat] :
% 12.17/7.89 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1254_Suc__mod__mult__self4,axiom,
% 12.17/7.89 ! [N: nat,K: nat,M: nat] :
% 12.17/7.89 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1255_gbinomial__Suc0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ( semiring_char_0 @ A )
% 12.17/7.89 & ( semidom_divide @ A ) )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1256_of__nat__of__bool,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring_1 @ A )
% 12.17/7.89 => ! [P: $o] :
% 12.17/7.89 ( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
% 12.17/7.89 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1257_add__neg__numeral__special_I8_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( neg_numeral @ A )
% 12.17/7.89 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1258_add__neg__numeral__special_I7_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( neg_numeral @ A )
% 12.17/7.89 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1259_div__mult__self4,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1260_div__mult__self3,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,C2: A,A3: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1261_div__mult__self2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1262_div__mult__self1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1263_of__nat__Suc,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring_1 @ A )
% 12.17/7.89 => ! [M: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ A @ ( suc @ M ) )
% 12.17/7.89 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1264_sgn__mult__self__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
% 12.17/7.89 = ( zero_neq_one_of_bool @ A
% 12.17/7.89 @ ( A3
% 12.17/7.89 != ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1265_ceiling__add__one,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.89 => ! [X: A] :
% 12.17/7.89 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 12.17/7.89 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1266_Suc__0__mod__eq,axiom,
% 12.17/7.89 ! [N: nat] :
% 12.17/7.89 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.17/7.89 = ( zero_neq_one_of_bool @ nat
% 12.17/7.89 @ ( N
% 12.17/7.89 != ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1267_sgn__of__nat,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [N: nat] :
% 12.17/7.89 ( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.17/7.89 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1268_of__nat__gbinomial,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [N: nat,K: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
% 12.17/7.89 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1269_int__plus,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
% 12.17/7.89 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1270_int__ops_I5_J,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
% 12.17/7.89 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1271_zadd__int__left,axiom,
% 12.17/7.89 ! [M: nat,N: nat,Z3: int] :
% 12.17/7.89 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z3 ) )
% 12.17/7.89 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1272_Euclid__induct,axiom,
% 12.17/7.89 ! [P: nat > nat > $o,A3: nat,B2: nat] :
% 12.17/7.89 ( ! [A8: nat,B5: nat] :
% 12.17/7.89 ( ( P @ A8 @ B5 )
% 12.17/7.89 = ( P @ B5 @ A8 ) )
% 12.17/7.89 => ( ! [A8: nat] : ( P @ A8 @ ( zero_zero @ nat ) )
% 12.17/7.89 => ( ! [A8: nat,B5: nat] :
% 12.17/7.89 ( ( P @ A8 @ B5 )
% 12.17/7.89 => ( P @ A8 @ ( plus_plus @ nat @ A8 @ B5 ) ) )
% 12.17/7.89 => ( P @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1273_of__bool__eq__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ! [P5: $o,Q4: $o] :
% 12.17/7.89 ( ( ( zero_neq_one_of_bool @ A @ P5 )
% 12.17/7.89 = ( zero_neq_one_of_bool @ A @ Q4 ) )
% 12.17/7.89 = ( P5 = Q4 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1274_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_semigroup_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1275_is__num__normalize_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( neg_numeral @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1276_add__mono__thms__linordered__semiring_I4_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( I = J )
% 12.17/7.89 & ( K = L ) )
% 12.17/7.89 => ( ( plus_plus @ A @ I @ K )
% 12.17/7.89 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1277_group__cancel_Oadd1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_monoid_add @ A )
% 12.17/7.89 => ! [A6: A,K: A,A3: A,B2: A] :
% 12.17/7.89 ( ( A6
% 12.17/7.89 = ( plus_plus @ A @ K @ A3 ) )
% 12.17/7.89 => ( ( plus_plus @ A @ A6 @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1278_group__cancel_Oadd2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_monoid_add @ A )
% 12.17/7.89 => ! [B7: A,K: A,B2: A,A3: A] :
% 12.17/7.89 ( ( B7
% 12.17/7.89 = ( plus_plus @ A @ K @ B2 ) )
% 12.17/7.89 => ( ( plus_plus @ A @ A3 @ B7 )
% 12.17/7.89 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1279_add_Oassoc,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semigroup_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1280_add_Oleft__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ C2 ) )
% 12.17/7.89 = ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1281_add_Oright__cancel,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ B2 @ A3 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ A3 ) )
% 12.17/7.89 = ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1282_add_Ocommute,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_semigroup_add @ A )
% 12.17/7.89 => ( ( plus_plus @ A )
% 12.17/7.89 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1283_add_Oleft__commute,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_semigroup_add @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1284_add__left__imp__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel_semigroup_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ A3 @ C2 ) )
% 12.17/7.89 => ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1285_add__right__imp__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel_semigroup_add @ A )
% 12.17/7.89 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ B2 @ A3 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ A3 ) )
% 12.17/7.89 => ( B2 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1286_gbinomial__Suc__Suc,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 12.17/7.89 = ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1287_nat__int__add,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
% 12.17/7.89 = ( plus_plus @ nat @ A3 @ B2 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1288_ceiling__add__le,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.89 => ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y3 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1289_of__bool__conj,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring_1 @ A )
% 12.17/7.89 => ! [P: $o,Q: $o] :
% 12.17/7.89 ( ( zero_neq_one_of_bool @ A
% 12.17/7.89 @ ( P
% 12.17/7.89 & Q ) )
% 12.17/7.89 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1290_add__mono__thms__linordered__semiring_I3_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less_eq @ A @ I @ J )
% 12.17/7.89 & ( K = L ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1291_add__mono__thms__linordered__semiring_I2_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( I = J )
% 12.17/7.89 & ( ord_less_eq @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1292_add__mono__thms__linordered__semiring_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less_eq @ A @ I @ J )
% 12.17/7.89 & ( ord_less_eq @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1293_add__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1294_add__left__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1295_less__eqE,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ~ ! [C3: A] :
% 12.17/7.89 ( B2
% 12.17/7.89 != ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1296_add__right__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6658533253407199908up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1297_le__iff__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.89 => ( ( ord_less_eq @ A )
% 12.17/7.89 = ( ^ [A5: A,B3: A] :
% 12.17/7.89 ? [C4: A] :
% 12.17/7.89 ( B3
% 12.17/7.89 = ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1298_add__le__imp__le__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 12.17/7.89 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1299_add__le__imp__le__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1300_comm__monoid__add__class_Oadd__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_monoid_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1301_add_Ocomm__neutral,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_monoid_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1302_add_Ogroup__left__neutral,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1303_verit__sum__simplify,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 = A3 ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1304_add__mono__thms__linordered__field_I5_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less @ A @ I @ J )
% 12.17/7.89 & ( ord_less @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1305_add__mono__thms__linordered__field_I2_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( I = J )
% 12.17/7.89 & ( ord_less @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1306_add__mono__thms__linordered__field_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less @ A @ I @ J )
% 12.17/7.89 & ( K = L ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1307_add__strict__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( strict9044650504122735259up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less @ A @ C2 @ D2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1308_add__strict__left__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1309_add__strict__right__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1310_add__less__imp__less__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 12.17/7.89 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1311_add__less__imp__less__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere2412721322843649153imp_le @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1312_combine__common__factor,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring @ A )
% 12.17/7.89 => ! [A3: A,E: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1313_distrib__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1314_distrib__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( semiring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1315_comm__semiring__class_Odistrib,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_semiring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1316_ring__class_Oring__distribs_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1317_ring__class_Oring__distribs_I2_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1318_group__cancel_Osub1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A6: A,K: A,A3: A,B2: A] :
% 12.17/7.89 ( ( A6
% 12.17/7.89 = ( plus_plus @ A @ K @ A3 ) )
% 12.17/7.89 => ( ( minus_minus @ A @ A6 @ B2 )
% 12.17/7.89 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1319_diff__eq__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ( minus_minus @ A @ A3 @ B2 )
% 12.17/7.89 = C2 )
% 12.17/7.89 = ( A3
% 12.17/7.89 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1320_eq__diff__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 = ( minus_minus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1321_add__diff__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1322_diff__diff__eq2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1323_diff__add__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1324_diff__add__eq__diff__diff__swap,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1325_add__implies__diff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel1802427076303600483id_add @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ C2 @ B2 )
% 12.17/7.89 = A3 )
% 12.17/7.89 => ( C2
% 12.17/7.89 = ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1326_diff__diff__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( cancel2418104881723323429up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1327_is__num__normalize_I8_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( neg_numeral @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1328_group__cancel_Oneg1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A6: A,K: A,A3: A] :
% 12.17/7.89 ( ( A6
% 12.17/7.89 = ( plus_plus @ A @ K @ A3 ) )
% 12.17/7.89 => ( ( uminus_uminus @ A @ A6 )
% 12.17/7.89 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1329_add_Oinverse__distrib__swap,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1330_dvd__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_semiring_1 @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.17/7.89 => ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1331_dvd__add__left__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.17/7.89 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1332_dvd__add__right__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_s4317794764714335236cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.17/7.89 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1333_add__divide__distrib,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1334_nat__arith_Osuc1,axiom,
% 12.17/7.89 ! [A6: nat,K: nat,A3: nat] :
% 12.17/7.89 ( ( A6
% 12.17/7.89 = ( plus_plus @ nat @ K @ A3 ) )
% 12.17/7.89 => ( ( suc @ A6 )
% 12.17/7.89 = ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1335_add__Suc,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 12.17/7.89 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1336_add__Suc__shift,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 12.17/7.89 = ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1337_plus__nat_Oadd__0,axiom,
% 12.17/7.89 ! [N: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 12.17/7.89 = N ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1338_add__eq__self__zero,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ( plus_plus @ nat @ M @ N )
% 12.17/7.89 = M )
% 12.17/7.89 => ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1339_mod__add__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1340_mod__add__cong,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
% 12.17/7.89 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ A4 @ C2 ) )
% 12.17/7.89 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 12.17/7.89 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1341_mod__add__left__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1342_mod__add__right__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1343_plus__int__code_I2_J,axiom,
% 12.17/7.89 ! [L: int] :
% 12.17/7.89 ( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
% 12.17/7.89 = L ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1344_plus__int__code_I1_J,axiom,
% 12.17/7.89 ! [K: int] :
% 12.17/7.89 ( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
% 12.17/7.89 = K ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1345_add__lessD1,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 12.17/7.89 => ( ord_less @ nat @ I @ K ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1346_add__less__mono,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ J )
% 12.17/7.89 => ( ( ord_less @ nat @ K @ L )
% 12.17/7.89 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1347_not__add__less1,axiom,
% 12.17/7.89 ! [I: nat,J: nat] :
% 12.17/7.89 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1348_not__add__less2,axiom,
% 12.17/7.89 ! [J: nat,I: nat] :
% 12.17/7.89 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1349_add__less__mono1,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ J )
% 12.17/7.89 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1350_trans__less__add1,axiom,
% 12.17/7.89 ! [I: nat,J: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ J )
% 12.17/7.89 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1351_trans__less__add2,axiom,
% 12.17/7.89 ! [I: nat,J: nat,M: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ J )
% 12.17/7.89 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1352_less__add__eq__less,axiom,
% 12.17/7.89 ! [K: nat,L: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ K @ L )
% 12.17/7.89 => ( ( ( plus_plus @ nat @ M @ L )
% 12.17/7.89 = ( plus_plus @ nat @ K @ N ) )
% 12.17/7.89 => ( ord_less @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1353_nat__le__iff__add,axiom,
% 12.17/7.89 ( ( ord_less_eq @ nat )
% 12.17/7.89 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.89 ? [K3: nat] :
% 12.17/7.89 ( N3
% 12.17/7.89 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1354_trans__le__add2,axiom,
% 12.17/7.89 ! [I: nat,J: nat,M: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.89 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1355_trans__le__add1,axiom,
% 12.17/7.89 ! [I: nat,J: nat,M: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.89 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1356_add__le__mono1,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1357_add__le__mono,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat,L: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ K @ L )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1358_le__Suc__ex,axiom,
% 12.17/7.89 ! [K: nat,L: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ L )
% 12.17/7.89 => ? [N4: nat] :
% 12.17/7.89 ( L
% 12.17/7.89 = ( plus_plus @ nat @ K @ N4 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1359_add__leD2,axiom,
% 12.17/7.89 ! [M: nat,K: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 12.17/7.89 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1360_add__leD1,axiom,
% 12.17/7.89 ! [M: nat,K: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 12.17/7.89 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1361_le__add2,axiom,
% 12.17/7.89 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1362_le__add1,axiom,
% 12.17/7.89 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1363_add__leE,axiom,
% 12.17/7.89 ! [M: nat,K: nat,N: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 12.17/7.89 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.89 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1364_Nat_Odiff__cancel,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 12.17/7.89 = ( minus_minus @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1365_diff__cancel2,axiom,
% 12.17/7.89 ! [M: nat,K: nat,N: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 12.17/7.89 = ( minus_minus @ nat @ M @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1366_diff__add__inverse,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 12.17/7.89 = M ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1367_diff__add__inverse2,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 12.17/7.89 = M ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1368_left__add__mult__distrib,axiom,
% 12.17/7.89 ! [I: nat,U: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1369_add__mult__distrib2,axiom,
% 12.17/7.89 ! [K: nat,M: nat,N: nat] :
% 12.17/7.89 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1370_add__mult__distrib,axiom,
% 12.17/7.89 ! [M: nat,N: nat,K: nat] :
% 12.17/7.89 ( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1371_int__distrib_I2_J,axiom,
% 12.17/7.89 ! [W2: int,Z1: int,Z22: int] :
% 12.17/7.89 ( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
% 12.17/7.89 = ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1372_int__distrib_I1_J,axiom,
% 12.17/7.89 ! [Z1: int,Z22: int,W2: int] :
% 12.17/7.89 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
% 12.17/7.89 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1373_div__mod__decomp__int,axiom,
% 12.17/7.89 ! [A6: int,N: int] :
% 12.17/7.89 ( A6
% 12.17/7.89 = ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A6 @ N ) @ N ) @ ( modulo_modulo @ int @ A6 @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1374_same__sgn__sgn__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [B2: A,A3: A] :
% 12.17/7.89 ( ( ( sgn_sgn @ A @ B2 )
% 12.17/7.89 = ( sgn_sgn @ A @ A3 ) )
% 12.17/7.89 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.17/7.89 = ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1375_gbinomial__addition__formula,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 12.17/7.89 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1376_bezout__lemma__nat,axiom,
% 12.17/7.89 ! [D2: nat,A3: nat,B2: nat,X: nat,Y3: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 12.17/7.89 => ( ( dvd_dvd @ nat @ D2 @ B2 )
% 12.17/7.89 => ( ( ( ( times_times @ nat @ A3 @ X )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) )
% 12.17/7.89 | ( ( times_times @ nat @ B2 @ X )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) )
% 12.17/7.89 => ? [X3: nat,Y5: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 12.17/7.89 & ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A3 @ B2 ) )
% 12.17/7.89 & ( ( ( times_times @ nat @ A3 @ X3 )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y5 ) @ D2 ) )
% 12.17/7.89 | ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X3 )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1377_bezout__add__nat,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ? [D6: nat,X3: nat,Y5: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 12.17/7.89 & ( dvd_dvd @ nat @ D6 @ B2 )
% 12.17/7.89 & ( ( ( times_times @ nat @ A3 @ X3 )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) )
% 12.17/7.89 | ( ( times_times @ nat @ B2 @ X3 )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D6 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1378_gbinomial__mult__1_H,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
% 12.17/7.89 = ( 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 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1379_gbinomial__mult__1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
% 12.17/7.89 = ( 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 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1380_nat__add__distrib,axiom,
% 12.17/7.89 ! [Z3: int,Z6: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.89 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
% 12.17/7.89 => ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z6 ) )
% 12.17/7.89 = ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1381_Suc__times__gbinomial,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [K: nat,A3: A] :
% 12.17/7.89 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
% 12.17/7.89 = ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1382_zero__less__eq__of__bool,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1383_of__bool__less__eq__one,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1384_of__bool__def,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ( ( zero_neq_one_of_bool @ A )
% 12.17/7.89 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1385_split__of__bool,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ! [P: A > $o,P5: $o] :
% 12.17/7.89 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 12.17/7.89 = ( ( P5
% 12.17/7.89 => ( P @ ( one_one @ A ) ) )
% 12.17/7.89 & ( ~ P5
% 12.17/7.89 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1386_split__of__bool__asm,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( zero_neq_one @ A )
% 12.17/7.89 => ! [P: A > $o,P5: $o] :
% 12.17/7.89 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 12.17/7.89 = ( ~ ( ( P5
% 12.17/7.89 & ~ ( P @ ( one_one @ A ) ) )
% 12.17/7.89 | ( ~ P5
% 12.17/7.89 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1387_add__decreasing,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1388_add__increasing,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.17/7.89 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1389_add__decreasing2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1390_add__increasing2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.17/7.89 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1391_add__nonneg__nonneg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1392_add__nonpos__nonpos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1393_add__nonneg__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.17/7.89 => ( ( ( plus_plus @ A @ X @ Y3 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( ( X
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 & ( Y3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1394_add__nonpos__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ( plus_plus @ A @ X @ Y3 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( ( X
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 & ( Y3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1395_add__mono__thms__linordered__field_I4_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less_eq @ A @ I @ J )
% 12.17/7.89 & ( ord_less @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1396_add__mono__thms__linordered__field_I3_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [I: A,J: A,K: A,L: A] :
% 12.17/7.89 ( ( ( ord_less @ A @ I @ J )
% 12.17/7.89 & ( ord_less_eq @ A @ K @ L ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1397_add__le__less__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less @ A @ C2 @ D2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1398_add__less__le__mono,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere580206878836729694up_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1399_add__neg__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1400_add__pos__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1401_canonically__ordered__monoid__add__class_OlessE,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( canoni5634975068530333245id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ~ ! [C3: A] :
% 12.17/7.89 ( ( B2
% 12.17/7.89 = ( plus_plus @ A @ A3 @ C3 ) )
% 12.17/7.89 => ( C3
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1402_pos__add__strict,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( strict7427464778891057005id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ B2 @ C2 )
% 12.17/7.89 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1403_add__less__zeroD,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_idom @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 12.17/7.89 | ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1404_add__le__imp__le__diff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [I: A,K: A,N: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 12.17/7.89 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1405_add__le__add__imp__diff__le,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [I: A,K: A,N: A,J: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 12.17/7.89 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 12.17/7.89 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 12.17/7.89 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1406_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ( minus_minus @ A @ B2 @ A3 )
% 12.17/7.89 = C2 )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1407_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
% 12.17/7.89 = B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1408_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1409_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
% 12.17/7.89 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1410_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1411_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
% 12.17/7.89 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1412_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1413_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1414_le__add__diff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1415_diff__add,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere1170586879665033532d_diff @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
% 12.17/7.89 = B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1416_le__diff__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1417_diff__le__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1418_less__add__one,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1419_add__mono1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord181362715937106298miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1420_diff__less__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1421_less__diff__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ab_group_add @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 12.17/7.89 = ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1422_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_semidom @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ~ ( ord_less @ A @ A3 @ B2 )
% 12.17/7.89 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.17/7.89 = A3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1423_add__eq__0__iff,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 = ( B2
% 12.17/7.89 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1424_ab__group__add__class_Oab__left__minus,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [A3: A] :
% 12.17/7.89 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1425_add_Oinverse__unique,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( ( uminus_uminus @ A @ A3 )
% 12.17/7.89 = B2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1426_eq__neg__iff__add__eq__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 = ( uminus_uminus @ A @ B2 ) )
% 12.17/7.89 = ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1427_neg__eq__iff__add__eq__0,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ( uminus_uminus @ A @ A3 )
% 12.17/7.89 = B2 )
% 12.17/7.89 = ( ( plus_plus @ A @ A3 @ B2 )
% 12.17/7.89 = ( zero_zero @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1428_square__diff__square__factored,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( comm_ring @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
% 12.17/7.89 = ( times_times @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ A @ X @ Y3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1429_eq__add__iff2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( C2
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1430_eq__add__iff1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 )
% 12.17/7.89 = D2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1431_pinf_I9_J,axiom,
% 12.17/7.89 ! [B: $tType] :
% 12.17/7.89 ( ( ( plus @ B )
% 12.17/7.89 & ( linorder @ B )
% 12.17/7.89 & ( dvd @ B ) )
% 12.17/7.89 => ! [D2: B,S: B] :
% 12.17/7.89 ? [Z4: B] :
% 12.17/7.89 ! [X4: B] :
% 12.17/7.89 ( ( ord_less @ B @ Z4 @ X4 )
% 12.17/7.89 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
% 12.17/7.89 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1432_pinf_I10_J,axiom,
% 12.17/7.89 ! [B: $tType] :
% 12.17/7.89 ( ( ( plus @ B )
% 12.17/7.89 & ( linorder @ B )
% 12.17/7.89 & ( dvd @ B ) )
% 12.17/7.89 => ! [D2: B,S: B] :
% 12.17/7.89 ? [Z4: B] :
% 12.17/7.89 ! [X4: B] :
% 12.17/7.89 ( ( ord_less @ B @ Z4 @ X4 )
% 12.17/7.89 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
% 12.17/7.89 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1433_minf_I9_J,axiom,
% 12.17/7.89 ! [B: $tType] :
% 12.17/7.89 ( ( ( plus @ B )
% 12.17/7.89 & ( linorder @ B )
% 12.17/7.89 & ( dvd @ B ) )
% 12.17/7.89 => ! [D2: B,S: B] :
% 12.17/7.89 ? [Z4: B] :
% 12.17/7.89 ! [X4: B] :
% 12.17/7.89 ( ( ord_less @ B @ X4 @ Z4 )
% 12.17/7.89 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
% 12.17/7.89 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1434_minf_I10_J,axiom,
% 12.17/7.89 ! [B: $tType] :
% 12.17/7.89 ( ( ( plus @ B )
% 12.17/7.89 & ( linorder @ B )
% 12.17/7.89 & ( dvd @ B ) )
% 12.17/7.89 => ! [D2: B,S: B] :
% 12.17/7.89 ? [Z4: B] :
% 12.17/7.89 ! [X4: B] :
% 12.17/7.89 ( ( ord_less @ B @ X4 @ Z4 )
% 12.17/7.89 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
% 12.17/7.89 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1435_group__cancel_Osub2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ! [B7: A,K: A,B2: A,A3: A] :
% 12.17/7.89 ( ( B7
% 12.17/7.89 = ( plus_plus @ A @ K @ B2 ) )
% 12.17/7.89 => ( ( minus_minus @ A @ A3 @ B7 )
% 12.17/7.89 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1436_diff__conv__add__uminus,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( group_add @ A )
% 12.17/7.89 => ( ( minus_minus @ A )
% 12.17/7.89 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1437_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ab_group_add @ A )
% 12.17/7.89 => ( ( minus_minus @ A )
% 12.17/7.89 = ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1438_div__plus__div__distrib__dvd__left,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [C2: A,A3: A,B2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1439_div__plus__div__distrib__dvd__right,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.89 => ! [C2: A,B2: A,A3: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.17/7.89 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1440_mod__eqE,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid8851590272496341667cancel @ A )
% 12.17/7.89 => ! [A3: A,C2: A,B2: A] :
% 12.17/7.89 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 12.17/7.89 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 12.17/7.89 => ~ ! [D6: A] :
% 12.17/7.89 ( B2
% 12.17/7.89 != ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D6 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1441_div__add1__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.89 = ( 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 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1442_add__is__1,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ( plus_plus @ nat @ M @ N )
% 12.17/7.89 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.89 = ( ( ( M
% 12.17/7.89 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.89 & ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) )
% 12.17/7.89 | ( ( M
% 12.17/7.89 = ( zero_zero @ nat ) )
% 12.17/7.89 & ( N
% 12.17/7.89 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1443_one__is__add,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ( suc @ ( zero_zero @ nat ) )
% 12.17/7.89 = ( plus_plus @ nat @ M @ N ) )
% 12.17/7.89 = ( ( ( M
% 12.17/7.89 = ( suc @ ( zero_zero @ nat ) ) )
% 12.17/7.89 & ( N
% 12.17/7.89 = ( zero_zero @ nat ) ) )
% 12.17/7.89 | ( ( M
% 12.17/7.89 = ( zero_zero @ nat ) )
% 12.17/7.89 & ( N
% 12.17/7.89 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1444_less__natE,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ M @ N )
% 12.17/7.89 => ~ ! [Q5: nat] :
% 12.17/7.89 ( N
% 12.17/7.89 != ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1445_less__add__Suc1,axiom,
% 12.17/7.89 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1446_less__add__Suc2,axiom,
% 12.17/7.89 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1447_less__iff__Suc__add,axiom,
% 12.17/7.89 ( ( ord_less @ nat )
% 12.17/7.89 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.89 ? [K3: nat] :
% 12.17/7.89 ( N3
% 12.17/7.89 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1448_less__imp__Suc__add,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ M @ N )
% 12.17/7.89 => ? [K2: nat] :
% 12.17/7.89 ( N
% 12.17/7.89 = ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1449_less__imp__add__positive,axiom,
% 12.17/7.89 ! [I: nat,J: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ J )
% 12.17/7.89 => ? [K2: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
% 12.17/7.89 & ( ( plus_plus @ nat @ I @ K2 )
% 12.17/7.89 = J ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1450_mono__nat__linear__lb,axiom,
% 12.17/7.89 ! [F: nat > nat,M: nat,K: nat] :
% 12.17/7.89 ( ! [M6: nat,N4: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ M6 @ N4 )
% 12.17/7.89 => ( ord_less @ nat @ ( F @ M6 ) @ ( F @ N4 ) ) )
% 12.17/7.89 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1451_ex__has__greatest__nat__lemma,axiom,
% 12.17/7.89 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 12.17/7.89 ( ( P @ K )
% 12.17/7.89 => ( ! [X3: A] :
% 12.17/7.89 ( ( P @ X3 )
% 12.17/7.89 => ? [Y4: A] :
% 12.17/7.89 ( ( P @ Y4 )
% 12.17/7.89 & ~ ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) )
% 12.17/7.89 => ? [Y5: A] :
% 12.17/7.89 ( ( P @ Y5 )
% 12.17/7.89 & ~ ( ord_less @ nat @ ( F @ Y5 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1452_bezout__add__strong__nat,axiom,
% 12.17/7.89 ! [A3: nat,B2: nat] :
% 12.17/7.89 ( ( A3
% 12.17/7.89 != ( zero_zero @ nat ) )
% 12.17/7.89 => ? [D6: nat,X3: nat,Y5: nat] :
% 12.17/7.89 ( ( dvd_dvd @ nat @ D6 @ A3 )
% 12.17/7.89 & ( dvd_dvd @ nat @ D6 @ B2 )
% 12.17/7.89 & ( ( times_times @ nat @ A3 @ X3 )
% 12.17/7.89 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1453_diff__add__0,axiom,
% 12.17/7.89 ! [N: nat,M: nat] :
% 12.17/7.89 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 12.17/7.89 = ( zero_zero @ nat ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1454_odd__nonzero,axiom,
% 12.17/7.89 ! [Z3: int] :
% 12.17/7.89 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
% 12.17/7.89 != ( zero_zero @ int ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1455_mult__Suc,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ( times_times @ nat @ ( suc @ M ) @ N )
% 12.17/7.89 = ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1456_less__diff__conv,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 12.17/7.89 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1457_add__diff__inverse__nat,axiom,
% 12.17/7.89 ! [M: nat,N: nat] :
% 12.17/7.89 ( ~ ( ord_less @ nat @ M @ N )
% 12.17/7.89 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 12.17/7.89 = M ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1458_Suc__eq__plus1__left,axiom,
% 12.17/7.89 ( suc
% 12.17/7.89 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1459_plus__1__eq__Suc,axiom,
% 12.17/7.89 ( ( plus_plus @ nat @ ( one_one @ nat ) )
% 12.17/7.89 = suc ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1460_Suc__eq__plus1,axiom,
% 12.17/7.89 ( suc
% 12.17/7.89 = ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1461_Nat_Ole__imp__diff__is__add,axiom,
% 12.17/7.89 ! [I: nat,J: nat,K: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.89 => ( ( ( minus_minus @ nat @ J @ I )
% 12.17/7.89 = K )
% 12.17/7.89 = ( J
% 12.17/7.89 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1462_Nat_Odiff__add__assoc2,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 12.17/7.89 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1463_Nat_Odiff__add__assoc,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 12.17/7.89 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1464_Nat_Ole__diff__conv2,axiom,
% 12.17/7.89 ! [K: nat,J: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.89 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 12.17/7.89 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1465_le__diff__conv,axiom,
% 12.17/7.89 ! [J: nat,K: nat,I: nat] :
% 12.17/7.89 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 12.17/7.89 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1466_int__ge__induct,axiom,
% 12.17/7.89 ! [K: int,I: int,P: int > $o] :
% 12.17/7.89 ( ( ord_less_eq @ int @ K @ I )
% 12.17/7.89 => ( ( P @ K )
% 12.17/7.89 => ( ! [I4: int] :
% 12.17/7.89 ( ( ord_less_eq @ int @ K @ I4 )
% 12.17/7.89 => ( ( P @ I4 )
% 12.17/7.89 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.89 => ( P @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1467_zless__add1__eq,axiom,
% 12.17/7.89 ! [W2: int,Z3: int] :
% 12.17/7.89 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
% 12.17/7.89 = ( ( ord_less @ int @ W2 @ Z3 )
% 12.17/7.89 | ( W2 = Z3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1468_int__gr__induct,axiom,
% 12.17/7.89 ! [K: int,I: int,P: int > $o] :
% 12.17/7.89 ( ( ord_less @ int @ K @ I )
% 12.17/7.89 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 12.17/7.89 => ( ! [I4: int] :
% 12.17/7.89 ( ( ord_less @ int @ K @ I4 )
% 12.17/7.89 => ( ( P @ I4 )
% 12.17/7.89 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.89 => ( P @ I ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1469_zle__iff__zadd,axiom,
% 12.17/7.89 ( ( ord_less_eq @ int )
% 12.17/7.89 = ( ^ [W3: int,Z7: int] :
% 12.17/7.89 ? [N3: nat] :
% 12.17/7.89 ( Z7
% 12.17/7.89 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1470_zdvd__period,axiom,
% 12.17/7.89 ! [A3: int,D2: int,X: int,T2: int,C2: int] :
% 12.17/7.89 ( ( dvd_dvd @ int @ A3 @ D2 )
% 12.17/7.89 => ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
% 12.17/7.89 = ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1471_zdvd__reduce,axiom,
% 12.17/7.89 ! [K: int,N: int,M: int] :
% 12.17/7.89 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
% 12.17/7.89 = ( dvd_dvd @ int @ K @ N ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1472_nat__mod__eq__iff,axiom,
% 12.17/7.89 ! [X: nat,N: nat,Y3: nat] :
% 12.17/7.89 ( ( ( modulo_modulo @ nat @ X @ N )
% 12.17/7.89 = ( modulo_modulo @ nat @ Y3 @ N ) )
% 12.17/7.89 = ( ? [Q1: nat,Q22: nat] :
% 12.17/7.89 ( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
% 12.17/7.89 = ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1473_frac__1__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( archim2362893244070406136eiling @ A )
% 12.17/7.89 => ! [X: A] :
% 12.17/7.89 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 12.17/7.89 = ( archimedean_frac @ A @ X ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1474_gbinomial__factors,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 12.17/7.89 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1475_gbinomial__rec,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [A3: A,K: nat] :
% 12.17/7.89 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 12.17/7.89 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1476_dbl__inc__def,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( neg_numeral @ A )
% 12.17/7.89 => ( ( neg_numeral_dbl_inc @ A )
% 12.17/7.89 = ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1477_gbinomial__reduce__nat,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field_char_0 @ A )
% 12.17/7.89 => ! [K: nat,A3: A] :
% 12.17/7.89 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.17/7.89 => ( ( gbinomial @ A @ A3 @ K )
% 12.17/7.89 = ( 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 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1478_field__le__epsilon,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_field @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ( ! [E2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 12.17/7.89 => ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y3 @ E2 ) ) )
% 12.17/7.89 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1479_add__strict__increasing2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere8940638589300402666id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ B2 @ C2 )
% 12.17/7.89 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1480_add__strict__increasing,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere8940638589300402666id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.17/7.89 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1481_add__pos__nonneg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1482_add__nonpos__neg,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1483_add__nonneg__pos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.17/7.89 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1484_add__neg__nonpos,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordere6911136660526730532id_add @ A )
% 12.17/7.89 => ! [A3: A,B2: A] :
% 12.17/7.89 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 12.17/7.89 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1485_sum__squares__ge__zero,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_ring @ A )
% 12.17/7.89 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1486_discrete,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( unique1627219031080169319umeral @ A )
% 12.17/7.89 => ( ( ord_less @ A )
% 12.17/7.89 = ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1487_not__sum__squares__lt__zero,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linordered_ring @ A )
% 12.17/7.89 => ! [X: A,Y3: A] :
% 12.17/7.89 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1488_zero__less__two,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( linord181362715937106298miring @ A )
% 12.17/7.89 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1489_le__add__iff2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1490_le__add__iff1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1491_less__add__iff2,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1492_less__add__iff1,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ordered_ring @ A )
% 12.17/7.89 => ! [A3: A,E: A,C2: A,B2: A,D2: A] :
% 12.17/7.89 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
% 12.17/7.89 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1493_unity__coeff__ex,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( ( dvd @ A )
% 12.17/7.89 & ( semiring_0 @ A ) )
% 12.17/7.89 => ! [P: A > $o,L: A] :
% 12.17/7.89 ( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
% 12.17/7.89 = ( ? [X2: A] :
% 12.17/7.89 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
% 12.17/7.89 & ( P @ X2 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1494_add__divide__eq__if__simps_I2_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ( Z3
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 12.17/7.89 = B2 ) )
% 12.17/7.89 & ( ( Z3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
% 12.17/7.89 = ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1495_add__divide__eq__if__simps_I1_J,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( division_ring @ A )
% 12.17/7.89 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.89 ( ( ( Z3
% 12.17/7.89 = ( zero_zero @ A ) )
% 12.17/7.89 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 12.17/7.89 = A3 ) )
% 12.17/7.89 & ( ( Z3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
% 12.17/7.89 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1496_add__frac__eq,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field @ A )
% 12.17/7.89 => ! [Y3: A,Z3: A,X: A,W2: A] :
% 12.17/7.89 ( ( Y3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( Z3
% 12.17/7.89 != ( zero_zero @ A ) )
% 12.17/7.89 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
% 12.17/7.89 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% 12.17/7.89
% 12.17/7.89 thf(fact_1497_add__frac__num,axiom,
% 12.17/7.89 ! [A: $tType] :
% 12.17/7.89 ( ( field @ A )
% 12.17/7.89 => ! [Y3: A,X: A,Z3: A] :
% 12.17/7.90 ( ( Y3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1498_add__num__frac,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( field @ A )
% 12.17/7.90 => ! [Y3: A,Z3: A,X: A] :
% 12.17/7.90 ( ( Y3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1499_add__divide__eq__iff,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( division_ring @ A )
% 12.17/7.90 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.90 ( ( Z3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1500_divide__add__eq__iff,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( division_ring @ A )
% 12.17/7.90 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.90 ( ( Z3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1501_div__add__self1,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.90 => ! [B2: A,A3: A] :
% 12.17/7.90 ( ( B2
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 12.17/7.90 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1502_div__add__self2,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( euclid4440199948858584721cancel @ A )
% 12.17/7.90 => ! [B2: A,A3: A] :
% 12.17/7.90 ( ( B2
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.17/7.90 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1503_square__diff__one__factored,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( ring_1 @ A )
% 12.17/7.90 => ! [X: A] :
% 12.17/7.90 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
% 12.17/7.90 = ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1504_gt__half__sum,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( linordered_field @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.90 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1505_less__half__sum,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( linordered_field @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( ( ord_less @ A @ A3 @ B2 )
% 12.17/7.90 => ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1506_inf__period_I4_J,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( ( comm_ring @ A )
% 12.17/7.90 & ( dvd @ A ) )
% 12.17/7.90 => ! [D2: A,D4: A,T2: A] :
% 12.17/7.90 ( ( dvd_dvd @ A @ D2 @ D4 )
% 12.17/7.90 => ! [X4: A,K4: A] :
% 12.17/7.90 ( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
% 12.17/7.90 = ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1507_inf__period_I3_J,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( ( comm_ring @ A )
% 12.17/7.90 & ( dvd @ A ) )
% 12.17/7.90 => ! [D2: A,D4: A,T2: A] :
% 12.17/7.90 ( ( dvd_dvd @ A @ D2 @ D4 )
% 12.17/7.90 => ! [X4: A,K4: A] :
% 12.17/7.90 ( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
% 12.17/7.90 = ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1508_gbinomial__of__nat__symmetric,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( field_char_0 @ A )
% 12.17/7.90 => ! [K: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ K @ N )
% 12.17/7.90 => ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
% 12.17/7.90 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1509_div__mult1__eq,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( euclid3128863361964157862miring @ A )
% 12.17/7.90 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.90 ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.17/7.90 = ( 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 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1510_mult__div__mod__eq,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semiring_modulo @ A )
% 12.17/7.90 => ! [B2: A,A3: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.90 = A3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1511_mod__mult__div__eq,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semiring_modulo @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 12.17/7.90 = A3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1512_mod__div__mult__eq,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semiring_modulo @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 12.17/7.90 = A3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1513_div__mult__mod__eq,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semiring_modulo @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 12.17/7.90 = A3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1514_mod__div__decomp,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semiring_modulo @ A )
% 12.17/7.90 => ! [A3: A,B2: A] :
% 12.17/7.90 ( A3
% 12.17/7.90 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1515_cancel__div__mod__rules_I1_J,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semidom_modulo @ A )
% 12.17/7.90 => ! [A3: A,B2: A,C2: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 12.17/7.90 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1516_cancel__div__mod__rules_I2_J,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( semidom_modulo @ A )
% 12.17/7.90 => ! [B2: A,A3: A,C2: A] :
% 12.17/7.90 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 12.17/7.90 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1517_nat__diff__split__asm,axiom,
% 12.17/7.90 ! [P: nat > $o,A3: nat,B2: nat] :
% 12.17/7.90 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 12.17/7.90 = ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
% 12.17/7.90 & ~ ( P @ ( zero_zero @ nat ) ) )
% 12.17/7.90 | ? [D5: nat] :
% 12.17/7.90 ( ( A3
% 12.17/7.90 = ( plus_plus @ nat @ B2 @ D5 ) )
% 12.17/7.90 & ~ ( P @ D5 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1518_nat__diff__split,axiom,
% 12.17/7.90 ! [P: nat > $o,A3: nat,B2: nat] :
% 12.17/7.90 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 12.17/7.90 = ( ( ( ord_less @ nat @ A3 @ B2 )
% 12.17/7.90 => ( P @ ( zero_zero @ nat ) ) )
% 12.17/7.90 & ! [D5: nat] :
% 12.17/7.90 ( ( A3
% 12.17/7.90 = ( plus_plus @ nat @ B2 @ D5 ) )
% 12.17/7.90 => ( P @ D5 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1519_mod__pos__neg__trivial,axiom,
% 12.17/7.90 ! [K: int,L: int] :
% 12.17/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.90 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 12.17/7.90 => ( ( modulo_modulo @ int @ K @ L )
% 12.17/7.90 = ( plus_plus @ int @ K @ L ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1520_less__diff__conv2,axiom,
% 12.17/7.90 ! [K: nat,J: nat,I: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ K @ J )
% 12.17/7.90 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 12.17/7.90 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1521_odd__less__0__iff,axiom,
% 12.17/7.90 ! [Z3: int] :
% 12.17/7.90 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 ) @ ( zero_zero @ int ) )
% 12.17/7.90 = ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1522_zless__iff__Suc__zadd,axiom,
% 12.17/7.90 ( ( ord_less @ int )
% 12.17/7.90 = ( ^ [W3: int,Z7: int] :
% 12.17/7.90 ? [N3: nat] :
% 12.17/7.90 ( Z7
% 12.17/7.90 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1523_int__ops_I4_J,axiom,
% 12.17/7.90 ! [A3: nat] :
% 12.17/7.90 ( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
% 12.17/7.90 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1524_int__Suc,axiom,
% 12.17/7.90 ! [N: nat] :
% 12.17/7.90 ( ( semiring_1_of_nat @ int @ ( suc @ N ) )
% 12.17/7.90 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1525_zmod__zmult2__eq,axiom,
% 12.17/7.90 ! [C2: int,A3: int,B2: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 12.17/7.90 => ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 12.17/7.90 = ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1526_zless__imp__add1__zle,axiom,
% 12.17/7.90 ! [W2: int,Z3: int] :
% 12.17/7.90 ( ( ord_less @ int @ W2 @ Z3 )
% 12.17/7.90 => ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1527_add1__zle__eq,axiom,
% 12.17/7.90 ! [W2: int,Z3: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 )
% 12.17/7.90 = ( ord_less @ int @ W2 @ Z3 ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1528_nat__eq__add__iff1,axiom,
% 12.17/7.90 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ J @ I )
% 12.17/7.90 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
% 12.17/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
% 12.17/7.90 = N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1529_nat__eq__add__iff2,axiom,
% 12.17/7.90 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.90 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
% 12.17/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( M
% 12.17/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1530_nat__le__add__iff1,axiom,
% 12.17/7.90 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ J @ I )
% 12.17/7.90 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1531_nat__le__add__iff2,axiom,
% 12.17/7.90 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.90 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1532_nat__diff__add__eq1,axiom,
% 12.17/7.90 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ J @ I )
% 12.17/7.90 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1533_nat__diff__add__eq2,axiom,
% 12.17/7.90 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.90 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1534_fact__fact__dvd__fact,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( linordered_semidom @ A )
% 12.17/7.90 => ! [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 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1535_int__induct,axiom,
% 12.17/7.90 ! [P: int > $o,K: int,I: int] :
% 12.17/7.90 ( ( P @ K )
% 12.17/7.90 => ( ! [I4: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ K @ I4 )
% 12.17/7.90 => ( ( P @ I4 )
% 12.17/7.90 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.90 => ( ! [I4: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ I4 @ K )
% 12.17/7.90 => ( ( P @ I4 )
% 12.17/7.90 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
% 12.17/7.90 => ( P @ I ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1536_mod__eq__nat1E,axiom,
% 12.17/7.90 ! [M: nat,Q4: nat,N: nat] :
% 12.17/7.90 ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 12.17/7.90 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 12.17/7.90 => ( ( ord_less_eq @ nat @ N @ M )
% 12.17/7.90 => ~ ! [S2: nat] :
% 12.17/7.90 ( M
% 12.17/7.90 != ( plus_plus @ nat @ N @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1537_mod__eq__nat2E,axiom,
% 12.17/7.90 ! [M: nat,Q4: nat,N: nat] :
% 12.17/7.90 ( ( ( modulo_modulo @ nat @ M @ Q4 )
% 12.17/7.90 = ( modulo_modulo @ nat @ N @ Q4 ) )
% 12.17/7.90 => ( ( ord_less_eq @ nat @ M @ N )
% 12.17/7.90 => ~ ! [S2: nat] :
% 12.17/7.90 ( N
% 12.17/7.90 != ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1538_nat__mod__eq__lemma,axiom,
% 12.17/7.90 ! [X: nat,N: nat,Y3: nat] :
% 12.17/7.90 ( ( ( modulo_modulo @ nat @ X @ N )
% 12.17/7.90 = ( modulo_modulo @ nat @ Y3 @ N ) )
% 12.17/7.90 => ( ( ord_less_eq @ nat @ Y3 @ X )
% 12.17/7.90 => ? [Q5: nat] :
% 12.17/7.90 ( X
% 12.17/7.90 = ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1539_mod__mult2__eq,axiom,
% 12.17/7.90 ! [M: nat,N: nat,Q4: nat] :
% 12.17/7.90 ( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
% 12.17/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1540_div__mod__decomp,axiom,
% 12.17/7.90 ! [A6: nat,N: nat] :
% 12.17/7.90 ( A6
% 12.17/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A6 @ N ) @ N ) @ ( modulo_modulo @ nat @ A6 @ N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1541_dbl__dec__def,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( neg_numeral @ A )
% 12.17/7.90 => ( ( neg_numeral_dbl_dec @ A )
% 12.17/7.90 = ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1542_gbinomial__absorb__comp,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( field_char_0 @ A )
% 12.17/7.90 => ! [A3: A,K: nat] :
% 12.17/7.90 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
% 12.17/7.90 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1543_convex__bound__le,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( linord6961819062388156250ring_1 @ A )
% 12.17/7.90 => ! [X: A,A3: A,Y3: A,U: A,V: A] :
% 12.17/7.90 ( ( ord_less_eq @ A @ X @ A3 )
% 12.17/7.90 => ( ( ord_less_eq @ A @ Y3 @ A3 )
% 12.17/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 12.17/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 12.17/7.90 => ( ( ( plus_plus @ A @ U @ V )
% 12.17/7.90 = ( one_one @ A ) )
% 12.17/7.90 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1544_divide__int__unfold,axiom,
% 12.17/7.90 ! [L: int,K: int,N: nat,M: nat] :
% 12.17/7.90 ( ( ( ( ( sgn_sgn @ int @ L )
% 12.17/7.90 = ( zero_zero @ int ) )
% 12.17/7.90 | ( ( sgn_sgn @ int @ K )
% 12.17/7.90 = ( zero_zero @ int ) )
% 12.17/7.90 | ( N
% 12.17/7.90 = ( zero_zero @ nat ) ) )
% 12.17/7.90 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.90 = ( zero_zero @ int ) ) )
% 12.17/7.90 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 12.17/7.90 = ( zero_zero @ int ) )
% 12.17/7.90 | ( ( sgn_sgn @ int @ K )
% 12.17/7.90 = ( zero_zero @ int ) )
% 12.17/7.90 | ( N
% 12.17/7.90 = ( zero_zero @ nat ) ) )
% 12.17/7.90 => ( ( ( ( sgn_sgn @ int @ K )
% 12.17/7.90 = ( sgn_sgn @ int @ L ) )
% 12.17/7.90 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.90 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
% 12.17/7.90 & ( ( ( sgn_sgn @ int @ K )
% 12.17/7.90 != ( sgn_sgn @ int @ L ) )
% 12.17/7.90 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.17/7.90 = ( uminus_uminus @ int
% 12.17/7.90 @ ( semiring_1_of_nat @ int
% 12.17/7.90 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
% 12.17/7.90 @ ( zero_neq_one_of_bool @ nat
% 12.17/7.90 @ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1545_add__divide__eq__if__simps_I3_J,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( division_ring @ A )
% 12.17/7.90 => ! [Z3: A,A3: A,B2: A] :
% 12.17/7.90 ( ( ( Z3
% 12.17/7.90 = ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 12.17/7.90 = B2 ) )
% 12.17/7.90 & ( ( Z3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1546_minus__divide__add__eq__iff,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( division_ring @ A )
% 12.17/7.90 => ! [Z3: A,X: A,Y3: A] :
% 12.17/7.90 ( ( Z3
% 12.17/7.90 != ( zero_zero @ A ) )
% 12.17/7.90 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
% 12.17/7.90 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1547_mod__mult2__eq_H,axiom,
% 12.17/7.90 ! [A: $tType] :
% 12.17/7.90 ( ( euclid5411537665997757685th_nat @ A )
% 12.17/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.17/7.90 ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 12.17/7.90 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1548_add__eq__if,axiom,
% 12.17/7.90 ( ( plus_plus @ nat )
% 12.17/7.90 = ( ^ [M2: nat,N3: nat] :
% 12.17/7.90 ( if @ nat
% 12.17/7.90 @ ( M2
% 12.17/7.90 = ( zero_zero @ nat ) )
% 12.17/7.90 @ N3
% 12.17/7.90 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1549_le__imp__0__less,axiom,
% 12.17/7.90 ! [Z3: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.17/7.90 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1550_verit__le__mono__div__int,axiom,
% 12.17/7.90 ! [A6: int,B7: int,N: int] :
% 12.17/7.90 ( ( ord_less @ int @ A6 @ B7 )
% 12.17/7.90 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.17/7.90 => ( ord_less_eq @ int
% 12.17/7.90 @ ( plus_plus @ int @ ( divide_divide @ int @ A6 @ N )
% 12.17/7.90 @ ( if @ int
% 12.17/7.90 @ ( ( modulo_modulo @ int @ B7 @ N )
% 12.17/7.90 = ( zero_zero @ int ) )
% 12.17/7.90 @ ( one_one @ int )
% 12.17/7.90 @ ( zero_zero @ int ) ) )
% 12.17/7.90 @ ( divide_divide @ int @ B7 @ N ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1551_nat__less__add__iff1,axiom,
% 12.17/7.90 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ J @ I )
% 12.17/7.90 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1552_nat__less__add__iff2,axiom,
% 12.17/7.90 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 12.17/7.90 ( ( ord_less_eq @ nat @ I @ J )
% 12.17/7.90 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 12.17/7.90 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1553_incr__mult__lemma,axiom,
% 12.17/7.90 ! [D2: int,P: int > $o,K: int] :
% 12.17/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.17/7.90 => ( ! [X3: int] :
% 12.17/7.90 ( ( P @ X3 )
% 12.17/7.90 => ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
% 12.17/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.17/7.90 => ! [X4: int] :
% 12.17/7.90 ( ( P @ X4 )
% 12.17/7.90 => ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1554_unique__quotient__lemma__neg,axiom,
% 12.17/7.90 ! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.17/7.90 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 12.17/7.90 => ( ( ord_less @ int @ B2 @ R )
% 12.17/7.90 => ( ( ord_less @ int @ B2 @ R4 )
% 12.17/7.90 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1555_unique__quotient__lemma,axiom,
% 12.17/7.90 ! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
% 12.17/7.90 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.17/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 12.17/7.90 => ( ( ord_less @ int @ R4 @ B2 )
% 12.17/7.90 => ( ( ord_less @ int @ R @ B2 )
% 12.17/7.90 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ).
% 12.17/7.90
% 12.17/7.90 thf(fact_1556_zdiv__mono2__neg__lemma,axiom,
% 12.17/7.90 ! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
% 12.24/7.90 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 12.24/7.90 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
% 12.24/7.90 => ( ( ord_less @ int @ R @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 12.24/7.90 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 12.24/7.90 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 12.24/7.90 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1557_zdiv__mono2__lemma,axiom,
% 12.24/7.90 ! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
% 12.24/7.90 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 12.24/7.90 => ( ( ord_less @ int @ R4 @ B4 )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 12.24/7.90 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 12.24/7.90 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 12.24/7.90 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1558_q__pos__lemma,axiom,
% 12.24/7.90 ! [B4: int,Q6: int,R4: int] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
% 12.24/7.90 => ( ( ord_less @ int @ R4 @ B4 )
% 12.24/7.90 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 12.24/7.90 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1559_int__div__pos__eq,axiom,
% 12.24/7.90 ! [A3: int,B2: int,Q4: int,R: int] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 12.24/7.90 => ( ( ord_less @ int @ R @ B2 )
% 12.24/7.90 => ( ( divide_divide @ int @ A3 @ B2 )
% 12.24/7.90 = Q4 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1560_int__div__neg__eq,axiom,
% 12.24/7.90 ! [A3: int,B2: int,Q4: int,R: int] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 12.24/7.90 => ( ( ord_less @ int @ B2 @ R )
% 12.24/7.90 => ( ( divide_divide @ int @ A3 @ B2 )
% 12.24/7.90 = Q4 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1561_split__zdiv,axiom,
% 12.24/7.90 ! [P: int > $o,N: int,K: int] :
% 12.24/7.90 ( ( P @ ( divide_divide @ int @ N @ K ) )
% 12.24/7.90 = ( ( ( K
% 12.24/7.90 = ( zero_zero @ int ) )
% 12.24/7.90 => ( P @ ( zero_zero @ int ) ) )
% 12.24/7.90 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.90 => ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 12.24/7.90 & ( ord_less @ int @ J3 @ K )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ I5 ) ) )
% 12.24/7.90 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.90 => ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less @ int @ K @ J3 )
% 12.24/7.90 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ I5 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1562_int__mod__pos__eq,axiom,
% 12.24/7.90 ! [A3: int,B2: int,Q4: int,R: int] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 12.24/7.90 => ( ( ord_less @ int @ R @ B2 )
% 12.24/7.90 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.24/7.90 = R ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1563_int__mod__neg__eq,axiom,
% 12.24/7.90 ! [A3: int,B2: int,Q4: int,R: int] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 12.24/7.90 => ( ( ord_less @ int @ B2 @ R )
% 12.24/7.90 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 12.24/7.90 = R ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1564_split__zmod,axiom,
% 12.24/7.90 ! [P: int > $o,N: int,K: int] :
% 12.24/7.90 ( ( P @ ( modulo_modulo @ int @ N @ K ) )
% 12.24/7.90 = ( ( ( K
% 12.24/7.90 = ( zero_zero @ int ) )
% 12.24/7.90 => ( P @ N ) )
% 12.24/7.90 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.90 => ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 12.24/7.90 & ( ord_less @ int @ J3 @ K )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ J3 ) ) )
% 12.24/7.90 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.90 => ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less @ int @ K @ J3 )
% 12.24/7.90 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ J3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1565_split__pos__lemma,axiom,
% 12.24/7.90 ! [K: int,P: int > int > $o,N: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.90 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 12.24/7.90 = ( ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
% 12.24/7.90 & ( ord_less @ int @ J3 @ K )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ I5 @ J3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1566_split__neg__lemma,axiom,
% 12.24/7.90 ! [K: int,P: int > int > $o,N: int] :
% 12.24/7.90 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.90 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 12.24/7.90 = ( ! [I5: int,J3: int] :
% 12.24/7.90 ( ( ( ord_less @ int @ K @ J3 )
% 12.24/7.90 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
% 12.24/7.90 & ( N
% 12.24/7.90 = ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
% 12.24/7.90 => ( P @ I5 @ J3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1567_frac__add,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 12.24/7.90 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) ) )
% 12.24/7.90 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 12.24/7.90 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 12.24/7.90 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1568_mult__eq__if,axiom,
% 12.24/7.90 ( ( times_times @ nat )
% 12.24/7.90 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.90 ( if @ nat
% 12.24/7.90 @ ( M2
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 @ ( zero_zero @ nat )
% 12.24/7.90 @ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1569_Suc__as__int,axiom,
% 12.24/7.90 ( suc
% 12.24/7.90 = ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1570_split__mod,axiom,
% 12.24/7.90 ! [P: nat > $o,M: nat,N: nat] :
% 12.24/7.90 ( ( P @ ( modulo_modulo @ nat @ M @ N ) )
% 12.24/7.90 = ( ( ( N
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 => ( P @ M ) )
% 12.24/7.90 & ( ( N
% 12.24/7.90 != ( zero_zero @ nat ) )
% 12.24/7.90 => ! [I5: nat,J3: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ J3 @ N )
% 12.24/7.90 => ( ( M
% 12.24/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
% 12.24/7.90 => ( P @ J3 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1571_dividend__less__times__div,axiom,
% 12.24/7.90 ! [N: nat,M: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1572_dividend__less__div__times,axiom,
% 12.24/7.90 ! [N: nat,M: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1573_split__div,axiom,
% 12.24/7.90 ! [P: nat > $o,M: nat,N: nat] :
% 12.24/7.90 ( ( P @ ( divide_divide @ nat @ M @ N ) )
% 12.24/7.90 = ( ( ( N
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 => ( P @ ( zero_zero @ nat ) ) )
% 12.24/7.90 & ( ( N
% 12.24/7.90 != ( zero_zero @ nat ) )
% 12.24/7.90 => ! [I5: nat,J3: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ J3 @ N )
% 12.24/7.90 => ( ( M
% 12.24/7.90 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
% 12.24/7.90 => ( P @ I5 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1574_dvd__minus__add,axiom,
% 12.24/7.90 ! [Q4: nat,N: nat,R: nat,M: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ Q4 @ N )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R @ M ) )
% 12.24/7.90 => ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q4 ) )
% 12.24/7.90 = ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M ) @ Q4 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1575_convex__bound__lt,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord715952674999750819strict @ A )
% 12.24/7.90 => ! [X: A,A3: A,Y3: A,U: A,V: A] :
% 12.24/7.90 ( ( ord_less @ A @ X @ A3 )
% 12.24/7.90 => ( ( ord_less @ A @ Y3 @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 12.24/7.90 => ( ( ( plus_plus @ A @ U @ V )
% 12.24/7.90 = ( one_one @ A ) )
% 12.24/7.90 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1576_gbinomial__absorption,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [K: nat,A3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
% 12.24/7.90 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1577_scaling__mono,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [U: A,V: A,R: A,S: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ U @ V )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 12.24/7.90 => ( ( ord_less_eq @ A @ R @ S )
% 12.24/7.90 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1578_gbinomial__trinomial__revision,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [K: nat,M: nat,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ K @ M )
% 12.24/7.90 => ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
% 12.24/7.90 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1579_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.90 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.90 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1580_Suc__nat__eq__nat__zadd1,axiom,
% 12.24/7.90 ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.24/7.90 => ( ( suc @ ( nat2 @ Z3 ) )
% 12.24/7.90 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1581_div__pos__neg__trivial,axiom,
% 12.24/7.90 ! [K: int,L: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 12.24/7.90 => ( ( divide_divide @ int @ K @ L )
% 12.24/7.90 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1582_div__pos__geq,axiom,
% 12.24/7.90 ! [L: int,K: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.24/7.90 => ( ( ord_less_eq @ int @ L @ K )
% 12.24/7.90 => ( ( divide_divide @ int @ K @ L )
% 12.24/7.90 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1583_div__eqI,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid3128863361964157862miring @ A )
% 12.24/7.90 => ! [B2: A,R: A,Q4: A,A3: A] :
% 12.24/7.90 ( ( B2
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 12.24/7.90 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.24/7.90 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.24/7.90 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R )
% 12.24/7.90 = A3 )
% 12.24/7.90 => ( ( divide_divide @ A @ A3 @ B2 )
% 12.24/7.90 = Q4 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1584_div__bounded,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid3128863361964157862miring @ A )
% 12.24/7.90 => ! [B2: A,R: A,Q4: A] :
% 12.24/7.90 ( ( B2
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 12.24/7.90 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.24/7.90 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.24/7.90 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R ) @ B2 )
% 12.24/7.90 = Q4 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1585_mod__eqI,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid3128863361964157862miring @ A )
% 12.24/7.90 => ! [B2: A,R: A,Q4: A,A3: A] :
% 12.24/7.90 ( ( B2
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 12.24/7.90 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 12.24/7.90 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 12.24/7.90 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R )
% 12.24/7.90 = A3 )
% 12.24/7.90 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 12.24/7.90 = R ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1586_verit__le__mono__div,axiom,
% 12.24/7.90 ! [A6: nat,B7: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ A6 @ B7 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less_eq @ nat
% 12.24/7.90 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A6 @ N )
% 12.24/7.90 @ ( if @ nat
% 12.24/7.90 @ ( ( modulo_modulo @ nat @ B7 @ N )
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 @ ( one_one @ nat )
% 12.24/7.90 @ ( zero_zero @ nat ) ) )
% 12.24/7.90 @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1587_division__decomp,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_gcd @ A )
% 12.24/7.90 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.90 ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.90 => ? [B8: A,C5: A] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( times_times @ A @ B8 @ C5 ) )
% 12.24/7.90 & ( dvd_dvd @ A @ B8 @ B2 )
% 12.24/7.90 & ( dvd_dvd @ A @ C5 @ C2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1588_dvd__productE,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_gcd @ A )
% 12.24/7.90 => ! [P5: A,A3: A,B2: A] :
% 12.24/7.90 ( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.90 => ~ ! [X3: A,Y5: A] :
% 12.24/7.90 ( ( P5
% 12.24/7.90 = ( times_times @ A @ X3 @ Y5 ) )
% 12.24/7.90 => ( ( dvd_dvd @ A @ X3 @ A3 )
% 12.24/7.90 => ~ ( dvd_dvd @ A @ Y5 @ B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1589_gcd__nat_Oextremum__uniqueI,axiom,
% 12.24/7.90 ! [A3: nat] :
% 12.24/7.90 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.90 => ( A3
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1590_gcd__nat_Onot__eq__extremum,axiom,
% 12.24/7.90 ! [A3: nat] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 != ( zero_zero @ nat ) )
% 12.24/7.90 = ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.90 & ( A3
% 12.24/7.90 != ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1591_gcd__nat_Oextremum__unique,axiom,
% 12.24/7.90 ! [A3: nat] :
% 12.24/7.90 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1592_gcd__nat_Oextremum__strict,axiom,
% 12.24/7.90 ! [A3: nat] :
% 12.24/7.90 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.90 & ( ( zero_zero @ nat )
% 12.24/7.90 != A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1593_sum__squares__gt__zero__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord4710134922213307826strict @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) )
% 12.24/7.90 = ( ( X
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 | ( Y3
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1594_sum__squares__le__zero__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord4710134922213307826strict @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( ( X
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 & ( Y3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1595_sum__squares__eq__zero__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord4710134922213307826strict @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( ( X
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 & ( Y3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1596_add__scale__eq__noteq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri1453513574482234551roduct @ A )
% 12.24/7.90 => ! [R: A,A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.90 ( ( R
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ( A3 = B2 )
% 12.24/7.90 & ( C2 != D2 ) )
% 12.24/7.90 => ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
% 12.24/7.90 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1597_divide__int__def,axiom,
% 12.24/7.90 ( ( divide_divide @ int )
% 12.24/7.90 = ( ^ [K3: int,L3: int] :
% 12.24/7.90 ( if @ int
% 12.24/7.90 @ ( L3
% 12.24/7.90 = ( zero_zero @ int ) )
% 12.24/7.90 @ ( zero_zero @ int )
% 12.24/7.90 @ ( if @ int
% 12.24/7.90 @ ( ( sgn_sgn @ int @ K3 )
% 12.24/7.90 = ( sgn_sgn @ int @ L3 ) )
% 12.24/7.90 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
% 12.24/7.90 @ ( uminus_uminus @ int
% 12.24/7.90 @ ( semiring_1_of_nat @ int
% 12.24/7.90 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
% 12.24/7.90 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.90 @ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1598_gbinomial__pochhammer_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ( ( gbinomial @ A )
% 12.24/7.90 = ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1599_abs__idempotent,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1600_abs__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1601_abs__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1602_abs__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1603_abs__eq__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1604_abs__0__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ( zero_zero @ A )
% 12.24/7.90 = ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1605_abs__mult__self__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( times_times @ A @ A3 @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1606_abs__add__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
% 12.24/7.90 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1607_abs__1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1608_abs__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.90 = ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1609_abs__minus__cancel,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.90 = ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1610_dvd__abs__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: A,K: A] :
% 12.24/7.90 ( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
% 12.24/7.90 = ( dvd_dvd @ A @ M @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1611_abs__dvd__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: A,K: A] :
% 12.24/7.90 ( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
% 12.24/7.90 = ( dvd_dvd @ A @ M @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1612_abs__divide,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1613_abs__of__nat,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.24/7.90 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1614_abs__bool__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [P: $o] :
% 12.24/7.90 ( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
% 12.24/7.90 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1615_pochhammer__1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1616_nat__of__bool,axiom,
% 12.24/7.90 ! [P: $o] :
% 12.24/7.90 ( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
% 12.24/7.90 = ( zero_neq_one_of_bool @ nat @ P ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1617_abs__le__zero__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1618_abs__le__self__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
% 12.24/7.90 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1619_abs__of__nonneg,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1620_zero__less__abs__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1621_abs__neg__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1622_pochhammer__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1623_pochhammer__Suc0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1624_zdvd1__eq,axiom,
% 12.24/7.90 ! [X: int] :
% 12.24/7.90 ( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
% 12.24/7.90 = ( ( abs_abs @ int @ X )
% 12.24/7.90 = ( one_one @ int ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1625_abs__division__segment,axiom,
% 12.24/7.90 ! [K: int] :
% 12.24/7.90 ( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
% 12.24/7.90 = ( one_one @ int ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1626_abs__of__nonpos,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1627_zero__le__divide__abs__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
% 12.24/7.90 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 | ( B2
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1628_divide__le__0__abs__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.90 | ( B2
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1629_abs__sgn__eq__1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1630_zabs__less__one__iff,axiom,
% 12.24/7.90 ! [Z3: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( abs_abs @ int @ Z3 ) @ ( one_one @ int ) )
% 12.24/7.90 = ( Z3
% 12.24/7.90 = ( zero_zero @ int ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1631_idom__abs__sgn__class_Oabs__sgn,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = ( zero_neq_one_of_bool @ A
% 12.24/7.90 @ ( A3
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1632_sgn__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 = ( zero_neq_one_of_bool @ A
% 12.24/7.90 @ ( A3
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1633_nat__abs__dvd__iff,axiom,
% 12.24/7.90 ! [K: int,N: nat] :
% 12.24/7.90 ( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 12.24/7.90 = ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1634_dvd__nat__abs__iff,axiom,
% 12.24/7.90 ! [N: nat,K: int] :
% 12.24/7.90 ( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 12.24/7.90 = ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1635_pochhammer__of__nat,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [X: nat,N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
% 12.24/7.90 = ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1636_abs__ge__self,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1637_abs__le__D1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 12.24/7.90 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1638_abs__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1639_abs__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1640_abs__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1641_abs__minus__commute,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1642_abs__eq__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_ring @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ( abs_abs @ A @ X )
% 12.24/7.90 = ( abs_abs @ A @ Y3 ) )
% 12.24/7.90 = ( ( X = Y3 )
% 12.24/7.90 | ( X
% 12.24/7.90 = ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1643_dvd__if__abs__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [L: A,K: A] :
% 12.24/7.90 ( ( ( abs_abs @ A @ L )
% 12.24/7.90 = ( abs_abs @ A @ K ) )
% 12.24/7.90 => ( dvd_dvd @ A @ L @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1644_zdvd__antisym__abs,axiom,
% 12.24/7.90 ! [A3: int,B2: int] :
% 12.24/7.90 ( ( dvd_dvd @ int @ A3 @ B2 )
% 12.24/7.90 => ( ( dvd_dvd @ int @ B2 @ A3 )
% 12.24/7.90 => ( ( abs_abs @ int @ A3 )
% 12.24/7.90 = ( abs_abs @ int @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1645_abs__ge__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1646_abs__not__less__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1647_abs__of__pos,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1648_abs__triangle__ineq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1649_abs__triangle__ineq2,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1650_abs__triangle__ineq3,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1651_abs__triangle__ineq2__sym,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1652_abs__mult__less,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
% 12.24/7.90 => ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
% 12.24/7.90 => ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1653_abs__ge__minus__self,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1654_abs__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 12.24/7.90 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.90 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1655_abs__le__D2,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1656_abs__leI,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1657_abs__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 12.24/7.90 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.90 & ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1658_nonzero__abs__divide,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,A3: A] :
% 12.24/7.90 ( ( B2
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1659_linordered__idom__class_Oabs__sgn,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ( ( abs_abs @ A )
% 12.24/7.90 = ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1660_abs__mult__sgn,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1661_sgn__mult__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom_abs_sgn @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1662_mult__sgn__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
% 12.24/7.90 = X ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1663_same__sgn__abs__add,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [B2: A,A3: A] :
% 12.24/7.90 ( ( ( sgn_sgn @ A @ B2 )
% 12.24/7.90 = ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1664_abs__zmult__eq__1,axiom,
% 12.24/7.90 ! [M: int,N: int] :
% 12.24/7.90 ( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
% 12.24/7.90 = ( one_one @ int ) )
% 12.24/7.90 => ( ( abs_abs @ int @ M )
% 12.24/7.90 = ( one_one @ int ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1665_nat__abs__triangle__ineq,axiom,
% 12.24/7.90 ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1666_pochhammer__pos,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.90 => ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1667_pochhammer__neq__0__mono,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1668_pochhammer__eq__0__mono,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [A3: A,N: nat,M: nat] :
% 12.24/7.90 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1669_pochhammer__fact,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( semiring_char_0 @ A )
% 12.24/7.90 & ( comm_semiring_1 @ A ) )
% 12.24/7.90 => ( ( semiring_char_0_fact @ A )
% 12.24/7.90 = ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1670_dense__eq0__I,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 & ( dense_linorder @ A ) )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ! [E2: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
% 12.24/7.90 => ( X
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1671_abs__eq__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordered_ring_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 | ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 12.24/7.90 & ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 | ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.90 = ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1672_abs__mult__pos,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.90 => ( ( times_times @ A @ ( abs_abs @ A @ Y3 ) @ X )
% 12.24/7.90 = ( abs_abs @ A @ ( times_times @ A @ Y3 @ X ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1673_abs__minus__le__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1674_eq__abs__iff_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_ring @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( abs_abs @ A @ B2 ) )
% 12.24/7.90 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 & ( ( B2 = A3 )
% 12.24/7.90 | ( B2
% 12.24/7.90 = ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1675_abs__eq__iff_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_ring @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = B2 )
% 12.24/7.90 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 & ( ( A3 = B2 )
% 12.24/7.90 | ( A3
% 12.24/7.90 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1676_abs__of__neg,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ A3 )
% 12.24/7.90 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1677_abs__if__raw,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( abs_if @ A )
% 12.24/7.90 => ( ( abs_abs @ A )
% 12.24/7.90 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1678_abs__if,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( abs_if @ A )
% 12.24/7.90 => ( ( abs_abs @ A )
% 12.24/7.90 = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1679_abs__diff__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A,A3: A,R: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 12.24/7.90 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 12.24/7.90 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1680_abs__diff__triangle__ineq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [A3: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1681_abs__triangle__ineq4,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.90 => ! [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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1682_abs__div__pos,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [Y3: A,X: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.90 => ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y3 )
% 12.24/7.90 = ( abs_abs @ A @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1683_abs__diff__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A,A3: A,R: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 12.24/7.90 = ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 12.24/7.90 & ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1684_div__noneq__sgn__abs,axiom,
% 12.24/7.90 ! [L: int,K: int] :
% 12.24/7.90 ( ( L
% 12.24/7.90 != ( zero_zero @ int ) )
% 12.24/7.90 => ( ( ( sgn_sgn @ int @ K )
% 12.24/7.90 != ( sgn_sgn @ int @ L ) )
% 12.24/7.90 => ( ( divide_divide @ int @ K @ L )
% 12.24/7.90 = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
% 12.24/7.90 @ ( zero_neq_one_of_bool @ int
% 12.24/7.90 @ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1685_abs__sgn__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( ( A3
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 = ( zero_zero @ A ) ) )
% 12.24/7.90 & ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1686_zabs__def,axiom,
% 12.24/7.90 ( ( abs_abs @ int )
% 12.24/7.90 = ( ^ [I5: int] : ( if @ int @ ( ord_less @ int @ I5 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I5 ) @ I5 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1687_dvd__imp__le__int,axiom,
% 12.24/7.90 ! [I: int,D2: int] :
% 12.24/7.90 ( ( I
% 12.24/7.90 != ( zero_zero @ int ) )
% 12.24/7.90 => ( ( dvd_dvd @ int @ D2 @ I )
% 12.24/7.90 => ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1688_nat__abs__mult__distrib,axiom,
% 12.24/7.90 ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z3 ) ) )
% 12.24/7.90 = ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1689_abs__mod__less,axiom,
% 12.24/7.90 ! [L: int,K: int] :
% 12.24/7.90 ( ( L
% 12.24/7.90 != ( zero_zero @ int ) )
% 12.24/7.90 => ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1690_pochhammer__nonneg,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.90 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1691_pochhammer__0__left,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( ( N
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 12.24/7.90 = ( one_one @ A ) ) )
% 12.24/7.90 & ( ( N
% 12.24/7.90 != ( zero_zero @ nat ) )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1692_pochhammer__product_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [Z3: A,N: nat,M: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N @ M ) )
% 12.24/7.90 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1693_abs__add__one__gt__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1694_zdvd__mult__cancel1,axiom,
% 12.24/7.90 ! [M: int,N: int] :
% 12.24/7.90 ( ( M
% 12.24/7.90 != ( zero_zero @ int ) )
% 12.24/7.90 => ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
% 12.24/7.90 = ( ( abs_abs @ int @ N )
% 12.24/7.90 = ( one_one @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1695_div__sgn__abs__cancel,axiom,
% 12.24/7.90 ! [V: int,K: int,L: int] :
% 12.24/7.90 ( ( V
% 12.24/7.90 != ( zero_zero @ int ) )
% 12.24/7.90 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ L ) ) )
% 12.24/7.90 = ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1696_div__abs__eq__div__nat,axiom,
% 12.24/7.90 ! [K: int,L: int] :
% 12.24/7.90 ( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
% 12.24/7.90 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1697_mod__abs__eq__div__nat,axiom,
% 12.24/7.90 ! [K: int,L: int] :
% 12.24/7.90 ( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
% 12.24/7.90 = ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1698_pochhammer__rec,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1699_pochhammer__Suc,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1700_pochhammer__rec_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [Z3: A,N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) )
% 12.24/7.90 = ( times_times @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1701_div__dvd__sgn__abs,axiom,
% 12.24/7.90 ! [L: int,K: int] :
% 12.24/7.90 ( ( dvd_dvd @ int @ L @ K )
% 12.24/7.90 => ( ( divide_divide @ int @ K @ L )
% 12.24/7.90 = ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1702_pochhammer__of__nat__eq__0__lemma,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( idom @ A )
% 12.24/7.90 => ! [N: nat,K: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ N @ K )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1703_pochhammer__of__nat__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( ring_char_0 @ A )
% 12.24/7.90 & ( idom @ A ) )
% 12.24/7.90 => ! [N: nat,K: nat] :
% 12.24/7.90 ( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( ord_less @ nat @ N @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1704_pochhammer__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( ? [K3: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ K3 @ N )
% 12.24/7.90 & ( A3
% 12.24/7.90 = ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1705_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( ring_char_0 @ A )
% 12.24/7.90 & ( idom @ A ) )
% 12.24/7.90 => ! [K: nat,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1706_modulo__int__def,axiom,
% 12.24/7.90 ( ( modulo_modulo @ int )
% 12.24/7.90 = ( ^ [K3: int,L3: int] :
% 12.24/7.90 ( if @ int
% 12.24/7.90 @ ( L3
% 12.24/7.90 = ( zero_zero @ int ) )
% 12.24/7.90 @ K3
% 12.24/7.90 @ ( if @ int
% 12.24/7.90 @ ( ( sgn_sgn @ int @ K3 )
% 12.24/7.90 = ( sgn_sgn @ int @ L3 ) )
% 12.24/7.90 @ ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) )
% 12.24/7.90 @ ( times_times @ int @ ( sgn_sgn @ int @ L3 )
% 12.24/7.90 @ ( minus_minus @ int
% 12.24/7.90 @ ( times_times @ int @ ( abs_abs @ int @ L3 )
% 12.24/7.90 @ ( zero_neq_one_of_bool @ int
% 12.24/7.90 @ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) )
% 12.24/7.90 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1707_incr__lemma,axiom,
% 12.24/7.90 ! [D2: int,Z3: int,X: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.24/7.90 => ( ord_less @ int @ Z3 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1708_decr__lemma,axiom,
% 12.24/7.90 ! [D2: int,X: int,Z3: int] :
% 12.24/7.90 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.24/7.90 => ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1709_nat__abs__int__diff,axiom,
% 12.24/7.90 ! [A3: nat,B2: nat] :
% 12.24/7.90 ( ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.90 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 12.24/7.90 = ( minus_minus @ nat @ B2 @ A3 ) ) )
% 12.24/7.90 & ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.90 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
% 12.24/7.90 = ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1710_pochhammer__product,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [M: nat,N: nat,Z3: A] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.90 => ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N )
% 12.24/7.90 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1711_nat__intermed__int__val,axiom,
% 12.24/7.90 ! [M: nat,N: nat,F: nat > int,K: int] :
% 12.24/7.90 ( ! [I4: nat] :
% 12.24/7.90 ( ( ( ord_less_eq @ nat @ M @ I4 )
% 12.24/7.90 & ( ord_less @ nat @ I4 @ N ) )
% 12.24/7.90 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( F @ M ) @ K )
% 12.24/7.90 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 12.24/7.90 => ? [I4: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ M @ I4 )
% 12.24/7.90 & ( ord_less_eq @ nat @ I4 @ N )
% 12.24/7.90 & ( ( F @ I4 )
% 12.24/7.90 = K ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1712_pochhammer__absorb__comp,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [R: A,K: nat] :
% 12.24/7.90 ( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
% 12.24/7.90 = ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1713_add__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri1453513574482234551roduct @ A )
% 12.24/7.90 => ! [B2: A,A3: A] :
% 12.24/7.90 ( ( B2
% 12.24/7.90 = ( plus_plus @ A @ B2 @ A3 ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1714_crossproduct__noteq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri1453513574482234551roduct @ A )
% 12.24/7.90 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.90 ( ( ( A3 != B2 )
% 12.24/7.90 & ( C2 != D2 ) )
% 12.24/7.90 = ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
% 12.24/7.90 != ( plus_plus @ A @ ( times_times @ A @ A3 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1715_crossproduct__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri1453513574482234551roduct @ A )
% 12.24/7.90 => ! [W2: A,Y3: A,X: A,Z3: A] :
% 12.24/7.90 ( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y3 ) @ ( times_times @ A @ X @ Z3 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( times_times @ A @ W2 @ Z3 ) @ ( times_times @ A @ X @ Y3 ) ) )
% 12.24/7.90 = ( ( W2 = X )
% 12.24/7.90 | ( Y3 = Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1716_nat__ivt__aux,axiom,
% 12.24/7.90 ! [N: nat,F: nat > int,K: int] :
% 12.24/7.90 ( ! [I4: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ I4 @ N )
% 12.24/7.90 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 12.24/7.90 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 12.24/7.90 => ? [I4: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ I4 @ N )
% 12.24/7.90 & ( ( F @ I4 )
% 12.24/7.90 = K ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1717_nat0__intermed__int__val,axiom,
% 12.24/7.90 ! [N: nat,F: nat > int,K: int] :
% 12.24/7.90 ( ! [I4: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ I4 @ N )
% 12.24/7.90 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
% 12.24/7.90 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 12.24/7.90 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 12.24/7.90 => ? [I4: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ I4 @ N )
% 12.24/7.90 & ( ( F @ I4 )
% 12.24/7.90 = K ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1718_gbinomial__pochhammer,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ( ( gbinomial @ A )
% 12.24/7.90 = ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1719_int__power__div__base,axiom,
% 12.24/7.90 ! [M: nat,K: int] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.24/7.90 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.90 => ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K )
% 12.24/7.90 = ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1720_pochhammer__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [B2: A,K: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
% 12.24/7.90 = ( 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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1721_pochhammer__minus_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [B2: A,K: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1722_gbinomial__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [A3: A,K: nat] :
% 12.24/7.90 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
% 12.24/7.90 = ( 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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1723_triangle__Suc,axiom,
% 12.24/7.90 ! [N: nat] :
% 12.24/7.90 ( ( nat_triangle @ ( suc @ N ) )
% 12.24/7.90 = ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1724_frac__unique__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,A3: A] :
% 12.24/7.90 ( ( ( archimedean_frac @ A @ X )
% 12.24/7.90 = A3 )
% 12.24/7.90 = ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
% 12.24/7.90 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 & ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1725_power__Suc__0,axiom,
% 12.24/7.90 ! [N: nat] :
% 12.24/7.90 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.90 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1726_nat__power__eq__Suc__0__iff,axiom,
% 12.24/7.90 ! [X: nat,M: nat] :
% 12.24/7.90 ( ( ( power_power @ nat @ X @ M )
% 12.24/7.90 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.90 = ( ( M
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 | ( X
% 12.24/7.90 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1727_power__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1728_nat__zero__less__power__iff,axiom,
% 12.24/7.90 ! [X: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 12.24/7.90 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 12.24/7.90 | ( N
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1729_of__nat__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [M: nat,N: nat] :
% 12.24/7.90 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 12.24/7.90 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1730_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [B2: nat,W2: nat,X: nat] :
% 12.24/7.90 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 )
% 12.24/7.90 = ( semiring_1_of_nat @ A @ X ) )
% 12.24/7.90 = ( ( power_power @ nat @ B2 @ W2 )
% 12.24/7.90 = X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1731_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [X: nat,B2: nat,W2: nat] :
% 12.24/7.90 ( ( ( semiring_1_of_nat @ A @ X )
% 12.24/7.90 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( X
% 12.24/7.90 = ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1732_power__one__right,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( one_one @ nat ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1733_frac__in__Ints__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
% 12.24/7.90 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1734_triangle__0,axiom,
% 12.24/7.90 ( ( nat_triangle @ ( zero_zero @ nat ) )
% 12.24/7.90 = ( zero_zero @ nat ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1735_power__inject__exp,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ( ( power_power @ A @ A3 @ M )
% 12.24/7.90 = ( power_power @ A @ A3 @ N ) )
% 12.24/7.90 = ( M = N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1736_power__0__Suc,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1737_power__Suc0__right,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1738_abs__power__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 12.24/7.90 = ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1739_frac__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( ( archimedean_frac @ A @ X )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1740_minus__one__mult__self,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1741_left__minus__one__mult__self,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [N: nat,A3: A] :
% 12.24/7.90 ( ( 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 ) )
% 12.24/7.90 = A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1742_power__strict__increasing__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: A,X: nat,Y3: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
% 12.24/7.90 = ( ord_less @ nat @ X @ Y3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1743_power__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri2026040879449505780visors @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ( power_power @ A @ A3 @ N )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( ( A3
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1744_of__nat__power__less__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: nat,B2: nat,W2: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1745_of__nat__less__of__nat__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: nat,W2: nat,X: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
% 12.24/7.90 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1746_of__nat__power__le__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: nat,B2: nat,W2: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1747_of__nat__le__of__nat__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: nat,W2: nat,X: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1748_pow__divides__pow__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_gcd @ A )
% 12.24/7.90 => ! [N: nat,A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1749_frac__gt__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
% 12.24/7.90 = ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1750_power__strict__decreasing__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 12.24/7.90 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 = ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1751_power__increasing__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: A,X: nat,Y3: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ X @ Y3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1752_power__mono__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1753_zero__less__power__abs__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
% 12.24/7.90 = ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 | ( N
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1754_power__decreasing__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [B2: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1755_of__nat__zero__less__power__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 12.24/7.90 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 12.24/7.90 | ( N
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1756_Ints__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1757_power__not__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiri2026040879449505780visors @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( power_power @ A @ A3 @ N )
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1758_power__commutes,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
% 12.24/7.90 = ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1759_power__mult__distrib,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1760_power__commuting__commutes,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [X: A,Y3: A,N: nat] :
% 12.24/7.90 ( ( ( times_times @ A @ X @ Y3 )
% 12.24/7.90 = ( times_times @ A @ Y3 @ X ) )
% 12.24/7.90 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y3 )
% 12.24/7.90 = ( times_times @ A @ Y3 @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1761_power__mod,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.90 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.90 ( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ N ) @ B2 )
% 12.24/7.90 = ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1762_power__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( times_times @ nat @ M @ N ) )
% 12.24/7.90 = ( power_power @ A @ ( power_power @ A @ A3 @ M ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1763_Ints__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1764_Ints__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1765_Ints__add,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1766_Ints__1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1767_Ints__diff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A,B2: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1768_minus__in__Ints__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
% 12.24/7.90 = ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1769_Ints__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1770_Ints__of__nat,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1771_Ints__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( member @ A @ ( abs_abs @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1772_fact__in__Ints,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( semiring_char_0 @ A )
% 12.24/7.90 & ( ring_1 @ A ) )
% 12.24/7.90 => ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1773_nat__power__eq,axiom,
% 12.24/7.90 ! [Z3: int,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.24/7.90 => ( ( nat2 @ ( power_power @ int @ Z3 @ N ) )
% 12.24/7.90 = ( power_power @ nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1774_power__mono,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1775_zero__le__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1776_zero__less__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1777_one__le__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1778_left__right__inverse__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [X: A,Y3: A,N: nat] :
% 12.24/7.90 ( ( ( times_times @ A @ X @ Y3 )
% 12.24/7.90 = ( one_one @ A ) )
% 12.24/7.90 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1779_power__Suc2,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1780_power__Suc,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( power @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1781_power__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( power @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1782_power__one__over,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( division_ring @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
% 12.24/7.90 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1783_power__add,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1784_le__imp__power__dvd,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [M: nat,N: nat,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.90 => ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1785_power__le__dvd,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [A3: A,N: nat,B2: A,M: nat] :
% 12.24/7.90 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.90 => ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1786_dvd__power__le,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [X: A,Y3: A,N: nat,M: nat] :
% 12.24/7.90 ( ( dvd_dvd @ A @ X @ Y3 )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ M ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1787_nat__power__less__imp__less,axiom,
% 12.24/7.90 ! [I: nat,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 12.24/7.90 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 12.24/7.90 => ( ord_less @ nat @ M @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1788_Ints__double__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( ( plus_plus @ A @ A3 @ A3 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1789_power__less__imp__less__base,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat,B2: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1790_power__le__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1791_power__inject__base,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat,B2: A] :
% 12.24/7.90 ( ( ( power_power @ A @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( power_power @ A @ B2 @ ( suc @ N ) ) )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( A3 = B2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1792_power__le__imp__le__base,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat,B2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1793_power__gt1__lemma,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1794_power__less__power__Suc,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1795_power__0__left,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( ( N
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 12.24/7.90 = ( one_one @ A ) ) )
% 12.24/7.90 & ( ( N
% 12.24/7.90 != ( zero_zero @ nat ) )
% 12.24/7.90 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1796_power__gt1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1797_power__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1798_power__less__imp__less__exp,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.90 => ( ord_less @ nat @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1799_power__strict__increasing,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat,N6: nat,A3: A] :
% 12.24/7.90 ( ( ord_less @ nat @ N @ N6 )
% 12.24/7.90 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1800_power__increasing,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat,N6: nat,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ N @ N6 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1801_zero__le__power__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1802_zero__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1803_is__unit__power__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( algebraic_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
% 12.24/7.90 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 | ( N
% 12.24/7.90 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1804_power__gt__expt,axiom,
% 12.24/7.90 ! [N: nat,K: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.90 => ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1805_nat__one__le__power,axiom,
% 12.24/7.90 ! [I: nat,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 12.24/7.90 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1806_fact__le__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1807_Ints__odd__nonzero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
% 12.24/7.90 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1808_bounded__Max__nat,axiom,
% 12.24/7.90 ! [P: nat > $o,X: nat,M7: nat] :
% 12.24/7.90 ( ( P @ X )
% 12.24/7.90 => ( ! [X3: nat] :
% 12.24/7.90 ( ( P @ X3 )
% 12.24/7.90 => ( ord_less_eq @ nat @ X3 @ M7 ) )
% 12.24/7.90 => ~ ! [M6: nat] :
% 12.24/7.90 ( ( P @ M6 )
% 12.24/7.90 => ~ ! [X4: nat] :
% 12.24/7.90 ( ( P @ X4 )
% 12.24/7.90 => ( ord_less_eq @ nat @ X4 @ M6 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1809_power__Suc__less,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1810_power__Suc__le__self,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1811_power__Suc__less__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1812_power__strict__decreasing,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat,N6: nat,A3: A] :
% 12.24/7.90 ( ( ord_less @ nat @ N @ N6 )
% 12.24/7.90 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1813_power__decreasing,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat,N6: nat,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ N @ N6 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1814_power__le__imp__le__exp,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.90 => ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1815_power__eq__imp__eq__base,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat,B2: A] :
% 12.24/7.90 ( ( ( power_power @ A @ A3 @ N )
% 12.24/7.90 = ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( A3 = B2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1816_power__eq__iff__eq__base,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [N: nat,A3: A,B2: A] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.90 => ( ( ( power_power @ A @ A3 @ N )
% 12.24/7.90 = ( power_power @ A @ B2 @ N ) )
% 12.24/7.90 = ( A3 = B2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1817_self__le__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1818_one__less__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1819_dvd__power__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( algebraic_semidom @ A )
% 12.24/7.90 => ! [X: A,M: nat,N: nat] :
% 12.24/7.90 ( ( X
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
% 12.24/7.90 = ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
% 12.24/7.90 | ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1820_dvd__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_semiring_1 @ A )
% 12.24/7.90 => ! [N: nat,X: A] :
% 12.24/7.90 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 | ( X
% 12.24/7.90 = ( one_one @ A ) ) )
% 12.24/7.90 => ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1821_power__diff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semidom_divide @ A )
% 12.24/7.90 => ! [A3: A,N: nat,M: nat] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
% 12.24/7.90 = ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1822_gbinomial__ge__n__over__k__pow__k,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [K: nat,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1823_power__dvd__imp__le,axiom,
% 12.24/7.90 ! [I: nat,M: nat,N: nat] :
% 12.24/7.90 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 12.24/7.90 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
% 12.24/7.90 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1824_Ints__odd__less__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: A] :
% 12.24/7.90 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1825_fact__div__fact__le__pow,axiom,
% 12.24/7.90 ! [R: nat,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ R @ N )
% 12.24/7.90 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R ) ) ) @ ( power_power @ nat @ N @ R ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1826_Ints__nonzero__abs__ge1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( X
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1827_Ints__nonzero__abs__less1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 12.24/7.90 => ( X
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1828_Ints__eq__abs__less1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: A,Y3: A] :
% 12.24/7.90 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( member @ A @ Y3 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( X = Y3 )
% 12.24/7.90 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1829_power__strict__mono,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.90 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.90 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1830_power__diff__power__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.90 => ! [A3: A,N: nat,M: nat] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.90 = ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
% 12.24/7.90 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 12.24/7.90 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.90 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1831_power__eq__if,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( power @ A )
% 12.24/7.90 => ( ( power_power @ A )
% 12.24/7.90 = ( ^ [P6: A,M2: nat] :
% 12.24/7.90 ( if @ A
% 12.24/7.90 @ ( M2
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 @ ( one_one @ A )
% 12.24/7.90 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1832_power__minus__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [N: nat,A3: A] :
% 12.24/7.90 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.90 => ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
% 12.24/7.90 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1833_frac__neg,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 12.24/7.90 = ( zero_zero @ A ) ) )
% 12.24/7.90 & ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 12.24/7.90 = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1834_gbinomial__negated__upper,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ( ( gbinomial @ A )
% 12.24/7.90 = ( ^ [A5: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A5 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1835_gbinomial__index__swap,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( field_char_0 @ A )
% 12.24/7.90 => ! [K: nat,N: nat] :
% 12.24/7.90 ( ( 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 ) )
% 12.24/7.90 = ( 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 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1836_pochhammer__same,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( semiring_char_0 @ A )
% 12.24/7.90 & ( comm_ring_1 @ A )
% 12.24/7.90 & ( semiri3467727345109120633visors @ A ) )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1837_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [K: nat,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.90 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
% 12.24/7.90 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1838_power__minus_H,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [X: A,N: nat] :
% 12.24/7.90 ( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
% 12.24/7.90 => ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1839_ceiling__divide__lower,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [Q4: A,P5: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 12.24/7.90 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P5 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1840_power_Opower__eq__if,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( power2 @ A )
% 12.24/7.90 = ( ^ [One: A,Times: A > A > A,P6: A,M2: nat] :
% 12.24/7.90 ( if @ A
% 12.24/7.90 @ ( M2
% 12.24/7.90 = ( zero_zero @ nat ) )
% 12.24/7.90 @ One
% 12.24/7.90 @ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1841_le__divide__eq__numeral_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.90 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.90 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 12.24/7.90 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.90 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 12.24/7.90 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1842_divide__le__eq__numeral_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.90 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 12.24/7.90 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.90 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 12.24/7.90 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1843_mult__ceiling__le__Ints,axiom,
% 12.24/7.90 ! [A: $tType,B: $tType] :
% 12.24/7.90 ( ( ( archim2362893244070406136eiling @ B )
% 12.24/7.90 & ( linordered_idom @ A ) )
% 12.24/7.90 => ! [A3: B,B2: B] :
% 12.24/7.90 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
% 12.24/7.90 => ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
% 12.24/7.90 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1844_numeral__eq__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ( numeral_numeral @ A @ M )
% 12.24/7.90 = ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( M = N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1845_int__eq__iff__numeral,axiom,
% 12.24/7.90 ! [M: nat,V: num] :
% 12.24/7.90 ( ( ( semiring_1_of_nat @ int @ M )
% 12.24/7.90 = ( numeral_numeral @ int @ V ) )
% 12.24/7.90 = ( M
% 12.24/7.90 = ( numeral_numeral @ nat @ V ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1846_of__int__eq__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ W2 )
% 12.24/7.90 = ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( W2 = Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1847_nat__numeral,axiom,
% 12.24/7.90 ! [K: num] :
% 12.24/7.90 ( ( nat2 @ ( numeral_numeral @ int @ K ) )
% 12.24/7.90 = ( numeral_numeral @ nat @ K ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1848_numeral__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1849_numeral__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( ord_less @ num @ M @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1850_mult__numeral__left__semiring__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_numeral @ A )
% 12.24/7.90 => ! [V: num,W2: num,Z3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
% 12.24/7.90 = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1851_numeral__times__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_numeral @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1852_add__numeral__left,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( numeral @ A )
% 12.24/7.90 => ! [V: num,W2: num,Z3: A] :
% 12.24/7.90 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1853_numeral__plus__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( numeral @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1854_neg__numeral__eq__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( M = N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1855_power__zero__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1856_power__add__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,M: num,N: num] :
% 12.24/7.90 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
% 12.24/7.90 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1857_power__add__numeral2,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( monoid_mult @ A )
% 12.24/7.90 => ! [A3: A,M: num,N: num,B2: A] :
% 12.24/7.90 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
% 12.24/7.90 = ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1858_of__nat__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_1 @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.90 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1859_abs__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1860_of__int__eq__numeral__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Z3: int,N: num] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ Z3 )
% 12.24/7.90 = ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( Z3
% 12.24/7.90 = ( numeral_numeral @ int @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1861_of__int__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
% 12.24/7.90 = ( numeral_numeral @ A @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1862_ceiling__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
% 12.24/7.90 = ( numeral_numeral @ int @ V ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1863_euclidean__size__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( euclid6346220572633701492n_size @ A @ ( numeral_numeral @ A @ K ) )
% 12.24/7.90 = ( numeral_numeral @ nat @ K ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1864_of__int__fact,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( semiring_char_0 @ A )
% 12.24/7.90 & ( ring_1 @ A ) )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( semiring_char_0_fact @ int @ N ) )
% 12.24/7.90 = ( semiring_char_0_fact @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1865_ceiling__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = Z3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1866_round__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( numeral_numeral @ int @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1867_round__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [N: int] :
% 12.24/7.90 ( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
% 12.24/7.90 = N ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1868_neg__numeral__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1869_distrib__left__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( numeral @ A )
% 12.24/7.90 & ( semiring @ A ) )
% 12.24/7.90 => ! [V: num,B2: A,C2: A] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1870_distrib__right__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( numeral @ A )
% 12.24/7.90 & ( semiring @ A ) )
% 12.24/7.90 => ! [A3: A,B2: A,V: num] :
% 12.24/7.90 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.90 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1871_neg__numeral__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( ord_less @ num @ N @ M ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1872_right__diff__distrib__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( numeral @ A )
% 12.24/7.90 & ( ring @ A ) )
% 12.24/7.90 => ! [V: num,B2: A,C2: A] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.24/7.90 = ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1873_left__diff__distrib__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ( numeral @ A )
% 12.24/7.90 & ( ring @ A ) )
% 12.24/7.90 => ! [A3: A,B2: A,V: num] :
% 12.24/7.90 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.90 = ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1874_mult__neg__numeral__simps_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1875_mult__neg__numeral__simps_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1876_mult__neg__numeral__simps_I3_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1877_semiring__norm_I172_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 12.24/7.90 = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1878_semiring__norm_I171_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 12.24/7.90 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1879_semiring__norm_I170_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
% 12.24/7.90 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1880_semiring__norm_I168_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.90 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1881_add__neg__numeral__simps_I3_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1882_diff__numeral__simps_I3_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1883_diff__numeral__simps_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1884_abs__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1885_of__int__0,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1886_of__int__0__eq__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ( zero_zero @ A )
% 12.24/7.90 = ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( Z3
% 12.24/7.90 = ( zero_zero @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1887_of__int__eq__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ Z3 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 = ( Z3
% 12.24/7.90 = ( zero_zero @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1888_of__int__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1889_of__int__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1890_of__int__eq__1__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ Z3 )
% 12.24/7.90 = ( one_one @ A ) )
% 12.24/7.90 = ( Z3
% 12.24/7.90 = ( one_one @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1891_of__int__1,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1892_of__int__mult,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z3 ) )
% 12.24/7.90 = ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1893_of__int__add,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z3 ) )
% 12.24/7.90 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1894_of__int__minus,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z3 ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1895_nat__neg__numeral,axiom,
% 12.24/7.90 ! [K: num] :
% 12.24/7.90 ( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
% 12.24/7.90 = ( zero_zero @ nat ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1896_of__int__diff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [W2: int,Z3: int] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z3 ) )
% 12.24/7.90 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1897_division__segment__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
% 12.24/7.90 = ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1898_of__int__of__nat__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [N: nat] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 12.24/7.90 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1899_diff__nat__numeral,axiom,
% 12.24/7.90 ! [V: num,V2: num] :
% 12.24/7.90 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V2 ) )
% 12.24/7.90 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1900_of__int__abs,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: int] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
% 12.24/7.90 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1901_of__int__power__eq__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [X: int,B2: int,W2: nat] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ X )
% 12.24/7.90 = ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( X
% 12.24/7.90 = ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1902_of__int__eq__of__int__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [B2: int,W2: nat,X: int] :
% 12.24/7.90 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
% 12.24/7.90 = ( ring_1_of_int @ A @ X ) )
% 12.24/7.90 = ( ( power_power @ int @ B2 @ W2 )
% 12.24/7.90 = X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1903_of__int__power,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Z3: int,N: nat] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z3 @ N ) )
% 12.24/7.90 = ( power_power @ A @ ( ring_1_of_int @ A @ Z3 ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1904_frac__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1905_nat__eq__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [Y3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ( nat2 @ Y3 )
% 12.24/7.90 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 12.24/7.90 = ( Y3
% 12.24/7.90 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1906_numeral__power__eq__nat__cancel__iff,axiom,
% 12.24/7.90 ! [X: num,N: nat,Y3: int] :
% 12.24/7.90 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 12.24/7.90 = ( nat2 @ Y3 ) )
% 12.24/7.90 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 12.24/7.90 = Y3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1907_of__int__of__bool,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [P: $o] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
% 12.24/7.90 = ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1908_le__divide__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1909_divide__le__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
% 12.24/7.90 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1910_divide__eq__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( division_ring @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 = A3 )
% 12.24/7.90 = ( ( ( ( numeral_numeral @ A @ W2 )
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( B2
% 12.24/7.90 = ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
% 12.24/7.90 & ( ( ( numeral_numeral @ A @ W2 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1911_eq__divide__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( division_ring @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = ( ( ( ( numeral_numeral @ A @ W2 )
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 = B2 ) )
% 12.24/7.90 & ( ( ( numeral_numeral @ A @ W2 )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1912_less__divide__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1913_divide__less__eq__numeral1_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
% 12.24/7.90 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1914_of__int__le__numeral__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int,N: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( ord_less_eq @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1915_of__int__numeral__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [N: num,Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1916_of__int__numeral__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [N: num,Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1917_of__int__less__numeral__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int,N: num] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.90 = ( ord_less @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1918_ceiling__le__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 12.24/7.90 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1919_numeral__less__ceiling,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num,X: A] :
% 12.24/7.90 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.90 = ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1920_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [Y3: nat,X: num,N: nat] :
% 12.24/7.90 ( ( ( semiring_1_of_nat @ A @ Y3 )
% 12.24/7.90 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 12.24/7.90 = ( Y3
% 12.24/7.90 = ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1921_numeral__power__eq__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [X: num,N: nat,Y3: nat] :
% 12.24/7.90 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 12.24/7.90 = ( semiring_1_of_nat @ A @ Y3 ) )
% 12.24/7.90 = ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
% 12.24/7.90 = Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1922_ceiling__add__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.90 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1923_ceiling__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.90 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1924_ceiling__diff__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.90 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1925_Suc__times__numeral__mod__eq,axiom,
% 12.24/7.90 ! [K: num,N: nat] :
% 12.24/7.90 ( ( ( numeral_numeral @ nat @ K )
% 12.24/7.90 != ( one_one @ nat ) )
% 12.24/7.90 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.90 = ( one_one @ nat ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1926_numeral__power__eq__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [X: num,N: nat,Y3: int] :
% 12.24/7.90 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
% 12.24/7.90 = ( ring_1_of_int @ A @ Y3 ) )
% 12.24/7.90 = ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
% 12.24/7.90 = Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1927_of__int__eq__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Y3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ Y3 )
% 12.24/7.90 = ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 12.24/7.90 = ( Y3
% 12.24/7.90 = ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1928_ceiling__add__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,Z3: int] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 12.24/7.90 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1929_ceiling__diff__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,Z3: int] :
% 12.24/7.90 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 12.24/7.90 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1930_round__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.90 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1931_le__divide__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 12.24/7.90 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1932_divide__le__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
% 12.24/7.90 = ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1933_divide__eq__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( division_ring @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = A3 )
% 12.24/7.90 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( B2
% 12.24/7.90 = ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
% 12.24/7.90 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1934_eq__divide__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( division_ring @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( A3
% 12.24/7.90 = ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 12.24/7.90 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 != ( zero_zero @ A ) )
% 12.24/7.90 => ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.90 = B2 ) )
% 12.24/7.90 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.90 = ( zero_zero @ A ) )
% 12.24/7.90 => ( A3
% 12.24/7.90 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1935_less__divide__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [A3: A,B2: A,W2: num] :
% 12.24/7.90 ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
% 12.24/7.90 = ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1936_divide__less__eq__numeral1_I2_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_field @ A )
% 12.24/7.90 => ! [B2: A,W2: num,A3: A] :
% 12.24/7.90 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
% 12.24/7.90 = ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1937_dbl__dec__simps_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1938_dbl__inc__simps_I1_J,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( neg_numeral @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1939_of__int__le__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1940_of__int__0__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1941_of__int__less__0__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
% 12.24/7.90 = ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1942_of__int__0__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1943_of__int__1__le__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1944_of__int__le__1__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
% 12.24/7.90 = ( ord_less_eq @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1945_of__int__1__less__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1946_of__int__less__1__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
% 12.24/7.90 = ( ord_less @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1947_of__nat__nat,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Z3: int] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.24/7.90 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z3 ) )
% 12.24/7.90 = ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1948_of__int__le__of__int__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [B2: int,W2: nat,X: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1949_of__int__power__le__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: int,B2: int,W2: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1950_of__int__power__less__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: int,B2: int,W2: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
% 12.24/7.90 = ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1951_of__int__less__of__int__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [B2: int,W2: nat,X: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
% 12.24/7.90 = ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1952_nat__numeral__diff__1,axiom,
% 12.24/7.90 ! [V: num] :
% 12.24/7.90 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
% 12.24/7.90 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1953_nat__less__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 12.24/7.90 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1954_numeral__power__less__nat__cancel__iff,axiom,
% 12.24/7.90 ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1955_numeral__power__le__nat__cancel__iff,axiom,
% 12.24/7.90 ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1956_nat__le__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
% 12.24/7.90 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1957_ceiling__less__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 12.24/7.90 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1958_numeral__le__ceiling,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num,X: A] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.90 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1959_numeral__power__less__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [I: num,N: nat,X: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 12.24/7.90 = ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1960_of__nat__less__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: nat,I: num,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
% 12.24/7.90 = ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1961_of__nat__le__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [X: nat,I: num,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1962_numeral__power__le__of__nat__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_semidom @ A )
% 12.24/7.90 => ! [I: num,N: nat,X: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
% 12.24/7.90 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1963_ceiling__le__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 12.24/7.90 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1964_neg__numeral__less__ceiling,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num,X: A] :
% 12.24/7.90 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.90 = ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1965_of__int__le__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 12.24/7.90 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1966_numeral__power__le__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1967_numeral__power__less__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1968_of__int__less__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
% 12.24/7.90 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1969_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [Y3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ( ring_1_of_int @ A @ Y3 )
% 12.24/7.90 = ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 12.24/7.90 = ( Y3
% 12.24/7.90 = ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1970_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [X: num,N: nat,Y3: int] :
% 12.24/7.90 ( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
% 12.24/7.90 = ( ring_1_of_int @ A @ Y3 ) )
% 12.24/7.90 = ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
% 12.24/7.90 = Y3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1971_ceiling__less__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A,V: num] :
% 12.24/7.90 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 12.24/7.90 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1972_neg__numeral__le__ceiling,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [V: num,X: A] :
% 12.24/7.90 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.90 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1973_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 12.24/7.90 = ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1974_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 12.24/7.90 = ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1975_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [X: num,N: nat,A3: int] :
% 12.24/7.90 ( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
% 12.24/7.90 = ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1976_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [A3: int,X: num,N: nat] :
% 12.24/7.90 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
% 12.24/7.90 = ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1977_of__int__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [K: num] :
% 12.24/7.90 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
% 12.24/7.90 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1978_nat__numeral__as__int,axiom,
% 12.24/7.90 ( ( numeral_numeral @ nat )
% 12.24/7.90 = ( ^ [I5: num] : ( nat2 @ ( numeral_numeral @ int @ I5 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1979_int__ops_I3_J,axiom,
% 12.24/7.90 ! [N: num] :
% 12.24/7.90 ( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.90 = ( numeral_numeral @ int @ N ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1980_ex__le__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim462609752435547400_field @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ? [Z4: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1981_ex__of__int__less,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim462609752435547400_field @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ? [Z4: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z4 ) @ X ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1982_ex__less__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim462609752435547400_field @ A )
% 12.24/7.90 => ! [X: A] :
% 12.24/7.90 ? [Z4: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1983_mult__of__int__commute,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [X: int,Y3: A] :
% 12.24/7.90 ( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y3 )
% 12.24/7.90 = ( times_times @ A @ Y3 @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1984_zero__neq__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( semiring_char_0 @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( zero_zero @ A )
% 12.24/7.90 != ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1985_numeral__neq__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( numeral_numeral @ A @ M )
% 12.24/7.90 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1986_neg__numeral__neq__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
% 12.24/7.90 != ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1987_div__mult2__numeral__eq,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.90 => ! [A3: A,K: num,L: num] :
% 12.24/7.90 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
% 12.24/7.90 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1988_pochhammer__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [X: int,N: nat] :
% 12.24/7.90 ( ( comm_s3205402744901411588hammer @ A @ ( ring_1_of_int @ A @ X ) @ N )
% 12.24/7.90 = ( ring_1_of_int @ A @ ( comm_s3205402744901411588hammer @ int @ X @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1989_Ints__of__int,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Z3: int] : ( member @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1990_Ints__induct,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Q4: A,P: A > $o] :
% 12.24/7.90 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ( ! [Z4: int] : ( P @ ( ring_1_of_int @ A @ Z4 ) )
% 12.24/7.90 => ( P @ Q4 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1991_Ints__cases,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [Q4: A] :
% 12.24/7.90 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
% 12.24/7.90 => ~ ! [Z4: int] :
% 12.24/7.90 ( Q4
% 12.24/7.90 != ( ring_1_of_int @ A @ Z4 ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1992_Ints__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_1 @ A )
% 12.24/7.90 => ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1993_not__numeral__le__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1994_zero__le__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1995_not__numeral__less__zero,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1996_zero__less__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1997_le__of__int__ceiling,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.90 => ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1998_one__le__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_1999_not__numeral__less__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linord181362715937106298miring @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2000_not__numeral__le__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2001_neg__numeral__le__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2002_zero__neq__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( zero_zero @ A )
% 12.24/7.90 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2003_not__numeral__less__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] :
% 12.24/7.90 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2004_neg__numeral__less__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( linordered_idom @ A )
% 12.24/7.90 => ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2005_one__plus__numeral__commute,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( numeral @ A )
% 12.24/7.90 => ! [X: num] :
% 12.24/7.90 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 12.24/7.90 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2006_numeral__times__minus__swap,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( comm_ring_1 @ A )
% 12.24/7.90 => ! [W2: num,X: A] :
% 12.24/7.90 ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
% 12.24/7.90 = ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2007_numeral__neq__neg__one,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( numeral_numeral @ A @ N )
% 12.24/7.90 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2008_one__neq__neg__numeral,axiom,
% 12.24/7.90 ! [A: $tType] :
% 12.24/7.90 ( ( ring_char_0 @ A )
% 12.24/7.90 => ! [N: num] :
% 12.24/7.90 ( ( one_one @ A )
% 12.24/7.90 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.90
% 12.24/7.90 thf(fact_2009_power_Opower_Opower__Suc,axiom,
% 12.24/7.90 ! [A: $tType,One2: A,Times2: A > A > A,A3: A,N: nat] :
% 12.24/7.90 ( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N ) )
% 12.24/7.90 = ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2010_power_Opower_Opower__0,axiom,
% 12.24/7.91 ! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
% 12.24/7.91 ( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.91 = One2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2011_ceiling__le__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
% 12.24/7.91 = ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2012_less__ceiling__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.91 = ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2013_neg__numeral__le__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2014_not__zero__le__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2015_neg__numeral__less__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2016_not__zero__less__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2017_not__one__le__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2018_not__numeral__le__neg__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2019_neg__numeral__le__neg__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2020_neg__one__le__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2021_neg__numeral__le__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2022_divide__eq__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( division_ring @ A )
% 12.24/7.91 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.91 ( ( ( divide_divide @ A @ B2 @ C2 )
% 12.24/7.91 = ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( ( ( C2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( B2
% 12.24/7.91 = ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 12.24/7.91 & ( ( C2
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 => ( ( numeral_numeral @ A @ W2 )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2023_eq__divide__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( division_ring @ A )
% 12.24/7.91 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.91 ( ( ( numeral_numeral @ A @ W2 )
% 12.24/7.91 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( ( ( C2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
% 12.24/7.91 = B2 ) )
% 12.24/7.91 & ( ( C2
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 => ( ( numeral_numeral @ A @ W2 )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2024_not__neg__one__less__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2025_not__one__less__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2026_not__numeral__less__neg__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2027_neg__one__less__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2028_neg__numeral__less__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2029_of__int__divide__in__Ints,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( idom_divide @ A )
% 12.24/7.91 => ! [B2: int,A3: int] :
% 12.24/7.91 ( ( dvd_dvd @ int @ B2 @ A3 )
% 12.24/7.91 => ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2030_of__int__nonneg,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [Z3: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 12.24/7.91 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2031_of__int__pos,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [Z3: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
% 12.24/7.91 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2032_of__int__leD,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: int,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 12.24/7.91 => ( ( N
% 12.24/7.91 = ( zero_zero @ int ) )
% 12.24/7.91 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2033_of__int__lessD,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: int,X: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 12.24/7.91 => ( ( N
% 12.24/7.91 = ( zero_zero @ int ) )
% 12.24/7.91 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2034_floor__exists1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim462609752435547400_field @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ? [X3: int] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
% 12.24/7.91 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
% 12.24/7.91 & ! [Y4: int] :
% 12.24/7.91 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
% 12.24/7.91 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
% 12.24/7.91 => ( Y4 = X3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2035_floor__exists,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim462609752435547400_field @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ? [Z4: int] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z4 ) @ X )
% 12.24/7.91 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2036_of__nat__less__of__int__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: nat,X: int] :
% 12.24/7.91 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
% 12.24/7.91 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2037_divide__less__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2038_less__divide__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2039_divide__eq__eq__numeral_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( division_ring @ A )
% 12.24/7.91 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.91 ( ( ( divide_divide @ A @ B2 @ C2 )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.91 = ( ( ( C2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( B2
% 12.24/7.91 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 12.24/7.91 & ( ( C2
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2040_eq__divide__eq__numeral_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( division_ring @ A )
% 12.24/7.91 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.91 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( ( ( C2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
% 12.24/7.91 = B2 ) )
% 12.24/7.91 & ( ( C2
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2041_round__diff__minimal,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z3 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2042_ceiling__correct,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
% 12.24/7.91 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2043_ceiling__unique,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X )
% 12.24/7.91 => ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.91 => ( ( archimedean_ceiling @ A @ X )
% 12.24/7.91 = Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2044_ceiling__eq__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,A3: int] :
% 12.24/7.91 ( ( ( archimedean_ceiling @ A @ X )
% 12.24/7.91 = A3 )
% 12.24/7.91 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
% 12.24/7.91 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2045_ceiling__split,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [P: int > $o,T2: A] :
% 12.24/7.91 ( ( P @ ( archimedean_ceiling @ A @ T2 ) )
% 12.24/7.91 = ( ! [I5: int] :
% 12.24/7.91 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) @ T2 )
% 12.24/7.91 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I5 ) ) )
% 12.24/7.91 => ( P @ I5 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2046_ceiling__less__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
% 12.24/7.91 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2047_le__ceiling__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
% 12.24/7.91 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2048_divide__le__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2049_le__divide__eq__numeral_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2050_divide__less__eq__numeral_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [B2: A,C2: A,W2: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2051_less__divide__eq__numeral_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [W2: num,B2: A,C2: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.91 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2052_distrib__left__NO__MATCH,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( semiring @ A )
% 12.24/7.91 => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
% 12.24/7.91 => ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2053_distrib__right__NO__MATCH,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( semiring @ A )
% 12.24/7.91 => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
% 12.24/7.91 => ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.91 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2054_left__diff__distrib__NO__MATCH,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( ring @ A )
% 12.24/7.91 => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
% 12.24/7.91 => ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.91 = ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2055_right__diff__distrib__NO__MATCH,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( ring @ A )
% 12.24/7.91 => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
% 12.24/7.91 => ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 12.24/7.91 = ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2056_ceiling__divide__upper,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Q4: A,P5: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 12.24/7.91 => ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2057_of__int__of__nat,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ( ( ring_1_of_int @ A )
% 12.24/7.91 = ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2058_div__add__self2__no__field,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.91 & ( field @ B ) )
% 12.24/7.91 => ! [X: B,B2: A,A3: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 12.24/7.91 => ( ( B2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.91 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2059_div__add__self1__no__field,axiom,
% 12.24/7.91 ! [B: $tType,A: $tType] :
% 12.24/7.91 ( ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.91 & ( field @ B ) )
% 12.24/7.91 => ! [X: B,B2: A,A3: A] :
% 12.24/7.91 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 12.24/7.91 => ( ( B2
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 12.24/7.91 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2060_floor__le__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,V: num] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2061_neg__numeral__less__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num,X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2062_diff__numeral__special_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2063_diff__numeral__special_I6_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2064_add__neg__numeral__special_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2065_floor__of__int,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z3 ) )
% 12.24/7.91 = Z3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2066_floor__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2067_floor__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
% 12.24/7.91 = ( numeral_numeral @ int @ V ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2068_floor__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2069_floor__of__nat,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.24/7.91 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2070_floor__uminus__of__int,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) )
% 12.24/7.91 = ( uminus_uminus @ int @ Z3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2071_floor__diff__of__int,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
% 12.24/7.91 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2072_floor__add2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 12.24/7.91 | ( member @ A @ Y3 @ ( ring_1_Ints @ A ) ) )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 12.24/7.91 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2073_zero__le__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2074_floor__less__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2075_numeral__le__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2076_zero__less__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2077_floor__le__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2078_floor__less__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,V: num] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2079_one__le__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2080_floor__less__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2081_floor__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.91 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2082_floor__diff__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,V: num] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.91 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2083_add__neg__numeral__special_I6_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2084_floor__diff__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 12.24/7.91 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2085_numeral__less__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num,X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2086_floor__le__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,V: num] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2087_neg__numeral__le__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [V: num,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2088_floor__less__neg__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,V: num] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2089_add__inc,axiom,
% 12.24/7.91 ! [X: num,Y3: num] :
% 12.24/7.91 ( ( plus_plus @ num @ X @ ( inc @ Y3 ) )
% 12.24/7.91 = ( inc @ ( plus_plus @ num @ X @ Y3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2090_mult__inc,axiom,
% 12.24/7.91 ! [X: num,Y3: num] :
% 12.24/7.91 ( ( times_times @ num @ X @ ( inc @ Y3 ) )
% 12.24/7.91 = ( plus_plus @ num @ ( times_times @ num @ X @ Y3 ) @ X ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2091_floor__mono,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.91 => ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2092_floor__less__cancel,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) )
% 12.24/7.91 => ( ord_less @ A @ X @ Y3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2093_of__int__floor__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2094_floor__le__ceiling,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2095_floor__le__round,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2096_le__floor__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2097_floor__less__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 12.24/7.91 = ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2098_le__floor__add,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2099_int__add__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( plus_plus @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2100_floor__add__int,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 12.24/7.91 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2101_floor__divide__of__int__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [K: int,L: int] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
% 12.24/7.91 = ( divide_divide @ int @ K @ L ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2102_ceiling__def,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archimedean_ceiling @ A )
% 12.24/7.91 = ( ^ [X2: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2103_floor__minus,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
% 12.24/7.91 = ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2104_ceiling__minus,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
% 12.24/7.91 = ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2105_frac__def,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archimedean_frac @ A )
% 12.24/7.91 = ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2106_of__nat__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [R: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 12.24/7.91 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R ) ) ) @ R ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2107_one__add__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 12.24/7.91 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2108_numeral__inc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ! [X: num] :
% 12.24/7.91 ( ( numeral_numeral @ A @ ( inc @ X ) )
% 12.24/7.91 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2109_floor__divide__of__nat__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 12.24/7.91 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2110_ceiling__diff__floor__le__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2111_ceiling__altdef,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archimedean_ceiling @ A )
% 12.24/7.91 = ( ^ [X2: A] :
% 12.24/7.91 ( if @ int
% 12.24/7.91 @ ( X2
% 12.24/7.91 = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
% 12.24/7.91 @ ( archim6421214686448440834_floor @ A @ X2 )
% 12.24/7.91 @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2112_floor__split,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [P: int > $o,T2: A] :
% 12.24/7.91 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
% 12.24/7.91 = ( ! [I5: int] :
% 12.24/7.91 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I5 ) @ T2 )
% 12.24/7.91 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) ) )
% 12.24/7.91 => ( P @ I5 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2113_floor__eq__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,A3: int] :
% 12.24/7.91 ( ( ( archim6421214686448440834_floor @ A @ X )
% 12.24/7.91 = A3 )
% 12.24/7.91 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
% 12.24/7.91 & ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2114_floor__unique,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
% 12.24/7.91 => ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ X )
% 12.24/7.91 = Z3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2115_le__mult__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [A3: A,B2: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.91 => ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2116_less__floor__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Z3: int,X: A] :
% 12.24/7.91 ( ( ord_less @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2117_floor__le__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Z3: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
% 12.24/7.91 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2118_floor__correct,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
% 12.24/7.91 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2119_floor__divide__lower,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Q4: A,P5: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 12.24/7.91 => ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2120_le__mult__floor__Ints,axiom,
% 12.24/7.91 ! [A: $tType,B: $tType] :
% 12.24/7.91 ( ( ( archim2362893244070406136eiling @ B )
% 12.24/7.91 & ( linordered_idom @ A ) )
% 12.24/7.91 => ! [A3: B,B2: B] :
% 12.24/7.91 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
% 12.24/7.91 => ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2121_floor__add,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 12.24/7.91 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) )
% 12.24/7.91 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
% 12.24/7.91 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
% 12.24/7.91 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2122_floor__divide__upper,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [Q4: A,P5: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
% 12.24/7.91 => ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2123_diff__numeral__special_I4_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2124_diff__numeral__special_I3_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2125_cpmi,axiom,
% 12.24/7.91 ! [D4: int,P: int > $o,P2: int > $o,B7: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ? [Z5: int] :
% 12.24/7.91 ! [X3: int] :
% 12.24/7.91 ( ( ord_less @ int @ X3 @ Z5 )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 = ( P2 @ X3 ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ B7 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int,K2: int] :
% 12.24/7.91 ( ( P2 @ X3 )
% 12.24/7.91 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
% 12.24/7.91 => ( ( ? [X7: int] : ( P @ X7 ) )
% 12.24/7.91 = ( ? [X2: int] :
% 12.24/7.91 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 & ( P2 @ X2 ) )
% 12.24/7.91 | ? [X2: int] :
% 12.24/7.91 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 & ? [Y6: int] :
% 12.24/7.91 ( ( member @ int @ Y6 @ B7 )
% 12.24/7.91 & ( P @ ( plus_plus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2126_cppi,axiom,
% 12.24/7.91 ! [D4: int,P: int > $o,P2: int > $o,A6: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ? [Z5: int] :
% 12.24/7.91 ! [X3: int] :
% 12.24/7.91 ( ( ord_less @ int @ Z5 @ X3 )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 = ( P2 @ X3 ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ A6 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int,K2: int] :
% 12.24/7.91 ( ( P2 @ X3 )
% 12.24/7.91 = ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
% 12.24/7.91 => ( ( ? [X7: int] : ( P @ X7 ) )
% 12.24/7.91 = ( ? [X2: int] :
% 12.24/7.91 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 & ( P2 @ X2 ) )
% 12.24/7.91 | ? [X2: int] :
% 12.24/7.91 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 & ? [Y6: int] :
% 12.24/7.91 ( ( member @ int @ Y6 @ A6 )
% 12.24/7.91 & ( P @ ( minus_minus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2127_aset_I8_J,axiom,
% 12.24/7.91 ! [D4: int,A6: set @ int,T2: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ int @ T2 @ X4 )
% 12.24/7.91 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2128_aset_I6_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,A6: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ int @ X4 @ T2 )
% 12.24/7.91 => ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2129_bset_I8_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,B7: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ int @ T2 @ X4 )
% 12.24/7.91 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2130_atLeastatMost__subset__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( preorder @ A )
% 12.24/7.91 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 12.24/7.91 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.91 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.91 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2131_atLeastAtMost__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ord @ A )
% 12.24/7.91 => ! [I: A,L: A,U: A] :
% 12.24/7.91 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 12.24/7.91 = ( ( ord_less_eq @ A @ L @ I )
% 12.24/7.91 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2132_Icc__eq__Icc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( order @ A )
% 12.24/7.91 => ! [L: A,H2: A,L4: A,H: A] :
% 12.24/7.91 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
% 12.24/7.91 = ( set_or1337092689740270186AtMost @ A @ L4 @ H ) )
% 12.24/7.91 = ( ( ( L = L4 )
% 12.24/7.91 & ( H2 = H ) )
% 12.24/7.91 | ( ~ ( ord_less_eq @ A @ L @ H2 )
% 12.24/7.91 & ~ ( ord_less_eq @ A @ L4 @ H ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2133_semiring__norm_I12_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( times_times @ num @ one2 @ N )
% 12.24/7.91 = N ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2134_semiring__norm_I11_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ( ( times_times @ num @ M @ one2 )
% 12.24/7.91 = M ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2135_semiring__norm_I68_J,axiom,
% 12.24/7.91 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2136_semiring__norm_I75_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ~ ( ord_less @ num @ M @ one2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2137_one__eq__numeral__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_char_0 @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ( one_one @ A )
% 12.24/7.91 = ( numeral_numeral @ A @ N ) )
% 12.24/7.91 = ( one2 = N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2138_numeral__eq__one__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_char_0 @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ( numeral_numeral @ A @ N )
% 12.24/7.91 = ( one_one @ A ) )
% 12.24/7.91 = ( N = one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2139_numeral__eq__neg__one__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_char_0 @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( N = one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2140_neg__one__eq__numeral__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_char_0 @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ( uminus_uminus @ A @ ( one_one @ A ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.91 = ( N = one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2141_Suc__numeral,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( suc @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.91 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2142_not__neg__one__le__neg__numeral__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
% 12.24/7.91 = ( M != one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2143_neg__numeral__less__neg__one__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [M: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( M != one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2144_one__plus__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2145_numeral__plus__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2146_numeral__le__one__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linord181362715937106298miring @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 12.24/7.91 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2147_one__less__numeral__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linord181362715937106298miring @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.91 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2148_add__One__commute,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ one2 @ N )
% 12.24/7.91 = ( plus_plus @ num @ N @ one2 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2149_num__induct,axiom,
% 12.24/7.91 ! [P: num > $o,X: num] :
% 12.24/7.91 ( ( P @ one2 )
% 12.24/7.91 => ( ! [X3: num] :
% 12.24/7.91 ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( inc @ X3 ) ) )
% 12.24/7.91 => ( P @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2150_le__num__One__iff,axiom,
% 12.24/7.91 ! [X: num] :
% 12.24/7.91 ( ( ord_less_eq @ num @ X @ one2 )
% 12.24/7.91 = ( X = one2 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2151_atLeastatMost__psubset__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( preorder @ A )
% 12.24/7.91 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 12.24/7.91 = ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.91 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.91 & ( ord_less_eq @ A @ B2 @ D2 )
% 12.24/7.91 & ( ( ord_less @ A @ C2 @ A3 )
% 12.24/7.91 | ( ord_less @ A @ B2 @ D2 ) ) ) )
% 12.24/7.91 & ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2152_mult__numeral__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_numeral @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2153_mult__numeral__1__right,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_numeral @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2154_numeral__One,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ( ( numeral_numeral @ A @ one2 )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2155_divide__numeral__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( field @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2156_numerals_I1_J,axiom,
% 12.24/7.91 ( ( numeral_numeral @ nat @ one2 )
% 12.24/7.91 = ( one_one @ nat ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2157_add__One,axiom,
% 12.24/7.91 ! [X: num] :
% 12.24/7.91 ( ( plus_plus @ num @ X @ one2 )
% 12.24/7.91 = ( inc @ X ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2158_bset_I1_J,axiom,
% 12.24/7.91 ! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
% 12.24/7.91 ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ B7 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ B7 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( Q @ X3 )
% 12.24/7.91 => ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ( P @ X4 )
% 12.24/7.91 & ( Q @ X4 ) )
% 12.24/7.91 => ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
% 12.24/7.91 & ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2159_bset_I2_J,axiom,
% 12.24/7.91 ! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
% 12.24/7.91 ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ B7 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ B7 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( plus_plus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( Q @ X3 )
% 12.24/7.91 => ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ( P @ X4 )
% 12.24/7.91 | ( Q @ X4 ) )
% 12.24/7.91 => ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
% 12.24/7.91 | ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2160_aset_I1_J,axiom,
% 12.24/7.91 ! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
% 12.24/7.91 ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ A6 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ A6 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( Q @ X3 )
% 12.24/7.91 => ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ( P @ X4 )
% 12.24/7.91 & ( Q @ X4 ) )
% 12.24/7.91 => ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
% 12.24/7.91 & ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2161_aset_I2_J,axiom,
% 12.24/7.91 ! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
% 12.24/7.91 ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ A6 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( P @ X3 )
% 12.24/7.91 => ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ( ! [X3: int] :
% 12.24/7.91 ( ! [Xa: int] :
% 12.24/7.91 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb: int] :
% 12.24/7.91 ( ( member @ int @ Xb @ A6 )
% 12.24/7.91 => ( X3
% 12.24/7.91 != ( minus_minus @ int @ Xb @ Xa ) ) ) )
% 12.24/7.91 => ( ( Q @ X3 )
% 12.24/7.91 => ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ( P @ X4 )
% 12.24/7.91 | ( Q @ X4 ) )
% 12.24/7.91 => ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
% 12.24/7.91 | ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2162_mult__1s__ring__1_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [B2: A] :
% 12.24/7.91 ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
% 12.24/7.91 = ( uminus_uminus @ A @ B2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2163_mult__1s__ring__1_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [B2: A] :
% 12.24/7.91 ( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ B2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2164_uminus__numeral__One,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2165_cong__exp__iff__simps_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2166_numeral__1__eq__Suc__0,axiom,
% 12.24/7.91 ( ( numeral_numeral @ nat @ one2 )
% 12.24/7.91 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2167_bset_I9_J,axiom,
% 12.24/7.91 ! [D2: int,D4: int,B7: set @ int,T2: int] :
% 12.24/7.91 ( ( dvd_dvd @ int @ D2 @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 12.24/7.91 => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2168_bset_I10_J,axiom,
% 12.24/7.91 ! [D2: int,D4: int,B7: set @ int,T2: int] :
% 12.24/7.91 ( ( dvd_dvd @ int @ D2 @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 12.24/7.91 => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2169_aset_I9_J,axiom,
% 12.24/7.91 ! [D2: int,D4: int,A6: set @ int,T2: int] :
% 12.24/7.91 ( ( dvd_dvd @ int @ D2 @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 12.24/7.91 => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2170_aset_I10_J,axiom,
% 12.24/7.91 ! [D2: int,D4: int,A6: set @ int,T2: int] :
% 12.24/7.91 ( ( dvd_dvd @ int @ D2 @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
% 12.24/7.91 => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2171_Suc__nat__number__of__add,axiom,
% 12.24/7.91 ! [V: num,N: nat] :
% 12.24/7.91 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 12.24/7.91 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2172_periodic__finite__ex,axiom,
% 12.24/7.91 ! [D2: int,P: int > $o] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
% 12.24/7.91 => ( ! [X3: int,K2: int] :
% 12.24/7.91 ( ( P @ X3 )
% 12.24/7.91 = ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
% 12.24/7.91 => ( ( ? [X7: int] : ( P @ X7 ) )
% 12.24/7.91 = ( ? [X2: int] :
% 12.24/7.91 ( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
% 12.24/7.91 & ( P @ X2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2173_aset_I7_J,axiom,
% 12.24/7.91 ! [D4: int,A6: set @ int,T2: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less @ int @ T2 @ X4 )
% 12.24/7.91 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2174_aset_I5_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,A6: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ T2 @ A6 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less @ int @ X4 @ T2 )
% 12.24/7.91 => ( ord_less @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2175_aset_I4_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,A6: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ T2 @ A6 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( X4 != T2 )
% 12.24/7.91 => ( ( plus_plus @ int @ X4 @ D4 )
% 12.24/7.91 != T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2176_aset_I3_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,A6: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ A6 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( X4 = T2 )
% 12.24/7.91 => ( ( plus_plus @ int @ X4 @ D4 )
% 12.24/7.91 = T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2177_bset_I7_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,B7: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ T2 @ B7 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less @ int @ T2 @ X4 )
% 12.24/7.91 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2178_bset_I5_J,axiom,
% 12.24/7.91 ! [D4: int,B7: set @ int,T2: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less @ int @ X4 @ T2 )
% 12.24/7.91 => ( ord_less @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2179_bset_I4_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,B7: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ T2 @ B7 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( X4 != T2 )
% 12.24/7.91 => ( ( minus_minus @ int @ X4 @ D4 )
% 12.24/7.91 != T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2180_bset_I3_J,axiom,
% 12.24/7.91 ! [D4: int,T2: int,B7: set @ int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( X4 = T2 )
% 12.24/7.91 => ( ( minus_minus @ int @ X4 @ D4 )
% 12.24/7.91 = T2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2181_bset_I6_J,axiom,
% 12.24/7.91 ! [D4: int,B7: set @ int,T2: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 12.24/7.91 => ! [X4: int] :
% 12.24/7.91 ( ! [Xa2: int] :
% 12.24/7.91 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 12.24/7.91 => ! [Xb2: int] :
% 12.24/7.91 ( ( member @ int @ Xb2 @ B7 )
% 12.24/7.91 => ( X4
% 12.24/7.91 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ int @ X4 @ T2 )
% 12.24/7.91 => ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2182_pochhammer__double,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( field_char_0 @ A )
% 12.24/7.91 => ! [Z3: A,N: nat] :
% 12.24/7.91 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( 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 @ Z3 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2183_even__succ__mod__exp,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2184_even__succ__div__exp,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2185_fact__double,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( field_char_0 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( 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 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2186_round__unique,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,Y3: int] :
% 12.24/7.91 ( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y3 ) )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
% 12.24/7.91 => ( ( archimedean_round @ A @ X )
% 12.24/7.91 = Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2187_even__mult__exp__div__exp__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [A3: A,M: nat,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 = ( ( ord_less @ nat @ N @ M )
% 12.24/7.91 | ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 | ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.91 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2188_divmod__digit__1_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [A3: A,B2: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.91 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 12.24/7.91 => ( ( 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 ) )
% 12.24/7.91 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2189_verit__eq__simplify_I8_J,axiom,
% 12.24/7.91 ! [X22: num,Y2: num] :
% 12.24/7.91 ( ( ( bit0 @ X22 )
% 12.24/7.91 = ( bit0 @ Y2 ) )
% 12.24/7.91 = ( X22 = Y2 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2190_semiring__norm_I87_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ( bit0 @ M )
% 12.24/7.91 = ( bit0 @ N ) )
% 12.24/7.91 = ( M = N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2191_semiring__norm_I85_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ( ( bit0 @ M )
% 12.24/7.91 != one2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2192_semiring__norm_I83_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( one2
% 12.24/7.91 != ( bit0 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2193_semiring__norm_I6_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2194_semiring__norm_I13_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2195_semiring__norm_I78_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( ord_less @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2196_semiring__norm_I71_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( ord_less_eq @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2197_semiring__norm_I2_J,axiom,
% 12.24/7.91 ( ( plus_plus @ num @ one2 @ one2 )
% 12.24/7.91 = ( bit0 @ one2 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2198_num__double,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( times_times @ num @ ( bit0 @ one2 ) @ N )
% 12.24/7.91 = ( bit0 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2199_semiring__norm_I69_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2200_semiring__norm_I76_J,axiom,
% 12.24/7.91 ! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2201_one__add__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2202_even__mult__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A,B2: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.91 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2203_bits__one__mod__two__eq__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2204_one__mod__two__eq__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2205_zero__eq__power2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiri2026040879449505780visors @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 = ( A3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2206_power2__minus,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2207_add__2__eq__Suc_H,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( suc @ ( suc @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2208_add__2__eq__Suc,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 = ( suc @ ( suc @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2209_Suc__1,axiom,
% 12.24/7.91 ( ( suc @ ( one_one @ nat ) )
% 12.24/7.91 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2210_even__Suc,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) )
% 12.24/7.91 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2211_even__Suc__Suc__iff,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N ) ) )
% 12.24/7.91 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2212_mod2__Suc__Suc,axiom,
% 12.24/7.91 ! [M: nat] :
% 12.24/7.91 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2213_div2__Suc__Suc,axiom,
% 12.24/7.91 ! [M: nat] :
% 12.24/7.91 ( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2214_fact__2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_char_0 @ A )
% 12.24/7.91 => ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2215_add__self__div__2,axiom,
% 12.24/7.91 ! [M: nat] :
% 12.24/7.91 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = M ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2216_zmod__numeral__Bit0,axiom,
% 12.24/7.91 ! [V: num,W2: num] :
% 12.24/7.91 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
% 12.24/7.91 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2217_one__div__two__eq__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2218_bits__1__div__2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2219_add__neg__numeral__special_I9_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2220_diff__numeral__special_I11_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2221_diff__numeral__special_I10_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2222_even__plus__one__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
% 12.24/7.91 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2223_not__mod__2__eq__0__eq__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2224_not__mod__2__eq__1__eq__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 != ( one_one @ A ) )
% 12.24/7.91 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2225_minus__1__div__2__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid8789492081693882211th_nat @ A )
% 12.24/7.91 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2226_power2__less__eq__zero__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( A3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2227_power2__eq__iff__nonneg,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_semidom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.91 => ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( X = Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2228_zero__less__power2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( A3
% 12.24/7.91 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2229_bits__minus__1__mod__2__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2230_minus__1__mod__2__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid8789492081693882211th_nat @ A )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2231_of__bool__half__eq__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [B2: $o] :
% 12.24/7.91 ( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2232_Power_Oring__1__class_Opower__minus__even,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2233_power__minus__odd,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.91 = ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2234_Parity_Oring__1__class_Opower__minus__even,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.91 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2235_not__mod2__eq__Suc__0__eq__0,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 != ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.91 = ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2236_add__self__mod__2,axiom,
% 12.24/7.91 ! [M: nat] :
% 12.24/7.91 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ nat ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2237_half__nonnegative__int__iff,axiom,
% 12.24/7.91 ! [K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2238_half__negative__int__iff,axiom,
% 12.24/7.91 ! [K: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2239_even__Suc__div__two,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2240_odd__Suc__div__two,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2241_even__succ__div__2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2242_even__succ__div__two,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2243_odd__succ__div__two,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2244_even__power,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.91 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2245_zero__le__power__eq__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,W2: num] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
% 12.24/7.91 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2246_power__less__zero__eq__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,W2: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2247_power__less__zero__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2248_power__minus1__even,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2249_neg__one__even__power,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 12.24/7.91 = ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2250_neg__one__odd__power,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2251_even__of__nat,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
% 12.24/7.91 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2252_odd__Suc__minus__one,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.91 = N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2253_one__less__floor,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2254_floor__le__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 12.24/7.91 = ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2255_even__diff__nat,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
% 12.24/7.91 = ( ( ord_less @ nat @ M @ N )
% 12.24/7.91 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2256_mod2__gr__0,axiom,
% 12.24/7.91 ! [M: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ nat ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2257_odd__two__times__div__two__succ,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
% 12.24/7.91 = A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2258_semiring__parity__class_Oeven__mask__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
% 12.24/7.91 = ( N
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2259_zero__less__power__eq__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,W2: num] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
% 12.24/7.91 = ( ( ( numeral_numeral @ nat @ W2 )
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( A3
% 12.24/7.91 != ( zero_zero @ A ) ) )
% 12.24/7.91 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2260_bits__1__div__exp,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( zero_neq_one_of_bool @ A
% 12.24/7.91 @ ( N
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2261_one__div__2__pow__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( zero_neq_one_of_bool @ A
% 12.24/7.91 @ ( N
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2262_odd__two__times__div__two__nat,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2263_power__le__zero__eq__numeral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,W2: num] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 12.24/7.91 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
% 12.24/7.91 & ( A3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2264_one__mod__2__pow__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2265_verit__eq__simplify_I10_J,axiom,
% 12.24/7.91 ! [X22: num] :
% 12.24/7.91 ( one2
% 12.24/7.91 != ( bit0 @ X22 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2266_ex__nat__less,axiom,
% 12.24/7.91 ! [N: nat,P: nat > $o] :
% 12.24/7.91 ( ( ? [M2: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ M2 @ N )
% 12.24/7.91 & ( P @ M2 ) ) )
% 12.24/7.91 = ( ? [X2: nat] :
% 12.24/7.91 ( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.91 & ( P @ X2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2267_all__nat__less,axiom,
% 12.24/7.91 ! [N: nat,P: nat > $o] :
% 12.24/7.91 ( ( ! [M2: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ M2 @ N )
% 12.24/7.91 => ( P @ M2 ) ) )
% 12.24/7.91 = ( ! [X2: nat] :
% 12.24/7.91 ( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.91 => ( P @ X2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2268_numeral__Bit0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( numeral @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 12.24/7.91 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2269_inc_Osimps_I1_J,axiom,
% 12.24/7.91 ( ( inc @ one2 )
% 12.24/7.91 = ( bit0 @ one2 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2270_cong__exp__iff__simps_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [N: num,Q4: num] :
% 12.24/7.91 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2271_numeral__Bit0__div__2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [N: num] :
% 12.24/7.91 ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2272_power__minus__Bit0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [X: A,K: num] :
% 12.24/7.91 ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
% 12.24/7.91 = ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2273_mult__2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_numeral @ A )
% 12.24/7.91 => ! [Z3: A] :
% 12.24/7.91 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 )
% 12.24/7.91 = ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2274_mult__2__right,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_numeral @ A )
% 12.24/7.91 => ! [Z3: A] :
% 12.24/7.91 ( ( times_times @ A @ Z3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2275_left__add__twice,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_numeral @ A )
% 12.24/7.91 => ! [A3: A,B2: A] :
% 12.24/7.91 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 12.24/7.91 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2276_even__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2277_evenE,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ~ ! [B5: A] :
% 12.24/7.91 ( A3
% 12.24/7.91 != ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2278_odd__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2279_even__minus,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.91 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2280_zero__power2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_1 @ A )
% 12.24/7.91 => ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2281_power2__eq__square,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( monoid_mult @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( times_times @ A @ A3 @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2282_power4__eq__xxxx,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( monoid_mult @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2283_numeral__2__eq__2,axiom,
% 12.24/7.91 ( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
% 12.24/7.91 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2284_one__power2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_1 @ A )
% 12.24/7.91 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2285_power2__eq__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ( X = Y3 )
% 12.24/7.91 | ( X
% 12.24/7.91 = ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2286_double__not__eq__Suc__double,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 12.24/7.91 != ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2287_Suc__double__not__eq__double,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
% 12.24/7.91 != ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2288_power__even__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( monoid_mult @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2289_nat__1__add__1,axiom,
% 12.24/7.91 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
% 12.24/7.91 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2290_less__exp,axiom,
% 12.24/7.91 ! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2291_self__le__ge2__pow,axiom,
% 12.24/7.91 ! [K: nat,M: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.91 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2292_power2__nat__le__eq__le,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2293_power2__nat__le__imp__le,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
% 12.24/7.91 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2294_even__even__mod__4__iff,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2295_div2__even__ext__nat,axiom,
% 12.24/7.91 ! [X: nat,Y3: nat] :
% 12.24/7.91 ( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( divide_divide @ nat @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
% 12.24/7.91 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y3 ) )
% 12.24/7.91 => ( X = Y3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2296_half__gt__zero__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2297_half__gt__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_field @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2298_even__two__times__div__two,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2299_zero__le__power2,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2300_power2__eq__imp__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_semidom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.91 => ( X = Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2301_power2__le__imp__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_semidom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.91 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2302_even__iff__mod__2__eq__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2303_power2__less__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2304_exp__add__not__zero__imp__left,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
% 12.24/7.91 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2305_exp__add__not__zero__imp__right,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2306_odd__iff__mod__2__eq__one,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 12.24/7.91 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2307_sum__power2__eq__zero__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 = ( ( X
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 & ( Y3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2308_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [N: nat,M: nat] :
% 12.24/7.91 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
% 12.24/7.91 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2309_power2__eq__1__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_15535105094025558882visors @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) )
% 12.24/7.91 = ( ( A3
% 12.24/7.91 = ( one_one @ A ) )
% 12.24/7.91 | ( A3
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2310_less__2__cases__iff,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( ( N
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 | ( N
% 12.24/7.91 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2311_less__2__cases,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 => ( ( N
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 | ( N
% 12.24/7.91 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2312_abs__le__square__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y3 ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2313_power__mono__odd,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: nat,A3: A,B2: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.91 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2314_abs__square__eq__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) )
% 12.24/7.91 = ( ( abs_abs @ A @ X )
% 12.24/7.91 = ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2315_odd__pos,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2316_uminus__power__if,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.91 = ( power_power @ A @ A3 @ N ) ) )
% 12.24/7.91 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.91 = ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2317_nat__2,axiom,
% 12.24/7.91 ( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2318_nat__induct2,axiom,
% 12.24/7.91 ! [P: nat > $o,N: nat] :
% 12.24/7.91 ( ( P @ ( zero_zero @ nat ) )
% 12.24/7.91 => ( ( P @ ( one_one @ nat ) )
% 12.24/7.91 => ( ! [N4: nat] :
% 12.24/7.91 ( ( P @ N4 )
% 12.24/7.91 => ( P @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 12.24/7.91 => ( P @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2319_not__exp__less__eq__0__int,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2320_diff__le__diff__pow,axiom,
% 12.24/7.91 ! [K: nat,M: nat,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.91 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2321_dvd__power__iff__le,axiom,
% 12.24/7.91 ! [K: nat,M: nat,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.91 => ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
% 12.24/7.91 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2322_divmod__digit__0_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [B2: A,A3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.91 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 12.24/7.91 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
% 12.24/7.91 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2323_oddE,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ~ ! [B5: A] :
% 12.24/7.91 ( A3
% 12.24/7.91 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2324_power2__less__imp__less,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_semidom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.91 => ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2325_sum__power2__le__zero__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( ( X
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 & ( Y3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2326_sum__power2__ge__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2327_mod2__eq__if,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) )
% 12.24/7.91 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2328_parity__cases,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_parity @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 != ( zero_zero @ A ) ) )
% 12.24/7.91 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 != ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2329_bits__stable__imp__add__self,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = A3 )
% 12.24/7.91 => ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2330_not__sum__power2__lt__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2331_sum__power2__gt__zero__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 12.24/7.91 = ( ( X
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 | ( Y3
% 12.24/7.91 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2332_power2__sum,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( comm_semiring_1 @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2333_square__le__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 12.24/7.91 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2334_zero__le__even__power_H,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2335_power2__le__iff__abs__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [Y3: A,X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2336_zero__le__power__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.91 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2337_zero__le__odd__power,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2338_zero__le__even__power,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2339_abs__square__le__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2340_abs__square__less__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2341_bits__induct,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [P: A > $o,A3: A] :
% 12.24/7.91 ( ! [A8: A] :
% 12.24/7.91 ( ( ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = A8 )
% 12.24/7.91 => ( P @ A8 ) )
% 12.24/7.91 => ( ! [A8: A,B5: $o] :
% 12.24/7.91 ( ( P @ A8 )
% 12.24/7.91 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = A8 )
% 12.24/7.91 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) ) ) )
% 12.24/7.91 => ( P @ A3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2342_minus__power__mult__self,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( comm_ring_1 @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 12.24/7.91 = ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2343_power__odd__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( monoid_mult @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 = ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2344_nat__bit__induct,axiom,
% 12.24/7.91 ! [P: nat > $o,N: nat] :
% 12.24/7.91 ( ( P @ ( zero_zero @ nat ) )
% 12.24/7.91 => ( ! [N4: nat] :
% 12.24/7.91 ( ( P @ N4 )
% 12.24/7.91 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.24/7.91 => ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
% 12.24/7.91 => ( ! [N4: nat] :
% 12.24/7.91 ( ( P @ N4 )
% 12.24/7.91 => ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
% 12.24/7.91 => ( P @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2345_minus__one__power__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 12.24/7.91 = ( one_one @ A ) ) )
% 12.24/7.91 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2346_power__mono__even,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [N: nat,A3: A,B2: A] :
% 12.24/7.91 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2347_Suc__n__div__2__gt__zero,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.91 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2348_div__2__gt__zero,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.91 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2349_minus__1__div__exp__eq__int,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2350_triangle__def,axiom,
% 12.24/7.91 ( nat_triangle
% 12.24/7.91 = ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2351_divmod__digit__0_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [B2: A,A3: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.91 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 12.24/7.91 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 12.24/7.91 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2352_power2__diff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( comm_ring_1 @ A )
% 12.24/7.91 => ! [X: A,Y3: A] :
% 12.24/7.91 ( ( power_power @ A @ ( minus_minus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.91 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2353_odd__0__le__power__imp__0__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/7.91 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2354_odd__power__less__zero,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.91 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2355_zero__less__power__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
% 12.24/7.91 = ( ( N
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( A3
% 12.24/7.91 != ( zero_zero @ A ) ) )
% 12.24/7.91 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2356_mult__exp__mod__exp__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [M: nat,N: nat,A3: A] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.91 => ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2357_power__minus1__odd,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2358_exp__mod__exp,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2359_int__bit__induct,axiom,
% 12.24/7.91 ! [P: int > $o,K: int] :
% 12.24/7.91 ( ( P @ ( zero_zero @ int ) )
% 12.24/7.91 => ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.91 => ( ! [K2: int] :
% 12.24/7.91 ( ( P @ K2 )
% 12.24/7.91 => ( ( K2
% 12.24/7.91 != ( zero_zero @ int ) )
% 12.24/7.91 => ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
% 12.24/7.91 => ( ! [K2: int] :
% 12.24/7.91 ( ( P @ K2 )
% 12.24/7.91 => ( ( K2
% 12.24/7.91 != ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.91 => ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
% 12.24/7.91 => ( P @ K ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2360_ex__power__ivl1,axiom,
% 12.24/7.91 ! [B2: nat,K: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 12.24/7.91 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
% 12.24/7.91 => ? [N4: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
% 12.24/7.91 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2361_ex__power__ivl2,axiom,
% 12.24/7.91 ! [B2: nat,K: nat] :
% 12.24/7.91 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 12.24/7.91 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.91 => ? [N4: nat] :
% 12.24/7.91 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
% 12.24/7.91 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2362_round__def,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archimedean_round @ A )
% 12.24/7.91 = ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2363_mod__double__modulus,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [M: A,X: A] :
% 12.24/7.91 ( ( ord_less @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.91 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.91 => ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
% 12.24/7.91 = ( modulo_modulo @ A @ X @ M ) )
% 12.24/7.91 | ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
% 12.24/7.91 = ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2364_divmod__digit__1_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.91 => ! [A3: A,B2: A] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.91 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 12.24/7.91 => ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
% 12.24/7.91 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2365_even__mask__div__iff_H,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2366_power__le__zero__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( linordered_idom @ A )
% 12.24/7.91 => ! [A3: A,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
% 12.24/7.91 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.91 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
% 12.24/7.91 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 & ( A3
% 12.24/7.91 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2367_of__int__round__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [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 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2368_of__int__round__ge,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [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 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2369_of__int__round__gt,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [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 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2370_even__mod__4__div__2,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.91 => ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2371_neg__zdiv__mult__2,axiom,
% 12.24/7.91 ! [A3: int,B2: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 12.24/7.91 => ( ( 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 ) )
% 12.24/7.91 = ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2372_pos__zdiv__mult__2,axiom,
% 12.24/7.91 ! [A3: int,B2: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 12.24/7.91 => ( ( 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 ) )
% 12.24/7.91 = ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2373_even__nat__iff,axiom,
% 12.24/7.91 ! [K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.91 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
% 12.24/7.91 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2374_pos__zmod__mult__2,axiom,
% 12.24/7.91 ! [A3: int,B2: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 12.24/7.91 => ( ( 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 ) )
% 12.24/7.91 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2375_even__mask__div__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 | ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2376_exp__div__exp__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_semiring_bits @ A )
% 12.24/7.91 => ! [M: nat,N: nat] :
% 12.24/7.91 ( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 = ( times_times @ A
% 12.24/7.91 @ ( zero_neq_one_of_bool @ A
% 12.24/7.91 @ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
% 12.24/7.91 != ( zero_zero @ A ) )
% 12.24/7.91 & ( ord_less_eq @ nat @ N @ M ) ) )
% 12.24/7.91 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2377_of__int__round__abs__le,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [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 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2378_round__unique_H,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ! [X: A,N: int] :
% 12.24/7.91 ( ( 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 ) ) ) )
% 12.24/7.91 => ( ( archimedean_round @ A @ X )
% 12.24/7.91 = N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2379_neg__zmod__mult__2,axiom,
% 12.24/7.91 ! [A3: int,B2: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 12.24/7.91 => ( ( 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 ) )
% 12.24/7.91 = ( 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 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2380_round__altdef,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.91 => ( ( archimedean_round @ A )
% 12.24/7.91 = ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2381_flip__bit__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.91 = ( 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 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2382_set__bit__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.91 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2383_psubsetI,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ( A6 != B7 )
% 12.24/7.91 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2384_unset__bit__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.91 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2385_unset__bit__Suc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.91 = ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2386_flip__bit__Suc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.91 = ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2387_set__bit__Suc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.91 = ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2388_Compl__subset__Compl__iff,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 12.24/7.91 = ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2389_Compl__anti__mono,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2390_unset__bit__nonnegative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
% 12.24/7.91 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2391_set__bit__nonnegative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
% 12.24/7.91 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2392_flip__bit__nonnegative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
% 12.24/7.91 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2393_unset__bit__negative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2394_set__bit__negative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2395_flip__bit__negative__int__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2396_unset__bit__nat__def,axiom,
% 12.24/7.91 ( ( bit_se2638667681897837118et_bit @ nat )
% 12.24/7.91 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M2 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2397_unset__bit__less__eq,axiom,
% 12.24/7.91 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2398_set__bit__greater__eq,axiom,
% 12.24/7.91 ! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2399_psubset__imp__ex__mem,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2400_psubset__trans,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
% 12.24/7.91 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2401_psubsetD,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A,C2: A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ( member @ A @ C2 @ A6 )
% 12.24/7.91 => ( member @ A @ C2 @ B7 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2402_psubsetE,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2403_psubset__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) )
% 12.24/7.91 = ( ^ [A7: set @ A,B6: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 12.24/7.91 & ( A7 != B6 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2404_psubset__imp__subset,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2405_psubset__subset__trans,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C6 )
% 12.24/7.91 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2406_subset__not__subset__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) )
% 12.24/7.91 = ( ^ [A7: set @ A,B6: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
% 12.24/7.91 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2407_subset__psubset__trans,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.91 => ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
% 12.24/7.91 => ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2408_subset__iff__psubset__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ A ) )
% 12.24/7.91 = ( ^ [A7: set @ A,B6: set @ A] :
% 12.24/7.91 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 12.24/7.91 | ( A7 = B6 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2409_even__set__bit__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [M: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 12.24/7.91 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 & ( M
% 12.24/7.91 != ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2410_even__flip__bit__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [M: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 12.24/7.91 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 != ( M
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2411_even__unset__bit__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [M: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 12.24/7.91 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.91 | ( M
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2412_signed__take__bit__rec,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 12.24/7.91 = ( ^ [N3: nat,A5: A] :
% 12.24/7.91 ( if @ A
% 12.24/7.91 @ ( N3
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.91 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2413_dbl__simps_I4_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2414_dbl__simps_I3_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2415_concat__bit__Suc,axiom,
% 12.24/7.91 ! [N: nat,K: int,L: int] :
% 12.24/7.91 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 12.24/7.91 = ( 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 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2416_set__decode__Suc,axiom,
% 12.24/7.91 ! [N: nat,X: nat] :
% 12.24/7.91 ( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 12.24/7.91 = ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2417_signed__take__bit__Suc,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.91 = ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2418_Compl__eq__Compl__iff,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.91 ( ( ( uminus_uminus @ ( set @ A ) @ A6 )
% 12.24/7.91 = ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 12.24/7.91 = ( A6 = B7 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2419_Compl__iff,axiom,
% 12.24/7.91 ! [A: $tType,C2: A,A6: set @ A] :
% 12.24/7.91 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.91 = ( ~ ( member @ A @ C2 @ A6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2420_ComplI,axiom,
% 12.24/7.91 ! [A: $tType,C2: A,A6: set @ A] :
% 12.24/7.91 ( ~ ( member @ A @ C2 @ A6 )
% 12.24/7.91 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2421_signed__take__bit__of__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2422_concat__bit__0,axiom,
% 12.24/7.91 ! [K: int,L: int] :
% 12.24/7.91 ( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
% 12.24/7.91 = L ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2423_dbl__simps_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2424_signed__take__bit__of__minus__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2425_signed__take__bit__Suc__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2426_signed__take__bit__numeral__of__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [K: num] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2427_concat__bit__nonnegative__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int,L: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
% 12.24/7.91 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2428_concat__bit__negative__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int,L: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2429_dbl__simps_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [K: num] :
% 12.24/7.91 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2430_dbl__simps_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [K: num] :
% 12.24/7.91 ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 12.24/7.91 = ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2431_signed__take__bit__Suc__bit0,axiom,
% 12.24/7.91 ! [N: nat,K: num] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
% 12.24/7.91 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2432_signed__take__bit__Suc__minus__bit0,axiom,
% 12.24/7.91 ! [N: nat,K: num] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.91 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2433_set__decode__0,axiom,
% 12.24/7.91 ! [X: nat] :
% 12.24/7.91 ( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
% 12.24/7.91 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2434_signed__take__bit__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.91 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2435_double__complement,axiom,
% 12.24/7.91 ! [A: $tType,A6: set @ A] :
% 12.24/7.91 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.91 = A6 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2436_ComplD,axiom,
% 12.24/7.91 ! [A: $tType,C2: A,A6: set @ A] :
% 12.24/7.91 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.91 => ~ ( member @ A @ C2 @ A6 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2437_signed__take__bit__mult,axiom,
% 12.24/7.91 ! [N: nat,K: int,L: int] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
% 12.24/7.91 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2438_signed__take__bit__minus,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
% 12.24/7.91 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2439_subset__decode__imp__le,axiom,
% 12.24/7.91 ! [M: nat,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 12.24/7.91 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2440_dbl__def,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( neg_numeral_dbl @ A )
% 12.24/7.91 = ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2441_signed__take__bit__int__less__exp,axiom,
% 12.24/7.91 ! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2442_signed__take__bit__int__greater__eq__self__iff,axiom,
% 12.24/7.91 ! [K: int,N: nat] :
% 12.24/7.91 ( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2443_signed__take__bit__int__less__self__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
% 12.24/7.91 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2444_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 12.24/7.91 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2445_signed__take__bit__int__less__eq__self__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
% 12.24/7.91 = ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2446_signed__take__bit__int__greater__self__iff,axiom,
% 12.24/7.91 ! [K: int,N: nat] :
% 12.24/7.91 ( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 12.24/7.91 = ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2447_signed__take__bit__int__less__eq,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
% 12.24/7.91 => ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2448_signed__take__bit__int__eq__self,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
% 12.24/7.91 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.91 => ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
% 12.24/7.91 = K ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2449_signed__take__bit__int__eq__self__iff,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
% 12.24/7.91 = K )
% 12.24/7.91 = ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
% 12.24/7.91 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2450_signed__take__bit__int__greater__eq,axiom,
% 12.24/7.91 ! [K: int,N: nat] :
% 12.24/7.91 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2451_signed__take__bit__Suc__minus__bit1,axiom,
% 12.24/7.91 ! [N: nat,K: num] :
% 12.24/7.91 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.91 = ( 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 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2452_fact__code,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( semiring_char_0 @ A )
% 12.24/7.91 => ( ( semiring_char_0_fact @ A )
% 12.24/7.91 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2453_take__bit__rec,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ( ( bit_se2584673776208193580ke_bit @ A )
% 12.24/7.91 = ( ^ [N3: nat,A5: A] :
% 12.24/7.91 ( if @ A
% 12.24/7.91 @ ( N3
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 @ ( zero_zero @ A )
% 12.24/7.91 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2454_num_Osize__gen_I2_J,axiom,
% 12.24/7.91 ! [X22: num] :
% 12.24/7.91 ( ( size_num @ ( bit0 @ X22 ) )
% 12.24/7.91 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2455_and__int__unfold,axiom,
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ int )
% 12.24/7.91 = ( ^ [K3: int,L3: int] :
% 12.24/7.91 ( if @ int
% 12.24/7.91 @ ( ( K3
% 12.24/7.91 = ( zero_zero @ int ) )
% 12.24/7.91 | ( L3
% 12.24/7.91 = ( zero_zero @ int ) ) )
% 12.24/7.91 @ ( zero_zero @ int )
% 12.24/7.91 @ ( if @ int
% 12.24/7.91 @ ( K3
% 12.24/7.91 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.91 @ L3
% 12.24/7.91 @ ( if @ int
% 12.24/7.91 @ ( L3
% 12.24/7.91 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.91 @ K3
% 12.24/7.91 @ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2456_xor__Suc__0__eq,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.91 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.91 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.91 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2457_semiring__norm_I90_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ( bit1 @ M )
% 12.24/7.91 = ( bit1 @ N ) )
% 12.24/7.91 = ( M = N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2458_verit__eq__simplify_I9_J,axiom,
% 12.24/7.91 ! [X32: num,Y32: num] :
% 12.24/7.91 ( ( ( bit1 @ X32 )
% 12.24/7.91 = ( bit1 @ Y32 ) )
% 12.24/7.91 = ( X32 = Y32 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2459_take__bit__of__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2460_and__zero__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2461_zero__and__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2462_bit_Oconj__zero__left,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2463_bit_Oconj__zero__right,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2464_semiring__norm_I88_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( bit0 @ M )
% 12.24/7.91 != ( bit1 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2465_semiring__norm_I89_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( bit1 @ M )
% 12.24/7.91 != ( bit0 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2466_semiring__norm_I84_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( one2
% 12.24/7.91 != ( bit1 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2467_semiring__norm_I86_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ( ( bit1 @ M )
% 12.24/7.91 != one2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2468_bit_Oxor__self,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2469_xor__self__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2470_xor_Oleft__neutral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2471_xor_Oright__neutral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2472_concat__bit__of__zero__2,axiom,
% 12.24/7.91 ! [N: nat,K: int] :
% 12.24/7.91 ( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
% 12.24/7.91 = ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2473_semiring__norm_I73_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( ord_less_eq @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2474_semiring__norm_I80_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( ord_less @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2475_take__bit__0,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2476_take__bit__Suc__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2477_bit_Oconj__one__right,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [X: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = X ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2478_and_Oright__neutral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2479_and_Oleft__neutral,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.91 => ! [A3: A] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
% 12.24/7.91 = A3 ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2480_take__bit__numeral__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [L: num] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2481_of__nat__nat__take__bit__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( ring_1 @ A )
% 12.24/7.91 => ! [N: nat,K: int] :
% 12.24/7.91 ( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
% 12.24/7.91 = ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2482_and__nonnegative__int__iff,axiom,
% 12.24/7.91 ! [K: int,L: int] :
% 12.24/7.91 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
% 12.24/7.91 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.91 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2483_and__negative__int__iff,axiom,
% 12.24/7.91 ! [K: int,L: int] :
% 12.24/7.91 ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
% 12.24/7.91 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.91 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2484_semiring__norm_I9_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2485_semiring__norm_I7_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2486_semiring__norm_I14_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2487_semiring__norm_I15_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2488_semiring__norm_I81_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( ord_less @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2489_semiring__norm_I72_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( ord_less_eq @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2490_semiring__norm_I77_J,axiom,
% 12.24/7.91 ! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2491_semiring__norm_I70_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2492_dbl__inc__simps_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ! [K: num] :
% 12.24/7.91 ( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2493_take__bit__of__1__eq__0__iff,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) )
% 12.24/7.91 = ( N
% 12.24/7.91 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2494_and__numerals_I8_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2495_and__numerals_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [Y3: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.91 = ( one_one @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2496_semiring__norm_I10_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2497_semiring__norm_I8_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
% 12.24/7.91 = ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2498_semiring__norm_I5_J,axiom,
% 12.24/7.91 ! [M: num] :
% 12.24/7.91 ( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
% 12.24/7.91 = ( bit1 @ M ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2499_semiring__norm_I4_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
% 12.24/7.91 = ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2500_semiring__norm_I3_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
% 12.24/7.91 = ( bit1 @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2501_take__bit__of__Suc__0,axiom,
% 12.24/7.91 ! [N: nat] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.91 = ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2502_semiring__norm_I16_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2503_semiring__norm_I74_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.91 = ( ord_less @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2504_semiring__norm_I79_J,axiom,
% 12.24/7.91 ! [M: num,N: num] :
% 12.24/7.91 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.91 = ( ord_less_eq @ num @ M @ N ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2505_and__numerals_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [Y3: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2506_and__numerals_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( zero_zero @ A ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2507_and__numerals_I3_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num,Y3: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.91 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2508_xor__numerals_I3_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num,Y3: num] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.91 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2509_xor__numerals_I8_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2510_xor__numerals_I5_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2511_xor__numerals_I2_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [Y3: num] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2512_xor__numerals_I1_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [Y3: num] :
% 12.24/7.91 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2513_take__bit__of__1,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat] :
% 12.24/7.91 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 12.24/7.91 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2514_and__minus__numerals_I2_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.91 = ( one_one @ int ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2515_and__minus__numerals_I6_J,axiom,
% 12.24/7.91 ! [N: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
% 12.24/7.91 = ( one_one @ int ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2516_dbl__inc__simps_I3_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( neg_numeral @ A )
% 12.24/7.91 => ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
% 12.24/7.91 = ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2517_even__take__bit__eq,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.91 => ! [N: nat,A3: A] :
% 12.24/7.91 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 12.24/7.91 = ( ( N
% 12.24/7.91 = ( zero_zero @ nat ) )
% 12.24/7.91 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 12.24/7.91
% 12.24/7.91 thf(fact_2518_and__numerals_I4_J,axiom,
% 12.24/7.91 ! [A: $tType] :
% 12.24/7.91 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.91 => ! [X: num,Y3: num] :
% 12.24/7.91 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2519_and__numerals_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2520_xor__numerals_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2521_mod__Suc__eq__mod__add3,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.24/7.92 = ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2522_Suc__mod__eq__add3__mod__numeral,axiom,
% 12.24/7.92 ! [M: nat,V: num] :
% 12.24/7.92 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 12.24/7.92 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2523_div__Suc__eq__div__add3,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 12.24/7.92 = ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2524_Suc__div__eq__add3__div__numeral,axiom,
% 12.24/7.92 ! [M: nat,V: num] :
% 12.24/7.92 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 12.24/7.92 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2525_and__minus__numerals_I5_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
% 12.24/7.92 = ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2526_and__minus__numerals_I1_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2527_xor__nat__numerals_I4_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2528_xor__nat__numerals_I3_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2529_xor__nat__numerals_I2_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2530_xor__nat__numerals_I1_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2531_dbl__dec__simps_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2532_take__bit__Suc__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
% 12.24/7.92 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2533_and__numerals_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2534_xor__numerals_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2535_xor__numerals_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2536_take__bit__of__exp,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2537_take__bit__of__2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2538_zmod__numeral__Bit1,axiom,
% 12.24/7.92 ! [V: num,W2: num] :
% 12.24/7.92 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2539_signed__take__bit__Suc__bit1,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2540_take__bit__of__nat,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 12.24/7.92 = ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2541_of__nat__xor__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
% 12.24/7.92 = ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2542_of__nat__and__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
% 12.24/7.92 = ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2543_take__bit__tightened,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A,B2: A,M: nat] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
% 12.24/7.92 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2544_take__bit__tightened__less__eq__nat,axiom,
% 12.24/7.92 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 => ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q4 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2545_take__bit__nat__less__eq__self,axiom,
% 12.24/7.92 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2546_verit__eq__simplify_I14_J,axiom,
% 12.24/7.92 ! [X22: num,X32: num] :
% 12.24/7.92 ( ( bit0 @ X22 )
% 12.24/7.92 != ( bit1 @ X32 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2547_verit__eq__simplify_I12_J,axiom,
% 12.24/7.92 ! [X32: num] :
% 12.24/7.92 ( one2
% 12.24/7.92 != ( bit1 @ X32 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2548_take__bit__mult,axiom,
% 12.24/7.92 ! [N: nat,K: int,L: int] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2549_take__bit__minus,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2550_take__bit__nat__eq,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
% 12.24/7.92 = ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2551_nat__take__bit__eq,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 => ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2552_and__eq__minus__1__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( ( A3
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 & ( B2
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2553_num_Oexhaust,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( Y3 != one2 )
% 12.24/7.92 => ( ! [X23: num] :
% 12.24/7.92 ( Y3
% 12.24/7.92 != ( bit0 @ X23 ) )
% 12.24/7.92 => ~ ! [X33: num] :
% 12.24/7.92 ( Y3
% 12.24/7.92 != ( bit1 @ X33 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2554_AND__upper2_H,axiom,
% 12.24/7.92 ! [Y3: int,Z3: int,X: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ( ord_less_eq @ int @ Y3 @ Z3 )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2555_AND__upper1_H,axiom,
% 12.24/7.92 ! [Y3: int,Z3: int,Ya: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ( ord_less_eq @ int @ Y3 @ Z3 )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2556_AND__upper2,axiom,
% 12.24/7.92 ! [Y3: int,X: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Y3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2557_AND__upper1,axiom,
% 12.24/7.92 ! [X: int,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2558_AND__lower,axiom,
% 12.24/7.92 ! [X: int,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2559_take__bit__int__less__eq__self__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
% 12.24/7.92 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2560_take__bit__nonnegative,axiom,
% 12.24/7.92 ! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2561_not__take__bit__negative,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2562_take__bit__int__greater__self__iff,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 12.24/7.92 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2563_take__bit__tightened__less__eq__int,axiom,
% 12.24/7.92 ! [M: nat,N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2564_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A,B2: A] :
% 12.24/7.92 ( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
% 12.24/7.92 = ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
% 12.24/7.92 = ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2565_signed__take__bit__take__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat,A3: A] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 12.24/7.92 = ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2566_num_Osize__gen_I3_J,axiom,
% 12.24/7.92 ! [X32: num] :
% 12.24/7.92 ( ( size_num @ ( bit1 @ X32 ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2567_take__bit__unset__bit__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat,A3: A] :
% 12.24/7.92 ( ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2568_take__bit__set__bit__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat,A3: A] :
% 12.24/7.92 ( ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2569_take__bit__flip__bit__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat,A3: A] :
% 12.24/7.92 ( ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
% 12.24/7.92 = ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2570_inc_Osimps_I3_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( inc @ ( bit1 @ X ) )
% 12.24/7.92 = ( bit0 @ ( inc @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2571_inc_Osimps_I2_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( inc @ ( bit0 @ X ) )
% 12.24/7.92 = ( bit1 @ X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2572_numeral__Bit1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 12.24/7.92 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2573_take__bit__signed__take__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
% 12.24/7.92 = ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2574_and__less__eq,axiom,
% 12.24/7.92 ! [L: int,K: int] :
% 12.24/7.92 ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2575_AND__upper1_H_H,axiom,
% 12.24/7.92 ! [Y3: int,Z3: int,Ya: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ( ord_less @ int @ Y3 @ Z3 )
% 12.24/7.92 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2576_AND__upper2_H_H,axiom,
% 12.24/7.92 ! [Y3: int,Z3: int,X: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ( ord_less @ int @ Y3 @ Z3 )
% 12.24/7.92 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2577_eval__nat__numeral_I3_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( numeral_numeral @ nat @ ( bit1 @ N ) )
% 12.24/7.92 = ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2578_power__minus__Bit1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: A,K: num] :
% 12.24/7.92 ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2579_take__bit__decr__eq,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 != ( zero_zero @ int ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
% 12.24/7.92 = ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2580_take__bit__Suc__bit1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2581_cong__exp__iff__simps_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.92 => ! [N: num,Q4: num] :
% 12.24/7.92 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 12.24/7.92 != ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2582_numeral__Bit1__div__2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.92 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2583_power3__eq__cube,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( monoid_mult @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2584_numeral__3__eq__3,axiom,
% 12.24/7.92 ( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
% 12.24/7.92 = ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2585_Suc3__eq__add__3,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2586_take__bit__Suc__minus__bit1,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.92 = ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2587_take__bit__Suc__bit0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 12.24/7.92 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2588_and__one__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.92 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2589_one__and__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.92 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2590_take__bit__nat__eq__self,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
% 12.24/7.92 = M ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2591_take__bit__nat__less__exp,axiom,
% 12.24/7.92 ! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2592_take__bit__nat__eq__self__iff,axiom,
% 12.24/7.92 ! [N: nat,M: nat] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
% 12.24/7.92 = M )
% 12.24/7.92 = ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2593_cong__exp__iff__simps_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.92 => ! [Q4: num,N: num] :
% 12.24/7.92 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 12.24/7.92 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
% 12.24/7.92 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2594_cong__exp__iff__simps_I11_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.92 => ! [M: num,Q4: num] :
% 12.24/7.92 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
% 12.24/7.92 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
% 12.24/7.92 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2595_take__bit__int__less__exp,axiom,
% 12.24/7.92 ! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2596_fold__atLeastAtMost__nat_Osimps,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( set_fo6178422350223883121st_nat @ A )
% 12.24/7.92 = ( ^ [F2: nat > A > A,A5: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2597_fold__atLeastAtMost__nat_Oelims,axiom,
% 12.24/7.92 ! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
% 12.24/7.92 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
% 12.24/7.92 = Y3 )
% 12.24/7.92 => ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
% 12.24/7.92 => ( Y3 = Xc ) )
% 12.24/7.92 & ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
% 12.24/7.92 => ( Y3
% 12.24/7.92 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2598_Suc__mod__eq__add3__mod,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.24/7.92 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2599_Suc__div__eq__add3__div,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 12.24/7.92 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2600_num_Osize__gen_I1_J,axiom,
% 12.24/7.92 ( ( size_num @ one2 )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2601_take__bit__eq__0__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2602_take__bit__nat__less__self__iff,axiom,
% 12.24/7.92 ! [N: nat,M: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
% 12.24/7.92 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2603_take__bit__Suc__minus__bit0,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2604_take__bit__int__less__self__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
% 12.24/7.92 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2605_take__bit__int__greater__eq__self__iff,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
% 12.24/7.92 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2606_take__bit__int__eq__self,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 = K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2607_take__bit__int__eq__self__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 = K )
% 12.24/7.92 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2608_signed__take__bit__eq__take__bit__shift,axiom,
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2609_take__bit__incr__eq,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 != ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2610_take__bit__Suc__minus__1__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2611_take__bit__Suc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.92 = ( 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 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2612_take__bit__numeral__minus__1__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2613_take__bit__int__less__eq,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
% 12.24/7.92 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2614_take__bit__int__greater__eq,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.92 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2615_xor__nat__unfold,axiom,
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( if @ nat
% 12.24/7.92 @ ( M2
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 @ N3
% 12.24/7.92 @ ( if @ nat
% 12.24/7.92 @ ( N3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 @ M2
% 12.24/7.92 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( 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 @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2616_and__int__rec,axiom,
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int )
% 12.24/7.92 = ( ^ [K3: int,L3: int] :
% 12.24/7.92 ( plus_plus @ int
% 12.24/7.92 @ ( zero_neq_one_of_bool @ int
% 12.24/7.92 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 12.24/7.92 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 12.24/7.92 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2617_xor__nat__rec,axiom,
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( plus_plus @ nat
% 12.24/7.92 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.92 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
% 12.24/7.92 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
% 12.24/7.92 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2618_xor__one__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.92 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 12.24/7.92 @ ( zero_neq_one_of_bool @ A
% 12.24/7.92 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2619_one__xor__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.92 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 12.24/7.92 @ ( zero_neq_one_of_bool @ A
% 12.24/7.92 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2620_stable__imp__take__bit__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.92 = A3 )
% 12.24/7.92 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 12.24/7.92 = ( zero_zero @ A ) ) )
% 12.24/7.92 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 12.24/7.92 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2621_take__bit__minus__small__eq,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 => ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
% 12.24/7.92 = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2622_odd__mod__4__div__2,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 12.24/7.92 => ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2623_Suc__0__xor__eq,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.92 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 12.24/7.92 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.92 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2624_signed__take__bit__numeral__minus__bit1,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.92 = ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2625_binomial__code,axiom,
% 12.24/7.92 ( binomial
% 12.24/7.92 = ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2626_signed__take__bit__numeral__bit1,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2627_take__bit__numeral__minus__bit1,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.92 = ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2628_take__bit__numeral__bit1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( 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 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2629_signed__take__bit__numeral__minus__bit0,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2630_xor__nonnegative__int__iff,axiom,
% 12.24/7.92 ! [K: int,L: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
% 12.24/7.92 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2631_xor__negative__int__iff,axiom,
% 12.24/7.92 ! [K: int,L: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.92 != ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2632_binomial__Suc__n,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ ( suc @ N ) @ N )
% 12.24/7.92 = ( suc @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2633_binomial__n__n,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ N @ N )
% 12.24/7.92 = ( one_one @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2634_binomial__1,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = N ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2635_binomial__0__Suc,axiom,
% 12.24/7.92 ! [K: nat] :
% 12.24/7.92 ( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2636_binomial__eq__0__iff,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( ( binomial @ N @ K )
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 = ( ord_less @ nat @ N @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2637_binomial__Suc__Suc,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2638_pred__numeral__simps_I1_J,axiom,
% 12.24/7.92 ( ( pred_numeral @ one2 )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2639_binomial__n__0,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ N @ ( zero_zero @ nat ) )
% 12.24/7.92 = ( one_one @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2640_eq__numeral__Suc,axiom,
% 12.24/7.92 ! [K: num,N: nat] :
% 12.24/7.92 ( ( ( numeral_numeral @ nat @ K )
% 12.24/7.92 = ( suc @ N ) )
% 12.24/7.92 = ( ( pred_numeral @ K )
% 12.24/7.92 = N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2641_Suc__eq__numeral,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( ( suc @ N )
% 12.24/7.92 = ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( N
% 12.24/7.92 = ( pred_numeral @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2642_pred__numeral__inc,axiom,
% 12.24/7.92 ! [K: num] :
% 12.24/7.92 ( ( pred_numeral @ ( inc @ K ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2643_zero__less__binomial__iff,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( ord_less_eq @ nat @ K @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2644_less__Suc__numeral,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2645_less__numeral__Suc,axiom,
% 12.24/7.92 ! [K: num,N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 12.24/7.92 = ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2646_le__Suc__numeral,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2647_le__numeral__Suc,axiom,
% 12.24/7.92 ! [K: num,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 12.24/7.92 = ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2648_diff__numeral__Suc,axiom,
% 12.24/7.92 ! [K: num,N: nat] :
% 12.24/7.92 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 12.24/7.92 = ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2649_diff__Suc__numeral,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2650_pred__numeral__simps_I3_J,axiom,
% 12.24/7.92 ! [K: num] :
% 12.24/7.92 ( ( pred_numeral @ ( bit1 @ K ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2651_and__nat__numerals_I3_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2652_and__nat__numerals_I1_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2653_and__nat__numerals_I2_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( one_one @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2654_and__nat__numerals_I4_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( one_one @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2655_Suc__0__and__eq,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.92 = ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2656_and__Suc__0__eq,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2657_signed__take__bit__numeral__bit0,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
% 12.24/7.92 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2658_choose__one,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ N @ ( one_one @ nat ) )
% 12.24/7.92 = N ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2659_binomial__eq__0,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ N @ K )
% 12.24/7.92 => ( ( binomial @ N @ K )
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2660_Suc__times__binomial__eq,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2661_Suc__times__binomial,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 12.24/7.92 = ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2662_XOR__lower,axiom,
% 12.24/7.92 ! [X: int,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2663_binomial__symmetric,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ( binomial @ N @ K )
% 12.24/7.92 = ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2664_choose__mult__lemma,axiom,
% 12.24/7.92 ! [M: nat,R: nat,K: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
% 12.24/7.92 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R ) @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2665_binomial__le__pow,axiom,
% 12.24/7.92 ! [R: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ R @ N )
% 12.24/7.92 => ( ord_less_eq @ nat @ ( binomial @ N @ R ) @ ( power_power @ nat @ N @ R ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2666_binomial__gbinomial,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( field_char_0 @ A )
% 12.24/7.92 => ! [N: nat,K: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2667_numeral__eq__Suc,axiom,
% 12.24/7.92 ( ( numeral_numeral @ nat )
% 12.24/7.92 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2668_zero__less__binomial,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2669_Suc__times__binomial__add,axiom,
% 12.24/7.92 ! [A3: nat,B2: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
% 12.24/7.92 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2670_choose__mult,axiom,
% 12.24/7.92 ! [K: nat,M: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ M )
% 12.24/7.92 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 => ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 12.24/7.92 = ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2671_binomial__absorb__comp,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2672_binomial__Suc__Suc__eq__times,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 12.24/7.92 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2673_and__nat__def,axiom,
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2674_xor__nat__def,axiom,
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2675_pred__numeral__def,axiom,
% 12.24/7.92 ( pred_numeral
% 12.24/7.92 = ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2676_binomial__absorption,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 12.24/7.92 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2677_binomial__fact__lemma,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( semiring_char_0_fact @ nat @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2678_fact__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( semiring_char_0 @ A )
% 12.24/7.92 => ! [K: num] :
% 12.24/7.92 ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2679_binomial__ge__n__over__k__pow__k,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_field @ A )
% 12.24/7.92 => ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2680_choose__reduce__nat,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.24/7.92 => ( ( binomial @ N @ K )
% 12.24/7.92 = ( 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 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2681_times__binomial__minus1__eq,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.24/7.92 => ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2682_binomial__le__pow2,axiom,
% 12.24/7.92 ! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2683_binomial__altdef__nat,axiom,
% 12.24/7.92 ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ( binomial @ N @ K )
% 12.24/7.92 = ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2684_binomial__addition__formula,axiom,
% 12.24/7.92 ! [N: nat,K: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( binomial @ N @ ( suc @ K ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2685_fact__binomial,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( field_char_0 @ A )
% 12.24/7.92 => ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
% 12.24/7.92 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2686_binomial__fact,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( field_char_0 @ A )
% 12.24/7.92 => ! [K: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.92 => ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 12.24/7.92 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2687_take__bit__numeral__bit0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 12.24/7.92 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2688_take__bit__numeral__minus__bit0,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2689_XOR__upper,axiom,
% 12.24/7.92 ! [X: int,N: nat,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2690_choose__two,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.92 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2691_and__nat__unfold,axiom,
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( if @ nat
% 12.24/7.92 @ ( ( M2
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 | ( N3
% 12.24/7.92 = ( zero_zero @ nat ) ) )
% 12.24/7.92 @ ( zero_zero @ nat )
% 12.24/7.92 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2692_and__nat__rec,axiom,
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( plus_plus @ nat
% 12.24/7.92 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.92 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
% 12.24/7.92 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
% 12.24/7.92 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2693_xor__int__rec,axiom,
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ int )
% 12.24/7.92 = ( ^ [K3: int,L3: int] :
% 12.24/7.92 ( plus_plus @ int
% 12.24/7.92 @ ( zero_neq_one_of_bool @ int
% 12.24/7.92 @ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
% 12.24/7.92 != ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 12.24/7.92 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2694_xor__int__unfold,axiom,
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ int )
% 12.24/7.92 = ( ^ [K3: int,L3: int] :
% 12.24/7.92 ( if @ int
% 12.24/7.92 @ ( K3
% 12.24/7.92 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.92 @ ( bit_ri4277139882892585799ns_not @ int @ L3 )
% 12.24/7.92 @ ( if @ int
% 12.24/7.92 @ ( L3
% 12.24/7.92 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.92 @ ( bit_ri4277139882892585799ns_not @ int @ K3 )
% 12.24/7.92 @ ( if @ int
% 12.24/7.92 @ ( K3
% 12.24/7.92 = ( zero_zero @ int ) )
% 12.24/7.92 @ L3
% 12.24/7.92 @ ( if @ int
% 12.24/7.92 @ ( L3
% 12.24/7.92 = ( zero_zero @ int ) )
% 12.24/7.92 @ K3
% 12.24/7.92 @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2695_signed__take__bit__eq__take__bit__minus,axiom,
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2696_mask__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.92 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2697_take__bit__eq__mask__iff__exp__dvd,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 12.24/7.92 = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2698_take__bit__Suc__from__most,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
% 12.24/7.92 = ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2699_push__bit__numeral__minus__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2700_mask__nat__positive__iff,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 12.24/7.92 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2701_push__bit__nonnegative__int__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
% 12.24/7.92 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2702_push__bit__negative__int__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2703_push__bit__of__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2704_push__bit__eq__0__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( A3
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2705_concat__bit__of__zero__1,axiom,
% 12.24/7.92 ! [N: nat,L: int] :
% 12.24/7.92 ( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2706_mask__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2707_mask__eq__0__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( ( bit_se2239418461657761734s_mask @ A @ N )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2708_bit_Oconj__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2709_bit_Oconj__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2710_bit_Ocompl__zero,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2711_bit_Ocompl__one,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2712_bit__numeral__Bit0__Suc__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [M: num,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2713_push__bit__Suc__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2714_bit__numeral__Bit1__Suc__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [M: num,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2715_mask__Suc__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2716_bit_Oxor__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2717_bit_Oxor__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2718_bit_Oxor__one__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2719_bit_Oxor__one__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2720_take__bit__minus__one__eq__mask,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2721_not__negative__int__iff,axiom,
% 12.24/7.92 ! [K: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2722_not__nonnegative__int__iff,axiom,
% 12.24/7.92 ! [K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
% 12.24/7.92 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2723_minus__not__numeral__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2724_signed__take__bit__nonnegative__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2725_signed__take__bit__negative__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2726_push__bit__Suc__minus__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2727_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 12.24/7.92 ! [W2: num,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2728_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 12.24/7.92 ! [W2: num,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2729_push__bit__minus__one__eq__not__mask,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2730_not__one__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2731_bit__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.92 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2732_push__bit__Suc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2733_push__bit__of__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
% 12.24/7.92 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2734_even__push__bit__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
% 12.24/7.92 = ( ( N
% 12.24/7.92 != ( zero_zero @ nat ) )
% 12.24/7.92 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2735_push__bit__of__Suc__0,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2736_push__bit__minus__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2737_bit__minus__numeral__int_I1_J,axiom,
% 12.24/7.92 ! [W2: num,N: num] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2738_bit__minus__numeral__int_I2_J,axiom,
% 12.24/7.92 ! [W2: num,N: num] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2739_bit__mod__2__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
% 12.24/7.92 = ( ( N
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2740_of__nat__mask__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 12.24/7.92 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2741_bit__of__nat__iff__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2742_push__bit__of__nat,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 12.24/7.92 = ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2743_of__nat__push__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
% 12.24/7.92 = ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2744_bit__push__bit__iff__int,axiom,
% 12.24/7.92 ! [M: nat,K: int,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
% 12.24/7.92 = ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2745_push__bit__minus,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2746_less__eq__mask,axiom,
% 12.24/7.92 ! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2747_bit__minus__int__iff,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2748_bit__iff__and__push__bit__not__eq__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A )
% 12.24/7.92 = ( ^ [A5: A,N3: nat] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
% 12.24/7.92 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2749_unset__bit__eq__and__not,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_se2638667681897837118et_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2750_unset__bit__int__def,axiom,
% 12.24/7.92 ( ( bit_se2638667681897837118et_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2751_take__bit__not__mask__eq__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2752_not__bit__1__Suc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2753_bit__1__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
% 12.24/7.92 = ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2754_bit__numeral__simps_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2755_bit__take__bit__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [M: nat,A3: A,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
% 12.24/7.92 = ( ( ord_less @ nat @ N @ M )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2756_bit__of__bool__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [B2: $o,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N )
% 12.24/7.92 = ( B2
% 12.24/7.92 & ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2757_bit__not__iff__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
% 12.24/7.92 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.92 != ( zero_zero @ A ) )
% 12.24/7.92 & ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2758_mask__nonnegative__int,axiom,
% 12.24/7.92 ! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2759_not__mask__negative__int,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2760_flip__bit__nat__def,axiom,
% 12.24/7.92 ( ( bit_se8732182000553998342ip_bit @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2761_flip__bit__int__def,axiom,
% 12.24/7.92 ( ( bit_se8732182000553998342ip_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2762_minus__exp__eq__not__mask,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2763_minus__eq__not__plus__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( uminus_uminus @ A )
% 12.24/7.92 = ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2764_minus__eq__not__minus__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( uminus_uminus @ A )
% 12.24/7.92 = ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2765_not__eq__complement,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_ri4277139882892585799ns_not @ A )
% 12.24/7.92 = ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2766_not__int__def,axiom,
% 12.24/7.92 ( ( bit_ri4277139882892585799ns_not @ int )
% 12.24/7.92 = ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2767_bit__not__int__iff_H,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2768_and__not__numerals_I1_J,axiom,
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2769_minus__numeral__inc__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2770_less__mask,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.92 => ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2771_flip__bit__eq__xor,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se8732182000553998342ip_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2772_and__not__numerals_I4_J,axiom,
% 12.24/7.92 ! [M: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2773_and__not__numerals_I2_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( one_one @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2774_bit__imp__take__bit__positive,axiom,
% 12.24/7.92 ! [N: nat,M: nat,K: int] :
% 12.24/7.92 ( ( ord_less @ nat @ N @ M )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
% 12.24/7.92 => ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2775_not__numeral__Bit0__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2776_bit__concat__bit__iff,axiom,
% 12.24/7.92 ! [M: nat,K: int,L: int,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 12.24/7.92 = ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
% 12.24/7.92 | ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2777_push__bit__double,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.92 = ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2778_signed__take__bit__eq__concat__bit,axiom,
% 12.24/7.92 ( ( bit_ri4674362597316999326ke_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2779_exp__eq__0__imp__not__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 => ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2780_bit__Suc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2781_take__bit__eq__mask__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 12.24/7.92 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 12.24/7.92 = ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 12.24/7.92 = ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2782_int__bit__bound,axiom,
% 12.24/7.92 ! [K: int] :
% 12.24/7.92 ~ ! [N4: nat] :
% 12.24/7.92 ( ! [M3: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ N4 @ M3 )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
% 12.24/7.92 = ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) )
% 12.24/7.92 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2783_and__not__numerals_I5_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2784_and__not__numerals_I7_J,axiom,
% 12.24/7.92 ! [M: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2785_and__not__numerals_I3_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( zero_zero @ int ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2786_push__bit__int__def,axiom,
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2787_push__bit__nat__def,axiom,
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ nat )
% 12.24/7.92 = ( ^ [N3: nat,M2: nat] : ( times_times @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2788_push__bit__eq__mult,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se4730199178511100633sh_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2789_and__exp__eq__0__iff__not__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2790_Suc__mask__eq__exp,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
% 12.24/7.92 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2791_mask__nat__less__exp,axiom,
% 12.24/7.92 ! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2792_and__not__numerals_I9_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2793_and__not__numerals_I6_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2794_push__bit__minus__one,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2795_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) )
% 12.24/7.92 = ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2796_even__bit__succ__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
% 12.24/7.92 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 12.24/7.92 | ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2797_odd__bit__iff__bit__pred,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 12.24/7.92 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
% 12.24/7.92 | ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2798_mask__half__int,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 12.24/7.92 = ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2799_mask__int__def,axiom,
% 12.24/7.92 ( ( bit_se2239418461657761734s_mask @ int )
% 12.24/7.92 = ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2800_mask__nat__def,axiom,
% 12.24/7.92 ( ( bit_se2239418461657761734s_mask @ nat )
% 12.24/7.92 = ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2801_bit__sum__mult__2__cases,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.92 ( ! [J2: nat] :
% 12.24/7.92 ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
% 12.24/7.92 = ( ( ( N
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 12.24/7.92 & ( ( N
% 12.24/7.92 != ( zero_zero @ nat ) )
% 12.24/7.92 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2802_bit__rec,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_semiring_bits @ A )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A )
% 12.24/7.92 = ( ^ [A5: A,N3: nat] :
% 12.24/7.92 ( ( ( N3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
% 12.24/7.92 & ( ( N3
% 12.24/7.92 != ( zero_zero @ nat ) )
% 12.24/7.92 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2803_mask__eq__exp__minus__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se2239418461657761734s_mask @ A )
% 12.24/7.92 = ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2804_and__not__numerals_I8_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2805_not__int__rec,axiom,
% 12.24/7.92 ( ( bit_ri4277139882892585799ns_not @ int )
% 12.24/7.92 = ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2806_set__bit__eq,axiom,
% 12.24/7.92 ( ( bit_se5668285175392031749et_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] :
% 12.24/7.92 ( plus_plus @ int @ K3
% 12.24/7.92 @ ( times_times @ int
% 12.24/7.92 @ ( zero_neq_one_of_bool @ int
% 12.24/7.92 @ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
% 12.24/7.92 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2807_unset__bit__eq,axiom,
% 12.24/7.92 ( ( bit_se2638667681897837118et_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2808_num_Osize_I6_J,axiom,
% 12.24/7.92 ! [X32: num] :
% 12.24/7.92 ( ( size_size @ num @ ( bit1 @ X32 ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2809_num_Osize_I5_J,axiom,
% 12.24/7.92 ! [X22: num] :
% 12.24/7.92 ( ( size_size @ num @ ( bit0 @ X22 ) )
% 12.24/7.92 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2810_or__int__rec,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int )
% 12.24/7.92 = ( ^ [K3: int,L3: int] :
% 12.24/7.92 ( plus_plus @ int
% 12.24/7.92 @ ( zero_neq_one_of_bool @ int
% 12.24/7.92 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 12.24/7.92 | ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 12.24/7.92 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2811_drop__bit__numeral__minus__bit1,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2812_or__numerals_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2813_or__numerals_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2814_or_Oright__neutral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.92 = A3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2815_or_Oleft__neutral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.92 = A3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2816_drop__bit__of__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2817_bit_Odisj__one__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2818_bit_Odisj__one__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2819_drop__bit__of__bool,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,B2: $o] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B2 ) )
% 12.24/7.92 = ( zero_neq_one_of_bool @ A
% 12.24/7.92 @ ( ( N
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 & B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2820_or__nonnegative__int__iff,axiom,
% 12.24/7.92 ! [K: int,L: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
% 12.24/7.92 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2821_or__negative__int__iff,axiom,
% 12.24/7.92 ! [K: int,L: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 12.24/7.92 | ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2822_drop__bit__nonnegative__int__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
% 12.24/7.92 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2823_drop__bit__negative__int__iff,axiom,
% 12.24/7.92 ! [N: nat,K: int] :
% 12.24/7.92 ( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
% 12.24/7.92 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2824_drop__bit__minus__one,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2825_or__nat__numerals_I4_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2826_or__nat__numerals_I2_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2827_drop__bit__Suc__bit0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2828_or__numerals_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2829_or__numerals_I8_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2830_bit_Odisj__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2831_bit_Odisj__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2832_drop__bit__Suc__bit1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2833_drop__bit__of__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
% 12.24/7.92 = ( zero_neq_one_of_bool @ A
% 12.24/7.92 @ ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2834_or__numerals_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2835_or__nat__numerals_I1_J,axiom,
% 12.24/7.92 ! [Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2836_or__nat__numerals_I3_J,axiom,
% 12.24/7.92 ! [X: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2837_or__numerals_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2838_or__numerals_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2839_or__minus__numerals_I6_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2840_or__minus__numerals_I2_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2841_drop__bit__Suc__minus__bit0,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2842_drop__bit__numeral__minus__bit0,axiom,
% 12.24/7.92 ! [L: num,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2843_drop__bit__Suc__minus__bit1,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2844_and__minus__minus__numerals,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2845_or__minus__minus__numerals,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2846_or__numerals_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
% 12.24/7.92 = ( 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 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2847_size__neq__size__imp__neq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( size @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ( size_size @ A @ X )
% 12.24/7.92 != ( size_size @ A @ Y3 ) )
% 12.24/7.92 => ( X != Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2848_or__nat__def,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2849_of__nat__or__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
% 12.24/7.92 = ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2850_drop__bit__of__nat,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [N: nat,M: nat] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
% 12.24/7.92 = ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2851_of__nat__drop__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [M: nat,N: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2852_bit_Odisj__zero__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2853_or__eq__0__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( ( A3
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 & ( B2
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2854_set__bit__nat__def,axiom,
% 12.24/7.92 ( ( bit_se5668285175392031749et_bit @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2855_bit__Suc__0__iff,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.92 = ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2856_not__bit__Suc__0__Suc,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2857_bit__push__bit__iff__nat,axiom,
% 12.24/7.92 ! [M: nat,Q4: nat,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N )
% 12.24/7.92 = ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2858_OR__lower,axiom,
% 12.24/7.92 ! [X: int,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.92 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2859_or__greater__eq,axiom,
% 12.24/7.92 ! [L: int,K: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
% 12.24/7.92 => ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2860_set__bit__int__def,axiom,
% 12.24/7.92 ( ( bit_se5668285175392031749et_bit @ int )
% 12.24/7.92 = ( ^ [N3: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2861_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 12.24/7.92 = A3 )
% 12.24/7.92 = ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2862_not__bit__Suc__0__numeral,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2863_or__not__numerals_I1_J,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2864_set__bit__eq__or,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se5668285175392031749et_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2865_div__push__bit__of__1__eq__drop__bit,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A,N: nat] :
% 12.24/7.92 ( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2866_bit__iff__and__drop__bit__eq__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se5641148757651400278ts_bit @ A )
% 12.24/7.92 = ( ^ [A5: A,N3: nat] :
% 12.24/7.92 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A5 ) @ ( one_one @ A ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2867_bit_Ocomplement__unique,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [A3: A,X: A,Y3: A] :
% 12.24/7.92 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 => ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y3 )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y3 )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 => ( X = Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2868_or__not__numerals_I2_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2869_or__not__numerals_I4_J,axiom,
% 12.24/7.92 ! [M: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2870_bit__nat__iff,axiom,
% 12.24/7.92 ! [K: int,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
% 12.24/7.92 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 12.24/7.92 & ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2871_num_Osize_I4_J,axiom,
% 12.24/7.92 ( ( size_size @ num @ one2 )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2872_bit_Ocompl__unique,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y3 )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 => ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y3 )
% 12.24/7.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 => ( ( bit_ri4277139882892585799ns_not @ A @ X )
% 12.24/7.92 = Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2873_or__not__numerals_I3_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2874_or__not__numerals_I7_J,axiom,
% 12.24/7.92 ! [M: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2875_drop__bit__Suc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat,A3: A] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
% 12.24/7.92 = ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2876_mask__Suc__exp,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2877_or__not__numerals_I6_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2878_one__or__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.92 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2879_or__one__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.92 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2880_mask__Suc__double,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 12.24/7.92 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2881_OR__upper,axiom,
% 12.24/7.92 ! [X: int,N: nat,Y3: int] :
% 12.24/7.92 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.92 => ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.92 => ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2882_or__not__numerals_I5_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2883_signed__take__bit__def,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2884_or__Suc__0__eq,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2885_Suc__0__or__eq,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.92 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2886_drop__bit__rec,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.92 => ( ( bit_se4197421643247451524op_bit @ A )
% 12.24/7.92 = ( ^ [N3: nat,A5: A] :
% 12.24/7.92 ( if @ A
% 12.24/7.92 @ ( N3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 @ A5
% 12.24/7.92 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2887_or__not__numerals_I8_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2888_or__not__numerals_I9_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2889_or__nat__rec,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( plus_plus @ nat
% 12.24/7.92 @ ( zero_neq_one_of_bool @ nat
% 12.24/7.92 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
% 12.24/7.92 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
% 12.24/7.92 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2890_or__int__unfold,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int )
% 12.24/7.92 = ( ^ [K3: int,L3: int] :
% 12.24/7.92 ( if @ int
% 12.24/7.92 @ ( ( K3
% 12.24/7.92 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.92 | ( L3
% 12.24/7.92 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 12.24/7.92 @ ( uminus_uminus @ int @ ( one_one @ int ) )
% 12.24/7.92 @ ( if @ int
% 12.24/7.92 @ ( K3
% 12.24/7.92 = ( zero_zero @ int ) )
% 12.24/7.92 @ L3
% 12.24/7.92 @ ( if @ int
% 12.24/7.92 @ ( L3
% 12.24/7.92 = ( zero_zero @ int ) )
% 12.24/7.92 @ K3
% 12.24/7.92 @ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( 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 @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2891_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( 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 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2892_or__minus__numerals_I4_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2893_or__minus__numerals_I8_J,axiom,
% 12.24/7.92 ! [N: num,M: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2894_or__nat__unfold,axiom,
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ nat )
% 12.24/7.92 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.92 ( if @ nat
% 12.24/7.92 @ ( M2
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 @ N3
% 12.24/7.92 @ ( if @ nat
% 12.24/7.92 @ ( N3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 @ M2
% 12.24/7.92 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2895_xor__minus__numerals_I1_J,axiom,
% 12.24/7.92 ! [N: num,K: int] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2896_max_Oright__idem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.92 = ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2897_max_Oleft__idem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
% 12.24/7.92 = ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2898_max_Oidem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( ord_max @ A @ A3 @ A3 )
% 12.24/7.92 = A3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2899_max_Oabsorb1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 => ( ( ord_max @ A @ A3 @ B2 )
% 12.24/7.92 = A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2900_max_Oabsorb2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.92 => ( ( ord_max @ A @ A3 @ B2 )
% 12.24/7.92 = B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2901_max_Obounded__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.92 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2902_max_Oabsorb3,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.92 => ( ( ord_max @ A @ A3 @ B2 )
% 12.24/7.92 = A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2903_max_Oabsorb4,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.92 => ( ( ord_max @ A @ A3 @ B2 )
% 12.24/7.92 = B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2904_max__less__iff__conj,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( ord_less @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.92 = ( ( ord_less @ A @ X @ Z3 )
% 12.24/7.92 & ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2905_of__bool__or__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_semidom @ A )
% 12.24/7.92 => ! [P: $o,Q: $o] :
% 12.24/7.92 ( ( zero_neq_one_of_bool @ A
% 12.24/7.92 @ ( P
% 12.24/7.92 | Q ) )
% 12.24/7.92 = ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2906_max__Suc__Suc,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.24/7.92 = ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2907_max__0R,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
% 12.24/7.92 = N ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2908_max__0L,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 = N ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2909_max__nat_Oright__neutral,axiom,
% 12.24/7.92 ! [A3: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.92 = A3 ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2910_max__nat_Oneutr__eq__iff,axiom,
% 12.24/7.92 ! [A3: nat,B2: nat] :
% 12.24/7.92 ( ( ( zero_zero @ nat )
% 12.24/7.92 = ( ord_max @ nat @ A3 @ B2 ) )
% 12.24/7.92 = ( ( A3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 & ( B2
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2911_max__nat_Oleft__neutral,axiom,
% 12.24/7.92 ! [A3: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.92 = A3 ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2912_max__nat_Oeq__neutr__iff,axiom,
% 12.24/7.92 ! [A3: nat,B2: nat] :
% 12.24/7.92 ( ( ( ord_max @ nat @ A3 @ B2 )
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 = ( ( A3
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 & ( B2
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2913_max__number__of_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ( numeral @ A )
% 12.24/7.92 & ( ord @ A ) )
% 12.24/7.92 => ! [U: num,V: num] :
% 12.24/7.92 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( numeral_numeral @ A @ U ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2914_max__0__1_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( numeral_numeral @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2915_max__0__1_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( numeral_numeral @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2916_max__0__1_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2917_max__0__1_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2918_max__0__1_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( numeral_numeral @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2919_max__0__1_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 12.24/7.92 = ( numeral_numeral @ A @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2920_sub__num__simps_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ( ( neg_numeral_sub @ A @ one2 @ one2 )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2921_diff__numeral__simps_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num,N: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2922_sub__num__simps_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num,L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
% 12.24/7.92 = ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2923_sub__num__simps_I9_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num,L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 12.24/7.92 = ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2924_max__number__of_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ( uminus @ A )
% 12.24/7.92 & ( numeral @ A )
% 12.24/7.92 & ( ord @ A ) )
% 12.24/7.92 => ! [U: num,V: num] :
% 12.24/7.92 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 = ( numeral_numeral @ A @ U ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2925_max__number__of_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ( uminus @ A )
% 12.24/7.92 & ( numeral @ A )
% 12.24/7.92 & ( ord @ A ) )
% 12.24/7.92 => ! [U: num,V: num] :
% 12.24/7.92 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2926_max__number__of_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ( uminus @ A )
% 12.24/7.92 & ( numeral @ A )
% 12.24/7.92 & ( ord @ A ) )
% 12.24/7.92 => ! [U: num,V: num] :
% 12.24/7.92 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2927_add__neg__numeral__simps_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num,N: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2928_add__neg__numeral__simps_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num,N: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2929_semiring__norm_I166_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.92 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
% 12.24/7.92 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W2 ) @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2930_semiring__norm_I167_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [V: num,W2: num,Y3: A] :
% 12.24/7.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
% 12.24/7.92 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V ) @ Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2931_diff__numeral__simps_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num,N: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2932_max__numeral__Suc,axiom,
% 12.24/7.92 ! [K: num,N: nat] :
% 12.24/7.92 ( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 12.24/7.92 = ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2933_max__Suc__numeral,axiom,
% 12.24/7.92 ! [N: nat,K: num] :
% 12.24/7.92 ( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.92 = ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2934_sub__num__simps_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num,L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
% 12.24/7.92 = ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2935_sub__num__simps_I8_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num,L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
% 12.24/7.92 = ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2936_drop__bit__of__Suc__0,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( bit_se4197421643247451524op_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.92 = ( zero_neq_one_of_bool @ nat
% 12.24/7.92 @ ( N
% 12.24/7.92 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2937_diff__numeral__special_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2938_diff__numeral__special_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2939_sub__num__simps_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2940_not__minus__numeral__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2941_add__neg__numeral__special_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2942_add__neg__numeral__special_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2943_add__neg__numeral__special_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2944_add__neg__numeral__special_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2945_minus__sub__one__diff__one,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2946_diff__numeral__special_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2947_diff__numeral__special_I8_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [M: num] :
% 12.24/7.92 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 12.24/7.92 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2948_sub__num__simps_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2949_xor__minus__numerals_I2_J,axiom,
% 12.24/7.92 ! [K: int,N: num] :
% 12.24/7.92 ( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2950_max_Ostrict__coboundedI2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [C2: A,B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less @ A @ C2 @ B2 )
% 12.24/7.92 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2951_max_Ostrict__coboundedI1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less @ A @ C2 @ A3 )
% 12.24/7.92 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2952_max_Ostrict__order__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ( ( ord_less @ A )
% 12.24/7.92 = ( ^ [B3: A,A5: A] :
% 12.24/7.92 ( ( A5
% 12.24/7.92 = ( ord_max @ A @ A5 @ B3 ) )
% 12.24/7.92 & ( A5 != B3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2953_max_Ostrict__boundedE,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.92 ( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.92 => ~ ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.92 => ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2954_less__max__iff__disj,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [Z3: A,X: A,Y3: A] :
% 12.24/7.92 ( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
% 12.24/7.92 = ( ( ord_less @ A @ Z3 @ X )
% 12.24/7.92 | ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2955_max__add__distrib__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ordere2412721322843649153imp_le @ A )
% 12.24/7.92 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( plus_plus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.92 = ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2956_max__add__distrib__right,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ordere2412721322843649153imp_le @ A )
% 12.24/7.92 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( plus_plus @ A @ X @ ( ord_max @ A @ Y3 @ Z3 ) )
% 12.24/7.92 = ( ord_max @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2957_max__diff__distrib__left,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ordered_ab_group_add @ A )
% 12.24/7.92 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( minus_minus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.92 = ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2958_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,Y3: A,Z3: A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Conj @ ( Xor @ Y3 @ Z3 ) @ X )
% 12.24/7.92 = ( Xor @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2959_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ ( Compl @ X ) )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2960_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Conj @ X @ ( Xor @ Y3 @ Z3 ) )
% 12.24/7.92 = ( Xor @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2961_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ ( Compl @ Y3 ) )
% 12.24/7.92 = ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2962_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ ( Compl @ X ) @ X )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2963_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ ( Compl @ X ) @ Y3 )
% 12.24/7.92 = ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2964_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ One2 )
% 12.24/7.92 = ( Compl @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2965_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ ( Xor @ X @ Y3 ) )
% 12.24/7.92 = Y3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2966_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ One2 @ X )
% 12.24/7.92 = ( Compl @ X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2967_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ X )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2968_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ Y3 )
% 12.24/7.92 = ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2969_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( ( Xor @ X @ Y3 )
% 12.24/7.92 = ( Disj @ ( Conj @ X @ ( Compl @ Y3 ) ) @ ( Conj @ ( Compl @ X ) @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2970_of__nat__max,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: nat,Y3: nat] :
% 12.24/7.92 ( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y3 ) )
% 12.24/7.92 = ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2971_max_Oleft__commute,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.92 ( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
% 12.24/7.92 = ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2972_max_Ocommute,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ( ( ord_max @ A )
% 12.24/7.92 = ( ^ [A5: A,B3: A] : ( ord_max @ A @ B3 @ A5 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2973_max_Oassoc,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.92 ( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.92 = ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2974_of__int__max,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_idom @ A )
% 12.24/7.92 => ! [X: int,Y3: int] :
% 12.24/7.92 ( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y3 ) )
% 12.24/7.92 = ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2975_max__absorb2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ord @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.92 => ( ( ord_max @ A @ X @ Y3 )
% 12.24/7.92 = Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2976_max__absorb1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( order @ A )
% 12.24/7.92 => ! [Y3: A,X: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.92 => ( ( ord_max @ A @ X @ Y3 )
% 12.24/7.92 = X ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2977_max_OcoboundedI2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [C2: A,B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ C2 @ B2 )
% 12.24/7.92 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2978_max_OcoboundedI1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.92 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2979_max_Oabsorb__iff2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ( ( ord_less_eq @ A )
% 12.24/7.92 = ( ^ [A5: A,B3: A] :
% 12.24/7.92 ( ( ord_max @ A @ A5 @ B3 )
% 12.24/7.92 = B3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2980_max_Oabsorb__iff1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ( ( ord_less_eq @ A )
% 12.24/7.92 = ( ^ [B3: A,A5: A] :
% 12.24/7.92 ( ( ord_max @ A @ A5 @ B3 )
% 12.24/7.92 = A5 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2981_le__max__iff__disj,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [Z3: A,X: A,Y3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
% 12.24/7.92 = ( ( ord_less_eq @ A @ Z3 @ X )
% 12.24/7.92 | ( ord_less_eq @ A @ Z3 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2982_max_Ocobounded2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2983_max_Ocobounded1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2984_max_Oorder__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ( ( ord_less_eq @ A )
% 12.24/7.92 = ( ^ [B3: A,A5: A] :
% 12.24/7.92 ( A5
% 12.24/7.92 = ( ord_max @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2985_max_OboundedI,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.92 => ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2986_max_OboundedE,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.92 => ~ ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2987_max_OorderI,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( A3
% 12.24/7.92 = ( ord_max @ A @ A3 @ B2 ) )
% 12.24/7.92 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2988_max_OorderE,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 => ( A3
% 12.24/7.92 = ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2989_max_Omono,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [C2: A,A3: A,D2: A,B2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.92 => ( ( ord_less_eq @ A @ D2 @ B2 )
% 12.24/7.92 => ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2990_max__def,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ord @ A )
% 12.24/7.92 => ( ( ord_max @ A )
% 12.24/7.92 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2991_nat__add__max__left,axiom,
% 12.24/7.92 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.92 ( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
% 12.24/7.92 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N @ Q4 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2992_nat__add__max__right,axiom,
% 12.24/7.92 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.92 ( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
% 12.24/7.92 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2993_nat__mult__max__left,axiom,
% 12.24/7.92 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.92 ( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
% 12.24/7.92 = ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2994_nat__mult__max__right,axiom,
% 12.24/7.92 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.92 ( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
% 12.24/7.92 = ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2995_length__induct,axiom,
% 12.24/7.92 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 12.24/7.92 ( ! [Xs2: list @ A] :
% 12.24/7.92 ( ! [Ys: list @ A] :
% 12.24/7.92 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 12.24/7.92 => ( P @ Ys ) )
% 12.24/7.92 => ( P @ Xs2 ) )
% 12.24/7.92 => ( P @ Xs ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2996_nat__minus__add__max,axiom,
% 12.24/7.92 ! [N: nat,M: nat] :
% 12.24/7.92 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
% 12.24/7.92 = ( ord_max @ nat @ N @ M ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2997_neg__numeral__class_Osub__def,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ( ( neg_numeral_sub @ A )
% 12.24/7.92 = ( ^ [K3: num,L3: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2998_Heap_Osize__neq,axiom,
% 12.24/7.92 ! [A: $tType,X: heap_Time_Heap @ A] :
% 12.24/7.92 ( ( size_size @ ( heap_Time_Heap @ A ) @ X )
% 12.24/7.92 != ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_2999_sub__non__negative,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_idom @ A )
% 12.24/7.92 => ! [N: num,M: num] :
% 12.24/7.92 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 12.24/7.92 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3000_sub__non__positive,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_idom @ A )
% 12.24/7.92 => ! [N: num,M: num] :
% 12.24/7.92 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( ord_less_eq @ num @ N @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3001_sub__negative,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_idom @ A )
% 12.24/7.92 => ! [N: num,M: num] :
% 12.24/7.92 ( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( ord_less @ num @ N @ M ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3002_sub__positive,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linordered_idom @ A )
% 12.24/7.92 => ! [N: num,M: num] :
% 12.24/7.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
% 12.24/7.92 = ( ord_less @ num @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3003_sub__inc__One__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
% 12.24/7.92 = ( numeral_numeral @ A @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3004_minus__numeral__eq__not__sub__one,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 12.24/7.92 = ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3005_int__numeral__or__not__num__neg,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3006_int__numeral__not__or__num__neg,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3007_numeral__or__not__num__eq,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3008_or__minus__numerals_I1_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3009_or__minus__numerals_I5_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3010_horner__sum__of__bool__2__less,axiom,
% 12.24/7.92 ! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3011_or__minus__numerals_I7_J,axiom,
% 12.24/7.92 ! [N: num,M: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3012_or__minus__numerals_I3_J,axiom,
% 12.24/7.92 ! [M: num,N: num] :
% 12.24/7.92 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.92 = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3013_dbl__dec__simps_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num] :
% 12.24/7.92 ( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3014_pred__numeral__simps_I2_J,axiom,
% 12.24/7.92 ! [K: num] :
% 12.24/7.92 ( ( pred_numeral @ ( bit0 @ K ) )
% 12.24/7.92 = ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3015_sub__num__simps_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [K: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
% 12.24/7.92 = ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3016_sub__num__simps_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [L: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3017_semiring__norm_I26_J,axiom,
% 12.24/7.92 ( ( bitM @ one2 )
% 12.24/7.92 = one2 ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3018_semiring__norm_I27_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bitM @ ( bit0 @ N ) )
% 12.24/7.92 = ( bit1 @ ( bitM @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3019_semiring__norm_I28_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bitM @ ( bit1 @ N ) )
% 12.24/7.92 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3020_inc__BitM__eq,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( inc @ ( bitM @ N ) )
% 12.24/7.92 = ( bit0 @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3021_BitM__inc__eq,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( bitM @ ( inc @ N ) )
% 12.24/7.92 = ( bit1 @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3022_eval__nat__numeral_I2_J,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( numeral_numeral @ nat @ ( bit0 @ N ) )
% 12.24/7.92 = ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3023_one__plus__BitM,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
% 12.24/7.92 = ( bit0 @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3024_BitM__plus__one,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
% 12.24/7.92 = ( bit0 @ N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3025_numeral__BitM,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( neg_numeral @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( numeral_numeral @ A @ ( bitM @ N ) )
% 12.24/7.92 = ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3026_not__numeral__BitM__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ! [N: num] :
% 12.24/7.92 ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3027_sub__BitM__One__eq,axiom,
% 12.24/7.92 ! [N: num] :
% 12.24/7.92 ( ( neg_numeral_sub @ int @ ( bitM @ N ) @ one2 )
% 12.24/7.92 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3028_bit__horner__sum__bit__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.92 => ! [Bs: list @ $o,N: nat] :
% 12.24/7.92 ( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
% 12.24/7.92 = ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
% 12.24/7.92 & ( nth @ $o @ Bs @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3029_bit_Oabstract__boolean__algebra__axioms,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.92 => ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3030_num__of__nat_Osimps_I2_J,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( num_of_nat @ ( suc @ N ) )
% 12.24/7.92 = ( inc @ ( num_of_nat @ N ) ) ) )
% 12.24/7.92 & ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( num_of_nat @ ( suc @ N ) )
% 12.24/7.92 = one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3031_eq__numeral__iff__iszero_I7_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( one_one @ A ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3032_eq__numeral__iff__iszero_I8_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( ( one_one @ A )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3033_num__of__nat__numeral__eq,axiom,
% 12.24/7.92 ! [Q4: num] :
% 12.24/7.92 ( ( num_of_nat @ ( numeral_numeral @ nat @ Q4 ) )
% 12.24/7.92 = Q4 ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3034_iszero__neg__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [W2: num] :
% 12.24/7.92 ( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3035_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ ( Conj @ Y3 @ Z3 ) @ X )
% 12.24/7.92 = ( Conj @ ( Disj @ Y3 @ X ) @ ( Disj @ Z3 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3036_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ ( Disj @ Y3 @ Z3 ) @ X )
% 12.24/7.92 = ( Disj @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3037_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( ( Compl @ X )
% 12.24/7.92 = ( Compl @ Y3 ) )
% 12.24/7.92 = ( X = Y3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3038_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ X @ ( Conj @ Y3 @ Z3 ) )
% 12.24/7.92 = ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ X @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3039_abstract__boolean__algebra_Odisj__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ X @ ( Compl @ X ) )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3040_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ X @ ( Disj @ Y3 @ Z3 ) )
% 12.24/7.92 = ( Disj @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3041_abstract__boolean__algebra_Oconj__cancel__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ X @ ( Compl @ X ) )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3042_abstract__boolean__algebra_Ocomplement__unique,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,A3: A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( ( Conj @ A3 @ X )
% 12.24/7.92 = Zero )
% 12.24/7.92 => ( ( ( Disj @ A3 @ X )
% 12.24/7.92 = One2 )
% 12.24/7.92 => ( ( ( Conj @ A3 @ Y3 )
% 12.24/7.92 = Zero )
% 12.24/7.92 => ( ( ( Disj @ A3 @ Y3 )
% 12.24/7.92 = One2 )
% 12.24/7.92 => ( X = Y3 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3043_abstract__boolean__algebra_Odisj__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ ( Compl @ X ) @ X )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3044_abstract__boolean__algebra_Oconj__cancel__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ ( Compl @ X ) @ X )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3045_abstract__boolean__algebra_Odisj__zero__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ X @ Zero )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3046_abstract__boolean__algebra_Oconj__zero__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ X @ Zero )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3047_abstract__boolean__algebra_Odisj__one__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ X @ One2 )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3048_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Compl @ ( Disj @ X @ Y3 ) )
% 12.24/7.92 = ( Conj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3049_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Compl @ ( Conj @ X @ Y3 ) )
% 12.24/7.92 = ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3050_abstract__boolean__algebra_Oconj__zero__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ Zero @ X )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3051_abstract__boolean__algebra_Oconj__one__right,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Conj @ X @ One2 )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3052_abstract__boolean__algebra_Odisj__one__left,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Disj @ One2 @ X )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3053_abstract__boolean__algebra_Odouble__compl,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Compl @ ( Compl @ X ) )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3054_abstract__boolean__algebra_Ocompl__unique,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( ( Conj @ X @ Y3 )
% 12.24/7.92 = Zero )
% 12.24/7.92 => ( ( ( Disj @ X @ Y3 )
% 12.24/7.92 = One2 )
% 12.24/7.92 => ( ( Compl @ X )
% 12.24/7.92 = Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3055_abstract__boolean__algebra_Ocompl__zero,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Compl @ Zero )
% 12.24/7.92 = One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3056_abstract__boolean__algebra_Ocompl__one,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/7.92 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.92 => ( ( Compl @ One2 )
% 12.24/7.92 = Zero ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3057_iszero__def,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ( ( ring_1_iszero @ A )
% 12.24/7.92 = ( ^ [Z7: A] :
% 12.24/7.92 ( Z7
% 12.24/7.92 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3058_iszero__0,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3059_not__iszero__numeral,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_char_0 @ A )
% 12.24/7.92 => ! [W2: num] :
% 12.24/7.92 ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3060_not__iszero__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3061_eq__iff__iszero__diff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ( ( ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.24/7.92 = ( ^ [X2: A,Y6: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3062_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
% 12.24/7.92 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 12.24/7.92 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.92 => ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3063_nth__equalityI,axiom,
% 12.24/7.92 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.92 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.92 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 12.24/7.92 => ( ! [I4: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.92 => ( ( nth @ A @ Xs @ I4 )
% 12.24/7.92 = ( nth @ A @ Ys2 @ I4 ) ) )
% 12.24/7.92 => ( Xs = Ys2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3064_Skolem__list__nth,axiom,
% 12.24/7.92 ! [A: $tType,K: nat,P: nat > A > $o] :
% 12.24/7.92 ( ( ! [I5: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ I5 @ K )
% 12.24/7.92 => ? [X7: A] : ( P @ I5 @ X7 ) ) )
% 12.24/7.92 = ( ? [Xs3: list @ A] :
% 12.24/7.92 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.92 = K )
% 12.24/7.92 & ! [I5: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ I5 @ K )
% 12.24/7.92 => ( P @ I5 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3065_list__eq__iff__nth__eq,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ^ [Y: list @ A,Z2: list @ A] : ( Y = Z2 ) )
% 12.24/7.92 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.92 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.92 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.92 & ! [I5: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.92 => ( ( nth @ A @ Xs3 @ I5 )
% 12.24/7.92 = ( nth @ A @ Ys3 @ I5 ) ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3066_eq__numeral__iff__iszero_I10_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( ( zero_zero @ A )
% 12.24/7.92 = ( numeral_numeral @ A @ Y3 ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3067_eq__numeral__iff__iszero_I9_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ( numeral_numeral @ A @ X )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3068_num__of__nat_Osimps_I1_J,axiom,
% 12.24/7.92 ( ( num_of_nat @ ( zero_zero @ nat ) )
% 12.24/7.92 = one2 ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3069_not__iszero__Numeral1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3070_not__iszero__neg__1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3071_eq__numeral__iff__iszero_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( ( numeral_numeral @ A @ X )
% 12.24/7.92 = ( numeral_numeral @ A @ Y3 ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3072_eq__numeral__iff__iszero_I12_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( ( zero_zero @ A )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3073_eq__numeral__iff__iszero_I11_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( zero_zero @ A ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3074_not__iszero__neg__Numeral1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3075_numeral__num__of__nat,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
% 12.24/7.92 = N ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3076_num__of__nat__One,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
% 12.24/7.92 => ( ( num_of_nat @ N )
% 12.24/7.92 = one2 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3077_eq__numeral__iff__iszero_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( numeral_numeral @ A @ Y3 ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3078_eq__numeral__iff__iszero_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( ( numeral_numeral @ A @ X )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3079_eq__numeral__iff__iszero_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num,Y3: num] :
% 12.24/7.92 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y3 @ X ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3080_numeral__num__of__nat__unfold,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( semiring_1 @ A )
% 12.24/7.92 => ! [N: nat] :
% 12.24/7.92 ( ( ( N
% 12.24/7.92 = ( zero_zero @ nat ) )
% 12.24/7.92 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 12.24/7.92 = ( one_one @ A ) ) )
% 12.24/7.92 & ( ( N
% 12.24/7.92 != ( zero_zero @ nat ) )
% 12.24/7.92 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 12.24/7.92 = ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3081_num__of__nat__double,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
% 12.24/7.92 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3082_num__of__nat__plus__distrib,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.24/7.92 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 => ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
% 12.24/7.92 = ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3083_eq__numeral__iff__iszero_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ( numeral_numeral @ A @ X )
% 12.24/7.92 = ( one_one @ A ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3084_eq__numeral__iff__iszero_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ring_1 @ A )
% 12.24/7.92 => ! [Y3: num] :
% 12.24/7.92 ( ( ( one_one @ A )
% 12.24/7.92 = ( numeral_numeral @ A @ Y3 ) )
% 12.24/7.92 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3085_nth__rotate1,axiom,
% 12.24/7.92 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.92 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.92 => ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
% 12.24/7.92 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3086_nth__update__neq,axiom,
% 12.24/7.92 ! [B: $tType,A: $tType] :
% 12.24/7.92 ( ( ( heap @ A )
% 12.24/7.92 & ( heap @ B ) )
% 12.24/7.92 => ! [A3: array @ A,B2: array @ B,J: nat,V: B,H2: heap_ext @ product_unit,I: nat] :
% 12.24/7.92 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 12.24/7.92 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H2 ) @ A3 ) @ I )
% 12.24/7.92 = ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3087_mask__mod__exp,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.92 => ! [N: nat,M: nat] :
% 12.24/7.92 ( ( 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 ) ) @ M ) )
% 12.24/7.92 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3088_set__decode__plus__power__2,axiom,
% 12.24/7.92 ! [N: nat,Z3: nat] :
% 12.24/7.92 ( ~ ( member @ nat @ N @ ( nat_set_decode @ Z3 ) )
% 12.24/7.92 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z3 ) )
% 12.24/7.92 = ( insert @ nat @ N @ ( nat_set_decode @ Z3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3089_nth__rotate,axiom,
% 12.24/7.92 ! [A: $tType,N: nat,Xs: list @ A,M: nat] :
% 12.24/7.92 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.92 => ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N )
% 12.24/7.92 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3090_min_Oidem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A] :
% 12.24/7.92 ( ( ord_min @ A @ A3 @ A3 )
% 12.24/7.92 = A3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3091_min_Oleft__idem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
% 12.24/7.92 = ( ord_min @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3092_min_Oright__idem,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.92 = ( ord_min @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3093_min_Obounded__iff,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 12.24/7.92 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.92 & ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3094_min_Oabsorb2,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.92 => ( ( ord_min @ A @ A3 @ B2 )
% 12.24/7.92 = B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3095_min_Oabsorb1,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.92 => ( ( ord_min @ A @ A3 @ B2 )
% 12.24/7.92 = A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3096_min__less__iff__conj,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [Z3: A,X: A,Y3: A] :
% 12.24/7.92 ( ( ord_less @ A @ Z3 @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.92 = ( ( ord_less @ A @ Z3 @ X )
% 12.24/7.92 & ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3097_min_Oabsorb4,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [B2: A,A3: A] :
% 12.24/7.92 ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.92 => ( ( ord_min @ A @ A3 @ B2 )
% 12.24/7.92 = B2 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3098_min_Oabsorb3,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [A3: A,B2: A] :
% 12.24/7.92 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.92 => ( ( ord_min @ A @ A3 @ B2 )
% 12.24/7.92 = A3 ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3099_max__min__same_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [Y3: A,X: A] :
% 12.24/7.92 ( ( ord_max @ A @ Y3 @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.92 = Y3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3100_max__min__same_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ Y3 )
% 12.24/7.92 = Y3 ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3101_max__min__same_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ X )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3102_max__min__same_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linorder @ A )
% 12.24/7.92 => ! [X: A,Y3: A] :
% 12.24/7.92 ( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.92 = X ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3103_min__Suc__Suc,axiom,
% 12.24/7.92 ! [M: nat,N: nat] :
% 12.24/7.92 ( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 12.24/7.92 = ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3104_min__0L,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3105_min__0R,axiom,
% 12.24/7.92 ! [N: nat] :
% 12.24/7.92 ( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
% 12.24/7.92 = ( zero_zero @ nat ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3106_min__number__of_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( ( numeral @ A )
% 12.24/7.92 & ( ord @ A ) )
% 12.24/7.92 => ! [U: num,V: num] :
% 12.24/7.92 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( numeral_numeral @ A @ U ) ) )
% 12.24/7.92 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.92 = ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3107_min__0__1_I3_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3108_min__0__1_I4_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3109_min__0__1_I1_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3110_min__0__1_I2_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 12.24/7.92 = ( zero_zero @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3111_min__0__1_I6_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3112_min__0__1_I5_J,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( linord181362715937106298miring @ A )
% 12.24/7.92 => ! [X: num] :
% 12.24/7.92 ( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 12.24/7.92 = ( one_one @ A ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3113_rotate__Suc,axiom,
% 12.24/7.92 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.92 ( ( rotate @ A @ ( suc @ N ) @ Xs )
% 12.24/7.92 = ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% 12.24/7.92
% 12.24/7.92 thf(fact_3114_get__update__elem__neqIndex,axiom,
% 12.24/7.92 ! [A: $tType] :
% 12.24/7.92 ( ( heap @ A )
% 12.24/7.92 => ! [I: nat,J: nat,A3: array @ A,V: A,H2: heap_ext @ product_unit] :
% 12.24/7.92 ( ( I != J )
% 12.24/7.92 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H2 ) @ A3 ) @ I )
% 12.24/7.93 = ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3115_min__number__of_I4_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ( uminus @ A )
% 12.24/7.93 & ( numeral @ A )
% 12.24/7.93 & ( ord @ A ) )
% 12.24/7.93 => ! [U: num,V: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3116_min__number__of_I3_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ( uminus @ A )
% 12.24/7.93 & ( numeral @ A )
% 12.24/7.93 & ( ord @ A ) )
% 12.24/7.93 => ! [U: num,V: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
% 12.24/7.93 = ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3117_min__number__of_I2_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ( uminus @ A )
% 12.24/7.93 & ( numeral @ A )
% 12.24/7.93 & ( ord @ A ) )
% 12.24/7.93 => ! [U: num,V: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 = ( numeral_numeral @ A @ U ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3118_rotate__length01,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 12.24/7.93 => ( ( rotate @ A @ N @ Xs )
% 12.24/7.93 = Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3119_rotate__id,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( rotate @ A @ N @ Xs )
% 12.24/7.93 = Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3120_min__Suc__numeral,axiom,
% 12.24/7.93 ! [N: nat,K: num] :
% 12.24/7.93 ( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.93 = ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3121_min__numeral__Suc,axiom,
% 12.24/7.93 ! [K: num,N: nat] :
% 12.24/7.93 ( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
% 12.24/7.93 = ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3122_rotate1__length01,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 12.24/7.93 => ( ( rotate1 @ A @ Xs )
% 12.24/7.93 = Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3123_min__max__distrib2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
% 12.24/7.93 = ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3124_min__max__distrib1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.93 ( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.93 = ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3125_max__min__distrib2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 12.24/7.93 = ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3126_max__min__distrib1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.93 ( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.93 = ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3127_min__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ord @ A )
% 12.24/7.93 => ( ( ord_min @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3128_min_Omono,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 12.24/7.93 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3129_min_OorderE,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.93 => ( A3
% 12.24/7.93 = ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3130_min_OorderI,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 = ( ord_min @ A @ A3 @ B2 ) )
% 12.24/7.93 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3131_min_OboundedE,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 12.24/7.93 => ~ ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.93 => ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3132_min_OboundedI,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.93 => ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3133_min_Oorder__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ( ( ord_less_eq @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] :
% 12.24/7.93 ( A5
% 12.24/7.93 = ( ord_min @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3134_min_Ocobounded1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3135_min_Ocobounded2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3136_min_Oabsorb__iff1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ( ( ord_less_eq @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] :
% 12.24/7.93 ( ( ord_min @ A @ A5 @ B3 )
% 12.24/7.93 = A5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3137_min_Oabsorb__iff2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ( ( ord_less_eq @ A )
% 12.24/7.93 = ( ^ [B3: A,A5: A] :
% 12.24/7.93 ( ( ord_min @ A @ A5 @ B3 )
% 12.24/7.93 = B3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3138_min_OcoboundedI1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.93 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3139_min_OcoboundedI2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.24/7.93 => ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3140_min__le__iff__disj,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.93 = ( ( ord_less_eq @ A @ X @ Z3 )
% 12.24/7.93 | ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3141_min__absorb1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ord @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.93 => ( ( ord_min @ A @ X @ Y3 )
% 12.24/7.93 = X ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3142_min__absorb2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order @ A )
% 12.24/7.93 => ! [Y3: A,X: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.93 => ( ( ord_min @ A @ X @ Y3 )
% 12.24/7.93 = Y3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3143_of__int__min,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_idom @ A )
% 12.24/7.93 => ! [X: int,Y3: int] :
% 12.24/7.93 ( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3144_min_Oassoc,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.93 = ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3145_min_Ocommute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ( ( ord_min @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] : ( ord_min @ A @ B3 @ A5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3146_min_Oleft__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.93 ( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
% 12.24/7.93 = ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3147_of__nat__min,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linord181362715937106298miring @ A )
% 12.24/7.93 => ! [X: nat,Y3: nat] :
% 12.24/7.93 ( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3148_min__diff__distrib__left,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ordered_ab_group_add @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( minus_minus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.93 = ( ord_min @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3149_min__add__distrib__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ordere2412721322843649153imp_le @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( plus_plus @ A @ X @ ( ord_min @ A @ Y3 @ Z3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3150_min__add__distrib__left,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ordere2412721322843649153imp_le @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( plus_plus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.93 = ( ord_min @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3151_min__less__iff__disj,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.93 = ( ( ord_less @ A @ X @ Z3 )
% 12.24/7.93 | ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3152_min_Ostrict__boundedE,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
% 12.24/7.93 => ~ ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3153_min_Ostrict__order__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ( ( ord_less @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] :
% 12.24/7.93 ( ( A5
% 12.24/7.93 = ( ord_min @ A @ A5 @ B3 ) )
% 12.24/7.93 & ( A5 != B3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3154_min_Ostrict__coboundedI1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ C2 )
% 12.24/7.93 => ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3155_min_Ostrict__coboundedI2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ B2 @ C2 )
% 12.24/7.93 => ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3156_nat__mult__min__left,axiom,
% 12.24/7.93 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.93 ( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q4 )
% 12.24/7.93 = ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3157_nat__mult__min__right,axiom,
% 12.24/7.93 ! [M: nat,N: nat,Q4: nat] :
% 12.24/7.93 ( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q4 ) )
% 12.24/7.93 = ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3158_min__diff,axiom,
% 12.24/7.93 ! [M: nat,I: nat,N: nat] :
% 12.24/7.93 ( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
% 12.24/7.93 = ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3159_minus__min__eq__max,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linord5086331880401160121up_add @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3160_minus__max__eq__min,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linord5086331880401160121up_add @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3161_atLeast0__atMost__Suc,axiom,
% 12.24/7.93 ! [N: nat] :
% 12.24/7.93 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 12.24/7.93 = ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3162_atLeastAtMost__insertL,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 => ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
% 12.24/7.93 = ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3163_atLeastAtMostSuc__conv,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/7.93 => ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
% 12.24/7.93 = ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3164_Icc__eq__insert__lb__nat,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 => ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
% 12.24/7.93 = ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3165_length__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( heap @ A )
% 12.24/7.93 => ( ( array_length @ A )
% 12.24/7.93 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H4 @ A5 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3166_max__mult__distrib__left,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_idom @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3167_min__mult__distrib__left,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_idom @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3168_max__mult__distrib__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_idom @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3169_min__mult__distrib__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_idom @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3170_min__divide__distrib__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3171_max__divide__distrib__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [P5: A,X: A,Y3: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
% 12.24/7.93 => ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
% 12.24/7.93 = ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3172_and__int_Oelims,axiom,
% 12.24/7.93 ! [X: int,Xa3: int,Y3: int] :
% 12.24/7.93 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( uminus_uminus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) ) ) ) )
% 12.24/7.93 & ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( plus_plus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) )
% 12.24/7.93 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3173_and__int_Osimps,axiom,
% 12.24/7.93 ( ( bit_se5824344872417868541ns_and @ int )
% 12.24/7.93 = ( ^ [K3: int,L3: int] :
% 12.24/7.93 ( if @ int
% 12.24/7.93 @ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 @ ( uminus_uminus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 12.24/7.93 @ ( plus_plus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 12.24/7.93 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3174_nth__Cons__pos,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.93 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 12.24/7.93 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3175_get__set__neq,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( ( heap @ A )
% 12.24/7.93 & ( heap @ B ) )
% 12.24/7.93 => ! [R: array @ A,S: array @ B,X: list @ B,H2: heap_ext @ product_unit] :
% 12.24/7.93 ( ( array_noteq @ A @ B @ R @ S )
% 12.24/7.93 => ( ( array_get @ A @ ( array_set @ B @ S @ X @ H2 ) @ R )
% 12.24/7.93 = ( array_get @ A @ H2 @ R ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3176_divmod__step__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [L: num,R: A,Q4: A] :
% 12.24/7.93 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 12.24/7.93 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 12.24/7.93 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3177_bot__apply,axiom,
% 12.24/7.93 ! [C: $tType,D: $tType] :
% 12.24/7.93 ( ( bot @ C )
% 12.24/7.93 => ( ( bot_bot @ ( D > C ) )
% 12.24/7.93 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3178_set__same,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( heap @ A )
% 12.24/7.93 => ! [R: array @ A,X: list @ A,Y3: list @ A,H2: heap_ext @ product_unit] :
% 12.24/7.93 ( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y3 @ H2 ) )
% 12.24/7.93 = ( array_set @ A @ R @ X @ H2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3179_max__bot2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_max @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3180_max__bot,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_max @ A @ ( bot_bot @ A ) @ X )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3181_min__bot2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_min @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.93 = ( bot_bot @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3182_min__bot,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_min @ A @ ( bot_bot @ A ) @ X )
% 12.24/7.93 = ( bot_bot @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3183_get__set__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( heap @ A )
% 12.24/7.93 => ! [R: array @ A,X: list @ A,H2: heap_ext @ product_unit] :
% 12.24/7.93 ( ( array_get @ A @ ( array_set @ A @ R @ X @ H2 ) @ R )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3184_atLeastatMost__empty__iff2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( preorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( bot_bot @ ( set @ A ) )
% 12.24/7.93 = ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3185_atLeastatMost__empty__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( preorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.93 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3186_atLeastatMost__empty,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order @ A )
% 12.24/7.93 => ! [B2: A,A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.93 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3187_nth__Cons__Suc,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
% 12.24/7.93 = ( nth @ A @ Xs @ N ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3188_nth__Cons__0,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
% 12.24/7.93 = X ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3189_subset__Compl__singleton,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.93 = ( ~ ( member @ A @ B2 @ A6 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3190_nth__Cons__numeral,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A,V: num] :
% 12.24/7.93 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
% 12.24/7.93 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3191_bot__fun__def,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( bot @ B )
% 12.24/7.93 => ( ( bot_bot @ ( A > B ) )
% 12.24/7.93 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3192_bot_Onot__eq__extremum,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( bot_bot @ A ) )
% 12.24/7.93 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3193_bot_Oextremum__strict,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3194_bot_Oextremum__uniqueI,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 12.24/7.93 => ( A3
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3195_bot_Oextremum__unique,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 12.24/7.93 = ( A3
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3196_bot_Oextremum,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order_bot @ A )
% 12.24/7.93 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3197_not__psubset__empty,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A] :
% 12.24/7.93 ~ ( ord_less @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3198_diff__shunt__var,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( ( minus_minus @ A @ X @ Y3 )
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3199_simp__from__to,axiom,
% 12.24/7.93 ( ( set_or1337092689740270186AtMost @ int )
% 12.24/7.93 = ( ^ [I5: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I5 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3200_length__Suc__conv,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.93 = ( suc @ N ) )
% 12.24/7.93 = ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.93 ( ( Xs
% 12.24/7.93 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.93 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 12.24/7.93 = N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3201_Suc__length__conv,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ( suc @ N )
% 12.24/7.93 = ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 = ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.93 ( ( Xs
% 12.24/7.93 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.93 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 12.24/7.93 = N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3202_impossible__Cons,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 12.24/7.93 => ( Xs
% 12.24/7.93 != ( cons @ A @ X @ Ys2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3203_subset__Compl__self__eq,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.93 = ( A6
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3204_set__set__swap,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( ( heap @ A )
% 12.24/7.93 & ( heap @ B ) )
% 12.24/7.93 => ! [R: array @ A,R4: array @ B,X: list @ A,X5: list @ B,H2: heap_ext @ product_unit] :
% 12.24/7.93 ( ( array_noteq @ A @ B @ R @ R4 )
% 12.24/7.93 => ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X5 @ H2 ) )
% 12.24/7.93 = ( array_set @ B @ R4 @ X5 @ ( array_set @ A @ R @ X @ H2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3205_Suc__le__length__iff,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 = ( ? [X2: A,Ys3: list @ A] :
% 12.24/7.93 ( ( Xs
% 12.24/7.93 = ( cons @ A @ X2 @ Ys3 ) )
% 12.24/7.93 & ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3206_Compl__insert,axiom,
% 12.24/7.93 ! [A: $tType,X: A,A6: set @ A] :
% 12.24/7.93 ( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) )
% 12.24/7.93 = ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3207_list_Osize_I4_J,axiom,
% 12.24/7.93 ! [A: $tType,X21: A,X222: list @ A] :
% 12.24/7.93 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
% 12.24/7.93 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3208_nth__Cons_H,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( ( N
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 12.24/7.93 = X ) )
% 12.24/7.93 & ( ( N
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 12.24/7.93 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3209_atLeastAtMostPlus1__int__conv,axiom,
% 12.24/7.93 ! [M: int,N: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 12.24/7.93 => ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 12.24/7.93 = ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3210_psubset__insert__iff,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,X: A,B7: set @ A] :
% 12.24/7.93 ( ( ord_less @ ( set @ A ) @ A6 @ ( insert @ A @ X @ B7 ) )
% 12.24/7.93 = ( ( ( member @ A @ X @ B7 )
% 12.24/7.93 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.93 & ( ~ ( member @ A @ X @ B7 )
% 12.24/7.93 => ( ( ( member @ A @ X @ A6 )
% 12.24/7.93 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 12.24/7.93 & ( ~ ( member @ A @ X @ A6 )
% 12.24/7.93 => ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3211_nth__non__equal__first__eq,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Y3: A,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( X != Y3 )
% 12.24/7.93 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 12.24/7.93 = Y3 )
% 12.24/7.93 = ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/7.93 = Y3 )
% 12.24/7.93 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3212_horner__sum__simps_I2_J,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType] :
% 12.24/7.93 ( ( comm_semiring_0 @ A )
% 12.24/7.93 => ! [F: B > A,A3: A,X: B,Xs: list @ B] :
% 12.24/7.93 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
% 12.24/7.93 = ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3213_divmod__algorithm__code_I8_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less @ num @ M @ N )
% 12.24/7.93 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less @ num @ M @ N )
% 12.24/7.93 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.93 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3214_divmod__algorithm__code_I7_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ num @ M @ N )
% 12.24/7.93 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ num @ M @ N )
% 12.24/7.93 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.93 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3215_divides__aux__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [Q4: A,R: A] :
% 12.24/7.93 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R ) )
% 12.24/7.93 = ( R
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3216_product__nth,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
% 12.24/7.93 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
% 12.24/7.93 = ( 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 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3217_bit__0__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
% 12.24/7.93 = ( bot_bot @ ( nat > $o ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3218_set__decode__zero,axiom,
% 12.24/7.93 ( ( nat_set_decode @ ( zero_zero @ nat ) )
% 12.24/7.93 = ( bot_bot @ ( set @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3219_length__product,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 12.24/7.93 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys2 ) )
% 12.24/7.93 = ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3220_divmod__algorithm__code_I2_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [M: num] :
% 12.24/7.93 ( ( unique8689654367752047608divmod @ A @ M @ one2 )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3221_divmod__algorithm__code_I3_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [N: num] :
% 12.24/7.93 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3222_divmod__algorithm__code_I4_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [N: num] :
% 12.24/7.93 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
% 12.24/7.93 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3223_divmod__BitM__2__eq,axiom,
% 12.24/7.93 ! [M: num] :
% 12.24/7.93 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3224_bot__nat__def,axiom,
% 12.24/7.93 ( ( bot_bot @ nat )
% 12.24/7.93 = ( zero_zero @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3225_divmod__divmod__step,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ( ( unique8689654367752047608divmod @ A )
% 12.24/7.93 = ( ^ [M2: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3226_minus__one__div__numeral,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3227_one__div__minus__numeral,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3228_neg__eucl__rel__int__mult__2,axiom,
% 12.24/7.93 ! [B2: int,A3: int,Q4: int,R: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
% 12.24/7.93 => ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 12.24/7.93 => ( 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 @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3229_Divides_Oadjust__div__eq,axiom,
% 12.24/7.93 ! [Q4: int,R: int] :
% 12.24/7.93 ( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 12.24/7.93 = ( plus_plus @ int @ Q4
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( R
% 12.24/7.93 != ( zero_zero @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3230_minus__numeral__div__numeral,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3231_numeral__div__minus__numeral,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3232_eucl__rel__int__by0,axiom,
% 12.24/7.93 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3233_eucl__rel__int__dividesI,axiom,
% 12.24/7.93 ! [L: int,K: int,Q4: int] :
% 12.24/7.93 ( ( L
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 => ( ( K
% 12.24/7.93 = ( times_times @ int @ Q4 @ L ) )
% 12.24/7.93 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3234_zminus1__lemma,axiom,
% 12.24/7.93 ! [A3: int,B2: int,Q4: int,R: int] :
% 12.24/7.93 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 => ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
% 12.24/7.93 @ ( product_Pair @ int @ int
% 12.24/7.93 @ ( if @ int
% 12.24/7.93 @ ( R
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 @ ( uminus_uminus @ int @ Q4 )
% 12.24/7.93 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
% 12.24/7.93 @ ( if @ int
% 12.24/7.93 @ ( R
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 @ ( zero_zero @ int )
% 12.24/7.93 @ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3235_eucl__rel__int__iff,axiom,
% 12.24/7.93 ! [K: int,L: int,Q4: int,R: int] :
% 12.24/7.93 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 12.24/7.93 = ( ( K
% 12.24/7.93 = ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R ) )
% 12.24/7.93 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.24/7.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 12.24/7.93 & ( ord_less @ int @ R @ L ) ) )
% 12.24/7.93 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 12.24/7.93 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 12.24/7.93 => ( ( ord_less @ int @ L @ R )
% 12.24/7.93 & ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 12.24/7.93 => ( Q4
% 12.24/7.93 = ( zero_zero @ int ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3236_eucl__rel__int__remainderI,axiom,
% 12.24/7.93 ! [R: int,L: int,K: int,Q4: int] :
% 12.24/7.93 ( ( ( sgn_sgn @ int @ R )
% 12.24/7.93 = ( sgn_sgn @ int @ L ) )
% 12.24/7.93 => ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
% 12.24/7.93 => ( ( K
% 12.24/7.93 = ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R ) )
% 12.24/7.93 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3237_eucl__rel__int_Ocases,axiom,
% 12.24/7.93 ! [A1: int,A22: int,A32: product_prod @ int @ int] :
% 12.24/7.93 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 12.24/7.93 => ( ( ( A22
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 => ( A32
% 12.24/7.93 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
% 12.24/7.93 => ( ! [Q5: int] :
% 12.24/7.93 ( ( A32
% 12.24/7.93 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
% 12.24/7.93 => ( ( A22
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 => ( A1
% 12.24/7.93 != ( times_times @ int @ Q5 @ A22 ) ) ) )
% 12.24/7.93 => ~ ! [R3: int,Q5: int] :
% 12.24/7.93 ( ( A32
% 12.24/7.93 = ( product_Pair @ int @ int @ Q5 @ R3 ) )
% 12.24/7.93 => ( ( ( sgn_sgn @ int @ R3 )
% 12.24/7.93 = ( sgn_sgn @ int @ A22 ) )
% 12.24/7.93 => ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
% 12.24/7.93 => ( A1
% 12.24/7.93 != ( plus_plus @ int @ ( times_times @ int @ Q5 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3238_eucl__rel__int_Osimps,axiom,
% 12.24/7.93 ( eucl_rel_int
% 12.24/7.93 = ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
% 12.24/7.93 ( ? [K3: int] :
% 12.24/7.93 ( ( A12 = K3 )
% 12.24/7.93 & ( A23
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 & ( A33
% 12.24/7.93 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
% 12.24/7.93 | ? [L3: int,K3: int,Q3: int] :
% 12.24/7.93 ( ( A12 = K3 )
% 12.24/7.93 & ( A23 = L3 )
% 12.24/7.93 & ( A33
% 12.24/7.93 = ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
% 12.24/7.93 & ( L3
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 & ( K3
% 12.24/7.93 = ( times_times @ int @ Q3 @ L3 ) ) )
% 12.24/7.93 | ? [R5: int,L3: int,K3: int,Q3: int] :
% 12.24/7.93 ( ( A12 = K3 )
% 12.24/7.93 & ( A23 = L3 )
% 12.24/7.93 & ( A33
% 12.24/7.93 = ( product_Pair @ int @ int @ Q3 @ R5 ) )
% 12.24/7.93 & ( ( sgn_sgn @ int @ R5 )
% 12.24/7.93 = ( sgn_sgn @ int @ L3 ) )
% 12.24/7.93 & ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L3 ) )
% 12.24/7.93 & ( K3
% 12.24/7.93 = ( plus_plus @ int @ ( times_times @ int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3239_pos__eucl__rel__int__mult__2,axiom,
% 12.24/7.93 ! [B2: int,A3: int,Q4: int,R: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.93 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R ) )
% 12.24/7.93 => ( 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 @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3240_and__int_Opelims,axiom,
% 12.24/7.93 ! [X: int,Xa3: int,Y3: int] :
% 12.24/7.93 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
% 12.24/7.93 => ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( uminus_uminus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) ) ) ) )
% 12.24/7.93 & ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( plus_plus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa3 ) ) )
% 12.24/7.93 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3241_and__int_Opsimps,axiom,
% 12.24/7.93 ! [K: int,L: int] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
% 12.24/7.93 => ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 12.24/7.93 = ( uminus_uminus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
% 12.24/7.93 & ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 12.24/7.93 = ( plus_plus @ int
% 12.24/7.93 @ ( zero_neq_one_of_bool @ int
% 12.24/7.93 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.93 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
% 12.24/7.93 @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3242_length__Cons,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
% 12.24/7.93 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3243_normalize__negative,axiom,
% 12.24/7.93 ! [Q4: int,P5: int] :
% 12.24/7.93 ( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
% 12.24/7.93 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 12.24/7.93 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3244_upto__aux__rec,axiom,
% 12.24/7.93 ( upto_aux
% 12.24/7.93 = ( ^ [I5: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ Js @ ( upto_aux @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3245_normalize__denom__zero,axiom,
% 12.24/7.93 ! [P5: int] :
% 12.24/7.93 ( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3246_normalize__denom__pos,axiom,
% 12.24/7.93 ! [R: product_prod @ int @ int,P5: int,Q4: int] :
% 12.24/7.93 ( ( ( normalize @ R )
% 12.24/7.93 = ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 12.24/7.93 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3247_normalize__crossproduct,axiom,
% 12.24/7.93 ! [Q4: int,S: int,P5: int,R: int] :
% 12.24/7.93 ( ( Q4
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 => ( ( S
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 => ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 12.24/7.93 = ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
% 12.24/7.93 => ( ( times_times @ int @ P5 @ S )
% 12.24/7.93 = ( times_times @ int @ R @ Q4 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3248_and__int_Opinduct,axiom,
% 12.24/7.93 ! [A0: int,A1: int,P: int > int > $o] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 12.24/7.93 => ( ! [K2: int,L2: int] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L2 ) )
% 12.24/7.93 => ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 12.24/7.93 & ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 12.24/7.93 => ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 12.24/7.93 => ( P @ K2 @ L2 ) ) )
% 12.24/7.93 => ( P @ A0 @ A1 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3249_upto_Opinduct,axiom,
% 12.24/7.93 ! [A0: int,A1: int,P: int > int > $o] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 12.24/7.93 => ( ! [I4: int,J2: int] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ I4 @ J2 )
% 12.24/7.93 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) )
% 12.24/7.93 => ( P @ I4 @ J2 ) ) )
% 12.24/7.93 => ( P @ A0 @ A1 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3250_bit__cut__integer__def,axiom,
% 12.24/7.93 ( code_bit_cut_integer
% 12.24/7.93 = ( ^ [K3: code_integer] :
% 12.24/7.93 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
% 12.24/7.93 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3251_bezw__0,axiom,
% 12.24/7.93 ! [X: nat] :
% 12.24/7.93 ( ( bezw @ X @ ( zero_zero @ nat ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3252_nth__zip,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
% 12.24/7.93 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.93 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
% 12.24/7.93 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3253_Gcd__0__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_Gcd @ A )
% 12.24/7.93 => ! [A6: set @ A] :
% 12.24/7.93 ( ( ( gcd_Gcd @ A @ A6 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 = ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3254_Gcd__empty,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_Gcd @ A )
% 12.24/7.93 => ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3255_divmod__integer_H__def,axiom,
% 12.24/7.93 ( ( unique8689654367752047608divmod @ code_integer )
% 12.24/7.93 = ( ^ [M2: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3256_Gcd__nat__eq__one,axiom,
% 12.24/7.93 ! [N6: set @ nat] :
% 12.24/7.93 ( ( member @ nat @ ( one_one @ nat ) @ N6 )
% 12.24/7.93 => ( ( gcd_Gcd @ nat @ N6 )
% 12.24/7.93 = ( one_one @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3257_Gcd__1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_Gcd @ A )
% 12.24/7.93 => ! [A6: set @ A] :
% 12.24/7.93 ( ( member @ A @ ( one_one @ A ) @ A6 )
% 12.24/7.93 => ( ( gcd_Gcd @ A @ A6 )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3258_Gcd__int__greater__eq__0,axiom,
% 12.24/7.93 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3259_Gcd__eq__1__I,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_Gcd @ A )
% 12.24/7.93 => ! [A3: A,A6: set @ A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.93 => ( ( gcd_Gcd @ A @ A6 )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3260_Code__Numeral_Opositive__def,axiom,
% 12.24/7.93 ( code_positive
% 12.24/7.93 = ( numeral_numeral @ code_integer ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3261_integer__of__num__triv_I2_J,axiom,
% 12.24/7.93 ( ( code_integer_of_num @ ( bit0 @ one2 ) )
% 12.24/7.93 = ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3262_card__atLeastAtMost__int,axiom,
% 12.24/7.93 ! [L: int,U: int] :
% 12.24/7.93 ( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
% 12.24/7.93 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3263_possible__bit__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ( ( bit_se6407376104438227557le_bit @ A )
% 12.24/7.93 = ( ^ [Tyrep: itself @ A,N3: nat] :
% 12.24/7.93 ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
% 12.24/7.93 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3264_abstract__boolean__algebra__sym__diff_Ointro,axiom,
% 12.24/7.93 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 12.24/7.93 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/7.93 => ( ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor )
% 12.24/7.93 => ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3265_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
% 12.24/7.93 ! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
% 12.24/7.93 ( ! [X3: A,Y5: A] :
% 12.24/7.93 ( ( Xor @ X3 @ Y5 )
% 12.24/7.93 = ( Disj @ ( Conj @ X3 @ ( Compl @ Y5 ) ) @ ( Conj @ ( Compl @ X3 ) @ Y5 ) ) )
% 12.24/7.93 => ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3266_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( boolea5476839437570043046axioms @ A )
% 12.24/7.93 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
% 12.24/7.93 ! [X2: A,Y6: A] :
% 12.24/7.93 ( ( Xor2 @ X2 @ Y6 )
% 12.24/7.93 = ( Disj2 @ ( Conj2 @ X2 @ ( Compl2 @ Y6 ) ) @ ( Conj2 @ ( Compl2 @ X2 ) @ Y6 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3267_possible__bit__0,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3268_possible__bit__less__imp,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [Tyrep2: itself @ A,I: nat,J: nat] :
% 12.24/7.93 ( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ J @ I )
% 12.24/7.93 => ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3269_integer__of__num__triv_I1_J,axiom,
% 12.24/7.93 ( ( code_integer_of_num @ one2 )
% 12.24/7.93 = ( one_one @ code_integer ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3270_integer__of__num__def,axiom,
% 12.24/7.93 ( code_integer_of_num
% 12.24/7.93 = ( numeral_numeral @ code_integer ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3271_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
% 12.24/7.93 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
% 12.24/7.93 ( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
% 12.24/7.93 => ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3272_abstract__boolean__algebra__sym__diff__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( boolea3799213064322606851m_diff @ A )
% 12.24/7.93 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A,Xor2: A > A > A] :
% 12.24/7.93 ( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One )
% 12.24/7.93 & ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3273_card__Diff__insert,axiom,
% 12.24/7.93 ! [A: $tType,A3: A,A6: set @ A,B7: set @ A] :
% 12.24/7.93 ( ( member @ A @ A3 @ A6 )
% 12.24/7.93 => ( ~ ( member @ A @ A3 @ B7 )
% 12.24/7.93 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ B7 ) ) )
% 12.24/7.93 = ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3274_card_Oempty,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.93 = ( zero_zero @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3275_card__insert__le__m1,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Y3: set @ A,X: A] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/7.93 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y3 ) ) @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3276_card__Diff__singleton__if,axiom,
% 12.24/7.93 ! [A: $tType,X: A,A6: set @ A] :
% 12.24/7.93 ( ( ( member @ A @ X @ A6 )
% 12.24/7.93 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.93 = ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) )
% 12.24/7.93 & ( ~ ( member @ A @ X @ A6 )
% 12.24/7.93 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.93 = ( finite_card @ A @ A6 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3277_card__Diff__singleton,axiom,
% 12.24/7.93 ! [A: $tType,X: A,A6: set @ A] :
% 12.24/7.93 ( ( member @ A @ X @ A6 )
% 12.24/7.93 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.93 = ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3278_card__atLeastAtMost,axiom,
% 12.24/7.93 ! [L: nat,U: nat] :
% 12.24/7.93 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
% 12.24/7.93 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3279_sgn__integer__code,axiom,
% 12.24/7.93 ( ( sgn_sgn @ code_integer )
% 12.24/7.93 = ( ^ [K3: code_integer] :
% 12.24/7.93 ( if @ code_integer
% 12.24/7.93 @ ( K3
% 12.24/7.93 = ( zero_zero @ code_integer ) )
% 12.24/7.93 @ ( zero_zero @ code_integer )
% 12.24/7.93 @ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3280_card__insert__le,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ ( insert @ A @ X @ A6 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3281_card__1__singletonE,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A] :
% 12.24/7.93 ( ( ( finite_card @ A @ A6 )
% 12.24/7.93 = ( one_one @ nat ) )
% 12.24/7.93 => ~ ! [X3: A] :
% 12.24/7.93 ( A6
% 12.24/7.93 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3282_card__1__singleton__iff,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A] :
% 12.24/7.93 ( ( ( finite_card @ A @ A6 )
% 12.24/7.93 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.93 = ( ? [X2: A] :
% 12.24/7.93 ( A6
% 12.24/7.93 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3283_card__eq__SucD,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,K: nat] :
% 12.24/7.93 ( ( ( finite_card @ A @ A6 )
% 12.24/7.93 = ( suc @ K ) )
% 12.24/7.93 => ? [B5: A,B9: set @ A] :
% 12.24/7.93 ( ( A6
% 12.24/7.93 = ( insert @ A @ B5 @ B9 ) )
% 12.24/7.93 & ~ ( member @ A @ B5 @ B9 )
% 12.24/7.93 & ( ( finite_card @ A @ B9 )
% 12.24/7.93 = K )
% 12.24/7.93 & ( ( K
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( B9
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3284_card__Suc__eq,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,K: nat] :
% 12.24/7.93 ( ( ( finite_card @ A @ A6 )
% 12.24/7.93 = ( suc @ K ) )
% 12.24/7.93 = ( ? [B3: A,B6: set @ A] :
% 12.24/7.93 ( ( A6
% 12.24/7.93 = ( insert @ A @ B3 @ B6 ) )
% 12.24/7.93 & ~ ( member @ A @ B3 @ B6 )
% 12.24/7.93 & ( ( finite_card @ A @ B6 )
% 12.24/7.93 = K )
% 12.24/7.93 & ( ( K
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( B6
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3285_card__Diff1__le,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3286_drop__bit__exp__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.93 => ! [M: nat,N: nat] :
% 12.24/7.93 ( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 12.24/7.93 = ( times_times @ A
% 12.24/7.93 @ ( zero_neq_one_of_bool @ A
% 12.24/7.93 @ ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
% 12.24/7.93 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3287_bit__minus__2__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.93 => ! [N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3288_bit__double__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [A3: A,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
% 12.24/7.93 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/7.93 & ( N
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3289_bit__mask__sub__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [M: nat,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( ord_less @ nat @ N @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3290_bit__minus__exp__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.93 => ! [M: nat,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3291_bit__minus__1__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.93 => ! [N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 12.24/7.93 = ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3292_plus__integer__code_I1_J,axiom,
% 12.24/7.93 ! [K: code_integer] :
% 12.24/7.93 ( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 12.24/7.93 = K ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3293_plus__integer__code_I2_J,axiom,
% 12.24/7.93 ! [L: code_integer] :
% 12.24/7.93 ( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 12.24/7.93 = L ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3294_less__integer__code_I1_J,axiom,
% 12.24/7.93 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3295_abs__integer__code,axiom,
% 12.24/7.93 ( ( abs_abs @ code_integer )
% 12.24/7.93 = ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3296_uminus__integer__code_I1_J,axiom,
% 12.24/7.93 ( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
% 12.24/7.93 = ( zero_zero @ code_integer ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3297_minus__integer__code_I1_J,axiom,
% 12.24/7.93 ! [K: code_integer] :
% 12.24/7.93 ( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 12.24/7.93 = K ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3298_minus__integer__code_I2_J,axiom,
% 12.24/7.93 ! [L: code_integer] :
% 12.24/7.93 ( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 12.24/7.93 = ( uminus_uminus @ code_integer @ L ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3299_bit__mask__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.93 => ! [M: nat,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( ord_less @ nat @ N @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3300_bit__of__nat__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [M: nat,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3301_bit__minus__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.93 => ! [A3: A,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3302_bit__push__bit__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.93 => ! [M: nat,A3: A,N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) @ N )
% 12.24/7.93 = ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 12.24/7.93 & ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3303_fold__possible__bit,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [N: nat] :
% 12.24/7.93 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 = ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3304_bit__2__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_semiring_bits @ A )
% 12.24/7.93 => ! [N: nat] :
% 12.24/7.93 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.93 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
% 12.24/7.93 & ( N
% 12.24/7.93 = ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3305_CHAR__eq0__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 = ( ! [N3: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/7.93 => ( ( semiring_1_of_nat @ A @ N3 )
% 12.24/7.93 != ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3306_CHAR__eq__posI,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ! [C2: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
% 12.24/7.93 => ( ( ( semiring_1_of_nat @ A @ C2 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 => ( ! [X3: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
% 12.24/7.93 => ( ( ord_less @ nat @ X3 @ C2 )
% 12.24/7.93 => ( ( semiring_1_of_nat @ A @ X3 )
% 12.24/7.93 != ( zero_zero @ A ) ) ) )
% 12.24/7.93 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 12.24/7.93 = C2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3307_CHAR__pos__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
% 12.24/7.93 = ( ? [N3: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/7.93 & ( ( semiring_1_of_nat @ A @ N3 )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3308_of__nat__CHAR,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3309_CHAR__eqI,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ! [C2: nat] :
% 12.24/7.93 ( ( ( semiring_1_of_nat @ A @ C2 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 => ( ! [X3: nat] :
% 12.24/7.93 ( ( ( semiring_1_of_nat @ A @ X3 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 => ( dvd_dvd @ nat @ C2 @ X3 ) )
% 12.24/7.93 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 12.24/7.93 = C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3310_CHAR__eq__0,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_char_0 @ A )
% 12.24/7.93 => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3311_of__nat__eq__0__iff__char__dvd,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_1 @ A )
% 12.24/7.93 => ! [N: nat] :
% 12.24/7.93 ( ( ( semiring_1_of_nat @ A @ N )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 = ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3312_divmod__integer__def,axiom,
% 12.24/7.93 ( code_divmod_integer
% 12.24/7.93 = ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L3 ) @ ( modulo_modulo @ code_integer @ K3 @ L3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3313_divmod__abs__def,axiom,
% 12.24/7.93 ( code_divmod_abs
% 12.24/7.93 = ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3314_card__greaterThanLessThan__int,axiom,
% 12.24/7.93 ! [L: int,U: int] :
% 12.24/7.93 ( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
% 12.24/7.93 = ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3315_nth__enumerate__eq,axiom,
% 12.24/7.93 ! [A: $tType,M: nat,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
% 12.24/7.93 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3316_count__list_Osimps_I2_J,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Y3: A,Xs: list @ A] :
% 12.24/7.93 ( ( ( X = Y3 )
% 12.24/7.93 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
% 12.24/7.93 = ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y3 ) @ ( one_one @ nat ) ) ) )
% 12.24/7.93 & ( ( X != Y3 )
% 12.24/7.93 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
% 12.24/7.93 = ( count_list @ A @ Xs @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3317_greaterThanLessThan__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ord @ A )
% 12.24/7.93 => ! [I: A,L: A,U: A] :
% 12.24/7.93 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 12.24/7.93 = ( ( ord_less @ A @ L @ I )
% 12.24/7.93 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3318_greaterThanLessThan__empty,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order @ A )
% 12.24/7.93 => ! [L: A,K: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ L @ K )
% 12.24/7.93 => ( ( set_or5935395276787703475ssThan @ A @ K @ L )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3319_greaterThanLessThan__empty__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3320_greaterThanLessThan__empty__iff2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( bot_bot @ ( set @ A ) )
% 12.24/7.93 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3321_enumerate__simps_I2_J,axiom,
% 12.24/7.93 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 12.24/7.93 ( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
% 12.24/7.93 = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3322_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3323_divmod__abs__code_I6_J,axiom,
% 12.24/7.93 ! [J: code_integer] :
% 12.24/7.93 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
% 12.24/7.93 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3324_count__le__length,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3325_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3326_divmod__abs__code_I5_J,axiom,
% 12.24/7.93 ! [J: code_integer] :
% 12.24/7.93 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
% 12.24/7.93 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3327_sqr_Osimps_I3_J,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( sqr @ ( bit1 @ N ) )
% 12.24/7.93 = ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3328_quotient__of__number_I5_J,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3329_prod__decode__aux_Oelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
% 12.24/7.93 ( ( ( nat_prod_decode_aux @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( ord_less_eq @ nat @ Xa3 @ X )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3330_prod__decode__aux_Osimps,axiom,
% 12.24/7.93 ( nat_prod_decode_aux
% 12.24/7.93 = ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3331_nat__of__integer__non__positive,axiom,
% 12.24/7.93 ! [K: code_integer] :
% 12.24/7.93 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 12.24/7.93 => ( ( code_nat_of_integer @ K )
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3332_nat__of__integer__of__nat,axiom,
% 12.24/7.93 ! [N: nat] :
% 12.24/7.93 ( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 12.24/7.93 = N ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3333_of__nat__of__integer,axiom,
% 12.24/7.93 ! [K: code_integer] :
% 12.24/7.93 ( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
% 12.24/7.93 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3334_card__greaterThanLessThan,axiom,
% 12.24/7.93 ! [L: nat,U: nat] :
% 12.24/7.93 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
% 12.24/7.93 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3335_rat__one__code,axiom,
% 12.24/7.93 ( ( quotient_of @ ( one_one @ rat ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3336_rat__zero__code,axiom,
% 12.24/7.93 ( ( quotient_of @ ( zero_zero @ rat ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3337_quotient__of__number_I4_J,axiom,
% 12.24/7.93 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3338_quotient__of__number_I3_J,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3339_prod__decode__aux_Ocases,axiom,
% 12.24/7.93 ! [X: product_prod @ nat @ nat] :
% 12.24/7.93 ~ ! [K2: nat,M6: nat] :
% 12.24/7.93 ( X
% 12.24/7.93 != ( product_Pair @ nat @ nat @ K2 @ M6 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3340_sgn__rat__def,axiom,
% 12.24/7.93 ( ( sgn_sgn @ rat )
% 12.24/7.93 = ( ^ [A5: rat] :
% 12.24/7.93 ( if @ rat
% 12.24/7.93 @ ( A5
% 12.24/7.93 = ( zero_zero @ rat ) )
% 12.24/7.93 @ ( zero_zero @ rat )
% 12.24/7.93 @ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3341_diff__rat__def,axiom,
% 12.24/7.93 ( ( minus_minus @ rat )
% 12.24/7.93 = ( ^ [Q3: rat,R5: rat] : ( plus_plus @ rat @ Q3 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3342_abs__rat__def,axiom,
% 12.24/7.93 ( ( abs_abs @ rat )
% 12.24/7.93 = ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3343_sqr_Osimps_I2_J,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( sqr @ ( bit0 @ N ) )
% 12.24/7.93 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3344_sqr_Osimps_I1_J,axiom,
% 12.24/7.93 ( ( sqr @ one2 )
% 12.24/7.93 = one2 ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3345_sqr__conv__mult,axiom,
% 12.24/7.93 ( sqr
% 12.24/7.93 = ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3346_quotient__of__denom__pos,axiom,
% 12.24/7.93 ! [R: rat,P5: int,Q4: int] :
% 12.24/7.93 ( ( ( quotient_of @ R )
% 12.24/7.93 = ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 12.24/7.93 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3347_numeral__sqr,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_numeral @ A )
% 12.24/7.93 => ! [K: num] :
% 12.24/7.93 ( ( numeral_numeral @ A @ ( sqr @ K ) )
% 12.24/7.93 = ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3348_quotient__of__int,axiom,
% 12.24/7.93 ! [A3: int] :
% 12.24/7.93 ( ( quotient_of @ ( of_int @ A3 ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3349_pow_Osimps_I3_J,axiom,
% 12.24/7.93 ! [X: num,Y3: num] :
% 12.24/7.93 ( ( pow @ X @ ( bit1 @ Y3 ) )
% 12.24/7.93 = ( times_times @ num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3350_Frct__code__post_I4_J,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
% 12.24/7.93 = ( numeral_numeral @ rat @ K ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3351_Frct__code__post_I5_J,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
% 12.24/7.93 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3352_prod__encode__prod__decode__aux,axiom,
% 12.24/7.93 ! [K: nat,M: nat] :
% 12.24/7.93 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 12.24/7.93 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3353_prod__encode__eq,axiom,
% 12.24/7.93 ! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
% 12.24/7.93 ( ( ( nat_prod_encode @ X )
% 12.24/7.93 = ( nat_prod_encode @ Y3 ) )
% 12.24/7.93 = ( X = Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3354_less__eq__rat__def,axiom,
% 12.24/7.93 ( ( ord_less_eq @ rat )
% 12.24/7.93 = ( ^ [X2: rat,Y6: rat] :
% 12.24/7.93 ( ( ord_less @ rat @ X2 @ Y6 )
% 12.24/7.93 | ( X2 = Y6 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3355_Frct__code__post_I9_J,axiom,
% 12.24/7.93 ! [Q4: product_prod @ int @ int] :
% 12.24/7.93 ( ( uminus_uminus @ rat @ ( uminus_uminus @ rat @ ( frct @ Q4 ) ) )
% 12.24/7.93 = ( frct @ Q4 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3356_pow_Osimps_I1_J,axiom,
% 12.24/7.93 ! [X: num] :
% 12.24/7.93 ( ( pow @ X @ one2 )
% 12.24/7.93 = X ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3357_le__prod__encode__2,axiom,
% 12.24/7.93 ! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3358_le__prod__encode__1,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3359_Frct__code__post_I2_J,axiom,
% 12.24/7.93 ! [A3: int] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
% 12.24/7.93 = ( zero_zero @ rat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3360_Frct__code__post_I1_J,axiom,
% 12.24/7.93 ! [A3: int] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
% 12.24/7.93 = ( zero_zero @ rat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3361_Frct__code__post_I3_J,axiom,
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
% 12.24/7.93 = ( one_one @ rat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3362_pow_Osimps_I2_J,axiom,
% 12.24/7.93 ! [X: num,Y3: num] :
% 12.24/7.93 ( ( pow @ X @ ( bit0 @ Y3 ) )
% 12.24/7.93 = ( sqr @ ( pow @ X @ Y3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3363_Frct__code__post_I8_J,axiom,
% 12.24/7.93 ! [A3: int,B2: int] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
% 12.24/7.93 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3364_Frct__code__post_I7_J,axiom,
% 12.24/7.93 ! [A3: int,B2: int] :
% 12.24/7.93 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
% 12.24/7.93 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3365_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 12.24/7.93 ! [N: nat,J: nat,I: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
% 12.24/7.93 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
% 12.24/7.93 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3366_rat__sgn__code,axiom,
% 12.24/7.93 ! [P5: rat] :
% 12.24/7.93 ( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3367_one__div__numeral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [N: num] :
% 12.24/7.93 ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.93 = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3368_in__measures_I2_J,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Y3: A,F: A > nat,Fs: list @ ( A > nat )] :
% 12.24/7.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
% 12.24/7.93 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.93 | ( ( ( F @ X )
% 12.24/7.93 = ( F @ Y3 ) )
% 12.24/7.93 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3369_Cons__lenlex__iff,axiom,
% 12.24/7.93 ! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 12.24/7.93 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
% 12.24/7.93 = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
% 12.24/7.93 | ( ( ( size_size @ ( list @ A ) @ Ms )
% 12.24/7.93 = ( size_size @ ( list @ A ) @ Ns ) )
% 12.24/7.93 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
% 12.24/7.93 | ( ( M = N )
% 12.24/7.93 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3370_fst__divmod__integer,axiom,
% 12.24/7.93 ! [K: code_integer,L: code_integer] :
% 12.24/7.93 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 12.24/7.93 = ( divide_divide @ code_integer @ K @ L ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3371_fst__divmod__abs,axiom,
% 12.24/7.93 ! [K: code_integer,L: code_integer] :
% 12.24/7.93 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 12.24/7.93 = ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3372_Suc__0__div__numeral,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.93 = ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3373_lenlex__length,axiom,
% 12.24/7.93 ! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 12.24/7.93 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
% 12.24/7.93 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3374_measures__less,axiom,
% 12.24/7.93 ! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
% 12.24/7.93 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.93 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3375_measures__lesseq,axiom,
% 12.24/7.93 ! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) )
% 12.24/7.93 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3376_sorted__list__of__set__greaterThanLessThan,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( suc @ I ) @ J )
% 12.24/7.93 => ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
% 12.24/7.93 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3377_bezw_Osimps,axiom,
% 12.24/7.93 ( bezw
% 12.24/7.93 = ( ^ [X2: nat,Y6: nat] :
% 12.24/7.93 ( if @ ( product_prod @ int @ int )
% 12.24/7.93 @ ( Y6
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
% 12.24/7.93 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3378_bezw_Oelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
% 12.24/7.93 ( ( ( bezw @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( Xa3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 12.24/7.93 & ( ( Xa3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3379_bezw__non__0,axiom,
% 12.24/7.93 ! [Y3: nat,X: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y3 )
% 12.24/7.93 => ( ( bezw @ X @ Y3 )
% 12.24/7.93 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3380_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 12.24/7.93 ! [N: nat,J: nat,I: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
% 12.24/7.93 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
% 12.24/7.93 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3381_listrel__iff__nth,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 12.24/7.93 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 12.24/7.93 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.93 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.93 & ! [N3: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N3 ) @ ( nth @ B @ Ys2 @ N3 ) ) @ R ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3382_greaterThanAtMost__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ord @ A )
% 12.24/7.93 => ! [I: A,L: A,U: A] :
% 12.24/7.93 ( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 12.24/7.93 = ( ( ord_less @ A @ L @ I )
% 12.24/7.93 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3383_greaterThanAtMost__empty,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( order @ A )
% 12.24/7.93 => ! [L: A,K: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ L @ K )
% 12.24/7.93 => ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3384_greaterThanAtMost__empty__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( preorder @ A )
% 12.24/7.93 => ! [K: A,L: A] :
% 12.24/7.93 ( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 12.24/7.93 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.93 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3385_greaterThanAtMost__empty__iff2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( preorder @ A )
% 12.24/7.93 => ! [K: A,L: A] :
% 12.24/7.93 ( ( ( bot_bot @ ( set @ A ) )
% 12.24/7.93 = ( set_or3652927894154168847AtMost @ A @ K @ L ) )
% 12.24/7.93 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3386_one__mod__numeral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ! [N: num] :
% 12.24/7.93 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.93 = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3387_Ioc__inj,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
% 12.24/7.93 = ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ D2 @ C2 ) )
% 12.24/7.93 | ( ( A3 = C2 )
% 12.24/7.93 & ( B2 = D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3388_atLeastSucAtMost__greaterThanAtMost,axiom,
% 12.24/7.93 ! [L: nat,U: nat] :
% 12.24/7.93 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
% 12.24/7.93 = ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3389_Ioc__subset__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.93 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3390_divides__aux__def,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.93 => ( ( unique5940410009612947441es_aux @ A )
% 12.24/7.93 = ( ^ [Qr: product_prod @ A @ A] :
% 12.24/7.93 ( ( product_snd @ A @ A @ Qr )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3391_quotient__of__denom__pos_H,axiom,
% 12.24/7.93 ! [R: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3392_sorted__list__of__set__greaterThanAtMost,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
% 12.24/7.93 => ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
% 12.24/7.93 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3393_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3394_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( dense_linorder @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 12.24/7.93 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3395_minus__one__mod__numeral,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3396_one__mod__minus__numeral,axiom,
% 12.24/7.93 ! [N: num] :
% 12.24/7.93 ( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3397_numeral__mod__minus__numeral,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3398_minus__numeral__mod__numeral,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3399_size__prod__simp,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( basic_BNF_size_prod @ A @ B )
% 12.24/7.93 = ( ^ [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 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3400_snd__divmod__integer,axiom,
% 12.24/7.93 ! [K: code_integer,L: code_integer] :
% 12.24/7.93 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 12.24/7.93 = ( modulo_modulo @ code_integer @ K @ L ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3401_snd__divmod__abs,axiom,
% 12.24/7.93 ! [K: code_integer,L: code_integer] :
% 12.24/7.93 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 12.24/7.93 = ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3402_Suc__0__mod__numeral,axiom,
% 12.24/7.93 ! [K: num] :
% 12.24/7.93 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.93 = ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3403_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 12.24/7.93 ! [L: int,U: int] :
% 12.24/7.93 ( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 12.24/7.93 = ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3404_Divides_Oadjust__mod__def,axiom,
% 12.24/7.93 ( adjust_mod
% 12.24/7.93 = ( ^ [L3: int,R5: int] :
% 12.24/7.93 ( if @ int
% 12.24/7.93 @ ( R5
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 @ ( zero_zero @ int )
% 12.24/7.93 @ ( minus_minus @ int @ L3 @ R5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3405_bezw_Opelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
% 12.24/7.93 ( ( ( bezw @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 12.24/7.93 => ~ ( ( ( ( Xa3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 12.24/7.93 & ( ( Xa3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3406_in__set__enumerate__eq,axiom,
% 12.24/7.93 ! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
% 12.24/7.93 = ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
% 12.24/7.93 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 12.24/7.93 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
% 12.24/7.93 = ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3407_list__encode_Osimps_I2_J,axiom,
% 12.24/7.93 ! [X: nat,Xs: list @ nat] :
% 12.24/7.93 ( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
% 12.24/7.93 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3408_bezw__aux,axiom,
% 12.24/7.93 ! [X: nat,Y3: nat] :
% 12.24/7.93 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y3 ) )
% 12.24/7.93 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3409_in__set__zip,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Xs: list @ A,Ys2: list @ B] :
% 12.24/7.93 ( ( member @ ( product_prod @ A @ B ) @ P5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 12.24/7.93 = ( ? [N3: nat] :
% 12.24/7.93 ( ( ( nth @ A @ Xs @ N3 )
% 12.24/7.93 = ( product_fst @ A @ B @ P5 ) )
% 12.24/7.93 & ( ( nth @ B @ Ys2 @ N3 )
% 12.24/7.93 = ( product_snd @ A @ B @ P5 ) )
% 12.24/7.93 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3410_gcd__eq__0__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 = ( ( A3
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 & ( B2
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3411_gcd_Obottom__left__bottom,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.93 = ( one_one @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3412_gcd_Obottom__right__bottom,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 = ( one_one @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3413_gcd__neg2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.24/7.93 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3414_gcd__neg1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.24/7.93 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3415_gcd__0__left__nat,axiom,
% 12.24/7.93 ! [X: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
% 12.24/7.93 = X ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3416_gcd__0__nat,axiom,
% 12.24/7.93 ! [X: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
% 12.24/7.93 = X ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3417_gcd__nat_Oright__neutral,axiom,
% 12.24/7.93 ! [A3: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
% 12.24/7.93 = A3 ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3418_gcd__nat_Oneutr__eq__iff,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat] :
% 12.24/7.93 ( ( ( zero_zero @ nat )
% 12.24/7.93 = ( gcd_gcd @ nat @ A3 @ B2 ) )
% 12.24/7.93 = ( ( A3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 & ( B2
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3419_gcd__nat_Oleft__neutral,axiom,
% 12.24/7.93 ! [A3: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
% 12.24/7.93 = A3 ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3420_gcd__nat_Oeq__neutr__iff,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat] :
% 12.24/7.93 ( ( ( gcd_gcd @ nat @ A3 @ B2 )
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 = ( ( A3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 & ( B2
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3421_gcd__1__nat,axiom,
% 12.24/7.93 ! [M: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
% 12.24/7.93 = ( one_one @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3422_gcd__neg__numeral__2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ring_gcd @ A )
% 12.24/7.93 => ! [A3: A,N: num] :
% 12.24/7.93 ( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.93 = ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3423_gcd__neg__numeral__1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( ring_gcd @ A )
% 12.24/7.93 => ! [N: num,A3: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
% 12.24/7.93 = ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3424_is__unit__gcd__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 12.24/7.93 = ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3425_gcd__Suc__0,axiom,
% 12.24/7.93 ! [M: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.93 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3426_gcd__pos__nat,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N ) )
% 12.24/7.93 = ( ( M
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 | ( N
% 12.24/7.93 != ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3427_count__notin,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.93 ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( ( count_list @ A @ Xs @ X )
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3428_list__encode__eq,axiom,
% 12.24/7.93 ! [X: list @ nat,Y3: list @ nat] :
% 12.24/7.93 ( ( ( nat_list_encode @ X )
% 12.24/7.93 = ( nat_list_encode @ Y3 ) )
% 12.24/7.93 = ( X = Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3429_gcd__mult__distrib__nat,axiom,
% 12.24/7.93 ! [K: nat,M: nat,N: nat] :
% 12.24/7.93 ( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N ) )
% 12.24/7.93 = ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3430_gcd__add__mult,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [M: A,K: A,N: A] :
% 12.24/7.93 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N ) )
% 12.24/7.93 = ( gcd_gcd @ A @ M @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3431_gcd__dvd__prod,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3432_gcd__le1__nat,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3433_gcd__le2__nat,axiom,
% 12.24/7.93 ! [B2: nat,A3: nat] :
% 12.24/7.93 ( ( B2
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3434_gcd__diff2__nat,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M ) @ N )
% 12.24/7.93 = ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3435_gcd__diff1__nat,axiom,
% 12.24/7.93 ! [N: nat,M: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.93 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N ) @ N )
% 12.24/7.93 = ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3436_gcd__non__0__nat,axiom,
% 12.24/7.93 ! [Y3: nat,X: nat] :
% 12.24/7.93 ( ( Y3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( gcd_gcd @ nat @ X @ Y3 )
% 12.24/7.93 = ( gcd_gcd @ nat @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3437_gcd__nat_Osimps,axiom,
% 12.24/7.93 ( ( gcd_gcd @ nat )
% 12.24/7.93 = ( ^ [X2: nat,Y6: nat] :
% 12.24/7.93 ( if @ nat
% 12.24/7.93 @ ( Y6
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 @ X2
% 12.24/7.93 @ ( gcd_gcd @ nat @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3438_gcd__nat_Oelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: nat] :
% 12.24/7.93 ( ( ( gcd_gcd @ nat @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( Xa3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3 = X ) )
% 12.24/7.93 & ( ( Xa3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3439_gcd__mult__unit1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 12.24/7.93 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3440_gcd__mult__unit2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 12.24/7.93 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3441_gcd__div__unit1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 12.24/7.93 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3442_gcd__div__unit2,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semiring_gcd @ A )
% 12.24/7.93 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.93 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 12.24/7.93 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3443_bezout__nat,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ? [X3: nat,Y5: nat] :
% 12.24/7.93 ( ( times_times @ nat @ A3 @ X3 )
% 12.24/7.93 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3444_bezout__gcd__nat_H,axiom,
% 12.24/7.93 ! [B2: nat,A3: nat] :
% 12.24/7.93 ? [X3: nat,Y5: nat] :
% 12.24/7.93 ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( times_times @ nat @ A3 @ X3 ) )
% 12.24/7.93 & ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
% 12.24/7.93 = ( gcd_gcd @ nat @ A3 @ B2 ) ) )
% 12.24/7.93 | ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ ( times_times @ nat @ B2 @ X3 ) )
% 12.24/7.93 & ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
% 12.24/7.93 = ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3445_length__pos__if__in__set,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3446_card__length,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3447_all__set__conv__all__nth,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 12.24/7.93 ( ( ! [X2: A] :
% 12.24/7.93 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( P @ X2 ) ) )
% 12.24/7.93 = ( ! [I5: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( P @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3448_all__nth__imp__all__set,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
% 12.24/7.93 ( ! [I4: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( P @ ( nth @ A @ Xs @ I4 ) ) )
% 12.24/7.93 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( P @ X ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3449_in__set__conv__nth,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.93 = ( ? [I5: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 & ( ( nth @ A @ Xs @ I5 )
% 12.24/7.93 = X ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3450_list__ball__nth,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ! [X3: A] :
% 12.24/7.93 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( P @ X3 ) )
% 12.24/7.93 => ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3451_nth__mem,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3452_nth__equal__first__eq,axiom,
% 12.24/7.93 ! [A: $tType,X: A,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 12.24/7.93 = X )
% 12.24/7.93 = ( N
% 12.24/7.93 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3453_prod__decode__aux_Opelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
% 12.24/7.93 ( ( ( nat_prod_decode_aux @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 12.24/7.93 => ~ ( ( ( ( ord_less_eq @ nat @ Xa3 @ X )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3454_gcd__nat_Opelims,axiom,
% 12.24/7.93 ! [X: nat,Xa3: nat,Y3: nat] :
% 12.24/7.93 ( ( ( gcd_gcd @ nat @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
% 12.24/7.93 => ~ ( ( ( ( Xa3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3 = X ) )
% 12.24/7.93 & ( ( Xa3
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3455_list__encode_Oelims,axiom,
% 12.24/7.93 ! [X: list @ nat,Y3: nat] :
% 12.24/7.93 ( ( ( nat_list_encode @ X )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( X
% 12.24/7.93 = ( nil @ nat ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 != ( zero_zero @ nat ) ) )
% 12.24/7.93 => ~ ! [X3: nat,Xs2: list @ nat] :
% 12.24/7.93 ( ( X
% 12.24/7.93 = ( cons @ nat @ X3 @ Xs2 ) )
% 12.24/7.93 => ( Y3
% 12.24/7.93 != ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3456_zero__rat__def,axiom,
% 12.24/7.93 ( ( zero_zero @ rat )
% 12.24/7.93 = ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3457_pair__lessI2,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less @ nat @ S @ T2 )
% 12.24/7.93 => ( 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 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3458_gcd__1__int,axiom,
% 12.24/7.93 ! [M: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
% 12.24/7.93 = ( one_one @ int ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3459_gcd__neg2__int,axiom,
% 12.24/7.93 ! [X: int,Y3: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
% 12.24/7.93 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3460_gcd__neg1__int,axiom,
% 12.24/7.93 ! [X: int,Y3: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
% 12.24/7.93 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3461_gcd__pos__int,axiom,
% 12.24/7.93 ! [M: int,N: int] :
% 12.24/7.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N ) )
% 12.24/7.93 = ( ( M
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 | ( N
% 12.24/7.93 != ( zero_zero @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3462_gcd__neg__numeral__1__int,axiom,
% 12.24/7.93 ! [N: num,X: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
% 12.24/7.93 = ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3463_gcd__neg__numeral__2__int,axiom,
% 12.24/7.93 ! [X: int,N: num] :
% 12.24/7.93 ( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3464_gcd__0__left__int,axiom,
% 12.24/7.93 ! [X: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.93 = ( abs_abs @ int @ X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3465_gcd__0__int,axiom,
% 12.24/7.93 ! [X: int] :
% 12.24/7.93 ( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
% 12.24/7.93 = ( abs_abs @ int @ X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3466_gcd__int__int__eq,axiom,
% 12.24/7.93 ! [M: nat,N: nat] :
% 12.24/7.93 ( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 12.24/7.93 = ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3467_length__0__conv,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A] :
% 12.24/7.93 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 = ( Xs
% 12.24/7.93 = ( nil @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3468_horner__sum__simps_I1_J,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( comm_semiring_0 @ A )
% 12.24/7.93 => ! [F: B > A,A3: A] :
% 12.24/7.93 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( nil @ B ) )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3469_length__greater__0__conv,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 = ( Xs
% 12.24/7.93 != ( nil @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3470_gcd__nat__abs__right__eq,axiom,
% 12.24/7.93 ! [N: nat,K: int] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 12.24/7.93 = ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3471_gcd__nat__abs__left__eq,axiom,
% 12.24/7.93 ! [K: int,N: nat] :
% 12.24/7.93 ( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 12.24/7.93 = ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3472_pair__less__iff1,axiom,
% 12.24/7.93 ! [X: nat,Y3: nat,Z3: nat] :
% 12.24/7.93 ( ( 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 @ Y3 ) @ ( product_Pair @ nat @ nat @ X @ Z3 ) ) @ fun_pair_less )
% 12.24/7.93 = ( ord_less @ nat @ Y3 @ Z3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3473_list__encode_Ocases,axiom,
% 12.24/7.93 ! [X: list @ nat] :
% 12.24/7.93 ( ( X
% 12.24/7.93 != ( nil @ nat ) )
% 12.24/7.93 => ~ ! [X3: nat,Xs2: list @ nat] :
% 12.24/7.93 ( X
% 12.24/7.93 != ( cons @ nat @ X3 @ Xs2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3474_gcd__ge__0__int,axiom,
% 12.24/7.93 ! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3475_bezout__int,axiom,
% 12.24/7.93 ! [X: int,Y3: int] :
% 12.24/7.93 ? [U2: int,V4: int] :
% 12.24/7.93 ( ( plus_plus @ int @ ( times_times @ int @ U2 @ X ) @ ( times_times @ int @ V4 @ Y3 ) )
% 12.24/7.93 = ( gcd_gcd @ int @ X @ Y3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3476_gcd__mult__distrib__int,axiom,
% 12.24/7.93 ! [K: int,M: int,N: int] :
% 12.24/7.93 ( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N ) )
% 12.24/7.93 = ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3477_list_Osize_I3_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 12.24/7.93 = ( zero_zero @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3478_list__encode_Osimps_I1_J,axiom,
% 12.24/7.93 ( ( nat_list_encode @ ( nil @ nat ) )
% 12.24/7.93 = ( zero_zero @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3479_count__list_Osimps_I1_J,axiom,
% 12.24/7.93 ! [A: $tType,Y3: A] :
% 12.24/7.93 ( ( count_list @ A @ ( nil @ A ) @ Y3 )
% 12.24/7.93 = ( zero_zero @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3480_gcd__le1__int,axiom,
% 12.24/7.93 ! [A3: int,B2: int] :
% 12.24/7.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
% 12.24/7.93 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3481_gcd__le2__int,axiom,
% 12.24/7.93 ! [B2: int,A3: int] :
% 12.24/7.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.93 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ B2 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3482_gcd__cases__int,axiom,
% 12.24/7.93 ! [X: int,Y3: int,P: int > $o] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.93 => ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.93 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 12.24/7.93 => ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 12.24/7.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.93 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 12.24/7.93 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 12.24/7.93 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 12.24/7.93 => ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3483_gcd__unique__int,axiom,
% 12.24/7.93 ! [D2: int,A3: int,B2: int] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
% 12.24/7.93 & ( dvd_dvd @ int @ D2 @ A3 )
% 12.24/7.93 & ( dvd_dvd @ int @ D2 @ B2 )
% 12.24/7.93 & ! [E3: int] :
% 12.24/7.93 ( ( ( dvd_dvd @ int @ E3 @ A3 )
% 12.24/7.93 & ( dvd_dvd @ int @ E3 @ B2 ) )
% 12.24/7.93 => ( dvd_dvd @ int @ E3 @ D2 ) ) )
% 12.24/7.93 = ( D2
% 12.24/7.93 = ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3484_gcd__non__0__int,axiom,
% 12.24/7.93 ! [Y3: int,X: int] :
% 12.24/7.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.93 => ( ( gcd_gcd @ int @ X @ Y3 )
% 12.24/7.93 = ( gcd_gcd @ int @ Y3 @ ( modulo_modulo @ int @ X @ Y3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3485_gcd__code__int,axiom,
% 12.24/7.93 ( ( gcd_gcd @ int )
% 12.24/7.93 = ( ^ [K3: int,L3: int] :
% 12.24/7.93 ( abs_abs @ int
% 12.24/7.93 @ ( if @ int
% 12.24/7.93 @ ( L3
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 @ K3
% 12.24/7.93 @ ( gcd_gcd @ int @ L3 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3486_pair__lessI1,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ A3 @ B2 )
% 12.24/7.93 => ( 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 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3487_gcd__int__def,axiom,
% 12.24/7.93 ( ( gcd_gcd @ int )
% 12.24/7.93 = ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3488_one__rat__def,axiom,
% 12.24/7.93 ( ( one_one @ rat )
% 12.24/7.93 = ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3489_plus__rat_Oabs__eq,axiom,
% 12.24/7.93 ! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
% 12.24/7.93 ( ( ratrel @ Xa3 @ Xa3 )
% 12.24/7.93 => ( ( ratrel @ X @ X )
% 12.24/7.93 => ( ( plus_plus @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
% 12.24/7.93 = ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3490_pair__leqI2,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ S @ T2 )
% 12.24/7.93 => ( 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 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3491_pair__leqI1,axiom,
% 12.24/7.93 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ A3 @ B2 )
% 12.24/7.93 => ( 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 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3492_smax__insertI,axiom,
% 12.24/7.93 ! [Y3: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.93 ( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y7 )
% 12.24/7.93 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
% 12.24/7.93 => ( ( 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 ) ) @ X8 @ Y7 ) @ fun_max_strict )
% 12.24/7.93 => ( 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3493_smin__insertI,axiom,
% 12.24/7.93 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.93 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 12.24/7.93 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
% 12.24/7.93 => ( ( 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 )
% 12.24/7.93 => ( 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3494_ratrel__iff,axiom,
% 12.24/7.93 ( ratrel
% 12.24/7.93 = ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
% 12.24/7.93 ( ( ( product_snd @ int @ int @ X2 )
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 & ( ( product_snd @ int @ int @ Y6 )
% 12.24/7.93 != ( zero_zero @ int ) )
% 12.24/7.93 & ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
% 12.24/7.93 = ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3495_smin__emptyI,axiom,
% 12.24/7.93 ! [X8: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.93 ( ( X8
% 12.24/7.93 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 12.24/7.93 => ( 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3496_one__rat_Orsp,axiom,
% 12.24/7.93 ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3497_zero__rat_Orsp,axiom,
% 12.24/7.93 ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3498_times__rat_Oabs__eq,axiom,
% 12.24/7.93 ! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
% 12.24/7.93 ( ( ratrel @ Xa3 @ Xa3 )
% 12.24/7.93 => ( ( ratrel @ X @ X )
% 12.24/7.93 => ( ( times_times @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
% 12.24/7.93 = ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3499_uminus__rat_Oabs__eq,axiom,
% 12.24/7.93 ! [X: product_prod @ int @ int] :
% 12.24/7.93 ( ( ratrel @ X @ X )
% 12.24/7.93 => ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
% 12.24/7.93 = ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3500_wmin__insertI,axiom,
% 12.24/7.93 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.93 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 12.24/7.93 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
% 12.24/7.93 => ( ( 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 )
% 12.24/7.93 => ( 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3501_wmax__insertI,axiom,
% 12.24/7.93 ! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.93 ( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ YS )
% 12.24/7.93 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
% 12.24/7.93 => ( ( 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 )
% 12.24/7.93 => ( 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3502_positive_Oabs__eq,axiom,
% 12.24/7.93 ! [X: product_prod @ int @ int] :
% 12.24/7.93 ( ( ratrel @ X @ X )
% 12.24/7.93 => ( ( positive @ ( abs_Rat @ X ) )
% 12.24/7.93 = ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3503_upto_Opelims,axiom,
% 12.24/7.93 ! [X: int,Xa3: int,Y3: list @ int] :
% 12.24/7.93 ( ( ( upto @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
% 12.24/7.93 => ~ ( ( ( ( ord_less_eq @ int @ X @ Xa3 )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ X @ Xa3 )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( nil @ int ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3504_upto_Opsimps,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( upto @ I @ J )
% 12.24/7.93 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( upto @ I @ J )
% 12.24/7.93 = ( nil @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3505_upto__Nil,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( ( upto @ I @ J )
% 12.24/7.93 = ( nil @ int ) )
% 12.24/7.93 = ( ord_less @ int @ J @ I ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3506_upto__Nil2,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( ( nil @ int )
% 12.24/7.93 = ( upto @ I @ J ) )
% 12.24/7.93 = ( ord_less @ int @ J @ I ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3507_upto__empty,axiom,
% 12.24/7.93 ! [J: int,I: int] :
% 12.24/7.93 ( ( ord_less @ int @ J @ I )
% 12.24/7.93 => ( ( upto @ I @ J )
% 12.24/7.93 = ( nil @ int ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3508_nth__upto,axiom,
% 12.24/7.93 ! [I: int,K: nat,J: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
% 12.24/7.93 => ( ( nth @ int @ ( upto @ I @ J ) @ K )
% 12.24/7.93 = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3509_length__upto,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
% 12.24/7.93 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3510_upto__rec__numeral_I1_J,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( nil @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3511_upto__rec__numeral_I4_J,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( nil @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3512_upto__rec__numeral_I3_J,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.93 = ( nil @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3513_upto__rec__numeral_I2_J,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.93 = ( nil @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3514_less__by__empty,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
% 12.24/7.93 ( ( A6
% 12.24/7.93 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.93 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3515_wmin__emptyI,axiom,
% 12.24/7.93 ! [X8: 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3516_positive__minus,axiom,
% 12.24/7.93 ! [X: rat] :
% 12.24/7.93 ( ~ ( positive @ X )
% 12.24/7.93 => ( ( X
% 12.24/7.93 != ( zero_zero @ rat ) )
% 12.24/7.93 => ( positive @ ( uminus_uminus @ rat @ X ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3517_greaterThanAtMost__upto,axiom,
% 12.24/7.93 ( ( set_or3652927894154168847AtMost @ int )
% 12.24/7.93 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3518_less__rat__def,axiom,
% 12.24/7.93 ( ( ord_less @ rat )
% 12.24/7.93 = ( ^ [X2: rat,Y6: rat] : ( positive @ ( minus_minus @ rat @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3519_upto__rec1,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( upto @ I @ J )
% 12.24/7.93 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3520_upto_Oelims,axiom,
% 12.24/7.93 ! [X: int,Xa3: int,Y3: list @ int] :
% 12.24/7.93 ( ( ( upto @ X @ Xa3 )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( ( ord_less_eq @ int @ X @ Xa3 )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ int @ X @ Xa3 )
% 12.24/7.93 => ( Y3
% 12.24/7.93 = ( nil @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3521_upto_Osimps,axiom,
% 12.24/7.93 ( upto
% 12.24/7.93 = ( ^ [I5: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I5 @ J3 ) @ ( cons @ int @ I5 @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3522_greaterThanLessThan__upto,axiom,
% 12.24/7.93 ( ( set_or5935395276787703475ssThan @ int )
% 12.24/7.93 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3523_positive_Orep__eq,axiom,
% 12.24/7.93 ( positive
% 12.24/7.93 = ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3524_inverse__rat_Oabs__eq,axiom,
% 12.24/7.93 ! [X: product_prod @ int @ int] :
% 12.24/7.93 ( ( ratrel @ X @ X )
% 12.24/7.93 => ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
% 12.24/7.93 = ( abs_Rat
% 12.24/7.93 @ ( if @ ( product_prod @ int @ int )
% 12.24/7.93 @ ( ( product_fst @ int @ int @ X )
% 12.24/7.93 = ( zero_zero @ int ) )
% 12.24/7.93 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.93 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3525_min__rpair__set,axiom,
% 12.24/7.93 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 ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3526_max__rpair__set,axiom,
% 12.24/7.93 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 ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3527_upto__rec2,axiom,
% 12.24/7.93 ! [I: int,J: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( upto @ I @ J )
% 12.24/7.93 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3528_inverse__eq__iff__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( A3 = B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3529_inverse__inverse__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3530_inverse__zero,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3531_inverse__nonzero__iff__nonzero,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 = ( A3
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3532_inverse__mult__distrib,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3533_inverse__1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 12.24/7.93 = ( one_one @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3534_inverse__eq__1__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ X )
% 12.24/7.93 = ( one_one @ A ) )
% 12.24/7.93 = ( X
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3535_inverse__minus__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3536_inverse__divide,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( divide_divide @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3537_abs__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field_abs_sgn @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3538_sgn__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field_abs_sgn @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3539_inverse__sgn,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) )
% 12.24/7.93 = ( sgn_sgn @ A @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3540_inverse__nonnegative__iff__nonnegative,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3541_inverse__nonpositive__iff__nonpositive,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.24/7.93 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3542_inverse__less__iff__less,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.93 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3543_inverse__less__iff__less__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3544_inverse__negative__iff__negative,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.24/7.93 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3545_inverse__positive__iff__positive,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3546_inverse__le__iff__le,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3547_inverse__le__iff__le__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3548_left__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3549_right__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3550_inverse__eq__divide__numeral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [W2: num] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
% 12.24/7.93 = ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3551_inverse__eq__divide__neg__numeral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [W2: num] :
% 12.24/7.93 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
% 12.24/7.93 = ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3552_mult__commute__imp__mult__inverse__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [Y3: A,X: A] :
% 12.24/7.93 ( ( ( times_times @ A @ Y3 @ X )
% 12.24/7.93 = ( times_times @ A @ X @ Y3 ) )
% 12.24/7.93 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y3 ) @ X )
% 12.24/7.93 = ( times_times @ A @ X @ ( inverse_inverse @ A @ Y3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3553_inverse__eq__imp__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( A3 = B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3554_field__class_Ofield__inverse__zero,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 12.24/7.93 = ( zero_zero @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3555_inverse__zero__imp__zero,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( zero_zero @ A ) )
% 12.24/7.93 => ( A3
% 12.24/7.93 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3556_nonzero__inverse__eq__imp__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( A3 = B2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3557_nonzero__inverse__inverse__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3558_nonzero__imp__inverse__nonzero,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3559_inverse__less__imp__less,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3560_less__imp__inverse__less,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3561_inverse__less__imp__less__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3562_less__imp__inverse__less__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3563_inverse__negative__imp__negative,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3564_inverse__positive__imp__positive,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 => ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3565_negative__imp__inverse__negative,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3566_positive__imp__inverse__positive,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3567_nonzero__inverse__mult__distrib,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3568_nonzero__inverse__minus__eq,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
% 12.24/7.93 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3569_inverse__unique,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( times_times @ A @ A3 @ B2 )
% 12.24/7.93 = ( one_one @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3570_inverse__numeral__1,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
% 12.24/7.93 = ( numeral_numeral @ A @ one2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3571_divide__inverse__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ( ( divide_divide @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A5 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3572_divide__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ( ( divide_divide @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3573_field__class_Ofield__divide__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ( ( divide_divide @ A )
% 12.24/7.93 = ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3574_inverse__eq__divide,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ( ( inverse_inverse @ A )
% 12.24/7.93 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3575_power__mult__power__inverse__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [X: A,M: nat,N: nat] :
% 12.24/7.93 ( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
% 12.24/7.93 = ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3576_power__mult__inverse__distrib,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [X: A,M: nat] :
% 12.24/7.93 ( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
% 12.24/7.93 = ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3577_mult__inverse__of__nat__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [Xa3: nat,X: A] :
% 12.24/7.93 ( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) @ X )
% 12.24/7.93 = ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3578_nonzero__abs__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
% 12.24/7.93 = ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3579_mult__inverse__of__int__commute,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [Xa3: int,X: A] :
% 12.24/7.93 ( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) @ X )
% 12.24/7.93 = ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3580_inverse__le__imp__le,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3581_le__imp__inverse__le,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3582_inverse__le__imp__le__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3583_le__imp__inverse__le__neg,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3584_inverse__le__1__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 12.24/7.93 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.24/7.93 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3585_one__less__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3586_one__less__inverse__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 12.24/7.93 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.93 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3587_inverse__add,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3588_division__ring__inverse__add,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3589_field__class_Ofield__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 12.24/7.93 = ( one_one @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3590_division__ring__inverse__diff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( B2
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3591_nonzero__inverse__eq__divide,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( A3
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( inverse_inverse @ A @ A3 )
% 12.24/7.93 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3592_inverse__le__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.93 => ( ord_less_eq @ A @ B2 @ A3 ) )
% 12.24/7.93 & ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3593_inverse__less__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 12.24/7.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.93 => ( ord_less @ A @ B2 @ A3 ) )
% 12.24/7.93 & ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
% 12.24/7.93 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3594_one__le__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.93 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3595_inverse__less__1__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 12.24/7.93 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 12.24/7.93 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3596_one__le__inverse__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( linordered_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 12.24/7.93 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.93 & ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3597_reals__Archimedean,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( archim462609752435547400_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.93 => ? [N4: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ X ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3598_length__append__singleton,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,X: A] :
% 12.24/7.93 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 12.24/7.93 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3599_length__Suc__conv__rev,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.93 = ( suc @ N ) )
% 12.24/7.93 = ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.93 ( ( Xs
% 12.24/7.93 = ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
% 12.24/7.93 & ( ( size_size @ ( list @ A ) @ Ys3 )
% 12.24/7.93 = N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3600_nth__append,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.93 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
% 12.24/7.93 = ( nth @ A @ Xs @ N ) ) )
% 12.24/7.93 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
% 12.24/7.93 = ( nth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3601_upto__split2,axiom,
% 12.24/7.93 ! [I: int,J: int,K: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( ord_less_eq @ int @ J @ K )
% 12.24/7.93 => ( ( upto @ I @ K )
% 12.24/7.93 = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3602_upto__split1,axiom,
% 12.24/7.93 ! [I: int,J: int,K: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( ord_less_eq @ int @ J @ K )
% 12.24/7.93 => ( ( upto @ I @ K )
% 12.24/7.93 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3603_ex__inverse__of__nat__less,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( archim462609752435547400_field @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.93 => ? [N4: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.24/7.93 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) ) @ X ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3604_power__diff__conv__inverse,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( division_ring @ A )
% 12.24/7.93 => ! [X: A,M: nat,N: nat] :
% 12.24/7.93 ( ( X
% 12.24/7.93 != ( zero_zero @ A ) )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 => ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
% 12.24/7.93 = ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3605_horner__sum__append,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType] :
% 12.24/7.93 ( ( comm_semiring_1 @ A )
% 12.24/7.93 => ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
% 12.24/7.93 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
% 12.24/7.93 = ( 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 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3606_upto__split3,axiom,
% 12.24/7.93 ! [I: int,J: int,K: int] :
% 12.24/7.93 ( ( ord_less_eq @ int @ I @ J )
% 12.24/7.93 => ( ( ord_less_eq @ int @ J @ K )
% 12.24/7.93 => ( ( upto @ I @ K )
% 12.24/7.93 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3607_list__encode_Opelims,axiom,
% 12.24/7.93 ! [X: list @ nat,Y3: nat] :
% 12.24/7.93 ( ( ( nat_list_encode @ X )
% 12.24/7.93 = Y3 )
% 12.24/7.93 => ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
% 12.24/7.93 => ( ( ( X
% 12.24/7.93 = ( nil @ nat ) )
% 12.24/7.93 => ( ( Y3
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
% 12.24/7.93 => ~ ! [X3: nat,Xs2: list @ nat] :
% 12.24/7.93 ( ( X
% 12.24/7.93 = ( cons @ nat @ X3 @ Xs2 ) )
% 12.24/7.93 => ( ( Y3
% 12.24/7.93 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 12.24/7.93 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3608_rp__inv__image__rp,axiom,
% 12.24/7.93 ! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 12.24/7.93 ( ( fun_reduction_pair @ A @ P )
% 12.24/7.93 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3609_take__Suc__conv__app__nth,axiom,
% 12.24/7.93 ! [A: $tType,I: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( take @ A @ ( suc @ I ) @ Xs )
% 12.24/7.93 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3610_min__strict__def,axiom,
% 12.24/7.93 ( fun_min_strict
% 12.24/7.93 = ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3611_upt__rec__numeral,axiom,
% 12.24/7.93 ! [M: num,N: num] :
% 12.24/7.93 ( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.93 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.93 = ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.93 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.93 = ( nil @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3612_take__Suc__Cons,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
% 12.24/7.93 = ( cons @ A @ X @ ( take @ A @ N @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3613_take__eq__Nil2,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ( nil @ A )
% 12.24/7.93 = ( take @ A @ N @ Xs ) )
% 12.24/7.93 = ( ( N
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 | ( Xs
% 12.24/7.93 = ( nil @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3614_take__eq__Nil,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ( take @ A @ N @ Xs )
% 12.24/7.93 = ( nil @ A ) )
% 12.24/7.93 = ( ( N
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 | ( Xs
% 12.24/7.93 = ( nil @ A ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3615_take0,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( take @ A @ ( zero_zero @ nat ) )
% 12.24/7.93 = ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3616_take__all,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 12.24/7.93 => ( ( take @ A @ N @ Xs )
% 12.24/7.93 = Xs ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3617_take__all__iff,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ( take @ A @ N @ Xs )
% 12.24/7.93 = Xs )
% 12.24/7.93 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3618_take__upt,axiom,
% 12.24/7.93 ! [I: nat,M: nat,N: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N )
% 12.24/7.93 => ( ( take @ nat @ M @ ( upt @ I @ N ) )
% 12.24/7.93 = ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3619_nth__take,axiom,
% 12.24/7.93 ! [A: $tType,I: nat,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less @ nat @ I @ N )
% 12.24/7.93 => ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
% 12.24/7.93 = ( nth @ A @ Xs @ I ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3620_upt__conv__Nil,axiom,
% 12.24/7.93 ! [J: nat,I: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ J @ I )
% 12.24/7.93 => ( ( upt @ I @ J )
% 12.24/7.93 = ( nil @ nat ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3621_upt__eq__Nil__conv,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ( upt @ I @ J )
% 12.24/7.93 = ( nil @ nat ) )
% 12.24/7.93 = ( ( J
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 | ( ord_less_eq @ nat @ J @ I ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3622_nth__upt,axiom,
% 12.24/7.93 ! [I: nat,K: nat,J: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
% 12.24/7.93 => ( ( nth @ nat @ ( upt @ I @ J ) @ K )
% 12.24/7.93 = ( plus_plus @ nat @ I @ K ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3623_take__Cons__numeral,axiom,
% 12.24/7.93 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 12.24/7.93 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3624_upt__add__eq__append,axiom,
% 12.24/7.93 ! [I: nat,J: nat,K: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.93 => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
% 12.24/7.93 = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3625_take__0,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A] :
% 12.24/7.93 ( ( take @ A @ ( zero_zero @ nat ) @ Xs )
% 12.24/7.93 = ( nil @ A ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3626_atLeastAtMost__upt,axiom,
% 12.24/7.93 ( ( set_or1337092689740270186AtMost @ nat )
% 12.24/7.93 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3627_upt__Suc,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.93 => ( ( upt @ I @ ( suc @ J ) )
% 12.24/7.93 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
% 12.24/7.93 & ( ~ ( ord_less_eq @ nat @ I @ J )
% 12.24/7.93 => ( ( upt @ I @ ( suc @ J ) )
% 12.24/7.93 = ( nil @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3628_upt__Suc__append,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.93 => ( ( upt @ I @ ( suc @ J ) )
% 12.24/7.93 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3629_upt__conv__Cons__Cons,axiom,
% 12.24/7.93 ! [M: nat,N: nat,Ns: list @ nat,Q4: nat] :
% 12.24/7.93 ( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
% 12.24/7.93 = ( upt @ M @ Q4 ) )
% 12.24/7.93 = ( ( cons @ nat @ N @ Ns )
% 12.24/7.93 = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3630_upt__0,axiom,
% 12.24/7.93 ! [I: nat] :
% 12.24/7.93 ( ( upt @ I @ ( zero_zero @ nat ) )
% 12.24/7.93 = ( nil @ nat ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3631_greaterThanAtMost__upt,axiom,
% 12.24/7.93 ( ( set_or3652927894154168847AtMost @ nat )
% 12.24/7.93 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3632_greaterThanLessThan__upt,axiom,
% 12.24/7.93 ( ( set_or5935395276787703475ssThan @ nat )
% 12.24/7.93 = ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M2 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3633_set__take__subset__set__take,axiom,
% 12.24/7.93 ! [A: $tType,M: nat,N: nat,Xs: list @ A] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.93 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3634_upt__conv__Cons,axiom,
% 12.24/7.93 ! [I: nat,J: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I @ J )
% 12.24/7.93 => ( ( upt @ I @ J )
% 12.24/7.93 = ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3635_nth__take__lemma,axiom,
% 12.24/7.93 ! [A: $tType,K: nat,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.93 ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 12.24/7.93 => ( ! [I4: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I4 @ K )
% 12.24/7.93 => ( ( nth @ A @ Xs @ I4 )
% 12.24/7.93 = ( nth @ A @ Ys2 @ I4 ) ) )
% 12.24/7.93 => ( ( take @ A @ K @ Xs )
% 12.24/7.93 = ( take @ A @ K @ Ys2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3636_upt__eq__Cons__conv,axiom,
% 12.24/7.93 ! [I: nat,J: nat,X: nat,Xs: list @ nat] :
% 12.24/7.93 ( ( ( upt @ I @ J )
% 12.24/7.93 = ( cons @ nat @ X @ Xs ) )
% 12.24/7.93 = ( ( ord_less @ nat @ I @ J )
% 12.24/7.93 & ( I = X )
% 12.24/7.93 & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
% 12.24/7.93 = Xs ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3637_upt__rec,axiom,
% 12.24/7.93 ( upt
% 12.24/7.93 = ( ^ [I5: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I5 @ J3 ) @ ( cons @ nat @ I5 @ ( upt @ ( suc @ I5 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3638_take__Cons_H,axiom,
% 12.24/7.93 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.93 ( ( ( N
% 12.24/7.93 = ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 12.24/7.93 = ( nil @ A ) ) )
% 12.24/7.93 & ( ( N
% 12.24/7.93 != ( zero_zero @ nat ) )
% 12.24/7.93 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 12.24/7.93 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3639_horner__sum__bit__eq__take__bit,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.93 => ! [A3: A,N: nat] :
% 12.24/7.93 ( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 12.24/7.93 = ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3640_lexord__take__index__conv,axiom,
% 12.24/7.93 ! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
% 12.24/7.93 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
% 12.24/7.93 = ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) )
% 12.24/7.93 & ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y3 )
% 12.24/7.93 = X ) )
% 12.24/7.93 | ? [I5: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
% 12.24/7.93 & ( ( take @ A @ I5 @ X )
% 12.24/7.93 = ( take @ A @ I5 @ Y3 ) )
% 12.24/7.93 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I5 ) @ ( nth @ A @ Y3 @ I5 ) ) @ R ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3641_lex__take__index,axiom,
% 12.24/7.93 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 12.24/7.93 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 12.24/7.93 => ~ ! [I4: nat] :
% 12.24/7.93 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 12.24/7.93 => ( ( ( take @ A @ I4 @ Xs )
% 12.24/7.93 = ( take @ A @ I4 @ Ys2 ) )
% 12.24/7.93 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys2 @ I4 ) ) @ R ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3642_min__weak__def,axiom,
% 12.24/7.93 ( fun_min_weak
% 12.24/7.93 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3643_sup_Oidem,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( sup_sup @ A @ A3 @ A3 )
% 12.24/7.93 = A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3644_sup__idem,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( sup_sup @ A @ X @ X )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3645_sup_Oleft__idem,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3646_sup__left__idem,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.93 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3647_sup_Oright__idem,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.93 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3648_sup__apply,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ B )
% 12.24/7.93 => ( ( sup_sup @ ( A > B ) )
% 12.24/7.93 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3649_le__sup__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.93 = ( ( ord_less_eq @ A @ X @ Z3 )
% 12.24/7.93 & ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3650_sup_Obounded__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( semilattice_sup @ A )
% 12.24/7.93 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.93 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.93 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.93 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3651_sup__bot__left,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3652_sup__bot__right,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [X: A] :
% 12.24/7.93 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.93 = X ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3653_bot__eq__sup__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( ( bot_bot @ A )
% 12.24/7.93 = ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.93 = ( ( X
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 & ( Y3
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3654_sup__eq__bot__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 = ( ( X
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 & ( Y3
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3655_sup__bot_Oeq__neutr__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( sup_sup @ A @ A3 @ B2 )
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 = ( ( A3
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 & ( B2
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3656_sup__bot_Oleft__neutral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 12.24/7.93 = A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3657_sup__bot_Oneutr__eq__iff,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [A3: A,B2: A] :
% 12.24/7.93 ( ( ( bot_bot @ A )
% 12.24/7.93 = ( sup_sup @ A @ A3 @ B2 ) )
% 12.24/7.93 = ( ( A3
% 12.24/7.93 = ( bot_bot @ A ) )
% 12.24/7.93 & ( B2
% 12.24/7.93 = ( bot_bot @ A ) ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3658_sup__bot_Oright__neutral,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.93 => ! [A3: A] :
% 12.24/7.93 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 12.24/7.93 = A3 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3659_Compl__Diff__eq,axiom,
% 12.24/7.93 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.93 ( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.93 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ B7 ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3660_nth__map,axiom,
% 12.24/7.93 ! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: A > B] :
% 12.24/7.93 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.93 => ( ( nth @ B @ ( map @ A @ B @ F @ Xs ) @ N )
% 12.24/7.93 = ( F @ ( nth @ A @ Xs @ N ) ) ) ) ).
% 12.24/7.93
% 12.24/7.93 thf(fact_3661_inf__sup__aci_I8_J,axiom,
% 12.24/7.93 ! [A: $tType] :
% 12.24/7.93 ( ( lattice @ A )
% 12.24/7.93 => ! [X: A,Y3: A] :
% 12.24/7.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.93 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3662_inf__sup__aci_I7_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3663_inf__sup__aci_I6_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.94 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3664_inf__sup__aci_I5_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ( ( sup_sup @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3665_sup_Oassoc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.94 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3666_sup__assoc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.94 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3667_sup_Ocommute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( sup_sup @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3668_sup__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( sup_sup @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3669_sup_Oleft__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.94 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
% 12.24/7.94 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3670_sup__left__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3671_sup__fun__def,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ B )
% 12.24/7.94 => ( ( sup_sup @ ( A > B ) )
% 12.24/7.94 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3672_boolean__algebra__cancel_Osup1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A6: A,K: A,A3: A,B2: A] :
% 12.24/7.94 ( ( A6
% 12.24/7.94 = ( sup_sup @ A @ K @ A3 ) )
% 12.24/7.94 => ( ( sup_sup @ A @ A6 @ B2 )
% 12.24/7.94 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3673_boolean__algebra__cancel_Osup2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B7: A,K: A,B2: A,A3: A] :
% 12.24/7.94 ( ( B7
% 12.24/7.94 = ( sup_sup @ A @ K @ B2 ) )
% 12.24/7.94 => ( ( sup_sup @ A @ A3 @ B7 )
% 12.24/7.94 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3674_less__supI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ X @ A3 )
% 12.24/7.94 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3675_less__supI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ X @ B2 )
% 12.24/7.94 => ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3676_sup_Oabsorb3,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( sup_sup @ A @ A3 @ B2 )
% 12.24/7.94 = A3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3677_sup_Oabsorb4,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( sup_sup @ A @ A3 @ B2 )
% 12.24/7.94 = B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3678_sup_Ostrict__boundedE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.94 => ~ ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.94 => ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3679_sup_Ostrict__order__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( ord_less @ A )
% 12.24/7.94 = ( ^ [B3: A,A5: A] :
% 12.24/7.94 ( ( A5
% 12.24/7.94 = ( sup_sup @ A @ A5 @ B3 ) )
% 12.24/7.94 & ( A5 != B3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3680_sup_Ostrict__coboundedI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ C2 @ A3 )
% 12.24/7.94 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3681_sup_Ostrict__coboundedI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [C2: A,B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ C2 @ B2 )
% 12.24/7.94 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3682_sup__max,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ( semilattice_sup @ A )
% 12.24/7.94 & ( linorder @ A ) )
% 12.24/7.94 => ( ( sup_sup @ A )
% 12.24/7.94 = ( ord_max @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3683_sup_OcoboundedI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [C2: A,B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ C2 @ B2 )
% 12.24/7.94 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3684_sup_OcoboundedI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3685_sup_Oabsorb__iff2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ A5 @ B3 )
% 12.24/7.94 = B3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3686_sup_Oabsorb__iff1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [B3: A,A5: A] :
% 12.24/7.94 ( ( sup_sup @ A @ A5 @ B3 )
% 12.24/7.94 = A5 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3687_sup_Ocobounded2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3688_sup_Ocobounded1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3689_sup_Oorder__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [B3: A,A5: A] :
% 12.24/7.94 ( A5
% 12.24/7.94 = ( sup_sup @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3690_sup_OboundedI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3691_sup_OboundedE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
% 12.24/7.94 => ~ ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3692_sup__absorb2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 => ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.94 = Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3693_sup__absorb1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [Y3: A,X: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.94 => ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.94 = X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3694_sup_Oabsorb2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( sup_sup @ A @ A3 @ B2 )
% 12.24/7.94 = B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3695_sup_Oabsorb1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( sup_sup @ A @ A3 @ B2 )
% 12.24/7.94 = A3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3696_sup__unique,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [F: A > A > A,X: A,Y3: A] :
% 12.24/7.94 ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y5 ) )
% 12.24/7.94 => ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X3 @ Y5 ) )
% 12.24/7.94 => ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ Y5 @ X3 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ Z4 @ X3 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
% 12.24/7.94 => ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.94 = ( F @ X @ Y3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3697_sup_OorderI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( A3
% 12.24/7.94 = ( sup_sup @ A @ A3 @ B2 ) )
% 12.24/7.94 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3698_sup_OorderE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ( A3
% 12.24/7.94 = ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3699_le__iff__sup,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X2 @ Y6 )
% 12.24/7.94 = Y6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3700_sup__least,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [Y3: A,X: A,Z3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.94 => ( ( ord_less_eq @ A @ Z3 @ X )
% 12.24/7.94 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3701_sup__mono,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3702_sup_Omono,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [C2: A,A3: A,D2: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ D2 @ B2 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3703_le__supI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ B2 )
% 12.24/7.94 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3704_le__supI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ A3 )
% 12.24/7.94 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3705_sup__ge2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3706_sup__ge1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3707_le__supI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,X: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ X )
% 12.24/7.94 => ( ( ord_less_eq @ A @ B2 @ X )
% 12.24/7.94 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3708_le__supE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_sup @ A )
% 12.24/7.94 => ! [A3: A,B2: A,X: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
% 12.24/7.94 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 12.24/7.94 => ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3709_inf__sup__ord_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3710_inf__sup__ord_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3711_boolean__algebra_Odisj__zero__right,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3712_ivl__disj__un__two__touch_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3713_ivl__disj__un__two_I6_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3714_card__Un__le,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3715_ivl__disj__un__two_I8_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3716_ivl__disj__un__two__touch_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3717_ivl__disj__un__two_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3718_nth__map__upt,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,N: nat,M: nat,F: nat > A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
% 12.24/7.94 => ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M @ N ) ) @ I )
% 12.24/7.94 = ( F @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3719_lexord__sufI,axiom,
% 12.24/7.94 ! [A: $tType,U: list @ A,W2: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z3: list @ A] :
% 12.24/7.94 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R ) )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
% 12.24/7.94 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W2 @ Z3 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3720_ivl__disj__un__singleton_I5_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3721_ivl__disj__un__two_I5_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3722_ivl__disj__un__singleton_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.94 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3723_map__upt__eqI,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,N: nat,M: nat,F: nat > A] :
% 12.24/7.94 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.94 = ( minus_minus @ nat @ N @ M ) )
% 12.24/7.94 => ( ! [I4: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( nth @ A @ Xs @ I4 )
% 12.24/7.94 = ( F @ ( plus_plus @ nat @ M @ I4 ) ) ) )
% 12.24/7.94 => ( ( map @ nat @ A @ F @ ( upt @ M @ N ) )
% 12.24/7.94 = Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3724_max__weak__def,axiom,
% 12.24/7.94 ( fun_max_weak
% 12.24/7.94 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( 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 ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3725_sorted__list__of__set__atMost__Suc,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
% 12.24/7.94 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3726_sorted__list__of__set__lessThan__Suc,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
% 12.24/7.94 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3727_id__take__nth__drop,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( Xs
% 12.24/7.94 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3728_atMost__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ord @ A )
% 12.24/7.94 => ! [I: A,K: A] :
% 12.24/7.94 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 12.24/7.94 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3729_lessThan__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ord @ A )
% 12.24/7.94 => ! [I: A,K: A] :
% 12.24/7.94 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 12.24/7.94 = ( ord_less @ A @ I @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3730_drop0,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( drop @ A @ ( zero_zero @ nat ) )
% 12.24/7.94 = ( ^ [X2: list @ A] : X2 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3731_atMost__subset__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( preorder @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y3 ) )
% 12.24/7.94 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3732_lessThan__subset__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y3 ) )
% 12.24/7.94 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3733_drop__Suc__Cons,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.94 ( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
% 12.24/7.94 = ( drop @ A @ N @ Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3734_lessThan__0,axiom,
% 12.24/7.94 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
% 12.24/7.94 = ( bot_bot @ ( set @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3735_card__atMost,axiom,
% 12.24/7.94 ! [U: nat] :
% 12.24/7.94 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
% 12.24/7.94 = ( suc @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3736_Icc__subset__Iic__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( preorder @ A )
% 12.24/7.94 => ! [L: A,H2: A,H: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H ) )
% 12.24/7.94 = ( ~ ( ord_less_eq @ A @ L @ H2 )
% 12.24/7.94 | ( ord_less_eq @ A @ H2 @ H ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3737_drop__eq__Nil2,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ( nil @ A )
% 12.24/7.94 = ( drop @ A @ N @ Xs ) )
% 12.24/7.94 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3738_drop__eq__Nil,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ( drop @ A @ N @ Xs )
% 12.24/7.94 = ( nil @ A ) )
% 12.24/7.94 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3739_drop__all,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 12.24/7.94 => ( ( drop @ A @ N @ Xs )
% 12.24/7.94 = ( nil @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3740_atMost__0,axiom,
% 12.24/7.94 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
% 12.24/7.94 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3741_drop__Cons__numeral,axiom,
% 12.24/7.94 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 12.24/7.94 ( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 12.24/7.94 = ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3742_nth__drop,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 12.24/7.94 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3743_sup__nat__def,axiom,
% 12.24/7.94 ( ( sup_sup @ nat )
% 12.24/7.94 = ( ord_max @ nat ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3744_Iic__subset__Iio__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 12.24/7.94 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3745_drop__0,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
% 12.24/7.94 = Xs ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3746_lessThan__Suc__atMost,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 12.24/7.94 = ( set_ord_atMost @ nat @ K ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3747_ivl__disj__un__one_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 12.24/7.94 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3748_ivl__disj__un__one_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 12.24/7.94 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3749_sup__int__def,axiom,
% 12.24/7.94 ( ( sup_sup @ int )
% 12.24/7.94 = ( ord_max @ int ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3750_atMost__atLeast0,axiom,
% 12.24/7.94 ( ( set_ord_atMost @ nat )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3751_map__Suc__upt,axiom,
% 12.24/7.94 ! [M: nat,N: nat] :
% 12.24/7.94 ( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
% 12.24/7.94 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3752_lessThan__Suc,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 12.24/7.94 = ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3753_lessThan__empty__iff,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( ( set_ord_lessThan @ nat @ N )
% 12.24/7.94 = ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.94 = ( N
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3754_atMost__Suc,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( ( set_ord_atMost @ nat @ ( suc @ K ) )
% 12.24/7.94 = ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3755_lessThan__strict__subset__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [M: A,N: A] :
% 12.24/7.94 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 12.24/7.94 = ( ord_less @ A @ M @ N ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3756_set__drop__subset__set__drop,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,M: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.94 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3757_atLeast__upt,axiom,
% 12.24/7.94 ( ( set_ord_lessThan @ nat )
% 12.24/7.94 = ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3758_ivl__disj__un__one_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 12.24/7.94 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3759_drop__Cons_H,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 12.24/7.94 ( ( ( N
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 12.24/7.94 = ( cons @ A @ X @ Xs ) ) )
% 12.24/7.94 & ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 12.24/7.94 = ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3760_append__eq__append__conv__if,axiom,
% 12.24/7.94 ! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
% 12.24/7.94 ( ( ( append @ A @ Xs_1 @ Xs_2 )
% 12.24/7.94 = ( append @ A @ Ys_1 @ Ys_2 ) )
% 12.24/7.94 = ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
% 12.24/7.94 => ( ( Xs_1
% 12.24/7.94 = ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
% 12.24/7.94 & ( Xs_2
% 12.24/7.94 = ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
% 12.24/7.94 => ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
% 12.24/7.94 = Ys_1 )
% 12.24/7.94 & ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
% 12.24/7.94 = Ys_2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3761_atMost__upto,axiom,
% 12.24/7.94 ( ( set_ord_atMost @ nat )
% 12.24/7.94 = ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3762_max__strict__def,axiom,
% 12.24/7.94 ( fun_max_strict
% 12.24/7.94 = ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3763_Cons__nth__drop__Suc,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
% 12.24/7.94 = ( drop @ A @ I @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3764_atLeast1__atMost__eq__remove0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.94 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3765_take__hd__drop,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
% 12.24/7.94 = ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3766_upd__conv__take__nth__drop,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,A3: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( list_update @ A @ Xs @ I @ A3 )
% 12.24/7.94 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3767_effect__makeE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 12.24/7.94 ( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ R @ N2 )
% 12.24/7.94 => ~ ( ( R
% 12.24/7.94 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
% 12.24/7.94 => ( ( H
% 12.24/7.94 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
% 12.24/7.94 => ( ( ( array_get @ A @ H @ R )
% 12.24/7.94 = ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 12.24/7.94 => ( ( array_present @ A @ H @ R )
% 12.24/7.94 => ( ~ ( array_present @ A @ H2 @ R )
% 12.24/7.94 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 12.24/7.94 != N2 ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3768_n__lists__Nil,axiom,
% 12.24/7.94 ! [A: $tType,N: nat] :
% 12.24/7.94 ( ( ( N
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( n_lists @ A @ N @ ( nil @ A ) )
% 12.24/7.94 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
% 12.24/7.94 & ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( n_lists @ A @ N @ ( nil @ A ) )
% 12.24/7.94 = ( nil @ ( list @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3769_hd__upt,axiom,
% 12.24/7.94 ! [I: nat,J: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ J )
% 12.24/7.94 => ( ( hd @ nat @ ( upt @ I @ J ) )
% 12.24/7.94 = I ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3770_list__update__beyond,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,I: nat,X: A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 12.24/7.94 => ( ( list_update @ A @ Xs @ I @ X )
% 12.24/7.94 = Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3771_take__update__cancel,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,M: nat,Xs: list @ A,Y3: A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.94 => ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y3 ) )
% 12.24/7.94 = ( take @ A @ N @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3772_drop__update__cancel,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,M: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ N @ M )
% 12.24/7.94 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
% 12.24/7.94 = ( drop @ A @ M @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3773_get__update__eq,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [A3: array @ A,I: nat,V: A,H2: heap_ext @ product_unit] :
% 12.24/7.94 ( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H2 ) @ A3 )
% 12.24/7.94 = ( list_update @ A @ ( array_get @ A @ H2 @ A3 ) @ I @ V ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3774_nth__list__update__eq,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3775_hd__take,axiom,
% 12.24/7.94 ! [A: $tType,J: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
% 12.24/7.94 => ( ( hd @ A @ ( take @ A @ J @ Xs ) )
% 12.24/7.94 = ( hd @ A @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3776_set__swap,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
% 12.24/7.94 = ( set2 @ A @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3777_get__alloc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
% 12.24/7.94 ( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
% 12.24/7.94 = Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3778_set,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [Ls: list @ A,H2: heap_ext @ product_unit,New_ls: list @ A] :
% 12.24/7.94 ( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) )
% 12.24/7.94 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3779_not__present__alloc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [H2: heap_ext @ product_unit,Xs: list @ A] :
% 12.24/7.94 ~ ( array_present @ A @ H2 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3780_present__alloc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3781_list__update__code_I3_J,axiom,
% 12.24/7.94 ! [A: $tType,X: A,Xs: list @ A,I: nat,Y3: A] :
% 12.24/7.94 ( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y3 )
% 12.24/7.94 = ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3782_list__update__code_I2_J,axiom,
% 12.24/7.94 ! [A: $tType,X: A,Xs: list @ A,Y3: A] :
% 12.24/7.94 ( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y3 )
% 12.24/7.94 = ( cons @ A @ Y3 @ Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3783_set__update__memI,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3784_list__update__append1,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,Ys2: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ I @ X )
% 12.24/7.94 = ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3785_length__alloc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
% 12.24/7.94 ( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
% 12.24/7.94 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3786_list__update__same__conv,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ( list_update @ A @ Xs @ I @ X )
% 12.24/7.94 = Xs )
% 12.24/7.94 = ( ( nth @ A @ Xs @ I )
% 12.24/7.94 = X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3787_nth__list__update,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ( I = J )
% 12.24/7.94 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
% 12.24/7.94 = X ) )
% 12.24/7.94 & ( ( I != J )
% 12.24/7.94 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
% 12.24/7.94 = ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3788_hd__conv__nth,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( Xs
% 12.24/7.94 != ( nil @ A ) )
% 12.24/7.94 => ( ( hd @ A @ Xs )
% 12.24/7.94 = ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3789_drop__update__swap,axiom,
% 12.24/7.94 ! [A: $tType,M: nat,N: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
% 12.24/7.94 = ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3790_present__alloc__noteq,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( ( heap @ B )
% 12.24/7.94 & ( heap @ A ) )
% 12.24/7.94 => ! [H2: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
% 12.24/7.94 ( ( array_present @ A @ H2 @ A3 )
% 12.24/7.94 => ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3791_update__def,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ( ( array_update @ A )
% 12.24/7.94 = ( ^ [A5: array @ A,I5: nat,X2: A,H4: heap_ext @ product_unit] : ( array_set @ A @ A5 @ ( list_update @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 @ X2 ) @ H4 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3792_list__update__append,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A,X: A] :
% 12.24/7.94 ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
% 12.24/7.94 = ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys2 ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
% 12.24/7.94 = ( append @ A @ Xs @ ( list_update @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3793_n__lists_Osimps_I1_J,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
% 12.24/7.94 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3794_effect__makeI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [A3: array @ A,H: heap_ext @ product_unit,F: nat > A,N: nat,H2: heap_ext @ product_unit] :
% 12.24/7.94 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 12.24/7.94 = ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) )
% 12.24/7.94 => ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3795_hd__drop__conv__nth,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
% 12.24/7.94 = ( nth @ A @ Xs @ N ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3796_effect__of__listE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [Xs: list @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 12.24/7.94 ( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ R @ N2 )
% 12.24/7.94 => ~ ( ( R
% 12.24/7.94 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
% 12.24/7.94 => ( ( H
% 12.24/7.94 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
% 12.24/7.94 => ( ( ( array_get @ A @ H @ R )
% 12.24/7.94 = Xs )
% 12.24/7.94 => ( ( array_present @ A @ H @ R )
% 12.24/7.94 => ( ~ ( array_present @ A @ H2 @ R )
% 12.24/7.94 => ( N2
% 12.24/7.94 != ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3797_effect__of__listI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [A3: array @ A,H: heap_ext @ product_unit,Xs: list @ A,H2: heap_ext @ product_unit] :
% 12.24/7.94 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 12.24/7.94 = ( array_alloc @ A @ Xs @ H2 ) )
% 12.24/7.94 => ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3798_set__update__distinct,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,N: nat,X: A] :
% 12.24/7.94 ( ( distinct @ A @ Xs )
% 12.24/7.94 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
% 12.24/7.94 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3799_effect__newE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [N: nat,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 12.24/7.94 ( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ R @ N2 )
% 12.24/7.94 => ~ ( ( R
% 12.24/7.94 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
% 12.24/7.94 => ( ( H
% 12.24/7.94 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
% 12.24/7.94 => ( ( ( array_get @ A @ H @ R )
% 12.24/7.94 = ( replicate @ A @ N @ X ) )
% 12.24/7.94 => ( ( array_present @ A @ H @ R )
% 12.24/7.94 => ( ~ ( array_present @ A @ H2 @ R )
% 12.24/7.94 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 12.24/7.94 != N2 ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3800_replicate__eq__replicate,axiom,
% 12.24/7.94 ! [A: $tType,M: nat,X: A,N: nat,Y3: A] :
% 12.24/7.94 ( ( ( replicate @ A @ M @ X )
% 12.24/7.94 = ( replicate @ A @ N @ Y3 ) )
% 12.24/7.94 = ( ( M = N )
% 12.24/7.94 & ( ( M
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( X = Y3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3801_replicate__empty,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( ( replicate @ A @ N @ X )
% 12.24/7.94 = ( nil @ A ) )
% 12.24/7.94 = ( N
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3802_empty__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( ( nil @ A )
% 12.24/7.94 = ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( N
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3803_Ball__set__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,A3: A,P: A > $o] :
% 12.24/7.94 ( ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
% 12.24/7.94 => ( P @ X2 ) ) )
% 12.24/7.94 = ( ( P @ A3 )
% 12.24/7.94 | ( N
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3804_Bex__set__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,A3: A,P: A > $o] :
% 12.24/7.94 ( ( ? [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
% 12.24/7.94 & ( P @ X2 ) ) )
% 12.24/7.94 = ( ( P @ A3 )
% 12.24/7.94 & ( N
% 12.24/7.94 != ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3805_in__set__replicate,axiom,
% 12.24/7.94 ! [A: $tType,X: A,N: nat,Y3: A] :
% 12.24/7.94 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y3 ) ) )
% 12.24/7.94 = ( ( X = Y3 )
% 12.24/7.94 & ( N
% 12.24/7.94 != ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3806_nth__replicate,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,N: nat,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ N )
% 12.24/7.94 => ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3807_hd__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( hd @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3808_set__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3809_distinct__swap,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A,J: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
% 12.24/7.94 = ( distinct @ A @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3810_replicate__Suc,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( replicate @ A @ ( suc @ N ) @ X )
% 12.24/7.94 = ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3811_replicate__0,axiom,
% 12.24/7.94 ! [A: $tType,X: A] :
% 12.24/7.94 ( ( replicate @ A @ ( zero_zero @ nat ) @ X )
% 12.24/7.94 = ( nil @ A ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3812_distinct__conv__nth,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( distinct @ A )
% 12.24/7.94 = ( ^ [Xs3: list @ A] :
% 12.24/7.94 ! [I5: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.94 => ! [J3: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.94 => ( ( I5 != J3 )
% 12.24/7.94 => ( ( nth @ A @ Xs3 @ I5 )
% 12.24/7.94 != ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3813_nth__eq__iff__index__eq,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,I: nat,J: nat] :
% 12.24/7.94 ( ( distinct @ A @ Xs )
% 12.24/7.94 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ( nth @ A @ Xs @ I )
% 12.24/7.94 = ( nth @ A @ Xs @ J ) )
% 12.24/7.94 = ( I = J ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3814_effect__newI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( heap @ A )
% 12.24/7.94 => ! [A3: array @ A,H: heap_ext @ product_unit,N: nat,X: A,H2: heap_ext @ product_unit] :
% 12.24/7.94 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
% 12.24/7.94 = ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) )
% 12.24/7.94 => ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3815_distinct__Ex1,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( distinct @ A @ Xs )
% 12.24/7.94 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ? [X3: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 & ( ( nth @ A @ Xs @ X3 )
% 12.24/7.94 = X )
% 12.24/7.94 & ! [Y4: nat] :
% 12.24/7.94 ( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 & ( ( nth @ A @ Xs @ Y4 )
% 12.24/7.94 = X ) )
% 12.24/7.94 => ( Y4 = X3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3816_set__replicate__Suc,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
% 12.24/7.94 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3817_set__replicate__conv__if,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( ( N
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.94 & ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3818_Cons__replicate__eq,axiom,
% 12.24/7.94 ! [A: $tType,X: A,Xs: list @ A,N: nat,Y3: A] :
% 12.24/7.94 ( ( ( cons @ A @ X @ Xs )
% 12.24/7.94 = ( replicate @ A @ N @ Y3 ) )
% 12.24/7.94 = ( ( X = Y3 )
% 12.24/7.94 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.94 & ( Xs
% 12.24/7.94 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3819_atLeast1__lessThan__eq__remove0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 12.24/7.94 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3820_listrel1__iff__update,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 12.24/7.94 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 12.24/7.94 = ( ? [Y6: A,N3: nat] :
% 12.24/7.94 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N3 ) @ Y6 ) @ R )
% 12.24/7.94 & ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 & ( Ys2
% 12.24/7.94 = ( list_update @ A @ Xs @ N3 @ Y6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3821_Gcd__remove0__nat,axiom,
% 12.24/7.94 ! [M7: set @ nat] :
% 12.24/7.94 ( ( finite_finite2 @ nat @ M7 )
% 12.24/7.94 => ( ( gcd_Gcd @ nat @ M7 )
% 12.24/7.94 = ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3822_remdups__adj__singleton__iff,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
% 12.24/7.94 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.94 = ( ( Xs
% 12.24/7.94 != ( nil @ A ) )
% 12.24/7.94 & ( Xs
% 12.24/7.94 = ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3823_atLeastLessThan__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ord @ A )
% 12.24/7.94 => ! [I: A,L: A,U: A] :
% 12.24/7.94 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ L @ I )
% 12.24/7.94 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3824_infinite__Icc__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
% 12.24/7.94 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3825_atLeastLessThan__empty,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3826_atLeastLessThan__empty__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( preorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3827_atLeastLessThan__empty__iff2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( preorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ( bot_bot @ ( set @ A ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 12.24/7.94 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3828_ivl__subset,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [I: A,J: A,M: A,N: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ J @ I )
% 12.24/7.94 | ( ( ord_less_eq @ A @ M @ I )
% 12.24/7.94 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3829_infinite__Ico__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
% 12.24/7.94 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3830_card_Oinfinite,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ A6 )
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3831_ivl__diff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [I: A,N: A,M: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ I @ N )
% 12.24/7.94 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3832_infinite__Ioc__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
% 12.24/7.94 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3833_infinite__Ioo__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
% 12.24/7.94 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3834_card__0__eq,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ( finite_card @ A @ A6 )
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 = ( A6
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3835_card__insert__disjoint,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.94 = ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3836_atLeastLessThan__singleton,axiom,
% 12.24/7.94 ! [M: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
% 12.24/7.94 = ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3837_finite__psubset__induct,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ! [A9: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A9 )
% 12.24/7.94 => ( ! [B10: set @ A] :
% 12.24/7.94 ( ( ord_less @ ( set @ A ) @ B10 @ A9 )
% 12.24/7.94 => ( P @ B10 ) )
% 12.24/7.94 => ( P @ A9 ) ) )
% 12.24/7.94 => ( P @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3838_atLeastLessThan__eq__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less @ A @ C2 @ D2 )
% 12.24/7.94 => ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( ( A3 = C2 )
% 12.24/7.94 & ( B2 = D2 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3839_atLeastLessThan__inj_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 => ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less @ A @ C2 @ D2 )
% 12.24/7.94 => ( A3 = C2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3840_atLeastLessThan__inj_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 => ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less @ A @ C2 @ D2 )
% 12.24/7.94 => ( B2 = D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3841_infinite__Ico,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3842_finite__nat__set__iff__bounded,axiom,
% 12.24/7.94 ( ( finite_finite2 @ nat )
% 12.24/7.94 = ( ^ [N7: set @ nat] :
% 12.24/7.94 ? [M2: nat] :
% 12.24/7.94 ! [X2: nat] :
% 12.24/7.94 ( ( member @ nat @ X2 @ N7 )
% 12.24/7.94 => ( ord_less @ nat @ X2 @ M2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3843_bounded__nat__set__is__finite,axiom,
% 12.24/7.94 ! [N6: set @ nat,N: nat] :
% 12.24/7.94 ( ! [X3: nat] :
% 12.24/7.94 ( ( member @ nat @ X3 @ N6 )
% 12.24/7.94 => ( ord_less @ nat @ X3 @ N ) )
% 12.24/7.94 => ( finite_finite2 @ nat @ N6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3844_finite__nat__set__iff__bounded__le,axiom,
% 12.24/7.94 ( ( finite_finite2 @ nat )
% 12.24/7.94 = ( ^ [N7: set @ nat] :
% 12.24/7.94 ? [M2: nat] :
% 12.24/7.94 ! [X2: nat] :
% 12.24/7.94 ( ( member @ nat @ X2 @ N7 )
% 12.24/7.94 => ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3845_finite__has__minimal2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [A6: set @ A,A3: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.94 => ? [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ A6 )
% 12.24/7.94 & ( ord_less_eq @ A @ X3 @ A3 )
% 12.24/7.94 & ! [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 12.24/7.94 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3846_finite__has__maximal2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [A6: set @ A,A3: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.94 => ? [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ A6 )
% 12.24/7.94 & ( ord_less_eq @ A @ A3 @ X3 )
% 12.24/7.94 & ! [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 12.24/7.94 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3847_subset__eq__atLeast0__lessThan__finite,axiom,
% 12.24/7.94 ! [N6: set @ nat,N: nat] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 => ( finite_finite2 @ nat @ N6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3848_finite__set__decode,axiom,
% 12.24/7.94 ! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3849_atLeastLessThan__subset__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3850_ivl__disj__un__two_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3851_ex__nat__less__eq,axiom,
% 12.24/7.94 ! [N: nat,P: nat > $o] :
% 12.24/7.94 ( ( ? [M2: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ M2 @ N )
% 12.24/7.94 & ( P @ M2 ) ) )
% 12.24/7.94 = ( ? [X2: nat] :
% 12.24/7.94 ( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 & ( P @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3852_all__nat__less__eq,axiom,
% 12.24/7.94 ! [N: nat,P: nat > $o] :
% 12.24/7.94 ( ( ! [M2: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ M2 @ N )
% 12.24/7.94 => ( P @ M2 ) ) )
% 12.24/7.94 = ( ! [X2: nat] :
% 12.24/7.94 ( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 => ( P @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3853_atLeastLessThan__add__Un,axiom,
% 12.24/7.94 ! [I: nat,J: nat,K: nat] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
% 12.24/7.94 = ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3854_atLeastLessThanSuc__atLeastAtMost,axiom,
% 12.24/7.94 ! [L: nat,U: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3855_finite__has__minimal,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( A6
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ? [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ A6 )
% 12.24/7.94 & ! [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 12.24/7.94 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3856_finite__has__maximal,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( A6
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ? [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ A6 )
% 12.24/7.94 & ! [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 12.24/7.94 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3857_infinite__growing,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [X8: set @ A] :
% 12.24/7.94 ( ( X8
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ X8 )
% 12.24/7.94 => ? [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ X8 )
% 12.24/7.94 & ( ord_less @ A @ X3 @ Xa ) ) )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3858_ex__min__if__finite,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [S3: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.94 => ( ( S3
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ? [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ S3 )
% 12.24/7.94 & ~ ? [Xa: A] :
% 12.24/7.94 ( ( member @ A @ Xa @ S3 )
% 12.24/7.94 & ( ord_less @ A @ Xa @ X3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3859_infinite__Icc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3860_lessThan__atLeast0,axiom,
% 12.24/7.94 ( ( set_ord_lessThan @ nat )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3861_card__le__if__inj__on__rel,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType,B7: set @ A,A6: set @ B,R: B > A > $o] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ A6 )
% 12.24/7.94 => ? [B11: A] :
% 12.24/7.94 ( ( member @ A @ B11 @ B7 )
% 12.24/7.94 & ( R @ A8 @ B11 ) ) )
% 12.24/7.94 => ( ! [A13: B,A24: B,B5: A] :
% 12.24/7.94 ( ( member @ B @ A13 @ A6 )
% 12.24/7.94 => ( ( member @ B @ A24 @ A6 )
% 12.24/7.94 => ( ( member @ A @ B5 @ B7 )
% 12.24/7.94 => ( ( R @ A13 @ B5 )
% 12.24/7.94 => ( ( R @ A24 @ B5 )
% 12.24/7.94 => ( A13 = A24 ) ) ) ) ) )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3862_remdups__adj__length,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3863_infinite__Ioc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3864_atLeastLessThan0,axiom,
% 12.24/7.94 ! [M: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
% 12.24/7.94 = ( bot_bot @ ( set @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3865_infinite__Ioo,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3866_atLeastSucLessThan__greaterThanLessThan,axiom,
% 12.24/7.94 ! [L: nat,U: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3867_ivl__disj__un__two_I7_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3868_ivl__disj__un__one_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 12.24/7.94 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3869_finite__ranking__induct,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [S3: set @ B,P: ( set @ B ) > $o,F: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.94 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 12.24/7.94 => ( ! [X3: B,S4: set @ B] :
% 12.24/7.94 ( ( finite_finite2 @ B @ S4 )
% 12.24/7.94 => ( ! [Y4: B] :
% 12.24/7.94 ( ( member @ B @ Y4 @ S4 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 12.24/7.94 => ( ( P @ S4 )
% 12.24/7.94 => ( P @ ( insert @ B @ X3 @ S4 ) ) ) ) )
% 12.24/7.94 => ( P @ S3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3870_finite__linorder__max__induct,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A6: set @ A,P: ( set @ A ) > $o] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ! [B5: A,A9: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A9 )
% 12.24/7.94 => ( ! [X4: A] :
% 12.24/7.94 ( ( member @ A @ X4 @ A9 )
% 12.24/7.94 => ( ord_less @ A @ X4 @ B5 ) )
% 12.24/7.94 => ( ( P @ A9 )
% 12.24/7.94 => ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
% 12.24/7.94 => ( P @ A6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3871_finite__linorder__min__induct,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A6: set @ A,P: ( set @ A ) > $o] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ! [B5: A,A9: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A9 )
% 12.24/7.94 => ( ! [X4: A] :
% 12.24/7.94 ( ( member @ A @ X4 @ A9 )
% 12.24/7.94 => ( ord_less @ A @ B5 @ X4 ) )
% 12.24/7.94 => ( ( P @ A9 )
% 12.24/7.94 => ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
% 12.24/7.94 => ( P @ A6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3872_card__eq__0__iff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( ( finite_card @ A @ A6 )
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 = ( ( A6
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 | ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3873_atLeast0__lessThan__Suc,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3874_card__Suc__eq__finite,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,K: nat] :
% 12.24/7.94 ( ( ( finite_card @ A @ A6 )
% 12.24/7.94 = ( suc @ K ) )
% 12.24/7.94 = ( ? [B3: A,B6: set @ A] :
% 12.24/7.94 ( ( A6
% 12.24/7.94 = ( insert @ A @ B3 @ B6 ) )
% 12.24/7.94 & ~ ( member @ A @ B3 @ B6 )
% 12.24/7.94 & ( ( finite_card @ A @ B6 )
% 12.24/7.94 = K )
% 12.24/7.94 & ( finite_finite2 @ A @ B6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3875_card__insert__if,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.94 = ( finite_card @ A @ A6 ) ) )
% 12.24/7.94 & ( ~ ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.94 = ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3876_card__ge__0__finite,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
% 12.24/7.94 => ( finite_finite2 @ A @ A6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3877_obtain__subset__with__card__n,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,S3: set @ A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
% 12.24/7.94 => ~ ! [T4: set @ A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
% 12.24/7.94 => ( ( ( finite_card @ A @ T4 )
% 12.24/7.94 = N )
% 12.24/7.94 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3878_finite__if__finite__subsets__card__bdd,axiom,
% 12.24/7.94 ! [A: $tType,F4: set @ A,C6: nat] :
% 12.24/7.94 ( ! [G3: set @ A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
% 12.24/7.94 => ( ( finite_finite2 @ A @ G3 )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C6 ) ) )
% 12.24/7.94 => ( ( finite_finite2 @ A @ F4 )
% 12.24/7.94 & ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3879_card__seteq,axiom,
% 12.24/7.94 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B7 ) @ ( finite_card @ A @ A6 ) )
% 12.24/7.94 => ( A6 = B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3880_card__mono,axiom,
% 12.24/7.94 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3881_card__less__sym__Diff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 12.24/7.94 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3882_card__le__sym__Diff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3883_psubset__card__mono,axiom,
% 12.24/7.94 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 => ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3884_subset__eq__atLeast0__atMost__finite,axiom,
% 12.24/7.94 ! [N6: set @ nat,N: nat] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 => ( finite_finite2 @ nat @ N6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3885_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( preorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3886_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3887_ivl__disj__un__two__touch_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3888_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3889_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( dense_linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 12.24/7.94 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3890_ivl__disj__un__two_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less_eq @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3891_card__gt__0__iff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
% 12.24/7.94 = ( ( A6
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 & ( finite_finite2 @ A @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3892_card__le__Suc0__iff__eq,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.94 = ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ A6 )
% 12.24/7.94 => ! [Y6: A] :
% 12.24/7.94 ( ( member @ A @ Y6 @ A6 )
% 12.24/7.94 => ( X2 = Y6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3893_card__le__Suc__iff,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,A6: set @ A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A6 ) )
% 12.24/7.94 = ( ? [A5: A,B6: set @ A] :
% 12.24/7.94 ( ( A6
% 12.24/7.94 = ( insert @ A @ A5 @ B6 ) )
% 12.24/7.94 & ~ ( member @ A @ A5 @ B6 )
% 12.24/7.94 & ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
% 12.24/7.94 & ( finite_finite2 @ A @ B6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3894_ivl__disj__un__two__touch_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3895_finite__induct__select,axiom,
% 12.24/7.94 ! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
% 12.24/7.94 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.94 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ! [T4: set @ A] :
% 12.24/7.94 ( ( ord_less @ ( set @ A ) @ T4 @ S3 )
% 12.24/7.94 => ( ( P @ T4 )
% 12.24/7.94 => ? [X4: A] :
% 12.24/7.94 ( ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) )
% 12.24/7.94 & ( P @ ( insert @ A @ X4 @ T4 ) ) ) ) )
% 12.24/7.94 => ( P @ S3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3896_card__psubset,axiom,
% 12.24/7.94 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 => ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
% 12.24/7.94 => ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3897_diff__card__le__card__Diff,axiom,
% 12.24/7.94 ! [A: $tType,B7: set @ A,A6: set @ A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3898_remdups__adj__adjacent,axiom,
% 12.24/7.94 ! [A: $tType,I: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
% 12.24/7.94 => ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
% 12.24/7.94 != ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3899_remdups__adj__replicate,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,X: A] :
% 12.24/7.94 ( ( ( N
% 12.24/7.94 = ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( nil @ A ) ) )
% 12.24/7.94 & ( ( N
% 12.24/7.94 != ( zero_zero @ nat ) )
% 12.24/7.94 => ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.94 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3900_atLeastLessThanSuc,axiom,
% 12.24/7.94 ! [M: nat,N: nat] :
% 12.24/7.94 ( ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
% 12.24/7.94 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3901_subset__eq__atLeast0__lessThan__card,axiom,
% 12.24/7.94 ! [N6: set @ nat,N: nat] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3902_ivl__disj__un__singleton_I6_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3903_ivl__disj__un__two_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,M: A,U: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ L @ M )
% 12.24/7.94 => ( ( ord_less @ A @ M @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3904_ivl__disj__un__singleton_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [L: A,U: A] :
% 12.24/7.94 ( ( ord_less @ A @ L @ U )
% 12.24/7.94 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3905_card_Oremove,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ A6 )
% 12.24/7.94 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3906_card_Oinsert__remove,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.94 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3907_card__Suc__Diff1,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 12.24/7.94 = ( finite_card @ A @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3908_card__Diff1__less,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3909_card__Diff2__less,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A,Y3: A] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( member @ A @ X @ A6 )
% 12.24/7.94 => ( ( member @ A @ Y3 @ A6 )
% 12.24/7.94 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3910_card__Diff1__less__iff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) )
% 12.24/7.94 = ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 & ( member @ A @ X @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3911_remdups__adj__length__ge1,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( Xs
% 12.24/7.94 != ( nil @ A ) )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3912_atLeastLessThan__nat__numeral,axiom,
% 12.24/7.94 ! [M: nat,K: num] :
% 12.24/7.94 ( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.94 = ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
% 12.24/7.94 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3913_set__encode__insert,axiom,
% 12.24/7.94 ! [A6: set @ nat,N: nat] :
% 12.24/7.94 ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.94 => ( ~ ( member @ nat @ N @ A6 )
% 12.24/7.94 => ( ( nat_set_encode @ ( insert @ nat @ N @ A6 ) )
% 12.24/7.94 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3914_even__set__encode__iff,axiom,
% 12.24/7.94 ! [A6: set @ nat] :
% 12.24/7.94 ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.94 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A6 ) )
% 12.24/7.94 = ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3915_arg__min__if__finite_I2_J,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( order @ B )
% 12.24/7.94 => ! [S3: set @ A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.94 => ( ( S3
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ~ ? [X4: A] :
% 12.24/7.94 ( ( member @ A @ X4 @ S3 )
% 12.24/7.94 & ( ord_less @ B @ ( F @ X4 ) @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3916_arg__min__least,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linorder @ B )
% 12.24/7.94 => ! [S3: set @ A,Y3: A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.94 => ( ( S3
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ( member @ A @ Y3 @ S3 )
% 12.24/7.94 => ( ord_less_eq @ B @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3917_set__decode__inverse,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( nat_set_encode @ ( nat_set_decode @ N ) )
% 12.24/7.94 = N ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3918_set__encode__empty,axiom,
% 12.24/7.94 ( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.94 = ( zero_zero @ nat ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3919_set__encode__inverse,axiom,
% 12.24/7.94 ! [A6: set @ nat] :
% 12.24/7.94 ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.94 => ( ( nat_set_decode @ ( nat_set_encode @ A6 ) )
% 12.24/7.94 = A6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3920_finite__atLeastZeroLessThan__int,axiom,
% 12.24/7.94 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3921_finite__maxlen,axiom,
% 12.24/7.94 ! [A: $tType,M7: set @ ( list @ A )] :
% 12.24/7.94 ( ( finite_finite2 @ ( list @ A ) @ M7 )
% 12.24/7.94 => ? [N4: nat] :
% 12.24/7.94 ! [X4: list @ A] :
% 12.24/7.94 ( ( member @ ( list @ A ) @ X4 @ M7 )
% 12.24/7.94 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N4 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3922_set__encode__eq,axiom,
% 12.24/7.94 ! [A6: set @ nat,B7: set @ nat] :
% 12.24/7.94 ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ nat @ B7 )
% 12.24/7.94 => ( ( ( nat_set_encode @ A6 )
% 12.24/7.94 = ( nat_set_encode @ B7 ) )
% 12.24/7.94 = ( A6 = B7 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3923_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 12.24/7.94 ! [L: int,U: int] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3924_card__atLeastZeroLessThan__int,axiom,
% 12.24/7.94 ! [U: int] :
% 12.24/7.94 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
% 12.24/7.94 = ( nat2 @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3925_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 12.24/7.94 ! [L: int,U: int] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3926_set__encode__inf,axiom,
% 12.24/7.94 ! [A6: set @ nat] :
% 12.24/7.94 ( ~ ( finite_finite2 @ nat @ A6 )
% 12.24/7.94 => ( ( nat_set_encode @ A6 )
% 12.24/7.94 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3927_atLeastLessThan__upto,axiom,
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ int )
% 12.24/7.94 = ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3928_wmax__emptyI,axiom,
% 12.24/7.94 ! [X8: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.94 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 )
% 12.24/7.94 => ( 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 ) ) ) @ X8 ) @ fun_max_weak ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3929_smax__emptyI,axiom,
% 12.24/7.94 ! [Y7: set @ ( product_prod @ nat @ nat )] :
% 12.24/7.94 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 )
% 12.24/7.94 => ( ( Y7
% 12.24/7.94 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 12.24/7.94 => ( 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 ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3930_arg__min__if__finite_I1_J,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( order @ B )
% 12.24/7.94 => ! [S3: set @ A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.94 => ( ( S3
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) @ S3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3931_nth__image,axiom,
% 12.24/7.94 ! [A: $tType,L: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
% 12.24/7.94 = ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3932_fact__split,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semiring_char_0 @ A )
% 12.24/7.94 => ! [K: nat,N: nat] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.94 => ( ( semiring_char_0_fact @ A @ N )
% 12.24/7.94 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3933_sum__list__update,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordere1170586879665033532d_diff @ A )
% 12.24/7.94 => ! [K: nat,Xs: list @ A,X: A] :
% 12.24/7.94 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 12.24/7.94 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3934_set__take__disj__set__drop__if__distinct,axiom,
% 12.24/7.94 ! [A: $tType,Vs: list @ A,I: nat,J: nat] :
% 12.24/7.94 ( ( distinct @ A @ Vs )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.94 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3935_inf_Oidem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ A3 @ A3 )
% 12.24/7.94 = A3 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3936_inf__idem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ X )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3937_inf_Oleft__idem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
% 12.24/7.94 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3938_inf__left__idem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.94 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3939_inf_Oright__idem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.94 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3940_inf__right__idem,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 )
% 12.24/7.94 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3941_inf__apply,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ B )
% 12.24/7.94 => ( ( inf_inf @ ( A > B ) )
% 12.24/7.94 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3942_le__inf__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 & ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3943_inf_Obounded__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 & ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3944_boolean__algebra_Oconj__zero__right,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3945_boolean__algebra_Oconj__zero__left,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3946_inf__bot__right,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( bounded_lattice_bot @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3947_inf__bot__left,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( bounded_lattice_bot @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3948_inf__sup__absorb,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3949_sup__inf__absorb,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.94 = X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3950_image__Suc__atLeastAtMost,axiom,
% 12.24/7.94 ! [I: nat,J: nat] :
% 12.24/7.94 ( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3951_image__Suc__atLeastLessThan,axiom,
% 12.24/7.94 ! [I: nat,J: nat] :
% 12.24/7.94 ( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3952_Ints__prod,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( ( comm_monoid_mult @ B )
% 12.24/7.94 & ( ring_1 @ B ) )
% 12.24/7.94 => ! [A6: set @ A,F: A > B] :
% 12.24/7.94 ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ A6 )
% 12.24/7.94 => ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
% 12.24/7.94 => ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3953_image__add__0,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_add @ A )
% 12.24/7.94 => ! [S3: set @ A] :
% 12.24/7.94 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
% 12.24/7.94 = S3 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3954_inf__compl__bot__left1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3955_inf__compl__bot__left2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3956_inf__compl__bot__right,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ ( uminus_uminus @ A @ X ) ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3957_boolean__algebra_Oconj__cancel__left,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3958_boolean__algebra_Oconj__cancel__right,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3959_boolean__algebra_Ode__Morgan__conj,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.94 = ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3960_boolean__algebra_Ode__Morgan__disj,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.94 = ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3961_image__uminus__atLeastAtMost,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordered_ab_group_add @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3962_sum__list_ONil,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( monoid_add @ A )
% 12.24/7.94 => ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
% 12.24/7.94 = ( zero_zero @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3963_sum__list__eq__0__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( canoni5634975068530333245id_add @ A )
% 12.24/7.94 => ! [Ns: list @ A] :
% 12.24/7.94 ( ( ( groups8242544230860333062m_list @ A @ Ns )
% 12.24/7.94 = ( zero_zero @ A ) )
% 12.24/7.94 = ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
% 12.24/7.94 => ( X2
% 12.24/7.94 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3964_image__uminus__greaterThanLessThan,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordered_ab_group_add @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3965_Compl__disjoint,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3966_Compl__disjoint2,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A] :
% 12.24/7.94 ( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3967_Diff__Compl,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( minus_minus @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
% 12.24/7.94 = ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3968_image__uminus__greaterThanAtMost,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordered_ab_group_add @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3969_image__uminus__atLeastLessThan,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordered_ab_group_add @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3970_prod_OatMost__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: nat > A,N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3971_prod_OlessThan__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: nat > A,N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3972_image__mult__atLeastAtMost,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linordered_field @ A )
% 12.24/7.94 => ! [D2: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 12.24/7.94 => ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3973_prod_Ocl__ivl__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.94 ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3974_prod_Oop__ivl__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.94 ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3975_distrib__imp1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y5 @ Z4 ) )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( inf_inf @ A @ X3 @ Z4 ) ) )
% 12.24/7.94 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3976_distrib__imp2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y5 @ Z4 ) )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( sup_sup @ A @ X3 @ Z4 ) ) )
% 12.24/7.94 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3977_inf__sup__distrib1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( distrib_lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3978_inf__sup__distrib2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( distrib_lattice @ A )
% 12.24/7.94 => ! [Y3: A,Z3: A,X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3979_sup__inf__distrib1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( distrib_lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3980_sup__inf__distrib2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( distrib_lattice @ A )
% 12.24/7.94 => ! [Y3: A,Z3: A,X: A] :
% 12.24/7.94 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3981_boolean__algebra_Oconj__disj__distrib,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3982_boolean__algebra_Odisj__conj__distrib,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3983_boolean__algebra_Oconj__disj__distrib2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [Y3: A,Z3: A,X: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
% 12.24/7.94 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3984_boolean__algebra_Odisj__conj__distrib2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [Y3: A,Z3: A,X: A] :
% 12.24/7.94 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
% 12.24/7.94 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3985_inf__min,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ( semilattice_inf @ A )
% 12.24/7.94 & ( linorder @ A ) )
% 12.24/7.94 => ( ( inf_inf @ A )
% 12.24/7.94 = ( ord_min @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3986_inf__sup__aci_I4_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.94 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3987_inf__sup__aci_I3_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3988_inf__sup__aci_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.94 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3989_inf__sup__aci_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ( ( inf_inf @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3990_inf_Oassoc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.94 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3991_inf__assoc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.94 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3992_inf_Ocommute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( inf_inf @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] : ( inf_inf @ A @ B3 @ A5 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3993_inf__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( inf_inf @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3994_inf_Oleft__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,A3: A,C2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
% 12.24/7.94 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3995_inf__left__commute,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
% 12.24/7.94 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3996_inf__fun__def,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ B )
% 12.24/7.94 => ( ( inf_inf @ ( A > B ) )
% 12.24/7.94 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3997_boolean__algebra__cancel_Oinf1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A6: A,K: A,A3: A,B2: A] :
% 12.24/7.94 ( ( A6
% 12.24/7.94 = ( inf_inf @ A @ K @ A3 ) )
% 12.24/7.94 => ( ( inf_inf @ A @ A6 @ B2 )
% 12.24/7.94 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3998_boolean__algebra__cancel_Oinf2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B7: A,K: A,B2: A,A3: A] :
% 12.24/7.94 ( ( B7
% 12.24/7.94 = ( inf_inf @ A @ K @ B2 ) )
% 12.24/7.94 => ( ( inf_inf @ A @ A3 @ B7 )
% 12.24/7.94 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_3999_less__infI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,X: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ X )
% 12.24/7.94 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4000_less__infI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,X: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ B2 @ X )
% 12.24/7.94 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4001_inf_Oabsorb3,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( inf_inf @ A @ A3 @ B2 )
% 12.24/7.94 = A3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4002_inf_Oabsorb4,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( inf_inf @ A @ A3 @ B2 )
% 12.24/7.94 = B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4003_inf_Ostrict__boundedE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 12.24/7.94 => ~ ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4004_inf_Ostrict__order__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( ord_less @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] :
% 12.24/7.94 ( ( A5
% 12.24/7.94 = ( inf_inf @ A @ A5 @ B3 ) )
% 12.24/7.94 & ( A5 != B3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4005_inf_Ostrict__coboundedI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ A3 @ C2 )
% 12.24/7.94 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4006_inf_Ostrict__coboundedI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.94 ( ( ord_less @ A @ B2 @ C2 )
% 12.24/7.94 => ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4007_inf_OcoboundedI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ C2 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4008_inf_OcoboundedI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4009_inf_Oabsorb__iff2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [B3: A,A5: A] :
% 12.24/7.94 ( ( inf_inf @ A @ A5 @ B3 )
% 12.24/7.94 = B3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4010_inf_Oabsorb__iff1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] :
% 12.24/7.94 ( ( inf_inf @ A @ A5 @ B3 )
% 12.24/7.94 = A5 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4011_inf_Ocobounded2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4012_inf_Ocobounded1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4013_inf_Oorder__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [A5: A,B3: A] :
% 12.24/7.94 ( A5
% 12.24/7.94 = ( inf_inf @ A @ A5 @ B3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4014_inf__greatest,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ X @ Z3 )
% 12.24/7.94 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4015_inf_OboundedI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.94 => ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4016_inf_OboundedE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
% 12.24/7.94 => ~ ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4017_inf__absorb2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [Y3: A,X: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.94 => ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.94 = Y3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4018_inf__absorb1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 => ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.94 = X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4019_inf_Oabsorb2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 => ( ( inf_inf @ A @ A3 @ B2 )
% 12.24/7.94 = B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4020_inf_Oabsorb1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ( ( inf_inf @ A @ A3 @ B2 )
% 12.24/7.94 = A3 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4021_le__iff__inf,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ( ( ord_less_eq @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] :
% 12.24/7.94 ( ( inf_inf @ A @ X2 @ Y6 )
% 12.24/7.94 = X2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4022_inf__unique,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [F: A > A > A,X: A,Y3: A] :
% 12.24/7.94 ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ X3 )
% 12.24/7.94 => ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ Y5 )
% 12.24/7.94 => ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ X3 @ Z4 )
% 12.24/7.94 => ( ord_less_eq @ A @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
% 12.24/7.94 => ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.94 = ( F @ X @ Y3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4023_inf_OorderI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( A3
% 12.24/7.94 = ( inf_inf @ A @ A3 @ B2 ) )
% 12.24/7.94 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4024_inf_OorderE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.94 => ( A3
% 12.24/7.94 = ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4025_le__infI2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [B2: A,X: A,A3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ B2 @ X )
% 12.24/7.94 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4026_le__infI1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,X: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ X )
% 12.24/7.94 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4027_inf__mono,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ A3 @ C2 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4028_le__infI,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ A3 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ X @ B2 )
% 12.24/7.94 => ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4029_le__infE,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
% 12.24/7.94 => ~ ( ( ord_less_eq @ A @ X @ A3 )
% 12.24/7.94 => ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4030_inf__le2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4031_inf__le1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semilattice_inf @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4032_inf__sup__ord_I1_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4033_inf__sup__ord_I2_J,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4034_distrib__sup__le,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4035_distrib__inf__le,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( lattice @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4036_diff__eq,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ( ( minus_minus @ A )
% 12.24/7.94 = ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y6 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4037_inf__cancel__left1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4038_inf__cancel__left2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,A3: A,B2: A] :
% 12.24/7.94 ( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
% 12.24/7.94 = ( bot_bot @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4039_zero__notin__Suc__image,axiom,
% 12.24/7.94 ! [A6: set @ nat] :
% 12.24/7.94 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A6 ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4040_image__int__atLeastLessThan,axiom,
% 12.24/7.94 ! [A3: nat,B2: nat] :
% 12.24/7.94 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4041_member__le__sum__list,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( canoni5634975068530333245id_add @ A )
% 12.24/7.94 => ! [X: A,Xs: list @ A] :
% 12.24/7.94 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4042_Compl__Un,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.94 = ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4043_Compl__Int,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.94 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4044_Diff__eq,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( minus_minus @ ( set @ A ) )
% 12.24/7.94 = ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4045_prod_Oivl__cong,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( ( ord @ B )
% 12.24/7.94 & ( comm_monoid_mult @ A ) )
% 12.24/7.94 => ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 12.24/7.94 ( ( A3 = C2 )
% 12.24/7.94 => ( ( B2 = D2 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( ord_less_eq @ B @ C2 @ X3 )
% 12.24/7.94 => ( ( ord_less @ B @ X3 @ D2 )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( H2 @ X3 ) ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4046_prod_OatLeastLessThan__concat,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,P5: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ N @ P5 )
% 12.24/7.94 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4047_inf__shunt,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A] :
% 12.24/7.94 ( ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.94 = ( bot_bot @ A ) )
% 12.24/7.94 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4048_shunt1,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
% 12.24/7.94 = ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y3 ) @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4049_shunt2,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [X: A,Y3: A,Z3: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) @ Z3 )
% 12.24/7.94 = ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4050_sup__neg__inf,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.94 => ! [P5: A,Q4: A,R: A] :
% 12.24/7.94 ( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q4 @ R ) )
% 12.24/7.94 = ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q4 ) ) @ R ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4051_card__image__le,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( finite_card @ A @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4052_Groups__List_Osum__list__nonneg,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.94 => ! [Xs: list @ A] :
% 12.24/7.94 ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4053_sum__list__nonneg__eq__0__iff,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.94 => ! [Xs: list @ A] :
% 12.24/7.94 ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
% 12.24/7.94 => ( ( ( groups8242544230860333062m_list @ A @ Xs )
% 12.24/7.94 = ( zero_zero @ A ) )
% 12.24/7.94 = ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( X2
% 12.24/7.94 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4054_sum__list__nonpos,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.94 => ! [Xs: list @ A] :
% 12.24/7.94 ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4055_Ioc__disjoint,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linorder @ A )
% 12.24/7.94 => ! [A3: A,B2: A,C2: A,D2: A] :
% 12.24/7.94 ( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.94 | ( ord_less_eq @ A @ D2 @ C2 )
% 12.24/7.94 | ( ord_less_eq @ A @ B2 @ C2 )
% 12.24/7.94 | ( ord_less_eq @ A @ D2 @ A3 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4056_disjoint__eq__subset__Compl,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.94 ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 = ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4057_image__int__atLeastAtMost,axiom,
% 12.24/7.94 ! [A3: nat,B2: nat] :
% 12.24/7.94 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4058_sum__list__abs,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ordere166539214618696060dd_abs @ A )
% 12.24/7.94 => ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4059_sum__list__replicate,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semiring_1 @ A )
% 12.24/7.94 => ! [N: nat,C2: A] :
% 12.24/7.94 ( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
% 12.24/7.94 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4060_prod_OatLeast0__atMost__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: nat > A,N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4061_prod_OatLeast0__lessThan__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: nat > A,N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4062_prod_OatLeast__Suc__lessThan,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less @ nat @ M @ N )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4063_prod_OatLeast__Suc__atMost,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4064_prod_Onat__ivl__Suc_H,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4065_prod_OatLeastLessThan__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A3: nat,B2: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4066_prod_Olast__plus,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4067_surj__card__le,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ A @ B @ F @ A6 ) )
% 12.24/7.94 => ( ord_less_eq @ nat @ ( finite_card @ B @ B7 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4068_prod__Suc__fact,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.94 = ( semiring_char_0_fact @ nat @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4069_prod__Suc__Suc__fact,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 12.24/7.94 = ( semiring_char_0_fact @ nat @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4070_image__Suc__atMost,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4071_Iio__Int__singleton,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( order @ A )
% 12.24/7.94 => ! [X: A,K: A] :
% 12.24/7.94 ( ( ( ord_less @ A @ X @ K )
% 12.24/7.94 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.94 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ A @ X @ K )
% 12.24/7.94 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4072_image__Suc__lessThan,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4073_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4074_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4075_prod_Ohead,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4076_atMost__Suc__eq__insert__0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_ord_atMost @ nat @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4077_lessThan__Suc__eq__insert__0,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( set_ord_lessThan @ nat @ ( suc @ N ) )
% 12.24/7.94 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4078_sum__list__strict__mono,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( ( monoid_add @ B )
% 12.24/7.94 & ( strict9044650504122735259up_add @ B ) )
% 12.24/7.94 => ! [Xs: list @ A,F: A > B,G: A > B] :
% 12.24/7.94 ( ( Xs
% 12.24/7.94 != ( nil @ A ) )
% 12.24/7.94 => ( ! [X3: A] :
% 12.24/7.94 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.94 => ( ord_less @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.94 => ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4079_elem__le__sum__list,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( canoni5634975068530333245id_add @ A )
% 12.24/7.94 => ! [K: nat,Ns: list @ A] :
% 12.24/7.94 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4080_prod_Ohead__if,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.94 ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4081_prod_Oub__add__nat,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [M: nat,N: nat,G: nat > A,P5: nat] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4082_fact__prod__Suc,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( semiring_char_0 @ A )
% 12.24/7.94 => ( ( semiring_char_0_fact @ A )
% 12.24/7.94 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4083_image__atLeastZeroLessThan__int,axiom,
% 12.24/7.94 ! [U: int] :
% 12.24/7.94 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 12.24/7.94 => ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
% 12.24/7.94 = ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4084_image__mult__atLeastAtMost__if,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linordered_field @ A )
% 12.24/7.94 => ! [C2: A,X: A,Y3: A] :
% 12.24/7.94 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.94 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y3 ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.94 => ( ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y3 ) @ ( times_times @ A @ C2 @ X ) ) ) )
% 12.24/7.94 & ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.94 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.94 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4085_prod__pos__nat__iff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 ) )
% 12.24/7.94 = ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ A6 )
% 12.24/7.94 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4086_prod_Oinsert,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,X: B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ~ ( member @ B @ X @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4087_prod__eq__1__iff,axiom,
% 12.24/7.94 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.94 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.94 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 )
% 12.24/7.94 = ( one_one @ nat ) )
% 12.24/7.94 = ( ! [X2: A] :
% 12.24/7.94 ( ( member @ A @ X2 @ A6 )
% 12.24/7.94 => ( ( F @ X2 )
% 12.24/7.94 = ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4088_prod_Oinfinite,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,G: B > A] :
% 12.24/7.94 ( ~ ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( one_one @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4089_prod__zero__iff,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
% 12.24/7.94 = ( zero_zero @ A ) )
% 12.24/7.94 = ( ? [X2: B] :
% 12.24/7.94 ( ( member @ B @ X2 @ A6 )
% 12.24/7.94 & ( ( F @ X2 )
% 12.24/7.94 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4090_prod_Oempty,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: B > A] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 12.24/7.94 = ( one_one @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4091_inf__nat__def,axiom,
% 12.24/7.94 ( ( inf_inf @ nat )
% 12.24/7.94 = ( ord_min @ nat ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4092_inf__int__def,axiom,
% 12.24/7.94 ( ( inf_inf @ int )
% 12.24/7.94 = ( ord_min @ int ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4093_prod_Oneutral,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,G: B > A] :
% 12.24/7.94 ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ A6 )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( one_one @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4094_prod_Onot__neutral__contains__not__neutral,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [G: B > A,A6: set @ B] :
% 12.24/7.94 ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 != ( one_one @ A ) )
% 12.24/7.94 => ~ ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ A6 )
% 12.24/7.94 => ( ( G @ A8 )
% 12.24/7.94 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4095_prod__nonneg,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linordered_semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ A6 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4096_prod__mono,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linordered_semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A,G: B > A] :
% 12.24/7.94 ( ! [I4: B] :
% 12.24/7.94 ( ( member @ B @ I4 @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 12.24/7.94 & ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4097_prod__pos,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linordered_semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ A6 )
% 12.24/7.94 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 12.24/7.94 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4098_prod__ge__1,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linord181362715937106298miring @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ A6 )
% 12.24/7.94 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X3 ) ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4099_prod__zero,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_semiring_1 @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ? [X4: B] :
% 12.24/7.94 ( ( member @ B @ X4 @ A6 )
% 12.24/7.94 & ( ( F @ X4 )
% 12.24/7.94 = ( zero_zero @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
% 12.24/7.94 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4100_prod__le__1,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linord181362715937106298miring @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A] :
% 12.24/7.94 ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) )
% 12.24/7.94 & ( ord_less_eq @ A @ ( F @ X3 ) @ ( one_one @ A ) ) ) )
% 12.24/7.94 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4101_prod_Orelated,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.94 ( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
% 12.24/7.94 => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
% 12.24/7.94 ( ( ( R2 @ X1 @ X23 )
% 12.24/7.94 & ( R2 @ Y1 @ Y22 ) )
% 12.24/7.94 => ( R2 @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
% 12.24/7.94 => ( ( finite_finite2 @ B @ S3 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ S3 )
% 12.24/7.94 => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.94 => ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4102_prod_Oinsert__if,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,X: B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( ( member @ B @ X @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) )
% 12.24/7.94 & ( ~ ( member @ B @ X @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4103_prod_Oreindex__bij__witness__not__neutral,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ S5 )
% 12.24/7.94 => ( ( finite_finite2 @ C @ T5 )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 12.24/7.94 => ( ( I @ ( J @ A8 ) )
% 12.24/7.94 = A8 ) )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 12.24/7.94 => ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 12.24/7.94 => ( ! [B5: C] :
% 12.24/7.94 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/7.94 => ( ( J @ ( I @ B5 ) )
% 12.24/7.94 = B5 ) )
% 12.24/7.94 => ( ! [B5: C] :
% 12.24/7.94 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/7.94 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ S5 )
% 12.24/7.94 => ( ( G @ A8 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [B5: C] :
% 12.24/7.94 ( ( member @ C @ B5 @ T5 )
% 12.24/7.94 => ( ( H2 @ B5 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ S3 )
% 12.24/7.94 => ( ( H2 @ ( J @ A8 ) )
% 12.24/7.94 = ( G @ A8 ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4104_less__1__prod2,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linordered_idom @ B )
% 12.24/7.94 => ! [I6: set @ A,I: A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ I6 )
% 12.24/7.94 => ( ( member @ A @ I @ I6 )
% 12.24/7.94 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
% 12.24/7.94 => ( ! [I4: A] :
% 12.24/7.94 ( ( member @ A @ I4 @ I6 )
% 12.24/7.94 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
% 12.24/7.94 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4105_less__1__prod,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linordered_idom @ B )
% 12.24/7.94 => ! [I6: set @ A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ I6 )
% 12.24/7.94 => ( ( I6
% 12.24/7.94 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.94 => ( ! [I4: A] :
% 12.24/7.94 ( ( member @ A @ I4 @ I6 )
% 12.24/7.94 => ( ord_less @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
% 12.24/7.94 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4106_prod_Osubset__diff,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [B7: set @ B,A6: set @ B,G: B > A] :
% 12.24/7.94 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4107_prod_Omono__neutral__cong__right,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ S3 )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( H2 @ X3 ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4108_prod_Omono__neutral__cong__left,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.94 => ( ( H2 @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ S3 )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( H2 @ X3 ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4109_prod_Omono__neutral__right,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4110_prod_Omono__neutral__left,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4111_prod_Osame__carrierI,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ C6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 12.24/7.94 => ( ( G @ A8 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [B5: B] :
% 12.24/7.94 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 12.24/7.94 => ( ( H2 @ B5 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4112_prod_Osame__carrier,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ C6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 12.24/7.94 => ( ! [A8: B] :
% 12.24/7.94 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 12.24/7.94 => ( ( G @ A8 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [B5: B] :
% 12.24/7.94 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 12.24/7.94 => ( ( H2 @ B5 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) )
% 12.24/7.94 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4113_prod_Ounion__inter,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.94 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4114_prod_OInt__Diff,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4115_prod_Omono__neutral__cong,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ S3 )
% 12.24/7.94 => ( ! [I4: B] :
% 12.24/7.94 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.94 => ( ( H2 @ I4 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [I4: B] :
% 12.24/7.94 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
% 12.24/7.94 => ( ( G @ I4 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( H2 @ X3 ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4116_prod__mono__strict,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( linordered_semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ! [I4: B] :
% 12.24/7.94 ( ( member @ B @ I4 @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 12.24/7.94 & ( ord_less @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 12.24/7.94 => ( ( A6
% 12.24/7.94 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.94 => ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4117_prod_Oinsert__remove,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,G: B > A,X: B] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
% 12.24/7.94 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4118_prod_Oremove,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,X: B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( member @ B @ X @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
% 12.24/7.94 = ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4119_prod_Ounion__inter__neutral,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 => ( ( G @ X3 )
% 12.24/7.94 = ( one_one @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4120_prod_Ounion__disjoint,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.94 => ( ( ( inf_inf @ ( set @ B ) @ A6 @ B7 )
% 12.24/7.94 = ( bot_bot @ ( set @ B ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4121_prod_Ounion__diff2,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 = ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4122_prod__mono2,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linordered_idom @ B )
% 12.24/7.94 => ! [B7: set @ A,A6: set @ A,F: A > B] :
% 12.24/7.94 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.94 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.94 => ( ! [B5: A] :
% 12.24/7.94 ( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
% 12.24/7.94 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B5 ) ) )
% 12.24/7.94 => ( ! [A8: A] :
% 12.24/7.94 ( ( member @ A @ A8 @ A6 )
% 12.24/7.94 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A8 ) ) )
% 12.24/7.94 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4123_prod__le__power,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( linordered_semidom @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A,N: A,K: nat] :
% 12.24/7.94 ( ! [I4: B] :
% 12.24/7.94 ( ( member @ B @ I4 @ A6 )
% 12.24/7.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
% 12.24/7.94 & ( ord_less_eq @ A @ ( F @ I4 ) @ N ) ) )
% 12.24/7.94 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ K )
% 12.24/7.94 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
% 12.24/7.94 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4124_prod__diff1,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( semidom_divide @ A )
% 12.24/7.94 => ! [A6: set @ B,F: B > A,A3: B] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( ( F @ A3 )
% 12.24/7.94 != ( zero_zero @ A ) )
% 12.24/7.94 => ( ( ( member @ B @ A3 @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 12.24/7.94 = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( F @ A3 ) ) ) )
% 12.24/7.94 & ( ~ ( member @ B @ A3 @ A6 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4125_prod__Un,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( field @ A )
% 12.24/7.94 => ! [A6: set @ B,B7: set @ B,F: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.94 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.94 => ( ! [X3: B] :
% 12.24/7.94 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 => ( ( F @ X3 )
% 12.24/7.94 != ( zero_zero @ A ) ) )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.94 = ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4126_translation__Compl,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( ab_group_add @ A )
% 12.24/7.94 => ! [A3: A,T2: set @ A] :
% 12.24/7.94 ( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 12.24/7.94 = ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4127_remdups__adj__altdef,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.94 ( ( ( remdups_adj @ A @ Xs )
% 12.24/7.94 = Ys2 )
% 12.24/7.94 = ( ? [F2: nat > nat] :
% 12.24/7.94 ( ( order_mono @ nat @ nat @ F2 )
% 12.24/7.94 & ( ( image @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 12.24/7.94 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 12.24/7.94 & ! [I5: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( nth @ A @ Xs @ I5 )
% 12.24/7.94 = ( nth @ A @ Ys2 @ ( F2 @ I5 ) ) ) )
% 12.24/7.94 & ! [I5: nat] :
% 12.24/7.94 ( ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( ( nth @ A @ Xs @ I5 )
% 12.24/7.94 = ( nth @ A @ Xs @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) )
% 12.24/7.94 = ( ( F2 @ I5 )
% 12.24/7.94 = ( F2 @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4128_pochhammer__times__pochhammer__half,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( field_char_0 @ A )
% 12.24/7.94 => ! [Z3: A,N: nat] :
% 12.24/7.94 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.94 @ ^ [K3: nat] : ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 12.24/7.94 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4129_butlast__take,axiom,
% 12.24/7.94 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.94 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.94 => ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
% 12.24/7.94 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4130_prod_Oneutral__const,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [A6: set @ B] :
% 12.24/7.94 ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [Uu: B] : ( one_one @ A )
% 12.24/7.94 @ A6 )
% 12.24/7.94 = ( one_one @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4131_of__int__prod,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_ring_1 @ A )
% 12.24/7.94 => ! [F: B > int,A6: set @ B] :
% 12.24/7.94 ( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F @ A6 ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
% 12.24/7.94 @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4132_finite__Collect__less__nat,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( finite_finite2 @ nat
% 12.24/7.94 @ ( collect @ nat
% 12.24/7.94 @ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4133_finite__Collect__le__nat,axiom,
% 12.24/7.94 ! [K: nat] :
% 12.24/7.94 ( finite_finite2 @ nat
% 12.24/7.94 @ ( collect @ nat
% 12.24/7.94 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4134_card__Collect__less__nat,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( finite_card @ nat
% 12.24/7.94 @ ( collect @ nat
% 12.24/7.94 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) )
% 12.24/7.94 = N ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4135_finite__interval__int1,axiom,
% 12.24/7.94 ! [A3: int,B2: int] :
% 12.24/7.94 ( finite_finite2 @ int
% 12.24/7.94 @ ( collect @ int
% 12.24/7.94 @ ^ [I5: int] :
% 12.24/7.94 ( ( ord_less_eq @ int @ A3 @ I5 )
% 12.24/7.94 & ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4136_finite__interval__int4,axiom,
% 12.24/7.94 ! [A3: int,B2: int] :
% 12.24/7.94 ( finite_finite2 @ int
% 12.24/7.94 @ ( collect @ int
% 12.24/7.94 @ ^ [I5: int] :
% 12.24/7.94 ( ( ord_less @ int @ A3 @ I5 )
% 12.24/7.94 & ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4137_prod_Odelta,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [S3: set @ B,A3: B,B2: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.94 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 12.24/7.94 @ S3 )
% 12.24/7.94 = ( B2 @ A3 ) ) )
% 12.24/7.94 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 12.24/7.94 @ S3 )
% 12.24/7.94 = ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4138_prod_Odelta_H,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( comm_monoid_mult @ A )
% 12.24/7.94 => ! [S3: set @ B,A3: B,B2: B > A] :
% 12.24/7.94 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.94 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 12.24/7.94 @ S3 )
% 12.24/7.94 = ( B2 @ A3 ) ) )
% 12.24/7.94 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.94 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
% 12.24/7.94 @ S3 )
% 12.24/7.94 = ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4139_sum__list__0,axiom,
% 12.24/7.94 ! [B: $tType,A: $tType] :
% 12.24/7.94 ( ( monoid_add @ A )
% 12.24/7.94 => ! [Xs: list @ B] :
% 12.24/7.94 ( ( groups8242544230860333062m_list @ A
% 12.24/7.94 @ ( map @ B @ A
% 12.24/7.94 @ ^ [X2: B] : ( zero_zero @ A )
% 12.24/7.94 @ Xs ) )
% 12.24/7.94 = ( zero_zero @ A ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4140_card__Collect__le__nat,axiom,
% 12.24/7.94 ! [N: nat] :
% 12.24/7.94 ( ( finite_card @ nat
% 12.24/7.94 @ ( collect @ nat
% 12.24/7.94 @ ^ [I5: nat] : ( ord_less_eq @ nat @ I5 @ N ) ) )
% 12.24/7.94 = ( suc @ N ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4141_of__nat__prod,axiom,
% 12.24/7.94 ! [A: $tType,B: $tType] :
% 12.24/7.94 ( ( comm_semiring_1 @ A )
% 12.24/7.94 => ! [F: B > nat,A6: set @ B] :
% 12.24/7.94 ( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
% 12.24/7.94 = ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.94 @ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
% 12.24/7.94 @ A6 ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4142_finite__interval__int2,axiom,
% 12.24/7.94 ! [A3: int,B2: int] :
% 12.24/7.94 ( finite_finite2 @ int
% 12.24/7.94 @ ( collect @ int
% 12.24/7.94 @ ^ [I5: int] :
% 12.24/7.94 ( ( ord_less_eq @ int @ A3 @ I5 )
% 12.24/7.94 & ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4143_finite__interval__int3,axiom,
% 12.24/7.94 ! [A3: int,B2: int] :
% 12.24/7.94 ( finite_finite2 @ int
% 12.24/7.94 @ ( collect @ int
% 12.24/7.94 @ ^ [I5: int] :
% 12.24/7.94 ( ( ord_less @ int @ A3 @ I5 )
% 12.24/7.94 & ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4144_length__butlast,axiom,
% 12.24/7.94 ! [A: $tType,Xs: list @ A] :
% 12.24/7.94 ( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
% 12.24/7.94 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 12.24/7.94
% 12.24/7.94 thf(fact_4145_image__divide__atLeastAtMost,axiom,
% 12.24/7.94 ! [A: $tType] :
% 12.24/7.94 ( ( linordered_field @ A )
% 12.24/7.94 => ! [D2: A,A3: A,B2: A] :
% 12.24/7.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 12.24/7.94 => ( ( image @ A @ A
% 12.24/7.94 @ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
% 12.24/7.94 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.94 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4146_Gcd__int__eq,axiom,
% 12.24/7.95 ! [N6: set @ nat] :
% 12.24/7.95 ( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
% 12.24/7.95 = ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4147_prod_Ofinite__Collect__op,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [I6: set @ B,X: B > A,Y3: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( X @ I5 )
% 12.24/7.95 != ( one_one @ A ) ) ) ) )
% 12.24/7.95 => ( ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( Y3 @ I5 )
% 12.24/7.95 != ( one_one @ A ) ) ) ) )
% 12.24/7.95 => ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( times_times @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
% 12.24/7.95 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4148_prod_Oshift__bounds__Suc__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4149_nat__seg__image__imp__finite,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,F: nat > A,N: nat] :
% 12.24/7.95 ( ( A6
% 12.24/7.95 = ( image @ nat @ A @ F
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) ) )
% 12.24/7.95 => ( finite_finite2 @ A @ A6 ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4150_finite__conv__nat__seg__image,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( finite_finite2 @ A )
% 12.24/7.95 = ( ^ [A7: set @ A] :
% 12.24/7.95 ? [N3: nat,F2: nat > A] :
% 12.24/7.95 ( A7
% 12.24/7.95 = ( image @ nat @ A @ F2
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4151_sum_Ofinite__Collect__op,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [I6: set @ B,X: B > A,Y3: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( X @ I5 )
% 12.24/7.95 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.95 => ( ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( Y3 @ I5 )
% 12.24/7.95 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.95 => ( finite_finite2 @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [I5: B] :
% 12.24/7.95 ( ( member @ B @ I5 @ I6 )
% 12.24/7.95 & ( ( plus_plus @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
% 12.24/7.95 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4152_prod_Ointer__filter,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A,P: B > $o] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [X2: B] :
% 12.24/7.95 ( ( member @ B @ X2 @ A6 )
% 12.24/7.95 & ( P @ X2 ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
% 12.24/7.95 @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4153_prod_Odistrib,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: B > A,H2: B > A,A6: set @ B] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4154_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4155_mod__prod__eq,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.95 => ! [F: B > A,A3: A,A6: set @ B] :
% 12.24/7.95 ( ( modulo_modulo @ A
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
% 12.24/7.95 @ A6 )
% 12.24/7.95 @ A3 )
% 12.24/7.95 = ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4156_int__prod,axiom,
% 12.24/7.95 ! [B: $tType,F: B > nat,A6: set @ B] :
% 12.24/7.95 ( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ B @ int
% 12.24/7.95 @ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
% 12.24/7.95 @ A6 ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4157_prod__int__eq,axiom,
% 12.24/7.95 ! [I: nat,J: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ int @ int
% 12.24/7.95 @ ^ [X2: int] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4158_translation__subtract__Compl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [A3: A,T2: set @ A] :
% 12.24/7.95 ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
% 12.24/7.95 @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 12.24/7.95 = ( uminus_uminus @ ( set @ A )
% 12.24/7.95 @ ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
% 12.24/7.95 @ T2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4159_Collect__imp__eq,axiom,
% 12.24/7.95 ! [A: $tType,P: A > $o,Q: A > $o] :
% 12.24/7.95 ( ( collect @ A
% 12.24/7.95 @ ^ [X2: A] :
% 12.24/7.95 ( ( P @ X2 )
% 12.24/7.95 => ( Q @ X2 ) ) )
% 12.24/7.95 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4160_nat__less__as__int,axiom,
% 12.24/7.95 ( ( ord_less @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4161_len__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_len @ A )
% 12.24/7.95 = ( ^ [A5: array @ A] :
% 12.24/7.95 ( heap_Time_tap @ nat
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( array_length @ A @ H4 @ A5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4162_lambda__one,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( monoid_mult @ A )
% 12.24/7.95 => ( ( ^ [X2: A] : X2 )
% 12.24/7.95 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4163_lambda__zero,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( mult_zero @ A )
% 12.24/7.95 => ( ( ^ [H4: A] : ( zero_zero @ A ) )
% 12.24/7.95 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4164_Compl__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( uminus_uminus @ ( set @ A ) )
% 12.24/7.95 = ( ^ [A7: set @ A] :
% 12.24/7.95 ( collect @ A
% 12.24/7.95 @ ^ [X2: A] :
% 12.24/7.95 ~ ( member @ A @ X2 @ A7 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4165_Collect__neg__eq,axiom,
% 12.24/7.95 ! [A: $tType,P: A > $o] :
% 12.24/7.95 ( ( collect @ A
% 12.24/7.95 @ ^ [X2: A] :
% 12.24/7.95 ~ ( P @ X2 ) )
% 12.24/7.95 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4166_uminus__set__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( uminus_uminus @ ( set @ A ) )
% 12.24/7.95 = ( ^ [A7: set @ A] :
% 12.24/7.95 ( collect @ A
% 12.24/7.95 @ ( uminus_uminus @ ( A > $o )
% 12.24/7.95 @ ^ [X2: A] : ( member @ A @ X2 @ A7 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4167_min__of__mono,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( linorder @ A )
% 12.24/7.95 & ( linorder @ B ) )
% 12.24/7.95 => ! [F: A > B,M: A,N: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_min @ B @ ( F @ M ) @ ( F @ N ) )
% 12.24/7.95 = ( F @ ( ord_min @ A @ M @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4168_mono__Suc,axiom,
% 12.24/7.95 order_mono @ nat @ nat @ suc ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4169_strict__subset__divisors__dvd,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A3: A,B2: A] :
% 12.24/7.95 ( ( ord_less @ ( set @ A )
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
% 12.24/7.95 = ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.24/7.95 & ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4170_mono__add,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ordere6658533253407199908up_add @ A )
% 12.24/7.95 => ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4171_less__set__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord_less @ ( set @ A ) )
% 12.24/7.95 = ( ^ [A7: set @ A,B6: set @ A] :
% 12.24/7.95 ( ord_less @ ( A > $o )
% 12.24/7.95 @ ^ [X2: A] : ( member @ A @ X2 @ A7 )
% 12.24/7.95 @ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4172_mono__strict__invE,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( linorder @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.95 => ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4173_max__of__mono,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( linorder @ A )
% 12.24/7.95 & ( linorder @ B ) )
% 12.24/7.95 => ! [F: A > B,M: A,N: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_max @ B @ ( F @ M ) @ ( F @ N ) )
% 12.24/7.95 = ( F @ ( ord_max @ A @ M @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4174_numeral__code_I2_J,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( numeral @ A )
% 12.24/7.95 => ! [N: num] :
% 12.24/7.95 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4175_mono__def,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( order @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ( ( order_mono @ A @ B )
% 12.24/7.95 = ( ^ [F2: A > B] :
% 12.24/7.95 ! [X2: A,Y6: A] :
% 12.24/7.95 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4176_monoI,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( order @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ! [F: A > B] :
% 12.24/7.95 ( ! [X3: A,Y5: A] :
% 12.24/7.95 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.24/7.95 => ( order_mono @ A @ B @ F ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4177_monoE,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( order @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4178_monoD,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( order @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4179_min__def__raw,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord @ A )
% 12.24/7.95 => ( ( ord_min @ A )
% 12.24/7.95 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4180_subset__divisors__dvd,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A3: A,B2: A] :
% 12.24/7.95 ( ( ord_less_eq @ ( set @ A )
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
% 12.24/7.95 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4181_max__def__raw,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord @ A )
% 12.24/7.95 => ( ( ord_max @ A )
% 12.24/7.95 = ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4182_nat__leq__as__int,axiom,
% 12.24/7.95 ( ( ord_less_eq @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4183_effect__LetI,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: B,N: nat] :
% 12.24/7.95 ( ( X = T2 )
% 12.24/7.95 => ( ( heap_Time_effect @ B @ ( F @ X ) @ H2 @ H @ R @ N )
% 12.24/7.95 => ( heap_Time_effect @ B @ ( F @ T2 ) @ H2 @ H @ R @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4184_lessThan__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord @ A )
% 12.24/7.95 => ( ( set_ord_lessThan @ A )
% 12.24/7.95 = ( ^ [U3: A] :
% 12.24/7.95 ( collect @ A
% 12.24/7.95 @ ^ [X2: A] : ( ord_less @ A @ X2 @ U3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4185_atMost__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord @ A )
% 12.24/7.95 => ( ( set_ord_atMost @ A )
% 12.24/7.95 = ( ^ [U3: A] :
% 12.24/7.95 ( collect @ A
% 12.24/7.95 @ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4186_sum__list__const__mult,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_0 @ A )
% 12.24/7.95 => ! [C2: A,F: B > A,Xs: list @ B] :
% 12.24/7.95 ( ( groups8242544230860333062m_list @ A
% 12.24/7.95 @ ( map @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( times_times @ A @ C2 @ ( F @ X2 ) )
% 12.24/7.95 @ Xs ) )
% 12.24/7.95 = ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4187_sum__list__mult__const,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( semiring_0 @ A )
% 12.24/7.95 => ! [F: B > A,C2: A,Xs: list @ B] :
% 12.24/7.95 ( ( groups8242544230860333062m_list @ A
% 12.24/7.95 @ ( map @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ C2 )
% 12.24/7.95 @ Xs ) )
% 12.24/7.95 = ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ C2 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4188_finite__less__ub,axiom,
% 12.24/7.95 ! [F: nat > nat,U: nat] :
% 12.24/7.95 ( ! [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( F @ N4 ) )
% 12.24/7.95 => ( finite_finite2 @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4189_finite__int__segment,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.95 => ! [A3: A,B2: A] :
% 12.24/7.95 ( finite_finite2 @ A
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [X2: A] :
% 12.24/7.95 ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
% 12.24/7.95 & ( ord_less_eq @ A @ A3 @ X2 )
% 12.24/7.95 & ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4190_finite__M__bounded__by__nat,axiom,
% 12.24/7.95 ! [P: nat > $o,I: nat] :
% 12.24/7.95 ( finite_finite2 @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( P @ K3 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4191_finite__divisors__int,axiom,
% 12.24/7.95 ! [I: int] :
% 12.24/7.95 ( ( I
% 12.24/7.95 != ( zero_zero @ int ) )
% 12.24/7.95 => ( finite_finite2 @ int
% 12.24/7.95 @ ( collect @ int
% 12.24/7.95 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4192_integer__of__num_I2_J,axiom,
% 12.24/7.95 ! [N: num] :
% 12.24/7.95 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 12.24/7.95 = ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4193_numeral__code_I3_J,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( numeral @ A )
% 12.24/7.95 => ! [N: num] :
% 12.24/7.95 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4194_power__numeral__even,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( monoid_mult @ A )
% 12.24/7.95 => ! [Z3: A,W2: num] :
% 12.24/7.95 ( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
% 12.24/7.95 = ( times_times @ A @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4195_mono__ge2__power__minus__self,axiom,
% 12.24/7.95 ! [K: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 12.24/7.95 => ( order_mono @ nat @ nat
% 12.24/7.95 @ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4196_prod_Ointer__restrict,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( one_one @ A ) )
% 12.24/7.95 @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4197_prod_Osetdiff__irrelevant,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 12.24/7.95 @ ( minus_minus @ ( set @ B ) @ A6
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [X2: B] :
% 12.24/7.95 ( ( G @ X2 )
% 12.24/7.95 = ( one_one @ A ) ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4198_power__numeral__odd,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( monoid_mult @ A )
% 12.24/7.95 => ! [Z3: A,W2: num] :
% 12.24/7.95 ( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
% 12.24/7.95 = ( times_times @ A @ ( times_times @ A @ Z3 @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4199_card__length__sum__list__rec,axiom,
% 12.24/7.95 ! [M: nat,N6: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
% 12.24/7.95 => ( ( finite_card @ ( list @ nat )
% 12.24/7.95 @ ( collect @ ( list @ nat )
% 12.24/7.95 @ ^ [L3: list @ nat] :
% 12.24/7.95 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 12.24/7.95 = M )
% 12.24/7.95 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 12.24/7.95 = N6 ) ) ) )
% 12.24/7.95 = ( plus_plus @ nat
% 12.24/7.95 @ ( finite_card @ ( list @ nat )
% 12.24/7.95 @ ( collect @ ( list @ nat )
% 12.24/7.95 @ ^ [L3: list @ nat] :
% 12.24/7.95 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 12.24/7.95 = ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
% 12.24/7.95 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 12.24/7.95 = N6 ) ) ) )
% 12.24/7.95 @ ( finite_card @ ( list @ nat )
% 12.24/7.95 @ ( collect @ ( list @ nat )
% 12.24/7.95 @ ^ [L3: list @ nat] :
% 12.24/7.95 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 12.24/7.95 = M )
% 12.24/7.95 & ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
% 12.24/7.95 = N6 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4200_sum__list__mono,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( monoid_add @ B )
% 12.24/7.95 & ( ordere6658533253407199908up_add @ B ) )
% 12.24/7.95 => ! [Xs: list @ A,F: A > B,G: A > B] :
% 12.24/7.95 ( ! [X3: A] :
% 12.24/7.95 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.95 => ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4201_finite__abs__int__segment,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( archim2362893244070406136eiling @ A )
% 12.24/7.95 => ! [A3: A] :
% 12.24/7.95 ( finite_finite2 @ A
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [K3: A] :
% 12.24/7.95 ( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
% 12.24/7.95 & ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4202_nat__plus__as__int,axiom,
% 12.24/7.95 ( ( plus_plus @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4203_nat__times__as__int,axiom,
% 12.24/7.95 ( ( times_times @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4204_card__length__sum__list,axiom,
% 12.24/7.95 ! [M: nat,N6: nat] :
% 12.24/7.95 ( ( finite_card @ ( list @ nat )
% 12.24/7.95 @ ( collect @ ( list @ nat )
% 12.24/7.95 @ ^ [L3: list @ nat] :
% 12.24/7.95 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 12.24/7.95 = M )
% 12.24/7.95 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 12.24/7.95 = N6 ) ) ) )
% 12.24/7.95 = ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4205_nat__minus__as__int,axiom,
% 12.24/7.95 ( ( minus_minus @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4206_prod_Onat__diff__reindex,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4207_finite__divisors__nat,axiom,
% 12.24/7.95 ! [M: nat] :
% 12.24/7.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.24/7.95 => ( finite_finite2 @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4208_card__less,axiom,
% 12.24/7.95 ! [M7: set @ nat,I: nat] :
% 12.24/7.95 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 12.24/7.95 => ( ( finite_card @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( member @ nat @ K3 @ M7 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
% 12.24/7.95 != ( zero_zero @ nat ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4209_card__less__Suc,axiom,
% 12.24/7.95 ! [M7: set @ nat,I: nat] :
% 12.24/7.95 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 12.24/7.95 => ( ( suc
% 12.24/7.95 @ ( finite_card @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( member @ nat @ ( suc @ K3 ) @ M7 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ I ) ) ) ) )
% 12.24/7.95 = ( finite_card @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( member @ nat @ K3 @ M7 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4210_card__less__Suc2,axiom,
% 12.24/7.95 ! [M7: set @ nat,I: nat] :
% 12.24/7.95 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 12.24/7.95 => ( ( finite_card @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( member @ nat @ ( suc @ K3 ) @ M7 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ I ) ) ) )
% 12.24/7.95 = ( finite_card @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( ( member @ nat @ K3 @ M7 )
% 12.24/7.95 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4211_sum__list__Suc,axiom,
% 12.24/7.95 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 12.24/7.95 ( ( groups8242544230860333062m_list @ nat
% 12.24/7.95 @ ( map @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( suc @ ( F @ X2 ) )
% 12.24/7.95 @ Xs ) )
% 12.24/7.95 = ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4212_prod__int__plus__eq,axiom,
% 12.24/7.95 ! [I: nat,J: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ int @ int
% 12.24/7.95 @ ^ [X2: int] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4213_nat__div__as__int,axiom,
% 12.24/7.95 ( ( divide_divide @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4214_nat__mod__as__int,axiom,
% 12.24/7.95 ( ( modulo_modulo @ nat )
% 12.24/7.95 = ( ^ [A5: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4215_mono__invE,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( linorder @ A )
% 12.24/7.95 & ( order @ B ) )
% 12.24/7.95 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.95 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4216_mono__iff__le__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( order @ A )
% 12.24/7.95 => ( ( order_mono @ nat @ A )
% 12.24/7.95 = ( ^ [F2: nat > A] :
% 12.24/7.95 ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4217_image__add__int__atLeastLessThan,axiom,
% 12.24/7.95 ! [L: int,U: int] :
% 12.24/7.95 ( ( image @ int @ int
% 12.24/7.95 @ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
% 12.24/7.95 = ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4218_mono__inf,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( semilattice_inf @ A )
% 12.24/7.95 & ( semilattice_inf @ B ) )
% 12.24/7.95 => ! [F: A > B,A6: A,B7: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ ( inf_inf @ A @ A6 @ B7 ) ) @ ( inf_inf @ B @ ( F @ A6 ) @ ( F @ B7 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4219_map__replicate__trivial,axiom,
% 12.24/7.95 ! [A: $tType,X: A,I: nat] :
% 12.24/7.95 ( ( map @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : X
% 12.24/7.95 @ ( upt @ ( zero_zero @ nat ) @ I ) )
% 12.24/7.95 = ( replicate @ A @ I @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4220_mono__sup,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( semilattice_sup @ A )
% 12.24/7.95 & ( semilattice_sup @ B ) )
% 12.24/7.95 => ! [F: A > B,A6: A,B7: A] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A6 ) @ ( F @ B7 ) ) @ ( F @ ( sup_sup @ A @ A6 @ B7 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4221_semiring__char__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ( ( semiri4206861660011772517g_char @ A )
% 12.24/7.95 = ( ^ [Uu2: itself @ A] :
% 12.24/7.95 ( gcd_Gcd @ nat
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [N3: nat] :
% 12.24/7.95 ( ( semiring_1_of_nat @ A @ N3 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4222_map__add__upt,axiom,
% 12.24/7.95 ! [N: nat,M: nat] :
% 12.24/7.95 ( ( map @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ nat @ I5 @ N )
% 12.24/7.95 @ ( upt @ ( zero_zero @ nat ) @ M ) )
% 12.24/7.95 = ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4223_integer__of__num_I3_J,axiom,
% 12.24/7.95 ! [N: num] :
% 12.24/7.95 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 12.24/7.95 = ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4224_prod_OSuc__reindex__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.24/7.95 = ( times_times @ A @ ( G @ M )
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4225_prod_OIf__cases,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4226_sum__list__triv,axiom,
% 12.24/7.95 ! [C: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ B )
% 12.24/7.95 => ! [R: B,Xs: list @ C] :
% 12.24/7.95 ( ( groups8242544230860333062m_list @ B
% 12.24/7.95 @ ( map @ C @ B
% 12.24/7.95 @ ^ [X2: C] : R
% 12.24/7.95 @ Xs ) )
% 12.24/7.95 = ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4227_prod_OatMost__Suc__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4228_prod_OlessThan__Suc__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4229_prod_OatLeastLessThan__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat,M: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4230_prod_Onested__swap,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A3: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4231_prod_OatLeast1__atMost__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4232_prod_Onat__group,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,K: nat,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4233_map__upt__Suc,axiom,
% 12.24/7.95 ! [A: $tType,F: nat > A,N: nat] :
% 12.24/7.95 ( ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( cons @ A @ ( F @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( map @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
% 12.24/7.95 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4234_prod_Onested__swap_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [A3: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4235_prod__atLeastAtMost__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [F: nat > A,A3: nat,B2: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 12.24/7.95 = ( set_fo6178422350223883121st_nat @ A
% 12.24/7.95 @ ^ [A5: nat] : ( times_times @ A @ ( F @ A5 ) )
% 12.24/7.95 @ A3
% 12.24/7.95 @ B2
% 12.24/7.95 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4236_map__nth,axiom,
% 12.24/7.95 ! [A: $tType,Xs: list @ A] :
% 12.24/7.95 ( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 12.24/7.95 = Xs ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4237_ratrel__def,axiom,
% 12.24/7.95 ( ratrel
% 12.24/7.95 = ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
% 12.24/7.95 ( ( ( product_snd @ int @ int @ X2 )
% 12.24/7.95 != ( zero_zero @ int ) )
% 12.24/7.95 & ( ( product_snd @ int @ int @ Y6 )
% 12.24/7.95 != ( zero_zero @ int ) )
% 12.24/7.95 & ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
% 12.24/7.95 = ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4238_fact__prod,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( semiring_char_0 @ A )
% 12.24/7.95 => ( ( semiring_char_0_fact @ A )
% 12.24/7.95 = ( ^ [N3: nat] :
% 12.24/7.95 ( semiring_1_of_nat @ A
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4239_map__decr__upt,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( map @ nat @ nat
% 12.24/7.95 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.95 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( upt @ M @ N ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4240_prod_Odelta__remove,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 12.24/7.95 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4241_diff__nat__eq__if,axiom,
% 12.24/7.95 ! [Z6: int,Z3: int] :
% 12.24/7.95 ( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
% 12.24/7.95 => ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
% 12.24/7.95 = ( nat2 @ Z3 ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
% 12.24/7.95 => ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
% 12.24/7.95 = ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4242_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat,M: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4243_prod_OatMost__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4244_pochhammer__prod,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ( ( comm_s3205402744901411588hammer @ A )
% 12.24/7.95 = ( ^ [A5: A,N3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4245_mono__mult,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ordered_semiring @ A )
% 12.24/7.95 => ! [A3: A] :
% 12.24/7.95 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.95 => ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4246_mono__times__nat,axiom,
% 12.24/7.95 ! [N: nat] :
% 12.24/7.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.95 => ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4247_finite__lists__length__le,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,N: nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.95 => ( finite_finite2 @ ( list @ A )
% 12.24/7.95 @ ( collect @ ( list @ A )
% 12.24/7.95 @ ^ [Xs3: list @ A] :
% 12.24/7.95 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 12.24/7.95 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4248_mono__image__least,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ( order @ B )
% 12.24/7.95 & ( order @ A ) )
% 12.24/7.95 => ! [F: A > B,M: A,N: A,M5: B,N2: B] :
% 12.24/7.95 ( ( order_mono @ A @ B @ F )
% 12.24/7.95 => ( ( ( image @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
% 12.24/7.95 = ( set_or7035219750837199246ssThan @ B @ M5 @ N2 ) )
% 12.24/7.95 => ( ( ord_less @ A @ M @ N )
% 12.24/7.95 => ( ( F @ M )
% 12.24/7.95 = M5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4249_fact__prod__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( semiring_char_0 @ A )
% 12.24/7.95 => ( ( semiring_char_0_fact @ A )
% 12.24/7.95 = ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4250_fact__eq__fact__times,axiom,
% 12.24/7.95 ! [N: nat,M: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.95 => ( ( semiring_char_0_fact @ nat @ M )
% 12.24/7.95 = ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4251_image__mult__atLeastAtMost__if_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [X: A,Y3: A,C2: A] :
% 12.24/7.95 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.95 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y3 @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4252_image__affinity__atLeastAtMost,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [A3: A,B2: A,M: A,C2: A] :
% 12.24/7.95 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.95 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4253_image__affinity__atLeastAtMost__diff,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [A3: A,B2: A,M: A,C2: A] :
% 12.24/7.95 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.95 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4254_image__affinity__atLeastAtMost__div,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [A3: A,B2: A,M: A,C2: A] :
% 12.24/7.95 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.95 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4255_image__affinity__atLeastAtMost__div__diff,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [A3: A,B2: A,M: A,C2: A] :
% 12.24/7.95 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.95 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 12.24/7.95 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.95 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.95 => ( ( image @ A @ A
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 12.24/7.95 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4256_prod__gen__delta,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [S3: set @ B,A3: B,B2: B > A,C2: A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
% 12.24/7.95 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4257_pochhammer__Suc__prod,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [A3: A,N: nat] :
% 12.24/7.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4258_pochhammer__prod__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ( ( comm_s3205402744901411588hammer @ A )
% 12.24/7.95 = ( ^ [A5: A,N3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I5 ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4259_set__decode__def,axiom,
% 12.24/7.95 ( nat_set_decode
% 12.24/7.95 = ( ^ [X2: nat] :
% 12.24/7.95 ( collect @ nat
% 12.24/7.95 @ ^ [N3: nat] :
% 12.24/7.95 ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4260_image__minus__const__atLeastLessThan__nat,axiom,
% 12.24/7.95 ! [C2: nat,Y3: nat,X: nat] :
% 12.24/7.95 ( ( ( ord_less @ nat @ C2 @ Y3 )
% 12.24/7.95 => ( ( image @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 12.24/7.95 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y3 @ C2 ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ C2 @ Y3 )
% 12.24/7.95 => ( ( ( ord_less @ nat @ X @ Y3 )
% 12.24/7.95 => ( ( image @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 12.24/7.95 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ X @ Y3 )
% 12.24/7.95 => ( ( image @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
% 12.24/7.95 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4261_fact__div__fact,axiom,
% 12.24/7.95 ! [N: nat,M: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ N @ M )
% 12.24/7.95 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4262_prod_Oin__pairs,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4263_prod_Oin__pairs__0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4264_gbinomial__altdef__of__nat,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ( ( gbinomial @ A )
% 12.24/7.95 = ( ^ [A5: A,K3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I5 ) ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4265_signed__take__bit__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.95 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 12.24/7.95 = ( ^ [N3: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4266_pochhammer__Suc__prod__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [A3: A,N: nat] :
% 12.24/7.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4267_binomial__altdef__of__nat,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [K: nat,N: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ K @ N )
% 12.24/7.95 => ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I5 ) ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4268_gbinomial__mult__fact,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [K: nat,A3: A] :
% 12.24/7.95 ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4269_gbinomial__mult__fact_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [A3: A,K: nat] :
% 12.24/7.95 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4270_gbinomial__prod__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( semiring_char_0 @ A )
% 12.24/7.95 & ( semidom_divide @ A ) )
% 12.24/7.95 => ( ( gbinomial @ A )
% 12.24/7.95 = ( ^ [A5: A,K3: nat] :
% 12.24/7.95 ( divide_divide @ A
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
% 12.24/7.95 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4271_nth__butlast,axiom,
% 12.24/7.95 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.95 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
% 12.24/7.95 => ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
% 12.24/7.95 = ( nth @ A @ Xs @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4272_take__butlast,axiom,
% 12.24/7.95 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.95 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.95 => ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
% 12.24/7.95 = ( take @ A @ N @ Xs ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4273_prod_Ozero__middle,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 12.24/7.95 => ( ( ord_less_eq @ nat @ K @ P5 )
% 12.24/7.95 => ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ P5 ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4274_gbinomial__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( semiring_char_0 @ A )
% 12.24/7.95 & ( semidom_divide @ A ) )
% 12.24/7.95 => ! [A3: A,K: nat] :
% 12.24/7.95 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 12.24/7.95 = ( divide_divide @ A
% 12.24/7.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
% 12.24/7.95 @ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4275_pochhammer__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ( ( comm_s3205402744901411588hammer @ A )
% 12.24/7.95 = ( ^ [A5: A,N3: nat] :
% 12.24/7.95 ( if @ A
% 12.24/7.95 @ ( N3
% 12.24/7.95 = ( zero_zero @ nat ) )
% 12.24/7.95 @ ( one_one @ A )
% 12.24/7.95 @ ( set_fo6178422350223883121st_nat @ A
% 12.24/7.95 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
% 12.24/7.95 @ ( zero_zero @ nat )
% 12.24/7.95 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
% 12.24/7.95 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4276_normalize__def,axiom,
% 12.24/7.95 ( normalize
% 12.24/7.95 = ( ^ [P6: product_prod @ int @ int] :
% 12.24/7.95 ( 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 ) ) ) )
% 12.24/7.95 @ ( if @ ( product_prod @ int @ int )
% 12.24/7.95 @ ( ( product_snd @ int @ int @ P6 )
% 12.24/7.95 = ( zero_zero @ int ) )
% 12.24/7.95 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.95 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4277_butlast__conv__take,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( butlast @ A )
% 12.24/7.95 = ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4278_butlast__list__update,axiom,
% 12.24/7.95 ! [A: $tType,K: nat,Xs: list @ A,X: A] :
% 12.24/7.95 ( ( ( K
% 12.24/7.95 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 12.24/7.95 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 12.24/7.95 = ( butlast @ A @ Xs ) ) )
% 12.24/7.95 & ( ( K
% 12.24/7.95 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 12.24/7.95 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 12.24/7.95 = ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4279_gbinomial__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ( ( gbinomial @ A )
% 12.24/7.95 = ( ^ [A5: A,K3: nat] :
% 12.24/7.95 ( if @ A
% 12.24/7.95 @ ( K3
% 12.24/7.95 = ( zero_zero @ nat ) )
% 12.24/7.95 @ ( one_one @ A )
% 12.24/7.95 @ ( divide_divide @ A
% 12.24/7.95 @ ( set_fo6178422350223883121st_nat @ A
% 12.24/7.95 @ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L3 ) ) )
% 12.24/7.95 @ ( zero_zero @ nat )
% 12.24/7.95 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
% 12.24/7.95 @ ( one_one @ A ) )
% 12.24/7.95 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4280_card__lists__distinct__length__eq_H,axiom,
% 12.24/7.95 ! [A: $tType,K: nat,A6: set @ A] :
% 12.24/7.95 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A6 ) )
% 12.24/7.95 => ( ( finite_card @ ( list @ A )
% 12.24/7.95 @ ( collect @ ( list @ A )
% 12.24/7.95 @ ^ [Xs3: list @ A] :
% 12.24/7.95 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.95 = K )
% 12.24/7.95 & ( distinct @ A @ Xs3 )
% 12.24/7.95 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4281_card__lists__distinct__length__eq,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,K: nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.95 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A6 ) )
% 12.24/7.95 => ( ( finite_card @ ( list @ A )
% 12.24/7.95 @ ( collect @ ( list @ A )
% 12.24/7.95 @ ^ [Xs3: list @ A] :
% 12.24/7.95 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.95 = K )
% 12.24/7.95 & ( distinct @ A @ Xs3 )
% 12.24/7.95 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4282_bit__cut__integer__code,axiom,
% 12.24/7.95 ( code_bit_cut_integer
% 12.24/7.95 = ( ^ [K3: code_integer] :
% 12.24/7.95 ( if @ ( product_prod @ code_integer @ $o )
% 12.24/7.95 @ ( K3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
% 12.24/7.95 @ ^ [R5: code_integer,S6: code_integer] :
% 12.24/7.95 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S6 ) )
% 12.24/7.95 @ ( S6
% 12.24/7.95 = ( one_one @ code_integer ) ) )
% 12.24/7.95 @ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4283_integer__of__int__code,axiom,
% 12.24/7.95 ( code_integer_of_int
% 12.24/7.95 = ( ^ [K3: int] :
% 12.24/7.95 ( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
% 12.24/7.95 @ ( if @ code_integer
% 12.24/7.95 @ ( K3
% 12.24/7.95 = ( zero_zero @ int ) )
% 12.24/7.95 @ ( zero_zero @ code_integer )
% 12.24/7.95 @ ( if @ code_integer
% 12.24/7.95 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 12.24/7.95 = ( zero_zero @ int ) )
% 12.24/7.95 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 12.24/7.95 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4284_swap__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_swap @ A )
% 12.24/7.95 = ( ^ [I5: nat,X2: A,A5: array @ A] :
% 12.24/7.95 ( heap_Time_guard @ A
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4285_choose__even__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_ring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.95 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) )
% 12.24/7.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4286_integer__of__int__eq__of__int,axiom,
% 12.24/7.95 ( code_integer_of_int
% 12.24/7.95 = ( ring_1_of_int @ code_integer ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4287_Ints__sum,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ring_1 @ B )
% 12.24/7.95 => ! [A6: set @ A,F: A > B] :
% 12.24/7.95 ( ! [X3: A] :
% 12.24/7.95 ( ( member @ A @ X3 @ A6 )
% 12.24/7.95 => ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
% 12.24/7.95 => ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4288_sum_Oneutral__const,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [Uu: B] : ( zero_zero @ A )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4289_of__nat__sum,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ! [F: B > nat,A6: set @ B] :
% 12.24/7.95 ( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4290_of__int__sum,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ring_1 @ A )
% 12.24/7.95 => ! [F: B > int,A6: set @ B] :
% 12.24/7.95 ( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F @ A6 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4291_sum_Oempty,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: B > A] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4292_sum__eq__0__iff,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( canoni5634975068530333245id_add @ A )
% 12.24/7.95 => ! [F4: set @ B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ F4 )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ F4 )
% 12.24/7.95 = ( zero_zero @ A ) )
% 12.24/7.95 = ( ! [X2: B] :
% 12.24/7.95 ( ( member @ B @ X2 @ F4 )
% 12.24/7.95 => ( ( F @ X2 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4293_sum_Oinfinite,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A] :
% 12.24/7.95 ( ~ ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4294_sum_Odelta,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [S3: set @ B,A3: B,B2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( B2 @ A3 ) ) )
% 12.24/7.95 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4295_sum_Odelta_H,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [S3: set @ B,A3: B,B2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ( ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( B2 @ A3 ) ) )
% 12.24/7.95 & ( ~ ( member @ B @ A3 @ S3 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4296_sum__abs,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere166539214618696060dd_abs @ B )
% 12.24/7.95 => ! [F: A > B,A6: set @ A] :
% 12.24/7.95 ( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ A @ B
% 12.24/7.95 @ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4297_sum_OatMost__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4298_sum_OlessThan__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4299_sum__abs__ge__zero,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere166539214618696060dd_abs @ B )
% 12.24/7.95 => ! [F: A > B,A6: set @ A] :
% 12.24/7.95 ( ord_less_eq @ B @ ( zero_zero @ B )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ A @ B
% 12.24/7.95 @ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4300_sum__constant,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ! [Y3: A,A6: set @ B] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : Y3
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ Y3 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4301_sum__list__upt,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4302_sum__of__bool__mult__eq,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ! [A6: set @ B,P: B > $o,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4303_sum__mult__of__bool__eq,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,P: B > $o] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4304_sum_Ocl__ivl__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.95 ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4305_sum_Oop__ivl__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.95 ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4306_sum__of__bool__eq,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ! [A6: set @ B,P: B > $o] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4307_sum__zero__power,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( division_ring @ A )
% 12.24/7.95 => ! [A6: set @ nat,C2: nat > A] :
% 12.24/7.95 ( ( ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.95 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( C2 @ ( zero_zero @ nat ) ) ) )
% 12.24/7.95 & ( ~ ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.95 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4308_sum__zero__power_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field @ A )
% 12.24/7.95 => ! [A6: set @ nat,C2: nat > A,D2: nat > A] :
% 12.24/7.95 ( ( ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.95 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
% 12.24/7.95 & ( ~ ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.95 & ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4309_divmod__algorithm__code_I5_J,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.95 => ! [M: num,N: num] :
% 12.24/7.95 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.95 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 12.24/7.95 @ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
% 12.24/7.95 @ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4310_divmod__algorithm__code_I6_J,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.95 => ! [M: num,N: num] :
% 12.24/7.95 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.95 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 12.24/7.95 @ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
% 12.24/7.95 @ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4311_sum__mono,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [K5: set @ B,F: B > A,G: B > A] :
% 12.24/7.95 ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ K5 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4312_sum__negf,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [F: B > A,A6: set @ B] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( uminus_uminus @ A @ ( F @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4313_sum__distrib__left,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_0 @ A )
% 12.24/7.95 => ! [R: A,F: B > A,A6: set @ B] :
% 12.24/7.95 ( ( times_times @ A @ R @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [N3: B] : ( times_times @ A @ R @ ( F @ N3 ) )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4314_sum__distrib__right,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_0 @ A )
% 12.24/7.95 => ! [F: B > A,A6: set @ B,R: A] :
% 12.24/7.95 ( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ R )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [N3: B] : ( times_times @ A @ ( F @ N3 ) @ R )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4315_sum__product,axiom,
% 12.24/7.95 ! [B: $tType,C: $tType,A: $tType] :
% 12.24/7.95 ( ( semiring_0 @ B )
% 12.24/7.95 => ! [F: A > B,A6: set @ A,G: C > B,B7: set @ C] :
% 12.24/7.95 ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ A @ B
% 12.24/7.95 @ ^ [I5: A] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ C @ B
% 12.24/7.95 @ ^ [J3: C] : ( times_times @ B @ ( F @ I5 ) @ ( G @ J3 ) )
% 12.24/7.95 @ B7 )
% 12.24/7.95 @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4316_mod__sum__eq,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.95 => ! [F: B > A,A3: A,A6: set @ B] :
% 12.24/7.95 ( ( modulo_modulo @ A
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
% 12.24/7.95 @ A6 )
% 12.24/7.95 @ A3 )
% 12.24/7.95 = ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4317_unset__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se2638667681897837118et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4318_bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4319_or__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4320_and__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4321_euclidean__size__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( euclid6346220572633701492n_size @ int @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4322_not__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4323_take__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4324_xor__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4325_times__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( times_times @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( times_times @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4326_push__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4327_drop__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se4197421643247451524op_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4328_mask__integer_Oabs__eq,axiom,
% 12.24/7.95 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 12.24/7.95 = ( ^ [X2: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4329_split__cong,axiom,
% 12.24/7.95 ! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
% 12.24/7.95 ( ! [X3: A,Y5: B] :
% 12.24/7.95 ( ( ( product_Pair @ A @ B @ X3 @ Y5 )
% 12.24/7.95 = Q4 )
% 12.24/7.95 => ( ( F @ X3 @ Y5 )
% 12.24/7.95 = ( G @ X3 @ Y5 ) ) )
% 12.24/7.95 => ( ( P5 = Q4 )
% 12.24/7.95 => ( ( product_case_prod @ A @ B @ C @ F @ P5 )
% 12.24/7.95 = ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4330_set__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se5668285175392031749et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4331_flip__bit__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: nat,X: int] :
% 12.24/7.95 ( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4332_division__segment__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4333_sum_Oneutral,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A] :
% 12.24/7.95 ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ A6 )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4334_sum_Onot__neutral__contains__not__neutral,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: B > A,A6: set @ B] :
% 12.24/7.95 ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 12.24/7.95 != ( zero_zero @ A ) )
% 12.24/7.95 => ~ ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ A6 )
% 12.24/7.95 => ( ( G @ A8 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4335_sum__nonpos,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A] :
% 12.24/7.95 ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4336_sum__nonneg,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A] :
% 12.24/7.95 ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4337_sum__mono__inv,axiom,
% 12.24/7.95 ! [A: $tType,I7: $tType] :
% 12.24/7.95 ( ( ordere8940638589300402666id_add @ A )
% 12.24/7.95 => ! [F: I7 > A,I6: set @ I7,G: I7 > A,I: I7] :
% 12.24/7.95 ( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F @ I6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) )
% 12.24/7.95 => ( ! [I4: I7] :
% 12.24/7.95 ( ( member @ I7 @ I4 @ I6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 12.24/7.95 => ( ( member @ I7 @ I @ I6 )
% 12.24/7.95 => ( ( finite_finite2 @ I7 @ I6 )
% 12.24/7.95 => ( ( F @ I )
% 12.24/7.95 = ( G @ I ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4338_sum__cong__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ nat,F: nat > A,G: nat > A] :
% 12.24/7.95 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 12.24/7.95 => ( ! [X3: nat] :
% 12.24/7.95 ( ( member @ nat @ ( suc @ X3 ) @ A6 )
% 12.24/7.95 => ( ( F @ ( suc @ X3 ) )
% 12.24/7.95 = ( G @ ( suc @ X3 ) ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ A6 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4339_zero__integer__def,axiom,
% 12.24/7.95 ( ( zero_zero @ code_integer )
% 12.24/7.95 = ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4340_less__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( ord_less @ int @ Xa3 @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4341_plus__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( plus_plus @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4342_uminus__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4343_sum_Ointer__filter,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A,P: B > $o] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [X2: B] :
% 12.24/7.95 ( ( member @ B @ X2 @ A6 )
% 12.24/7.95 & ( P @ X2 ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4344_one__integer__def,axiom,
% 12.24/7.95 ( ( one_one @ code_integer )
% 12.24/7.95 = ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4345_less__eq__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( ord_less_eq @ int @ Xa3 @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4346_minus__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( minus_minus @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4347_abs__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4348_divide__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( divide_divide @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4349_nat__of__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( nat2 @ X ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4350_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4351_sum_Oshift__bounds__Suc__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4352_modulo__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [Xa3: int,X: int] :
% 12.24/7.95 ( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( modulo_modulo @ int @ Xa3 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4353_sgn__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4354_effect__guardE,axiom,
% 12.24/7.95 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.24/7.95 ( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N )
% 12.24/7.95 => ~ ( ( H
% 12.24/7.95 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( N
% 12.24/7.95 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( R
% 12.24/7.95 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 12.24/7.95 => ~ ( P @ H2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4355_effect__guardI,axiom,
% 12.24/7.95 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
% 12.24/7.95 ( ( P @ H2 )
% 12.24/7.95 => ( ( H
% 12.24/7.95 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( N
% 12.24/7.95 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( R
% 12.24/7.95 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 12.24/7.95 => ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4356_sum__le__included,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S )
% 12.24/7.95 => ( ( finite_finite2 @ C @ T2 )
% 12.24/7.95 => ( ! [X3: C] :
% 12.24/7.95 ( ( member @ C @ X3 @ T2 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ S )
% 12.24/7.95 => ? [Xa: C] :
% 12.24/7.95 ( ( member @ C @ Xa @ T2 )
% 12.24/7.95 & ( ( I @ Xa )
% 12.24/7.95 = X3 )
% 12.24/7.95 & ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4357_sum__nonneg__eq__0__iff,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 )
% 12.24/7.95 = ( zero_zero @ A ) )
% 12.24/7.95 = ( ! [X2: B] :
% 12.24/7.95 ( ( member @ B @ X2 @ A6 )
% 12.24/7.95 => ( ( F @ X2 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4358_sum__strict__mono__ex1,axiom,
% 12.24/7.95 ! [A: $tType,I7: $tType] :
% 12.24/7.95 ( ( ordere8940638589300402666id_add @ A )
% 12.24/7.95 => ! [A6: set @ I7,F: I7 > A,G: I7 > A] :
% 12.24/7.95 ( ( finite_finite2 @ I7 @ A6 )
% 12.24/7.95 => ( ! [X3: I7] :
% 12.24/7.95 ( ( member @ I7 @ X3 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.95 => ( ? [X4: I7] :
% 12.24/7.95 ( ( member @ I7 @ X4 @ A6 )
% 12.24/7.95 & ( ord_less @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 12.24/7.95 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4359_sum_Orelated,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.95 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
% 12.24/7.95 => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
% 12.24/7.95 ( ( ( R2 @ X1 @ X23 )
% 12.24/7.95 & ( R2 @ Y1 @ Y22 ) )
% 12.24/7.95 => ( R2 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
% 12.24/7.95 => ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ S3 )
% 12.24/7.95 => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.95 => ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4360_sum__strict__mono,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( strict7427464778891057005id_add @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( A6
% 12.24/7.95 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ A6 )
% 12.24/7.95 => ( ord_less @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.95 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4361_sum_Oivl__cong,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ( ord @ B )
% 12.24/7.95 & ( comm_monoid_add @ A ) )
% 12.24/7.95 => ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 12.24/7.95 ( ( A3 = C2 )
% 12.24/7.95 => ( ( B2 = D2 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( ord_less_eq @ B @ C2 @ X3 )
% 12.24/7.95 => ( ( ord_less @ B @ X3 @ D2 )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( H2 @ X3 ) ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4362_sum_Oreindex__bij__witness__not__neutral,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S5 )
% 12.24/7.95 => ( ( finite_finite2 @ C @ T5 )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 12.24/7.95 => ( ( I @ ( J @ A8 ) )
% 12.24/7.95 = A8 ) )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
% 12.24/7.95 => ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 12.24/7.95 => ( ! [B5: C] :
% 12.24/7.95 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/7.95 => ( ( J @ ( I @ B5 ) )
% 12.24/7.95 = B5 ) )
% 12.24/7.95 => ( ! [B5: C] :
% 12.24/7.95 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/7.95 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ S5 )
% 12.24/7.95 => ( ( G @ A8 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [B5: C] :
% 12.24/7.95 ( ( member @ C @ B5 @ T5 )
% 12.24/7.95 => ( ( H2 @ B5 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ S3 )
% 12.24/7.95 => ( ( H2 @ ( J @ A8 ) )
% 12.24/7.95 = ( G @ A8 ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4363_sum_OatLeastLessThan__concat,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,P5: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( ord_less_eq @ nat @ N @ P5 )
% 12.24/7.95 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4364_sum__diff__nat__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,P5: nat,F: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( ord_less_eq @ nat @ N @ P5 )
% 12.24/7.95 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4365_sum__nonneg__0,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [S: set @ B,F: B > A,I: B] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ S )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
% 12.24/7.95 = ( zero_zero @ A ) )
% 12.24/7.95 => ( ( member @ B @ I @ S )
% 12.24/7.95 => ( ( F @ I )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4366_sum__nonneg__leq__bound,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [S: set @ B,F: B > A,B7: A,I: B] :
% 12.24/7.95 ( ( finite_finite2 @ B @ S )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ S )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
% 12.24/7.95 = B7 )
% 12.24/7.95 => ( ( member @ B @ I @ S )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ I ) @ B7 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4367_sum_Ointer__restrict,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A,B7: set @ B] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
% 12.24/7.95 @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4368_sum_Osetdiff__irrelevant,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G
% 12.24/7.95 @ ( minus_minus @ ( set @ B ) @ A6
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [X2: B] :
% 12.24/7.95 ( ( G @ X2 )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4369_sum__power__add,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,M: nat,I6: set @ nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I5 ) )
% 12.24/7.95 @ I6 )
% 12.24/7.95 = ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4370_sum_Onat__diff__reindex,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4371_sum__choose__upper,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4372_sum__pos2,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [I6: set @ B,I: B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ I6 )
% 12.24/7.95 => ( ( member @ B @ I @ I6 )
% 12.24/7.95 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ I6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 12.24/7.95 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4373_sum__pos,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [I6: set @ B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ I6 )
% 12.24/7.95 => ( ( I6
% 12.24/7.95 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ I6 )
% 12.24/7.95 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
% 12.24/7.95 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4374_sum_Omono__neutral__cong__right,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ S3 )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( H2 @ X3 ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4375_sum_Omono__neutral__cong__left,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.95 => ( ( H2 @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ S3 )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( H2 @ X3 ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4376_sum_Omono__neutral__right,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4377_sum_Omono__neutral__left,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [T6: set @ B,S3: set @ B,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4378_sum_Osame__carrierI,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ C6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 12.24/7.95 => ( ( G @ A8 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [B5: B] :
% 12.24/7.95 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 12.24/7.95 => ( ( H2 @ B5 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4379_sum_Osame__carrier,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ C6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
% 12.24/7.95 => ( ! [A8: B] :
% 12.24/7.95 ( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
% 12.24/7.95 => ( ( G @ A8 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [B5: B] :
% 12.24/7.95 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
% 12.24/7.95 => ( ( H2 @ B5 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) )
% 12.24/7.95 = ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4380_sum__bounded__below,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 & ( semiring_1 @ A ) )
% 12.24/7.95 => ! [A6: set @ B,K5: A,F: B > A] :
% 12.24/7.95 ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ K5 @ ( F @ I4 ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4381_sum__bounded__above,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 & ( semiring_1 @ A ) )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,K5: A] :
% 12.24/7.95 ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ I4 ) @ K5 ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4382_sum_Omono__neutral__cong,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ( finite_finite2 @ B @ S3 )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.95 => ( ( H2 @ I4 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
% 12.24/7.95 => ( ( G @ I4 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( H2 @ X3 ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4383_sum__shift__lb__Suc0__0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [F: nat > A,K: nat] :
% 12.24/7.95 ( ( ( F @ ( zero_zero @ nat ) )
% 12.24/7.95 = ( zero_zero @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4384_sum__shift__lb__Suc0__0__upt,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [F: nat > A,K: nat] :
% 12.24/7.95 ( ( ( F @ ( zero_zero @ nat ) )
% 12.24/7.95 = ( zero_zero @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4385_sum_OatLeast0__atMost__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4386_sum_OatLeast0__lessThan__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4387_sum_OatLeast__Suc__lessThan,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4388_sum_OatLeast__Suc__atMost,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4389_sum_Onat__ivl__Suc_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4390_sum_OatLeastLessThan__Suc,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A3: nat,B2: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4391_sum_Olast__plus,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4392_sum_OIf__cases,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4393_sum_OSuc__reindex__ivl,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ M )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4394_sum__Suc__diff,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,F: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4395_sum__Suc__diff_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,F: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4396_sum_OatMost__Suc__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4397_sum__telescope,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [F: nat > A,I: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( minus_minus @ A @ ( F @ I5 ) @ ( F @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ I ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4398_sum_Ohead,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4399_sum_OlessThan__Suc__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4400_sum__lessThan__telescope_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [F: nat > A,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [N3: nat] : ( minus_minus @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ M ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4401_sum__lessThan__telescope,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [F: nat > A,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [N3: nat] : ( minus_minus @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ M ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4402_sum_OatLeastLessThan__rev,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4403_sum_Onested__swap,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A3: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4404_sum_OatLeast1__atMost__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4405_sum__bounds__lt__plus1,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [F: nat > A,Mm: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ Mm ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4406_sum_Onat__group,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,K: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4407_sum__choose__lower,axiom,
% 12.24/7.95 ! [R: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R @ K3 ) @ K3 )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4408_sum_Onested__swap_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A3: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4409_choose__rising__sum_I1_J,axiom,
% 12.24/7.95 ! [N: nat,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4410_choose__rising__sum_I2_J,axiom,
% 12.24/7.95 ! [N: nat,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4411_sum__atLeastAtMost__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [F: nat > A,A3: nat,B2: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 12.24/7.95 = ( set_fo6178422350223883121st_nat @ A
% 12.24/7.95 @ ^ [A5: nat] : ( plus_plus @ A @ ( F @ A5 ) )
% 12.24/7.95 @ A3
% 12.24/7.95 @ B2
% 12.24/7.95 @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4412_card__sum__le__nat__sum,axiom,
% 12.24/7.95 ! [S3: set @ nat] :
% 12.24/7.95 ( ord_less_eq @ nat
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ S3 ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4413_sum__mono2,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( ordere6911136660526730532id_add @ A )
% 12.24/7.95 => ! [B7: set @ B,A6: set @ B,F: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ B7 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 12.24/7.95 => ( ! [B5: B] :
% 12.24/7.95 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B5 ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B7 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4414_sum_Ounion__inter__neutral,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [A6: set @ B,B7: set @ B,G: B > A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.95 => ( ( G @ X3 )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4415_sum_Ohead__if,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [N: nat,M: nat,G: nat > A] :
% 12.24/7.95 ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4416_sum_Oub__add__nat,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,G: nat > A,P5: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4417_sum__div__partition,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( euclid4440199948858584721cancel @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,B2: A] :
% 12.24/7.95 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ B2 )
% 12.24/7.95 = ( plus_plus @ A
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ B @ A
% 12.24/7.95 @ ^ [A5: B] : ( divide_divide @ A @ ( F @ A5 ) @ B2 )
% 12.24/7.95 @ ( inf_inf @ ( set @ B ) @ A6
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
% 12.24/7.95 @ ( divide_divide @ A
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ B @ A @ F
% 12.24/7.95 @ ( inf_inf @ ( set @ B ) @ A6
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [A5: B] :
% 12.24/7.95 ~ ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
% 12.24/7.95 @ B2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4418_one__diff__power__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 12.24/7.95 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4419_power__diff__1__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
% 12.24/7.95 = ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4420_geometric__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field @ A )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( X
% 12.24/7.95 != ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4421_set__encode__def,axiom,
% 12.24/7.95 ( nat_set_encode
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4422_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4423_sum_OatMost__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4424_gbinomial__parallel__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [A3: A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4425_sum__list__sum__nth,axiom,
% 12.24/7.95 ! [B: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ B )
% 12.24/7.95 => ( ( groups8242544230860333062m_list @ B )
% 12.24/7.95 = ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4426_sum__choose__diagonal,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4427_vandermonde,axiom,
% 12.24/7.95 ! [M: nat,N: nat,R: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ R ) )
% 12.24/7.95 = ( binomial @ ( plus_plus @ nat @ M @ N ) @ R ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4428_sum__strict__mono2,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ordere8940638589300402666id_add @ B )
% 12.24/7.95 => ! [B7: set @ A,A6: set @ A,B2: A,F: A > B] :
% 12.24/7.95 ( ( finite_finite2 @ A @ B7 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.95 => ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
% 12.24/7.95 => ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
% 12.24/7.95 => ( ! [X3: A] :
% 12.24/7.95 ( ( member @ A @ X3 @ B7 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
% 12.24/7.95 => ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4429_member__le__sum,axiom,
% 12.24/7.95 ! [B: $tType,C: $tType] :
% 12.24/7.95 ( ( ( ordere6911136660526730532id_add @ B )
% 12.24/7.95 & ( semiring_1 @ B ) )
% 12.24/7.95 => ! [I: C,A6: set @ C,F: C > B] :
% 12.24/7.95 ( ( member @ C @ I @ A6 )
% 12.24/7.95 => ( ! [X3: C] :
% 12.24/7.95 ( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A6 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
% 12.24/7.95 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
% 12.24/7.95 => ( ( finite_finite2 @ C @ A6 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4430_sum__bounded__above__divide,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( linordered_field @ A )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,K5: A] :
% 12.24/7.95 ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ A @ ( F @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) ) ) )
% 12.24/7.95 => ( ( finite_finite2 @ B @ A6 )
% 12.24/7.95 => ( ( A6
% 12.24/7.95 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.95 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ K5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4431_sum__bounded__above__strict,axiom,
% 12.24/7.95 ! [B: $tType,A: $tType] :
% 12.24/7.95 ( ( ( ordere8940638589300402666id_add @ A )
% 12.24/7.95 & ( semiring_1 @ A ) )
% 12.24/7.95 => ! [A6: set @ B,F: B > A,K5: A] :
% 12.24/7.95 ( ! [I4: B] :
% 12.24/7.95 ( ( member @ B @ I4 @ A6 )
% 12.24/7.95 => ( ord_less @ A @ ( F @ I4 ) @ K5 ) )
% 12.24/7.95 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A6 ) )
% 12.24/7.95 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4432_convex__sum__bound__le,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( linordered_idom @ B )
% 12.24/7.95 => ! [I6: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
% 12.24/7.95 ( ! [I4: A] :
% 12.24/7.95 ( ( member @ A @ I4 @ I6 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I4 ) ) )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
% 12.24/7.95 = ( one_one @ B ) )
% 12.24/7.95 => ( ! [I4: A] :
% 12.24/7.95 ( ( member @ A @ I4 @ I6 )
% 12.24/7.95 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I4 ) @ B2 ) ) @ Delta ) )
% 12.24/7.95 => ( ord_less_eq @ B
% 12.24/7.95 @ ( abs_abs @ B
% 12.24/7.95 @ ( minus_minus @ B
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ A @ B
% 12.24/7.95 @ ^ [I5: A] : ( times_times @ B @ ( A3 @ I5 ) @ ( X @ I5 ) )
% 12.24/7.95 @ I6 )
% 12.24/7.95 @ B2 ) )
% 12.24/7.95 @ Delta ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4433_sum__fun__comp,axiom,
% 12.24/7.95 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.95 ( ( semiring_1 @ C )
% 12.24/7.95 => ! [S3: set @ A,R2: set @ B,G: A > B,F: B > C] :
% 12.24/7.95 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.95 => ( ( finite_finite2 @ B @ R2 )
% 12.24/7.95 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R2 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ A @ C
% 12.24/7.95 @ ^ [X2: A] : ( F @ ( G @ X2 ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ C
% 12.24/7.95 @ ^ [Y6: B] :
% 12.24/7.95 ( times_times @ C
% 12.24/7.95 @ ( semiring_1_of_nat @ C
% 12.24/7.95 @ ( finite_card @ A
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [X2: A] :
% 12.24/7.95 ( ( member @ A @ X2 @ S3 )
% 12.24/7.95 & ( ( G @ X2 )
% 12.24/7.95 = Y6 ) ) ) ) )
% 12.24/7.95 @ ( F @ Y6 ) )
% 12.24/7.95 @ R2 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4434_sum__gp__basic,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
% 12.24/7.95 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4435_sum__natinterval__diff,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,F: nat > A] :
% 12.24/7.95 ( ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
% 12.24/7.95 & ( ~ ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4436_sum__gp__strict,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( division_ring @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( ( X
% 12.24/7.95 = ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( semiring_1_of_nat @ A @ N ) ) )
% 12.24/7.95 & ( ( X
% 12.24/7.95 != ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/7.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4437_diff__power__eq__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat,Y3: A] :
% 12.24/7.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y3 @ ( suc @ N ) ) )
% 12.24/7.95 = ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P6 ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4438_power__diff__sumr2,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat,Y3: A] :
% 12.24/7.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
% 12.24/7.95 = ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) ) @ ( power_power @ A @ X @ I5 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4439_sum__telescope_H_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ab_group_add @ A )
% 12.24/7.95 => ! [M: nat,N: nat,F: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
% 12.24/7.95 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4440_sum__power__shift,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [M: nat,N: nat,X: A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4441_take__bit__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.95 => ( ( bit_se2584673776208193580ke_bit @ A )
% 12.24/7.95 = ( ^ [N3: nat,A5: A] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4442_binomial,axiom,
% 12.24/7.95 ! [A3: nat,B2: nat,N: nat] :
% 12.24/7.95 ( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4443_mask__eq__sum__exp,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( semiring_parity @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4444_sum__gp__multiplied,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [M: nat,N: nat,X: A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.95 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
% 12.24/7.95 = ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4445_sum_Oin__pairs,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4446_one__diff__power__eq_H,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( monoid_mult @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 12.24/7.95 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4447_sum_Oin__pairs__0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4448_gbinomial__sum__lower__neg,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [A3: A,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4449_binomial__ring,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.95 ( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4450_pochhammer__binomial__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_ring_1 @ A )
% 12.24/7.95 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.95 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4451_horner__sum__eq__sum,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ( ( groups4207007520872428315er_sum @ B @ A )
% 12.24/7.95 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) @ ( power_power @ A @ A5 @ N3 ) )
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4452_choose__square__sum,axiom,
% 12.24/7.95 ! [N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4453_sum_Ozero__middle,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 12.24/7.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 12.24/7.95 => ( ( ord_less_eq @ nat @ K @ P5 )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ P5 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4454_mask__eq__sum__exp__nat,axiom,
% 12.24/7.95 ! [N: nat] :
% 12.24/7.95 ( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.95 @ ( collect @ nat
% 12.24/7.95 @ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4455_gauss__sum__nat,axiom,
% 12.24/7.95 ! [N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4456_gbinomial__sum__up__index,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [K: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4457_gbinomial__partial__sum__poly,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [M: nat,A3: A,X: A,Y3: A] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4458_sum__power2,axiom,
% 12.24/7.95 ! [K: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
% 12.24/7.95 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4459_Sum__Ico__nat,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.95 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4460_card__lists__length__le,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,N: nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.95 => ( ( finite_card @ ( list @ A )
% 12.24/7.95 @ ( collect @ ( list @ A )
% 12.24/7.95 @ ^ [Xs3: list @ A] :
% 12.24/7.95 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 12.24/7.95 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A6 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4461_double__gauss__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 12.24/7.95 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4462_double__arith__series,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [A3: A,D2: A,N: nat] :
% 12.24/7.95 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 12.24/7.95 = ( 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 ) @ D2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4463_sum__gp0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( division_ring @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,N: nat] :
% 12.24/7.95 ( ( ( X
% 12.24/7.95 = ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
% 12.24/7.95 & ( ( X
% 12.24/7.95 != ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4464_choose__alternating__linear__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_ring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( N
% 12.24/7.95 != ( one_one @ nat ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4465_gbinomial__partial__sum__poly__xpos,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [M: nat,A3: A,X: A,Y3: A] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4466_gbinomial__sum__nat__pow2,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4467_arith__series__nat,axiom,
% 12.24/7.95 ! [A3: nat,D2: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I5 @ D2 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4468_Sum__Icc__nat,axiom,
% 12.24/7.95 ! [M: nat,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [X2: nat] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4469_binomial__r__part__sum,axiom,
% 12.24/7.95 ! [M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4470_choose__linear__sum,axiom,
% 12.24/7.95 ! [N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ nat @ I5 @ ( binomial @ N @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4471_double__gauss__sum__from__Suc__0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_semiring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( 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 ) ) )
% 12.24/7.95 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4472_arith__series,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.95 => ! [A3: A,D2: A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( 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 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4473_gauss__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/7.95 = ( 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 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4474_sum__gp__offset,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( division_ring @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [X: A,M: nat,N: nat] :
% 12.24/7.95 ( ( ( X
% 12.24/7.95 = ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
% 12.24/7.95 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
% 12.24/7.95 & ( ( X
% 12.24/7.95 != ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
% 12.24/7.95 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4475_choose__alternating__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_ring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/7.95 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4476_divmod__step__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( unique1627219031080169319umeral @ A )
% 12.24/7.95 => ( ( unique1321980374590559556d_step @ A )
% 12.24/7.95 = ( ^ [L3: num] :
% 12.24/7.95 ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 12.24/7.95 @ ^ [Q3: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4477_gauss__sum__from__Suc__0,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( euclid5411537665997757685th_nat @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 12.24/7.95 = ( 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 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4478_gchoose__row__sum__weighted,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [R: A,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
% 12.24/7.95 = ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R @ ( suc @ M ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4479_sum__gp,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ( division_ring @ A )
% 12.24/7.95 & ( comm_ring @ A ) )
% 12.24/7.95 => ! [N: nat,M: nat,X: A] :
% 12.24/7.95 ( ( ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( zero_zero @ A ) ) )
% 12.24/7.95 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/7.95 => ( ( ( X
% 12.24/7.95 = ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
% 12.24/7.95 & ( ( X
% 12.24/7.95 != ( one_one @ A ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4480_gbinomial__r__part__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4481_gbinomial__partial__row__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( field_char_0 @ A )
% 12.24/7.95 => ! [A3: A,M: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ M ) )
% 12.24/7.95 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4482_choose__odd__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_ring_1 @ A )
% 12.24/7.95 => ! [N: nat] :
% 12.24/7.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.95 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 12.24/7.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] :
% 12.24/7.95 ( if @ A
% 12.24/7.95 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 )
% 12.24/7.95 @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) )
% 12.24/7.95 @ ( zero_zero @ A ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) )
% 12.24/7.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4483_map__entry__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_map_entry @ A )
% 12.24/7.95 = ( ^ [I5: nat,F2: A > A,A5: array @ A] :
% 12.24/7.95 ( heap_Time_guard @ ( array @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) ) @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4484_Array__Time_Onth__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_nth @ A )
% 12.24/7.95 = ( ^ [A5: array @ A,I5: nat] :
% 12.24/7.95 ( heap_Time_guard @ A
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4485_upd__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_upd @ A )
% 12.24/7.95 = ( ^ [I5: nat,X2: A,A5: array @ A] :
% 12.24/7.95 ( heap_Time_guard @ ( array @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4486_nat__of__integer__code,axiom,
% 12.24/7.95 ( code_nat_of_integer
% 12.24/7.95 = ( ^ [K3: code_integer] :
% 12.24/7.95 ( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ nat
% 12.24/7.95 @ ^ [L3: code_integer,J3: code_integer] :
% 12.24/7.95 ( if @ nat
% 12.24/7.95 @ ( J3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
% 12.24/7.95 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
% 12.24/7.95 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4487_int__sum,axiom,
% 12.24/7.95 ! [B: $tType,F: B > nat,A6: set @ B] :
% 12.24/7.95 ( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ B @ int
% 12.24/7.95 @ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
% 12.24/7.95 @ A6 ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4488_sum_Otriangle__reindex__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/7.95 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.95 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4489_sum_Otriangle__reindex,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_add @ A )
% 12.24/7.95 => ! [G: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/7.95 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.95 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4490_sum__subtractf__nat,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,G: A > nat,F: A > nat] :
% 12.24/7.95 ( ! [X3: A] :
% 12.24/7.95 ( ( member @ A @ X3 @ A6 )
% 12.24/7.95 => ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4491_card__eq__sum,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( finite_card @ A )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4492_prod_Otriangle__reindex__eq,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/7.95 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.95 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4493_sum__eq__Suc0__iff,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 12.24/7.95 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.95 = ( ? [X2: A] :
% 12.24/7.95 ( ( member @ A @ X2 @ A6 )
% 12.24/7.95 & ( ( F @ X2 )
% 12.24/7.95 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.95 & ! [Y6: A] :
% 12.24/7.95 ( ( member @ A @ Y6 @ A6 )
% 12.24/7.95 => ( ( X2 != Y6 )
% 12.24/7.95 => ( ( F @ Y6 )
% 12.24/7.95 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4494_sum__SucD,axiom,
% 12.24/7.95 ! [A: $tType,F: A > nat,A6: set @ A,N: nat] :
% 12.24/7.95 ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 12.24/7.95 = ( suc @ N ) )
% 12.24/7.95 => ? [X3: A] :
% 12.24/7.95 ( ( member @ A @ X3 @ A6 )
% 12.24/7.95 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4495_sum__eq__1__iff,axiom,
% 12.24/7.95 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.95 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
% 12.24/7.95 = ( one_one @ nat ) )
% 12.24/7.95 = ( ? [X2: A] :
% 12.24/7.95 ( ( member @ A @ X2 @ A6 )
% 12.24/7.95 & ( ( F @ X2 )
% 12.24/7.95 = ( one_one @ nat ) )
% 12.24/7.95 & ! [Y6: A] :
% 12.24/7.95 ( ( member @ A @ Y6 @ A6 )
% 12.24/7.95 => ( ( X2 != Y6 )
% 12.24/7.95 => ( ( F @ Y6 )
% 12.24/7.95 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4496_prod_Otriangle__reindex,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( comm_monoid_mult @ A )
% 12.24/7.95 => ! [G: nat > nat > A,N: nat] :
% 12.24/7.95 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/7.95 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.95 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/7.95 = ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [K3: nat] :
% 12.24/7.95 ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.95 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/7.95 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4497_sum__Suc,axiom,
% 12.24/7.95 ! [A: $tType,F: A > nat,A6: set @ A] :
% 12.24/7.95 ( ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( suc @ ( F @ X2 ) )
% 12.24/7.95 @ A6 )
% 12.24/7.95 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( finite_card @ A @ A6 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4498_sum__multicount,axiom,
% 12.24/7.95 ! [A: $tType,B: $tType,S3: set @ A,T6: set @ B,R2: A > B > $o,K: nat] :
% 12.24/7.95 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.95 => ( ( finite_finite2 @ B @ T6 )
% 12.24/7.95 => ( ! [X3: B] :
% 12.24/7.95 ( ( member @ B @ X3 @ T6 )
% 12.24/7.95 => ( ( finite_card @ A
% 12.24/7.95 @ ( collect @ A
% 12.24/7.95 @ ^ [I5: A] :
% 12.24/7.95 ( ( member @ A @ I5 @ S3 )
% 12.24/7.95 & ( R2 @ I5 @ X3 ) ) ) )
% 12.24/7.95 = K ) )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [I5: A] :
% 12.24/7.95 ( finite_card @ B
% 12.24/7.95 @ ( collect @ B
% 12.24/7.95 @ ^ [J3: B] :
% 12.24/7.95 ( ( member @ B @ J3 @ T6 )
% 12.24/7.95 & ( R2 @ I5 @ J3 ) ) ) )
% 12.24/7.95 @ S3 )
% 12.24/7.95 = ( times_times @ nat @ K @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4499_sum__diff__distrib,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( ord @ A )
% 12.24/7.95 => ! [Q: A > nat,P: A > nat,N: A] :
% 12.24/7.95 ( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 12.24/7.95 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 12.24/7.95 @ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4500_prod__encode__def,axiom,
% 12.24/7.95 ( nat_prod_encode
% 12.24/7.95 = ( product_case_prod @ nat @ nat @ nat
% 12.24/7.95 @ ^ [M2: nat,N3: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N3 ) ) @ M2 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4501_rat__less__code,axiom,
% 12.24/7.95 ( ( ord_less @ rat )
% 12.24/7.95 = ( ^ [P6: rat,Q3: rat] :
% 12.24/7.95 ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
% 12.24/7.95 @ ( quotient_of @ Q3 ) )
% 12.24/7.95 @ ( quotient_of @ P6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4502_rat__less__eq__code,axiom,
% 12.24/7.95 ( ( ord_less_eq @ rat )
% 12.24/7.95 = ( ^ [P6: rat,Q3: rat] :
% 12.24/7.95 ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
% 12.24/7.95 @ ( quotient_of @ Q3 ) )
% 12.24/7.95 @ ( quotient_of @ P6 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4503_rat__uminus__code,axiom,
% 12.24/7.95 ! [P5: rat] :
% 12.24/7.95 ( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4504_lenlex__conv,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( lenlex @ A )
% 12.24/7.95 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 12.24/7.95 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 12.24/7.95 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 12.24/7.95 @ ^ [Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.95 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.95 | ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.95 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.95 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4505_Divides_Oadjust__div__def,axiom,
% 12.24/7.95 ( adjust_div
% 12.24/7.95 = ( product_case_prod @ int @ int @ int
% 12.24/7.95 @ ^ [Q3: int,R5: int] :
% 12.24/7.95 ( plus_plus @ int @ Q3
% 12.24/7.95 @ ( zero_neq_one_of_bool @ int
% 12.24/7.95 @ ( R5
% 12.24/7.95 != ( zero_zero @ int ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4506_sum__list__map__eq__sum__count,axiom,
% 12.24/7.95 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 12.24/7.95 ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
% 12.24/7.95 @ ( set2 @ A @ Xs ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4507_rat__divide__code,axiom,
% 12.24/7.95 ! [P5: rat,Q4: rat] :
% 12.24/7.95 ( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q4 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) ) )
% 12.24/7.95 @ ( quotient_of @ Q4 ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4508_rat__times__code,axiom,
% 12.24/7.95 ! [P5: rat,Q4: rat] :
% 12.24/7.95 ( ( quotient_of @ ( times_times @ rat @ P5 @ Q4 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B3 ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 12.24/7.95 @ ( quotient_of @ Q4 ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4509_rat__plus__code,axiom,
% 12.24/7.95 ! [P5: rat,Q4: rat] :
% 12.24/7.95 ( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q4 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 12.24/7.95 @ ( quotient_of @ Q4 ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4510_rat__minus__code,axiom,
% 12.24/7.95 ! [P5: rat,Q4: rat] :
% 12.24/7.95 ( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q4 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int,C4: int] :
% 12.24/7.95 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
% 12.24/7.95 @ ( quotient_of @ Q4 ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4511_sum__list__map__eq__sum__count2,axiom,
% 12.24/7.95 ! [A: $tType,Xs: list @ A,X8: set @ A,F: A > nat] :
% 12.24/7.95 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
% 12.24/7.95 => ( ( finite_finite2 @ A @ X8 )
% 12.24/7.95 => ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
% 12.24/7.95 = ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.95 @ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
% 12.24/7.95 @ X8 ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4512_divmod__step__nat__def,axiom,
% 12.24/7.95 ( ( unique1321980374590559556d_step @ nat )
% 12.24/7.95 = ( ^ [L3: num] :
% 12.24/7.95 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ^ [Q3: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4513_rat__inverse__code,axiom,
% 12.24/7.95 ! [P5: rat] :
% 12.24/7.95 ( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
% 12.24/7.95 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [A5: int,B3: int] :
% 12.24/7.95 ( if @ ( product_prod @ int @ int )
% 12.24/7.95 @ ( A5
% 12.24/7.95 = ( zero_zero @ int ) )
% 12.24/7.95 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.95 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B3 ) @ ( abs_abs @ int @ A5 ) ) )
% 12.24/7.95 @ ( quotient_of @ P5 ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4514_divmod__step__int__def,axiom,
% 12.24/7.95 ( ( unique1321980374590559556d_step @ int )
% 12.24/7.95 = ( ^ [L3: num] :
% 12.24/7.95 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 12.24/7.95 @ ^ [Q3: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4515_Sum__Icc__int,axiom,
% 12.24/7.95 ! [M: int,N: int] :
% 12.24/7.95 ( ( ord_less_eq @ int @ M @ N )
% 12.24/7.95 => ( ( groups7311177749621191930dd_sum @ int @ int
% 12.24/7.95 @ ^ [X2: int] : X2
% 12.24/7.95 @ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
% 12.24/7.95 = ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4516_divmod__step__integer__def,axiom,
% 12.24/7.95 ( ( unique1321980374590559556d_step @ code_integer )
% 12.24/7.95 = ( ^ [L3: num] :
% 12.24/7.95 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ^ [Q3: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4517_num__of__integer__code,axiom,
% 12.24/7.95 ( code_num_of_integer
% 12.24/7.95 = ( ^ [K3: code_integer] :
% 12.24/7.95 ( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ num
% 12.24/7.95 @ ^ [L3: code_integer,J3: code_integer] :
% 12.24/7.95 ( if @ num
% 12.24/7.95 @ ( J3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
% 12.24/7.95 @ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
% 12.24/7.95 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4518_Array__Time_Omake__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_make @ A )
% 12.24/7.95 = ( ^ [N3: nat,F2: nat > A] :
% 12.24/7.95 ( heap_Time_heap @ ( array @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] :
% 12.24/7.95 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.95 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
% 12.24/7.95 @ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) @ H4 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4519_new__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_new @ A )
% 12.24/7.95 = ( ^ [N3: nat,X2: A] :
% 12.24/7.95 ( heap_Time_heap @ ( array @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] :
% 12.24/7.95 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.95 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
% 12.24/7.95 @ ( array_alloc @ A @ ( replicate @ A @ N3 @ X2 ) @ H4 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4520_of__list__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_of_list @ A )
% 12.24/7.95 = ( ^ [Xs3: list @ A] :
% 12.24/7.95 ( heap_Time_heap @ ( array @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] :
% 12.24/7.95 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.95 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
% 12.24/7.95 @ ( array_alloc @ A @ Xs3 @ H4 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4521_num__of__integer_Oabs__eq,axiom,
% 12.24/7.95 ! [X: int] :
% 12.24/7.95 ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
% 12.24/7.95 = ( num_of_nat @ ( nat2 @ X ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4522_return__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap_Time_return @ A )
% 12.24/7.95 = ( ^ [X2: A] :
% 12.24/7.95 ( heap_Time_heap @ A
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4523_ureturn__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap_Time_ureturn @ A )
% 12.24/7.95 = ( ^ [X2: A] :
% 12.24/7.95 ( heap_Time_heap @ A
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4524_effect__heapE,axiom,
% 12.24/7.95 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.24/7.95 ( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N )
% 12.24/7.95 => ~ ( ( H
% 12.24/7.95 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( N
% 12.24/7.95 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( R
% 12.24/7.95 != ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4525_effect__heapI,axiom,
% 12.24/7.95 ! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A] :
% 12.24/7.95 ( ( N
% 12.24/7.95 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( H
% 12.24/7.95 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
% 12.24/7.95 => ( ( R
% 12.24/7.95 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
% 12.24/7.95 => ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4526_divmod__nat__if,axiom,
% 12.24/7.95 ( divmod_nat
% 12.24/7.95 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.95 ( if @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ( ( N3
% 12.24/7.95 = ( zero_zero @ nat ) )
% 12.24/7.95 | ( ord_less @ nat @ M2 @ N3 ) )
% 12.24/7.95 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
% 12.24/7.95 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.95 @ ^ [Q3: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q3 ) )
% 12.24/7.95 @ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4527_int__ge__less__than__def,axiom,
% 12.24/7.95 ( int_ge_less_than
% 12.24/7.95 = ( ^ [D5: int] :
% 12.24/7.95 ( collect @ ( product_prod @ int @ int )
% 12.24/7.95 @ ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [Z8: int,Z7: int] :
% 12.24/7.95 ( ( ord_less_eq @ int @ D5 @ Z8 )
% 12.24/7.95 & ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4528_int__ge__less__than2__def,axiom,
% 12.24/7.95 ( int_ge_less_than2
% 12.24/7.95 = ( ^ [D5: int] :
% 12.24/7.95 ( collect @ ( product_prod @ int @ int )
% 12.24/7.95 @ ( product_case_prod @ int @ int @ $o
% 12.24/7.95 @ ^ [Z8: int,Z7: int] :
% 12.24/7.95 ( ( ord_less_eq @ int @ D5 @ Z7 )
% 12.24/7.95 & ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4529_divmod__integer__code,axiom,
% 12.24/7.95 ( code_divmod_integer
% 12.24/7.95 = ( ^ [K3: code_integer,L3: code_integer] :
% 12.24/7.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ( K3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L3 )
% 12.24/7.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ^ [R5: code_integer,S6: code_integer] :
% 12.24/7.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ( S6
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S6 ) ) )
% 12.24/7.95 @ ( code_divmod_abs @ K3 @ L3 ) ) )
% 12.24/7.95 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ( L3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
% 12.24/7.95 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
% 12.24/7.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L3 )
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ^ [R5: code_integer,S6: code_integer] :
% 12.24/7.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.95 @ ( S6
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S6 ) ) )
% 12.24/7.95 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4530_freeze__def,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( heap @ A )
% 12.24/7.95 => ( ( array_freeze @ A )
% 12.24/7.95 = ( ^ [A5: array @ A] :
% 12.24/7.95 ( heap_Time_heap @ ( list @ A )
% 12.24/7.95 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H4 @ A5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H4 @ A5 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4531_int__of__integer__code,axiom,
% 12.24/7.95 ( code_int_of_integer
% 12.24/7.95 = ( ^ [K3: code_integer] :
% 12.24/7.95 ( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
% 12.24/7.95 @ ( if @ int
% 12.24/7.95 @ ( K3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( zero_zero @ int )
% 12.24/7.95 @ ( product_case_prod @ code_integer @ code_integer @ int
% 12.24/7.95 @ ^ [L3: code_integer,J3: code_integer] :
% 12.24/7.95 ( if @ int
% 12.24/7.95 @ ( J3
% 12.24/7.95 = ( zero_zero @ code_integer ) )
% 12.24/7.95 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
% 12.24/7.95 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
% 12.24/7.95 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4532_of__nat__code,axiom,
% 12.24/7.95 ! [A: $tType] :
% 12.24/7.95 ( ( semiring_1 @ A )
% 12.24/7.95 => ( ( semiring_1_of_nat @ A )
% 12.24/7.95 = ( ^ [N3: nat] :
% 12.24/7.95 ( semiri8178284476397505188at_aux @ A
% 12.24/7.95 @ ^ [I5: A] : ( plus_plus @ A @ I5 @ ( one_one @ A ) )
% 12.24/7.95 @ N3
% 12.24/7.95 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.95
% 12.24/7.95 thf(fact_4533_inverse__rat__def,axiom,
% 12.24/7.95 ( ( inverse_inverse @ rat )
% 12.24/7.95 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 12.24/7.95 @ ^ [X2: product_prod @ int @ int] :
% 12.24/7.95 ( if @ ( product_prod @ int @ int )
% 12.24/7.95 @ ( ( product_fst @ int @ int @ X2 )
% 12.24/7.95 = ( zero_zero @ int ) )
% 12.24/7.95 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.95 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4534_int__of__integer__inverse,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
% 12.24/7.96 = X ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4535_int__of__integer__integer__of__int,axiom,
% 12.24/7.96 ! [K: int] :
% 12.24/7.96 ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
% 12.24/7.96 = K ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4536_integer__of__int__int__of__integer,axiom,
% 12.24/7.96 ! [K: code_integer] :
% 12.24/7.96 ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
% 12.24/7.96 = K ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4537_int__of__integer__sub,axiom,
% 12.24/7.96 ! [K: num,L: num] :
% 12.24/7.96 ( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
% 12.24/7.96 = ( neg_numeral_sub @ int @ K @ L ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4538_int__of__integer__min,axiom,
% 12.24/7.96 ! [K: code_integer,L: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
% 12.24/7.96 = ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4539_of__int__integer__of,axiom,
% 12.24/7.96 ! [K: code_integer] :
% 12.24/7.96 ( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
% 12.24/7.96 = K ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4540_int__of__integer__of__int,axiom,
% 12.24/7.96 ! [K: int] :
% 12.24/7.96 ( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
% 12.24/7.96 = K ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4541_zero__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
% 12.24/7.96 = ( zero_zero @ int ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4542_int__of__integer__numeral,axiom,
% 12.24/7.96 ! [K: num] :
% 12.24/7.96 ( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
% 12.24/7.96 = ( numeral_numeral @ int @ K ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4543_plus__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4544_uminus__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
% 12.24/7.96 = ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4545_times__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4546_one__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( code_int_of_integer @ ( one_one @ code_integer ) )
% 12.24/7.96 = ( one_one @ int ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4547_minus__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4548_abs__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
% 12.24/7.96 = ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4549_divide__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4550_modulo__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4551_sgn__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
% 12.24/7.96 = ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4552_int__of__integer__of__nat,axiom,
% 12.24/7.96 ! [N: nat] :
% 12.24/7.96 ( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 12.24/7.96 = ( semiring_1_of_nat @ int @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4553_division__segment__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
% 12.24/7.96 = ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4554_euclidean__size__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( euclid6346220572633701492n_size @ code_integer )
% 12.24/7.96 = ( ^ [X2: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4555_int__of__integer__max,axiom,
% 12.24/7.96 ! [K: code_integer,L: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
% 12.24/7.96 = ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4556_integer__eq__iff,axiom,
% 12.24/7.96 ( ( ^ [Y: code_integer,Z2: code_integer] : ( Y = Z2 ) )
% 12.24/7.96 = ( ^ [K3: code_integer,L3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ K3 )
% 12.24/7.96 = ( code_int_of_integer @ L3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4557_integer__eqI,axiom,
% 12.24/7.96 ! [K: code_integer,L: code_integer] :
% 12.24/7.96 ( ( ( code_int_of_integer @ K )
% 12.24/7.96 = ( code_int_of_integer @ L ) )
% 12.24/7.96 => ( K = L ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4558_int__of__integer__inject,axiom,
% 12.24/7.96 ! [X: code_integer,Y3: code_integer] :
% 12.24/7.96 ( ( ( code_int_of_integer @ X )
% 12.24/7.96 = ( code_int_of_integer @ Y3 ) )
% 12.24/7.96 = ( X = Y3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4559_less__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( ord_less @ code_integer )
% 12.24/7.96 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4560_integer__less__iff,axiom,
% 12.24/7.96 ( ( ord_less @ code_integer )
% 12.24/7.96 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4561_integer__less__eq__iff,axiom,
% 12.24/7.96 ( ( ord_less_eq @ code_integer )
% 12.24/7.96 = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4562_less__eq__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( ord_less_eq @ code_integer )
% 12.24/7.96 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4563_nat__of__integer_Orep__eq,axiom,
% 12.24/7.96 ( code_nat_of_integer
% 12.24/7.96 = ( ^ [X2: code_integer] : ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4564_take__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4565_not__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
% 12.24/7.96 = ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4566_and__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4567_bit__integer_Orep__eq,axiom,
% 12.24/7.96 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 12.24/7.96 = ( ^ [X2: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4568_or__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4569_xor__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: code_integer,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4570_push__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4571_drop__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4572_mask__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
% 12.24/7.96 = ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4573_unset__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4574_set__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4575_flip__bit__integer_Orep__eq,axiom,
% 12.24/7.96 ! [X: nat,Xa3: code_integer] :
% 12.24/7.96 ( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa3 ) )
% 12.24/7.96 = ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4576_of__nat__aux_Osimps_I2_J,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [Inc: A > A,N: nat,I: A] :
% 12.24/7.96 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 12.24/7.96 = ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4577_of__nat__aux_Osimps_I1_J,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [Inc: A > A,I: A] :
% 12.24/7.96 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 12.24/7.96 = I ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4578_num__of__integer_Orep__eq,axiom,
% 12.24/7.96 ( code_num_of_integer
% 12.24/7.96 = ( ^ [X2: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4579_uminus__rat__def,axiom,
% 12.24/7.96 ( ( uminus_uminus @ rat )
% 12.24/7.96 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 12.24/7.96 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4580_plus__rat__def,axiom,
% 12.24/7.96 ( ( plus_plus @ rat )
% 12.24/7.96 = ( 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 )
% 12.24/7.96 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4581_times__rat__def,axiom,
% 12.24/7.96 ( ( times_times @ rat )
% 12.24/7.96 = ( 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 )
% 12.24/7.96 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4582_transpose__rectangle,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
% 12.24/7.96 ( ( ( Xs
% 12.24/7.96 = ( nil @ ( list @ A ) ) )
% 12.24/7.96 => ( N
% 12.24/7.96 = ( zero_zero @ nat ) ) )
% 12.24/7.96 => ( ! [I4: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 12.24/7.96 => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) )
% 12.24/7.96 = N ) )
% 12.24/7.96 => ( ( transpose @ A @ Xs )
% 12.24/7.96 = ( map @ nat @ ( list @ A )
% 12.24/7.96 @ ^ [I5: nat] :
% 12.24/7.96 ( map @ nat @ A
% 12.24/7.96 @ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I5 )
% 12.24/7.96 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4583_sorted__wrt__less__sum__mono__lowerbound,axiom,
% 12.24/7.96 ! [B: $tType] :
% 12.24/7.96 ( ( ordere6911136660526730532id_add @ B )
% 12.24/7.96 => ! [F: nat > B,Ns: list @ nat] :
% 12.24/7.96 ( ! [X3: nat,Y5: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ X3 @ Y5 )
% 12.24/7.96 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 12.24/7.96 => ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
% 12.24/7.96 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F @ Ns ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4584_strict__sorted__imp__sorted,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4585_sorted2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [X: A,Y3: A,Zs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
% 12.24/7.96 = ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4586_sorted0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4587_strict__sorted__simps_I1_J,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4588_strict__sorted__equal,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,Ys2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
% 12.24/7.96 => ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 )
% 12.24/7.96 => ( ( ( set2 @ A @ Ys2 )
% 12.24/7.96 = ( set2 @ A @ Xs ) )
% 12.24/7.96 => ( Ys2 = Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4589_sorted__take,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,N: nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4590_sorted__replicate,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4591_sorted__drop,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,N: nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4592_sorted__remdups__adj,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4593_sorted__wrt__upt,axiom,
% 12.24/7.96 ! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4594_sorted__upt,axiom,
% 12.24/7.96 ! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4595_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4596_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4597_sorted__map,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [F: B > A,Xs: list @ B] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.96 = ( sorted_wrt @ B
% 12.24/7.96 @ ^ [X2: B,Y6: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y6 ) )
% 12.24/7.96 @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4598_sorted1,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4599_sorted__simps_I2_J,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [X: A,Ys2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys2 ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ X2 ) )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4600_strict__sorted__simps_I2_J,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [X: A,Ys2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys2 ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
% 12.24/7.96 => ( ord_less @ A @ X @ X2 ) )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4601_sorted__append,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,Ys2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys2 ) )
% 12.24/7.96 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
% 12.24/7.96 & ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 12.24/7.96 => ! [Y6: A] :
% 12.24/7.96 ( ( member @ A @ Y6 @ ( set2 @ A @ Ys2 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4602_strict__sorted__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [L: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
% 12.24/7.96 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 12.24/7.96 & ( distinct @ A @ L ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4603_sorted__distinct__set__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,Ys2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( ( distinct @ A @ Xs )
% 12.24/7.96 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
% 12.24/7.96 => ( ( distinct @ A @ Ys2 )
% 12.24/7.96 => ( ( ( set2 @ A @ Xs )
% 12.24/7.96 = ( set2 @ A @ Ys2 ) )
% 12.24/7.96 => ( Xs = Ys2 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4604_sorted__wrt01,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ A,P: A > A > $o] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 12.24/7.96 => ( sorted_wrt @ A @ P @ Xs ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4605_sorted__wrt__iff__nth__less,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( sorted_wrt @ A )
% 12.24/7.96 = ( ^ [P4: A > A > $o,Xs3: list @ A] :
% 12.24/7.96 ! [I5: nat,J3: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ J3 )
% 12.24/7.96 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.96 => ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4606_sorted__wrt__nth__less,axiom,
% 12.24/7.96 ! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ P @ Xs )
% 12.24/7.96 => ( ( ord_less @ nat @ I @ J )
% 12.24/7.96 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4607_sorted__butlast,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( Xs
% 12.24/7.96 != ( nil @ A ) )
% 12.24/7.96 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4608_sorted__enumerate,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4609_sorted01,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4610_finite__sorted__distinct__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.96 => ? [X3: list @ A] :
% 12.24/7.96 ( ( ( set2 @ A @ X3 )
% 12.24/7.96 = A6 )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
% 12.24/7.96 & ( distinct @ A @ X3 )
% 12.24/7.96 & ! [Y4: list @ A] :
% 12.24/7.96 ( ( ( ( set2 @ A @ Y4 )
% 12.24/7.96 = A6 )
% 12.24/7.96 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
% 12.24/7.96 & ( distinct @ A @ Y4 ) )
% 12.24/7.96 => ( Y4 = X3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4611_sorted__iff__nth__mono__less,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 = ( ! [I5: nat,J3: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ J3 )
% 12.24/7.96 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4612_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( ( distinct @ A @ Xs )
% 12.24/7.96 => ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
% 12.24/7.96 = Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4613_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.96 => ~ ! [L2: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
% 12.24/7.96 => ( ( ( set2 @ A @ L2 )
% 12.24/7.96 = A6 )
% 12.24/7.96 => ( ( size_size @ ( list @ A ) @ L2 )
% 12.24/7.96 != ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4614_sorted__iff__nth__Suc,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 = ( ! [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4615_sorted__nth__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,I: nat,J: nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.96 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4616_sorted__iff__nth__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 = ( ! [I5: nat,J3: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ I5 @ J3 )
% 12.24/7.96 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4617_sorted__wrt__less__idx,axiom,
% 12.24/7.96 ! [Ns: list @ nat,I: nat] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
% 12.24/7.96 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
% 12.24/7.96 => ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4618_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [A6: set @ A,L: list @ A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.96 => ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
% 12.24/7.96 & ( ( set2 @ A @ L )
% 12.24/7.96 = A6 )
% 12.24/7.96 & ( ( size_size @ ( list @ A ) @ L )
% 12.24/7.96 = ( finite_card @ A @ A6 ) ) )
% 12.24/7.96 = ( ( linord4507533701916653071of_set @ A @ A6 )
% 12.24/7.96 = L ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4619_length__transpose__sorted,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A )] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 => ( ( ( Xs
% 12.24/7.96 = ( nil @ ( list @ A ) ) )
% 12.24/7.96 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 12.24/7.96 = ( zero_zero @ nat ) ) )
% 12.24/7.96 & ( ( Xs
% 12.24/7.96 != ( nil @ ( list @ A ) ) )
% 12.24/7.96 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 12.24/7.96 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4620_sum__diff1_H__aux,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( ab_group_add @ B )
% 12.24/7.96 => ! [F4: set @ A,I6: set @ A,F: A > B,I: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ F4 )
% 12.24/7.96 => ( ( ord_less_eq @ ( set @ A )
% 12.24/7.96 @ ( collect @ A
% 12.24/7.96 @ ^ [I5: A] :
% 12.24/7.96 ( ( member @ A @ I5 @ I6 )
% 12.24/7.96 & ( ( F @ I5 )
% 12.24/7.96 != ( zero_zero @ B ) ) ) )
% 12.24/7.96 @ F4 )
% 12.24/7.96 => ( ( ( member @ A @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.96 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
% 12.24/7.96 & ( ~ ( member @ A @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4621_finite__mono__remains__stable__implies__strict__prefix,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order @ A )
% 12.24/7.96 => ! [F: nat > A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) )
% 12.24/7.96 => ( ( order_mono @ nat @ A @ F )
% 12.24/7.96 => ( ! [N4: nat] :
% 12.24/7.96 ( ( ( F @ N4 )
% 12.24/7.96 = ( F @ ( suc @ N4 ) ) )
% 12.24/7.96 => ( ( F @ ( suc @ N4 ) )
% 12.24/7.96 = ( F @ ( suc @ ( suc @ N4 ) ) ) ) )
% 12.24/7.96 => ? [N8: nat] :
% 12.24/7.96 ( ! [N5: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ N5 @ N8 )
% 12.24/7.96 => ! [M3: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M3 @ N8 )
% 12.24/7.96 => ( ( ord_less @ nat @ M3 @ N5 )
% 12.24/7.96 => ( ord_less @ A @ ( F @ M3 ) @ ( F @ N5 ) ) ) ) )
% 12.24/7.96 & ! [N5: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ N8 @ N5 )
% 12.24/7.96 => ( ( F @ N8 )
% 12.24/7.96 = ( F @ N5 ) ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4622_top__apply,axiom,
% 12.24/7.96 ! [C: $tType,D: $tType] :
% 12.24/7.96 ( ( top @ C )
% 12.24/7.96 => ( ( top_top @ ( D > C ) )
% 12.24/7.96 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4623_inf__top__left,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 = X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4624_inf__top__right,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4625_inf__eq__top__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ( X
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 & ( Y3
% 12.24/7.96 = ( top_top @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4626_top__eq__inf__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ( top_top @ A )
% 12.24/7.96 = ( inf_inf @ A @ X @ Y3 ) )
% 12.24/7.96 = ( ( X
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 & ( Y3
% 12.24/7.96 = ( top_top @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4627_inf__top_Oeq__neutr__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [A3: A,B2: A] :
% 12.24/7.96 ( ( ( inf_inf @ A @ A3 @ B2 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ( A3
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 & ( B2
% 12.24/7.96 = ( top_top @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4628_inf__top_Oleft__neutral,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
% 12.24/7.96 = A3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4629_inf__top_Oneutr__eq__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [A3: A,B2: A] :
% 12.24/7.96 ( ( ( top_top @ A )
% 12.24/7.96 = ( inf_inf @ A @ A3 @ B2 ) )
% 12.24/7.96 = ( ( A3
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 & ( B2
% 12.24/7.96 = ( top_top @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4630_inf__top_Oright__neutral,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
% 12.24/7.96 = A3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4631_boolean__algebra_Odisj__one__right,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4632_boolean__algebra_Odisj__one__left,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4633_sup__top__right,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounded_lattice_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4634_sup__top__left,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounded_lattice_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4635_max__top,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( ord_max @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4636_max__top2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( ord_max @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4637_min__top2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( ord_min @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4638_min__top,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( ord_min @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 = X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4639_boolean__algebra_Ocompl__zero,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4640_boolean__algebra_Ocompl__one,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 12.24/7.96 = ( bot_bot @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4643_sup__compl__top__left2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4644_sup__compl__top__left1,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y3 ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4645_finite__compl,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.96 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.96 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4646_Gcd__UNIV,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_Gcd @ A )
% 12.24/7.96 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
% 12.24/7.96 = ( one_one @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4647_sum_Oempty_H,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [P5: B > A] :
% 12.24/7.96 ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 = ( zero_zero @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4648_sum_Oinsert_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [I6: set @ B,P5: B > A,I: B] :
% 12.24/7.96 ( ( finite_finite2 @ B
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 & ( ( P5 @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 => ( ( ( member @ B @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
% 12.24/7.96 & ( ~ ( member @ B @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 12.24/7.96 = ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4649_boolean__algebra_Oconj__one__right,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 12.24/7.96 = X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4650_top__greatest,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4651_top_Oextremum__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 12.24/7.96 = ( A3
% 12.24/7.96 = ( top_top @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4652_top_Oextremum__uniqueI,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 12.24/7.96 => ( A3
% 12.24/7.96 = ( top_top @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4653_top_Oextremum__strict,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4654_top_Onot__eq__extremum,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [A3: A] :
% 12.24/7.96 ( ( A3
% 12.24/7.96 != ( top_top @ A ) )
% 12.24/7.96 = ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4655_sum_Onon__neutral_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: B > A,I6: set @ B] :
% 12.24/7.96 ( ( groups1027152243600224163dd_sum @ B @ A @ G
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 & ( ( G @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4656_finite__fun__UNIVD1,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 12.24/7.96 => ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
% 12.24/7.96 != ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.96 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4657_sorted__upto,axiom,
% 12.24/7.96 ! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4658_sorted__wrt__upto,axiom,
% 12.24/7.96 ! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4659_sup__cancel__left2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,A3: A,B2: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4660_sup__cancel__left1,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,A3: A,B2: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
% 12.24/7.96 = ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4661_Compl__empty__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4662_Compl__UNIV__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 12.24/7.96 = ( bot_bot @ ( set @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4663_Compl__partition2,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ A] :
% 12.24/7.96 ( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4664_Compl__partition,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ A] :
% 12.24/7.96 ( ( sup_sup @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4665_Compl__eq__Diff__UNIV,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( uminus_uminus @ ( set @ A ) )
% 12.24/7.96 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4666_sup__shunt,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4667_boolean__algebra_Ocomplement__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [A3: A,X: A,Y3: A] :
% 12.24/7.96 ( ( ( inf_inf @ A @ A3 @ X )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 => ( ( ( sup_sup @ A @ A3 @ X )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 => ( ( ( inf_inf @ A @ A3 @ Y3 )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 => ( ( ( sup_sup @ A @ A3 @ Y3 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 => ( X = Y3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4668_surj__Compl__image__subset,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) )
% 12.24/7.96 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A6 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4669_sum_Omono__neutral__left_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( zero_zero @ A ) ) )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4670_sum_Omono__neutral__right_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( zero_zero @ A ) ) )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4671_sum_Omono__neutral__cong__left_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.96 => ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.96 => ( ( H2 @ I4 )
% 12.24/7.96 = ( zero_zero @ A ) ) )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ S3 )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( H2 @ X3 ) ) )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4672_sum_Omono__neutral__cong__right_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( zero_zero @ A ) ) )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ S3 )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( H2 @ X3 ) ) )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4673_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ( inf_inf @ A @ X @ Y3 )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 => ( ( ( sup_sup @ A @ X @ Y3 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 => ( ( uminus_uminus @ A @ X )
% 12.24/7.96 = Y3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4674_sum_Odistrib_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 12.24/7.96 ( ( finite_finite2 @ B
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 & ( ( G @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 => ( ( finite_finite2 @ B
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 & ( ( H2 @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A
% 12.24/7.96 @ ^ [I5: B] : ( plus_plus @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 12.24/7.96 @ I6 )
% 12.24/7.96 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4675_sum_OG__def,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ B @ A )
% 12.24/7.96 = ( ^ [P6: B > A,I8: set @ B] :
% 12.24/7.96 ( if @ A
% 12.24/7.96 @ ( finite_finite2 @ B
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I8 )
% 12.24/7.96 & ( ( P6 @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 @ ( groups7311177749621191930dd_sum @ B @ A @ P6
% 12.24/7.96 @ ( collect @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I8 )
% 12.24/7.96 & ( ( P6 @ X2 )
% 12.24/7.96 != ( zero_zero @ A ) ) ) ) )
% 12.24/7.96 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4676_finite__UNIV__card__ge__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 12.24/7.96 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4677_sorted__transpose,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4678_rev__nth,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.96 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
% 12.24/7.96 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4679_rev__update,axiom,
% 12.24/7.96 ! [A: $tType,K: nat,Xs: list @ A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y3 ) )
% 12.24/7.96 = ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4680_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.96 => ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4681_card__range__greater__zero,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,F: B > A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 12.24/7.96 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4682_sorted__rev__iff__nth__Suc,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 12.24/7.96 = ( ! [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I5 ) ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4683_sorted__rev__iff__nth__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 12.24/7.96 = ( ! [I5: nat,J3: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ I5 @ J3 )
% 12.24/7.96 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4684_sorted__rev__nth__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,I: nat,J: nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 12.24/7.96 => ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.96 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4685_range__mod,axiom,
% 12.24/7.96 ! [N: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.96 => ( ( image @ nat @ nat
% 12.24/7.96 @ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
% 12.24/7.96 @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.96 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4686_sum__diff1_H,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( ab_group_add @ B )
% 12.24/7.96 => ! [I6: set @ A,F: A > B,I: A] :
% 12.24/7.96 ( ( finite_finite2 @ A
% 12.24/7.96 @ ( collect @ A
% 12.24/7.96 @ ^ [I5: A] :
% 12.24/7.96 ( ( member @ A @ I5 @ I6 )
% 12.24/7.96 & ( ( F @ I5 )
% 12.24/7.96 != ( zero_zero @ B ) ) ) ) )
% 12.24/7.96 => ( ( ( member @ A @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.96 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
% 12.24/7.96 & ( ~ ( member @ A @ I @ I6 )
% 12.24/7.96 => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.96 = ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4687_nth__nth__transpose__sorted,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
% 12.24/7.96 => ( ( ord_less @ nat @ J
% 12.24/7.96 @ ( size_size @ ( list @ ( list @ A ) )
% 12.24/7.96 @ ( filter @ ( list @ A )
% 12.24/7.96 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.96 @ Xs ) ) )
% 12.24/7.96 => ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
% 12.24/7.96 = ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4688_transpose__column,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 12.24/7.96 => ( ( map @ ( list @ A ) @ A
% 12.24/7.96 @ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
% 12.24/7.96 @ ( filter @ ( list @ A )
% 12.24/7.96 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.96 @ ( transpose @ A @ Xs ) ) )
% 12.24/7.96 = ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4689_transpose__column__length,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
% 12.24/7.96 => ( ( size_size @ ( list @ ( list @ A ) )
% 12.24/7.96 @ ( filter @ ( list @ A )
% 12.24/7.96 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.96 @ ( transpose @ A @ Xs ) ) )
% 12.24/7.96 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4690_card__UNIV__unit,axiom,
% 12.24/7.96 ( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
% 12.24/7.96 = ( one_one @ nat ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4691_infinite__UNIV__int,axiom,
% 12.24/7.96 ~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4692_integer__of__int__cases,axiom,
% 12.24/7.96 ! [X: code_integer] :
% 12.24/7.96 ~ ! [Y5: int] :
% 12.24/7.96 ( ( X
% 12.24/7.96 = ( code_integer_of_int @ Y5 ) )
% 12.24/7.96 => ~ ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4693_integer__of__int__induct,axiom,
% 12.24/7.96 ! [P: code_integer > $o,X: code_integer] :
% 12.24/7.96 ( ! [Y5: int] :
% 12.24/7.96 ( ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ( P @ ( code_integer_of_int @ Y5 ) ) )
% 12.24/7.96 => ( P @ X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4694_integer__of__int__inject,axiom,
% 12.24/7.96 ! [X: int,Y3: int] :
% 12.24/7.96 ( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ( ( ( code_integer_of_int @ X )
% 12.24/7.96 = ( code_integer_of_int @ Y3 ) )
% 12.24/7.96 = ( X = Y3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4695_int__of__integer,axiom,
% 12.24/7.96 ! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4696_int__of__integer__cases,axiom,
% 12.24/7.96 ! [Y3: int] :
% 12.24/7.96 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ~ ! [X3: code_integer] :
% 12.24/7.96 ( Y3
% 12.24/7.96 != ( code_int_of_integer @ X3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4697_int__of__integer__induct,axiom,
% 12.24/7.96 ! [Y3: int,P: int > $o] :
% 12.24/7.96 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ( ! [X3: code_integer] : ( P @ ( code_int_of_integer @ X3 ) )
% 12.24/7.96 => ( P @ Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4698_length__filter__le,axiom,
% 12.24/7.96 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4699_sorted__same,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [G: ( list @ A ) > A,Xs: list @ A] :
% 12.24/7.96 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 12.24/7.96 @ ( filter @ A
% 12.24/7.96 @ ^ [X2: A] :
% 12.24/7.96 ( X2
% 12.24/7.96 = ( G @ Xs ) )
% 12.24/7.96 @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4700_integer__of__int__inverse,axiom,
% 12.24/7.96 ! [Y3: int] :
% 12.24/7.96 ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 => ( ( code_int_of_integer @ ( code_integer_of_int @ Y3 ) )
% 12.24/7.96 = Y3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4701_length__filter__less,axiom,
% 12.24/7.96 ! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
% 12.24/7.96 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.96 => ( ~ ( P @ X )
% 12.24/7.96 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4702_surj__list__encode,axiom,
% 12.24/7.96 ( ( image @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
% 12.24/7.96 = ( top_top @ ( set @ nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4703_surj__prod__encode,axiom,
% 12.24/7.96 ( ( image @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 12.24/7.96 = ( top_top @ ( set @ nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4704_sorted__filter,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [F: B > A,Xs: list @ B,P: B > $o] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4705_sorted__map__same,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [F: B > A,G: ( list @ B ) > A,Xs: list @ B] :
% 12.24/7.96 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 12.24/7.96 @ ( map @ B @ A @ F
% 12.24/7.96 @ ( filter @ B
% 12.24/7.96 @ ^ [X2: B] :
% 12.24/7.96 ( ( F @ X2 )
% 12.24/7.96 = ( G @ Xs ) )
% 12.24/7.96 @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4706_sum__list__map__filter_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( monoid_add @ A )
% 12.24/7.96 => ! [F: B > A,P: B > $o,Xs: list @ B] :
% 12.24/7.96 ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
% 12.24/7.96 = ( groups8242544230860333062m_list @ A
% 12.24/7.96 @ ( map @ B @ A
% 12.24/7.96 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F @ X2 ) @ ( zero_zero @ A ) )
% 12.24/7.96 @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4707_sum__list__filter__le__nat,axiom,
% 12.24/7.96 ! [A: $tType,F: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ ( filter @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4708_sum__list__map__filter,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( monoid_add @ A )
% 12.24/7.96 => ! [Xs: list @ B,P: B > $o,F: B > A] :
% 12.24/7.96 ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 12.24/7.96 => ( ~ ( P @ X3 )
% 12.24/7.96 => ( ( F @ X3 )
% 12.24/7.96 = ( zero_zero @ A ) ) ) )
% 12.24/7.96 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
% 12.24/7.96 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4709_Ints__def,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ring_1 @ A )
% 12.24/7.96 => ( ( ring_1_Ints @ A )
% 12.24/7.96 = ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4710_int__in__range__abs,axiom,
% 12.24/7.96 ! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4711_length__filter__conv__card,axiom,
% 12.24/7.96 ! [A: $tType,P5: A > $o,Xs: list @ A] :
% 12.24/7.96 ( ( size_size @ ( list @ A ) @ ( filter @ A @ P5 @ Xs ) )
% 12.24/7.96 = ( finite_card @ nat
% 12.24/7.96 @ ( collect @ nat
% 12.24/7.96 @ ^ [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 & ( P5 @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4712_nth__transpose,axiom,
% 12.24/7.96 ! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
% 12.24/7.96 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
% 12.24/7.96 => ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
% 12.24/7.96 = ( map @ ( list @ A ) @ A
% 12.24/7.96 @ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
% 12.24/7.96 @ ( filter @ ( list @ A )
% 12.24/7.96 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
% 12.24/7.96 @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4713_transpose__max__length,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A )] :
% 12.24/7.96 ( ( foldr @ ( list @ A ) @ nat
% 12.24/7.96 @ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.96 @ ( transpose @ A @ Xs )
% 12.24/7.96 @ ( zero_zero @ nat ) )
% 12.24/7.96 = ( size_size @ ( list @ ( list @ A ) )
% 12.24/7.96 @ ( filter @ ( list @ A )
% 12.24/7.96 @ ^ [X2: list @ A] :
% 12.24/7.96 ( X2
% 12.24/7.96 != ( nil @ A ) )
% 12.24/7.96 @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4714_transpose__aux__max,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
% 12.24/7.96 ( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 @ ( foldr @ ( list @ B ) @ nat
% 12.24/7.96 @ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
% 12.24/7.96 @ Xss
% 12.24/7.96 @ ( zero_zero @ nat ) ) )
% 12.24/7.96 = ( suc
% 12.24/7.96 @ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.96 @ ( foldr @ ( list @ B ) @ nat
% 12.24/7.96 @ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.96 @ ( filter @ ( list @ B )
% 12.24/7.96 @ ^ [Ys3: list @ B] :
% 12.24/7.96 ( Ys3
% 12.24/7.96 != ( nil @ B ) )
% 12.24/7.96 @ Xss )
% 12.24/7.96 @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4715_transpose__transpose,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A )] :
% 12.24/7.96 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
% 12.24/7.96 => ( ( transpose @ A @ ( transpose @ A @ Xs ) )
% 12.24/7.96 = ( takeWhile @ ( list @ A )
% 12.24/7.96 @ ^ [X2: list @ A] :
% 12.24/7.96 ( X2
% 12.24/7.96 != ( nil @ A ) )
% 12.24/7.96 @ Xs ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4716_length__takeWhile__le,axiom,
% 12.24/7.96 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4717_sorted__takeWhile,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,P: A > $o] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4718_takeWhile__nth,axiom,
% 12.24/7.96 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 12.24/7.96 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
% 12.24/7.96 => ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
% 12.24/7.96 = ( nth @ A @ Xs @ J ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4719_nth__length__takeWhile,axiom,
% 12.24/7.96 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 12.24/7.96 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4720_sum__list_Oeq__foldr,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( monoid_add @ A )
% 12.24/7.96 => ( ( groups8242544230860333062m_list @ A )
% 12.24/7.96 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4721_length__takeWhile__less__P__nth,axiom,
% 12.24/7.96 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 12.24/7.96 ( ! [I4: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I4 @ J )
% 12.24/7.96 => ( P @ ( nth @ A @ Xs @ I4 ) ) )
% 12.24/7.96 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4722_takeWhile__eq__take__P__nth,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
% 12.24/7.96 ( ! [I4: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I4 @ N )
% 12.24/7.96 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) )
% 12.24/7.96 => ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ~ ( P @ ( nth @ A @ Xs @ N ) ) )
% 12.24/7.96 => ( ( takeWhile @ A @ P @ Xs )
% 12.24/7.96 = ( take @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4723_nths__shift__lemma__Suc,axiom,
% 12.24/7.96 ! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
% 12.24/7.96 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 12.24/7.96 @ ( filter @ ( product_prod @ A @ nat )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
% 12.24/7.96 @ ( zip @ A @ nat @ Xs @ Is ) ) )
% 12.24/7.96 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 12.24/7.96 @ ( filter @ ( product_prod @ A @ nat )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
% 12.24/7.96 @ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4724_horner__sum__foldr,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_semiring_0 @ A )
% 12.24/7.96 => ( ( groups4207007520872428315er_sum @ B @ A )
% 12.24/7.96 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 12.24/7.96 ( foldr @ B @ A
% 12.24/7.96 @ ^ [X2: B,B3: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A5 @ B3 ) )
% 12.24/7.96 @ Xs3
% 12.24/7.96 @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4725_nths__shift__lemma,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ nat,Xs: list @ A,I: nat] :
% 12.24/7.96 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 12.24/7.96 @ ( filter @ ( product_prod @ A @ nat )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A6 )
% 12.24/7.96 @ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
% 12.24/7.96 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 12.24/7.96 @ ( filter @ ( product_prod @ A @ nat )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A6 )
% 12.24/7.96 @ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4726_filter__equals__takeWhile__sorted__rev,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [F: B > A,Xs: list @ B,T2: A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 12.24/7.96 => ( ( filter @ B
% 12.24/7.96 @ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
% 12.24/7.96 @ Xs )
% 12.24/7.96 = ( takeWhile @ B
% 12.24/7.96 @ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
% 12.24/7.96 @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4727_length__transpose,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ ( list @ A )] :
% 12.24/7.96 ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
% 12.24/7.96 = ( foldr @ ( list @ A ) @ nat
% 12.24/7.96 @ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.96 @ Xs
% 12.24/7.96 @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4728_foldr__max__sorted,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,Y3: A] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
% 12.24/7.96 => ( ( ( Xs
% 12.24/7.96 = ( nil @ A ) )
% 12.24/7.96 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
% 12.24/7.96 = Y3 ) )
% 12.24/7.96 & ( ( Xs
% 12.24/7.96 != ( nil @ A ) )
% 12.24/7.96 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
% 12.24/7.96 = ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4729_rec__nat__add__eq__if,axiom,
% 12.24/7.96 ! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
% 12.24/7.96 ( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 12.24/7.96 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4730_case__nat__add__eq__if,axiom,
% 12.24/7.96 ! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
% 12.24/7.96 ( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 12.24/7.96 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4731_divmod__integer__eq__cases,axiom,
% 12.24/7.96 ( code_divmod_integer
% 12.24/7.96 = ( ^ [K3: code_integer,L3: code_integer] :
% 12.24/7.96 ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.96 @ ( K3
% 12.24/7.96 = ( zero_zero @ code_integer ) )
% 12.24/7.96 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 12.24/7.96 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.96 @ ( L3
% 12.24/7.96 = ( zero_zero @ code_integer ) )
% 12.24/7.96 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
% 12.24/7.96 @ ( 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
% 12.24/7.96 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.96 @ ( ( sgn_sgn @ code_integer @ K3 )
% 12.24/7.96 = ( sgn_sgn @ code_integer @ L3 ) )
% 12.24/7.96 @ ( code_divmod_abs @ K3 @ L3 )
% 12.24/7.96 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.96 @ ^ [R5: code_integer,S6: code_integer] :
% 12.24/7.96 ( if @ ( product_prod @ code_integer @ code_integer )
% 12.24/7.96 @ ( S6
% 12.24/7.96 = ( zero_zero @ code_integer ) )
% 12.24/7.96 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
% 12.24/7.96 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S6 ) ) )
% 12.24/7.96 @ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4732_old_Onat_Osimps_I7_J,axiom,
% 12.24/7.96 ! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
% 12.24/7.96 ( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
% 12.24/7.96 = ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4733_old_Onat_Osimps_I6_J,axiom,
% 12.24/7.96 ! [T: $tType,F1: T,F22: nat > T > T] :
% 12.24/7.96 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
% 12.24/7.96 = F1 ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4734_case__nat__numeral,axiom,
% 12.24/7.96 ! [A: $tType,A3: A,F: nat > A,V: num] :
% 12.24/7.96 ( ( case_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
% 12.24/7.96 = ( F @ ( pred_numeral @ V ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4735_rec__nat__numeral,axiom,
% 12.24/7.96 ! [A: $tType,A3: A,F: nat > A > A,V: num] :
% 12.24/7.96 ( ( rec_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
% 12.24/7.96 = ( F @ ( pred_numeral @ V ) @ ( rec_nat @ A @ A3 @ F @ ( pred_numeral @ V ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4736_comp__cong,axiom,
% 12.24/7.96 ! [C: $tType,B: $tType,D: $tType,A: $tType,E4: $tType,F: B > A,G: C > B,X: C,F5: D > A,G4: E4 > D,X5: E4] :
% 12.24/7.96 ( ( ( F @ ( G @ X ) )
% 12.24/7.96 = ( F5 @ ( G4 @ X5 ) ) )
% 12.24/7.96 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 12.24/7.96 = ( comp @ D @ A @ E4 @ F5 @ G4 @ X5 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4737_nat_Ocase__distrib,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
% 12.24/7.96 ( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 12.24/7.96 = ( case_nat @ B @ ( H2 @ F1 )
% 12.24/7.96 @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
% 12.24/7.96 @ Nat ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4738_old_Onat_Osimps_I5_J,axiom,
% 12.24/7.96 ! [A: $tType,F1: A,F22: nat > A,X22: nat] :
% 12.24/7.96 ( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
% 12.24/7.96 = ( F22 @ X22 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4739_old_Onat_Osimps_I4_J,axiom,
% 12.24/7.96 ! [A: $tType,F1: A,F22: nat > A] :
% 12.24/7.96 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
% 12.24/7.96 = F1 ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4740_nat_Odisc__eq__case_I2_J,axiom,
% 12.24/7.96 ! [Nat: nat] :
% 12.24/7.96 ( ( Nat
% 12.24/7.96 != ( zero_zero @ nat ) )
% 12.24/7.96 = ( case_nat @ $o @ $false
% 12.24/7.96 @ ^ [Uu: nat] : $true
% 12.24/7.96 @ Nat ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4741_nat_Odisc__eq__case_I1_J,axiom,
% 12.24/7.96 ! [Nat: nat] :
% 12.24/7.96 ( ( Nat
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 = ( case_nat @ $o @ $true
% 12.24/7.96 @ ^ [Uu: nat] : $false
% 12.24/7.96 @ Nat ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4742_rec__nat__Suc__imp,axiom,
% 12.24/7.96 ! [A: $tType,F: nat > A,F1: A,F22: nat > A > A,N: nat] :
% 12.24/7.96 ( ( F
% 12.24/7.96 = ( rec_nat @ A @ F1 @ F22 ) )
% 12.24/7.96 => ( ( F @ ( suc @ N ) )
% 12.24/7.96 = ( F22 @ N @ ( F @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4743_rec__nat__0__imp,axiom,
% 12.24/7.96 ! [A: $tType,F: nat > A,F1: A,F22: nat > A > A] :
% 12.24/7.96 ( ( F
% 12.24/7.96 = ( rec_nat @ A @ F1 @ F22 ) )
% 12.24/7.96 => ( ( F @ ( zero_zero @ nat ) )
% 12.24/7.96 = F1 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4744_sum__comp__morphism,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,C: $tType] :
% 12.24/7.96 ( ( ( comm_monoid_add @ B )
% 12.24/7.96 & ( comm_monoid_add @ A ) )
% 12.24/7.96 => ! [H2: B > A,G: C > B,A6: set @ C] :
% 12.24/7.96 ( ( ( H2 @ ( zero_zero @ B ) )
% 12.24/7.96 = ( zero_zero @ A ) )
% 12.24/7.96 => ( ! [X3: B,Y5: B] :
% 12.24/7.96 ( ( H2 @ ( plus_plus @ B @ X3 @ Y5 ) )
% 12.24/7.96 = ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.96 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A6 )
% 12.24/7.96 = ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4745_uminus__sum__list__map,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( ab_group_add @ A )
% 12.24/7.96 => ! [F: B > A,Xs: list @ B] :
% 12.24/7.96 ( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 12.24/7.96 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4746_less__eq__nat_Osimps_I2_J,axiom,
% 12.24/7.96 ! [M: nat,N: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 12.24/7.96 = ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4747_max__Suc1,axiom,
% 12.24/7.96 ! [N: nat,M: nat] :
% 12.24/7.96 ( ( ord_max @ nat @ ( suc @ N ) @ M )
% 12.24/7.96 = ( case_nat @ nat @ ( suc @ N )
% 12.24/7.96 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N @ M4 ) )
% 12.24/7.96 @ M ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4748_max__Suc2,axiom,
% 12.24/7.96 ! [M: nat,N: nat] :
% 12.24/7.96 ( ( ord_max @ nat @ M @ ( suc @ N ) )
% 12.24/7.96 = ( case_nat @ nat @ ( suc @ N )
% 12.24/7.96 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N ) )
% 12.24/7.96 @ M ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4749_sum_Oreindex__nontrivial,axiom,
% 12.24/7.96 ! [C: $tType,A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [A6: set @ B,H2: B > C,G: C > A] :
% 12.24/7.96 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.96 => ( ! [X3: B,Y5: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A6 )
% 12.24/7.96 => ( ( member @ B @ Y5 @ A6 )
% 12.24/7.96 => ( ( X3 != Y5 )
% 12.24/7.96 => ( ( ( H2 @ X3 )
% 12.24/7.96 = ( H2 @ Y5 ) )
% 12.24/7.96 => ( ( G @ ( H2 @ X3 ) )
% 12.24/7.96 = ( zero_zero @ A ) ) ) ) ) )
% 12.24/7.96 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4750_prod_Oreindex__nontrivial,axiom,
% 12.24/7.96 ! [C: $tType,A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [A6: set @ B,H2: B > C,G: C > A] :
% 12.24/7.96 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.96 => ( ! [X3: B,Y5: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A6 )
% 12.24/7.96 => ( ( member @ B @ Y5 @ A6 )
% 12.24/7.96 => ( ( X3 != Y5 )
% 12.24/7.96 => ( ( ( H2 @ X3 )
% 12.24/7.96 = ( H2 @ Y5 ) )
% 12.24/7.96 => ( ( G @ ( H2 @ X3 ) )
% 12.24/7.96 = ( one_one @ A ) ) ) ) ) )
% 12.24/7.96 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4751_sum__image__le,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,C: $tType] :
% 12.24/7.96 ( ( ordere6911136660526730532id_add @ B )
% 12.24/7.96 => ! [I6: set @ C,G: A > B,F: C > A] :
% 12.24/7.96 ( ( finite_finite2 @ C @ I6 )
% 12.24/7.96 => ( ! [I4: C] :
% 12.24/7.96 ( ( member @ C @ I4 @ I6 )
% 12.24/7.96 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I4 ) ) ) )
% 12.24/7.96 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4752_diff__Suc,axiom,
% 12.24/7.96 ! [M: nat,N: nat] :
% 12.24/7.96 ( ( minus_minus @ nat @ M @ ( suc @ N ) )
% 12.24/7.96 = ( case_nat @ nat @ ( zero_zero @ nat )
% 12.24/7.96 @ ^ [K3: nat] : K3
% 12.24/7.96 @ ( minus_minus @ nat @ M @ N ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4753_min__Suc1,axiom,
% 12.24/7.96 ! [N: nat,M: nat] :
% 12.24/7.96 ( ( ord_min @ nat @ ( suc @ N ) @ M )
% 12.24/7.96 = ( case_nat @ nat @ ( zero_zero @ nat )
% 12.24/7.96 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N @ M4 ) )
% 12.24/7.96 @ M ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4754_min__Suc2,axiom,
% 12.24/7.96 ! [M: nat,N: nat] :
% 12.24/7.96 ( ( ord_min @ nat @ M @ ( suc @ N ) )
% 12.24/7.96 = ( case_nat @ nat @ ( zero_zero @ nat )
% 12.24/7.96 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N ) )
% 12.24/7.96 @ M ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4755_Nitpick_Ocase__nat__unfold,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( case_nat @ A )
% 12.24/7.96 = ( ^ [X2: A,F2: nat > A,N3: nat] :
% 12.24/7.96 ( if @ A
% 12.24/7.96 @ ( N3
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 @ X2
% 12.24/7.96 @ ( F2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4756_nths__def,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( nths @ A )
% 12.24/7.96 = ( ^ [Xs3: list @ A,A7: set @ nat] :
% 12.24/7.96 ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 12.24/7.96 @ ( filter @ ( product_prod @ A @ nat )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
% 12.24/7.96 @ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4757_length__product__lists,axiom,
% 12.24/7.96 ! [B: $tType,Xss: list @ ( list @ B )] :
% 12.24/7.96 ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
% 12.24/7.96 = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4758_nat_Osplit__sels_I2_J,axiom,
% 12.24/7.96 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
% 12.24/7.96 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 12.24/7.96 = ( ~ ( ( ( Nat
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 & ~ ( P @ F1 ) )
% 12.24/7.96 | ( ( Nat
% 12.24/7.96 = ( suc @ ( pred @ Nat ) ) )
% 12.24/7.96 & ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4759_nths__singleton,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ nat,X: A] :
% 12.24/7.96 ( ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 12.24/7.96 => ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
% 12.24/7.96 = ( cons @ A @ X @ ( nil @ A ) ) ) )
% 12.24/7.96 & ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 12.24/7.96 => ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
% 12.24/7.96 = ( nil @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4760_card_Ocomp__fun__commute__on,axiom,
% 12.24/7.96 ( ( comp @ nat @ nat @ nat @ suc @ suc )
% 12.24/7.96 = ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4761_nths__all,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 12.24/7.96 ( ! [I4: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 => ( member @ nat @ I4 @ I6 ) )
% 12.24/7.96 => ( ( nths @ A @ Xs @ I6 )
% 12.24/7.96 = Xs ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4762_sorted__nths,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linorder @ A )
% 12.24/7.96 => ! [Xs: list @ A,I6: set @ nat] :
% 12.24/7.96 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.96 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4763_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4764_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4765_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4766_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4767_lift__def,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( heap_Time_lift @ A @ B )
% 12.24/7.96 = ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4768_drop__eq__nths,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( drop @ A )
% 12.24/7.96 = ( ^ [N3: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4769_euclidean__size__int__def,axiom,
% 12.24/7.96 ( ( euclid6346220572633701492n_size @ int )
% 12.24/7.96 = ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4770_pred__def,axiom,
% 12.24/7.96 ( pred
% 12.24/7.96 = ( case_nat @ nat @ ( zero_zero @ nat )
% 12.24/7.96 @ ^ [X24: nat] : X24 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4771_length__nths,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 12.24/7.96 ( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
% 12.24/7.96 = ( finite_card @ nat
% 12.24/7.96 @ ( collect @ nat
% 12.24/7.96 @ ^ [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 & ( member @ nat @ I5 @ I6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4772_sum_OatLeast0__atMost__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4773_sum_OatLeast0__lessThan__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4774_prod_OatLeast0__atMost__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4775_prod_OatLeast0__lessThan__Suc__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4776_sum_OatLeastLessThan__shift__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4777_prod_OatLeastLessThan__shift__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4778_sum_OatLeast__atMost__pred__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.96 @ ( comp @ nat @ A @ nat @ G
% 12.24/7.96 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.96 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4779_sum_OatLeast__lessThan__pred__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.96 @ ( comp @ nat @ A @ nat @ G
% 12.24/7.96 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.96 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4780_prod_OatLeast__atMost__pred__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.96 @ ( comp @ nat @ A @ nat @ G
% 12.24/7.96 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.96 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4781_prod_OatLeast__lessThan__pred__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: nat > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ nat @ A
% 12.24/7.96 @ ( comp @ nat @ A @ nat @ G
% 12.24/7.96 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/7.96 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4782_sum_OatLeast__int__atMost__int__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: int > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4783_filter__eq__nths,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( filter @ A )
% 12.24/7.96 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 12.24/7.96 ( nths @ A @ Xs3
% 12.24/7.96 @ ( collect @ nat
% 12.24/7.96 @ ^ [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.96 & ( P4 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4784_prod_OatLeast__int__atMost__int__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: int > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4785_sum_OatLeast__int__lessThan__int__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [G: int > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4786_sum_OatLeastAtMost__shift__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.96 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.96 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4787_prod_OatLeastAtMost__shift__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [M: nat,N: nat,G: nat > A] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.96 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4788_prod_OatLeast__int__lessThan__int__shift,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [G: int > A,M: nat,N: nat] :
% 12.24/7.96 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/7.96 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4789_Gcd__int__def,axiom,
% 12.24/7.96 ( ( gcd_Gcd @ int )
% 12.24/7.96 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4790_nths__Cons,axiom,
% 12.24/7.96 ! [A: $tType,X: A,L: list @ A,A6: set @ nat] :
% 12.24/7.96 ( ( nths @ A @ ( cons @ A @ X @ L ) @ A6 )
% 12.24/7.96 = ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
% 12.24/7.96 @ ( nths @ A @ L
% 12.24/7.96 @ ( collect @ nat
% 12.24/7.96 @ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4791_nat_Osplit__sels_I1_J,axiom,
% 12.24/7.96 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
% 12.24/7.96 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
% 12.24/7.96 = ( ( ( Nat
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 => ( P @ F1 ) )
% 12.24/7.96 & ( ( Nat
% 12.24/7.96 = ( suc @ ( pred @ Nat ) ) )
% 12.24/7.96 => ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4792_Code__Numeral_Onegative__def,axiom,
% 12.24/7.96 ( code_negative
% 12.24/7.96 = ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4793_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
% 12.24/7.96 ! [A: $tType,F: nat > ( set @ A ),S3: set @ A] :
% 12.24/7.96 ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I4 ) @ S3 )
% 12.24/7.96 => ( ( finite_finite2 @ A @ S3 )
% 12.24/7.96 => ( ? [N9: nat] :
% 12.24/7.96 ( ! [N4: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ N4 @ N9 )
% 12.24/7.96 => ! [M6: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M6 @ N9 )
% 12.24/7.96 => ( ( ord_less @ nat @ M6 @ N4 )
% 12.24/7.96 => ( ord_less @ ( set @ A ) @ ( F @ M6 ) @ ( F @ N4 ) ) ) ) )
% 12.24/7.96 & ! [N4: nat] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ N9 @ N4 )
% 12.24/7.96 => ( ( F @ N9 )
% 12.24/7.96 = ( F @ N4 ) ) ) )
% 12.24/7.96 => ( ( F @ ( finite_card @ A @ S3 ) )
% 12.24/7.96 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4794_range__abs__Nats,axiom,
% 12.24/7.96 ( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 = ( semiring_1_Nats @ int ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4795_Sup__atLeastAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4796_cSup__atLeastAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4797_cSup__atLeastLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4798_Sup__atLeastLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4799_cSup__greaterThanAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4800_Sup__greaterThanAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4801_Sup__greaterThanLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4802_cSup__greaterThanLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4803_Nats__0,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4804_cSup__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 & ( no_bot @ A ) )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ A3 @ Y5 ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ X8 )
% 12.24/7.96 = A3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4805_cSup__eq__maximum,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Z3: A,X8: set @ A] :
% 12.24/7.96 ( ( member @ A @ Z3 @ X8 )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ X8 )
% 12.24/7.96 = Z3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4806_Nats__1,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4807_Nats__add,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [A3: A,B2: A] :
% 12.24/7.96 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4808_Nats__mult,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [A3: A,B2: A] :
% 12.24/7.96 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4809_Nats__numeral,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4810_of__nat__in__Nats,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4811_Nats__induct,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [X: A,P: A > $o] :
% 12.24/7.96 ( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ A @ N4 ) )
% 12.24/7.96 => ( P @ X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4812_Nats__cases,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ~ ! [N4: nat] :
% 12.24/7.96 ( X
% 12.24/7.96 != ( semiring_1_of_nat @ A @ N4 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4813_fact__in__Nats,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_char_0 @ A )
% 12.24/7.96 => ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4814_cSup__eq__non__empty,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ A3 @ Y5 ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ X8 )
% 12.24/7.96 = A3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4815_cSup__least,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,Z3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4816_le__cSup__finite,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,X: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( member @ A @ X @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4817_less__cSupD,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,Z3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 12.24/7.96 => ? [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 & ( ord_less @ A @ Z3 @ X3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4818_less__cSupE,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [Y3: A,X8: set @ A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 12.24/7.96 => ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ~ ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4819_finite__imp__Sup__less,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,X: A,A3: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( member @ A @ X @ X8 )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less @ A @ X3 @ A3 ) )
% 12.24/7.96 => ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4820_inf__Sup,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [A3: A,B7: set @ A] :
% 12.24/7.96 ( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B7 ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( inf_inf @ A @ A3 ) @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4821_Sup__inf__eq__bot__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A3: A] :
% 12.24/7.96 ( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ B7 )
% 12.24/7.96 => ( ( inf_inf @ A @ X2 @ A3 )
% 12.24/7.96 = ( bot_bot @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4822_finite__subset__Union,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ A,B12: set @ ( set @ A )] :
% 12.24/7.96 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.96 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
% 12.24/7.96 => ~ ! [F6: set @ ( set @ A )] :
% 12.24/7.96 ( ( finite_finite2 @ ( set @ A ) @ F6 )
% 12.24/7.96 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F6 @ B12 )
% 12.24/7.96 => ~ ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ F6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4823_card__Union__le__sum__card,axiom,
% 12.24/7.96 ! [A: $tType,U4: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4824_SUP__inf,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [F: B > A,B7: set @ B,A3: A] :
% 12.24/7.96 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [B3: B] : ( inf_inf @ A @ ( F @ B3 ) @ A3 )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4825_Sup__inf,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A3: A] :
% 12.24/7.96 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ A @ A
% 12.24/7.96 @ ^ [B3: A] : ( inf_inf @ A @ B3 @ A3 )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4826_inf__SUP,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [A3: A,F: B > A,B7: set @ B] :
% 12.24/7.96 ( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [B3: B] : ( inf_inf @ A @ A3 @ ( F @ B3 ) )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4827_SUP__inf__distrib2,axiom,
% 12.24/7.96 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
% 12.24/7.96 ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [A5: B] :
% 12.24/7.96 ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ C @ A
% 12.24/7.96 @ ^ [B3: C] : ( inf_inf @ A @ ( F @ A5 ) @ ( G @ B3 ) )
% 12.24/7.96 @ B7 ) )
% 12.24/7.96 @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4828_cSUP__least,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [A6: set @ B,F: B > A,M7: A] :
% 12.24/7.96 ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ M7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4829_finite__Sup__less__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ X8 )
% 12.24/7.96 => ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4830_cSup__abs__le,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( linordered_idom @ A ) )
% 12.24/7.96 => ! [S3: set @ A,A3: A] :
% 12.24/7.96 ( ( S3
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ S3 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4831_Nats__diff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linordered_idom @ A )
% 12.24/7.96 => ! [A3: A,B2: A] :
% 12.24/7.96 ( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
% 12.24/7.96 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 12.24/7.96 => ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4832_sum_OUnion__comp,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_add @ A )
% 12.24/7.96 => ! [B7: set @ ( set @ B ),G: B > A] :
% 12.24/7.96 ( ! [X3: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ X3 @ B7 )
% 12.24/7.96 => ( finite_finite2 @ B @ X3 ) )
% 12.24/7.96 => ( ! [A14: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ A14 @ B7 )
% 12.24/7.96 => ! [A25: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ A25 @ B7 )
% 12.24/7.96 => ( ( A14 != A25 )
% 12.24/7.96 => ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A14 )
% 12.24/7.96 => ( ( member @ B @ X3 @ A25 )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( zero_zero @ A ) ) ) ) ) ) )
% 12.24/7.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
% 12.24/7.96 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4833_prod_OUnion__comp,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_monoid_mult @ A )
% 12.24/7.96 => ! [B7: set @ ( set @ B ),G: B > A] :
% 12.24/7.96 ( ! [X3: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ X3 @ B7 )
% 12.24/7.96 => ( finite_finite2 @ B @ X3 ) )
% 12.24/7.96 => ( ! [A14: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ A14 @ B7 )
% 12.24/7.96 => ! [A25: set @ B] :
% 12.24/7.96 ( ( member @ ( set @ B ) @ A25 @ B7 )
% 12.24/7.96 => ( ( A14 != A25 )
% 12.24/7.96 => ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A14 )
% 12.24/7.96 => ( ( member @ B @ X3 @ A25 )
% 12.24/7.96 => ( ( G @ X3 )
% 12.24/7.96 = ( one_one @ A ) ) ) ) ) ) )
% 12.24/7.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
% 12.24/7.96 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4834_card__Union__le__sum__card__weak,axiom,
% 12.24/7.96 ! [A: $tType,U4: set @ ( set @ A )] :
% 12.24/7.96 ( ! [X3: set @ A] :
% 12.24/7.96 ( ( member @ ( set @ A ) @ X3 @ U4 )
% 12.24/7.96 => ( finite_finite2 @ A @ X3 ) )
% 12.24/7.96 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4835_Nats__subset__Ints,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ring_1 @ A )
% 12.24/7.96 => ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4836_UN__UN__finite__eq,axiom,
% 12.24/7.96 ! [A: $tType,A6: nat > ( set @ A )] :
% 12.24/7.96 ( ( complete_Sup_Sup @ ( set @ A )
% 12.24/7.96 @ ( image @ nat @ ( set @ A )
% 12.24/7.96 @ ^ [N3: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 12.24/7.96 @ ( top_top @ ( set @ nat ) ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4837_cSup__asclose,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( linordered_idom @ A ) )
% 12.24/7.96 => ! [S3: set @ A,L: A,E: A] :
% 12.24/7.96 ( ( S3
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ S3 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4838_card__UN__le,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,I6: set @ A,A6: A > ( set @ B )] :
% 12.24/7.96 ( ( finite_finite2 @ A @ I6 )
% 12.24/7.96 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A6 @ I6 ) ) )
% 12.24/7.96 @ ( groups7311177749621191930dd_sum @ A @ nat
% 12.24/7.96 @ ^ [I5: A] : ( finite_card @ B @ ( A6 @ I5 ) )
% 12.24/7.96 @ I6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4839_UN__finite__subset,axiom,
% 12.24/7.96 ! [A: $tType,A6: nat > ( set @ A ),C6: set @ A] :
% 12.24/7.96 ( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ C6 )
% 12.24/7.96 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C6 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4840_UN__finite2__eq,axiom,
% 12.24/7.96 ! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
% 12.24/7.96 ( ! [N4: nat] :
% 12.24/7.96 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4841_Nats__def,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ( ( semiring_1_Nats @ A )
% 12.24/7.96 = ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4842_card__partition,axiom,
% 12.24/7.96 ! [A: $tType,C6: set @ ( set @ A ),K: nat] :
% 12.24/7.96 ( ( finite_finite2 @ ( set @ A ) @ C6 )
% 12.24/7.96 => ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
% 12.24/7.96 => ( ! [C3: set @ A] :
% 12.24/7.96 ( ( member @ ( set @ A ) @ C3 @ C6 )
% 12.24/7.96 => ( ( finite_card @ A @ C3 )
% 12.24/7.96 = K ) )
% 12.24/7.96 => ( ! [C1: set @ A,C22: set @ A] :
% 12.24/7.96 ( ( member @ ( set @ A ) @ C1 @ C6 )
% 12.24/7.96 => ( ( member @ ( set @ A ) @ C22 @ C6 )
% 12.24/7.96 => ( ( C1 != C22 )
% 12.24/7.96 => ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
% 12.24/7.96 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 12.24/7.96 => ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C6 ) )
% 12.24/7.96 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4843_Nats__altdef2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( linordered_idom @ A )
% 12.24/7.96 => ( ( semiring_1_Nats @ A )
% 12.24/7.96 = ( collect @ A
% 12.24/7.96 @ ^ [N3: A] :
% 12.24/7.96 ( ( member @ A @ N3 @ ( ring_1_Ints @ A ) )
% 12.24/7.96 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4844_UN__finite2__subset,axiom,
% 12.24/7.96 ! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
% 12.24/7.96 ( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
% 12.24/7.96 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4845_UN__le__eq__Un0,axiom,
% 12.24/7.96 ! [A: $tType,M7: nat > ( set @ A ),N: nat] :
% 12.24/7.96 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N ) ) )
% 12.24/7.96 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4846_SUP__eq__top__iff,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 12.24/7.96 => ? [Y6: B] :
% 12.24/7.96 ( ( member @ B @ Y6 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ X2 @ ( F @ Y6 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4847_Sup__eq__top__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( ( complete_Sup_Sup @ A @ A6 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 12.24/7.96 => ? [Y6: A] :
% 12.24/7.96 ( ( member @ A @ Y6 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4848_mono__SUP,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( comple6319245703460814977attice @ B ) )
% 12.24/7.96 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 12.24/7.96 ( ( order_mono @ A @ B @ F )
% 12.24/7.96 => ( ord_less_eq @ B
% 12.24/7.96 @ ( complete_Sup_Sup @ B
% 12.24/7.96 @ ( image @ C @ B
% 12.24/7.96 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 12.24/7.96 @ I6 ) )
% 12.24/7.96 @ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4849_Sup__nat__empty,axiom,
% 12.24/7.96 ( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.96 = ( zero_zero @ nat ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4850_Sup__upper2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [U: A,A6: set @ A,V: A] :
% 12.24/7.96 ( ( member @ A @ U @ A6 )
% 12.24/7.96 => ( ( ord_less_eq @ A @ V @ U )
% 12.24/7.96 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4851_Sup__le__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,B2: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ B2 )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4852_Sup__upper,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,A6: set @ A] :
% 12.24/7.96 ( ( member @ A @ X @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4853_Sup__least,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,Z3: A] :
% 12.24/7.96 ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4854_Sup__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.96 ( ! [A8: A] :
% 12.24/7.96 ( ( member @ A @ A8 @ A6 )
% 12.24/7.96 => ? [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ B7 )
% 12.24/7.96 & ( ord_less_eq @ A @ A8 @ X4 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4855_Sup__eqI,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,X: A] :
% 12.24/7.96 ( ! [Y5: A] :
% 12.24/7.96 ( ( member @ A @ Y5 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [Z5: A] :
% 12.24/7.96 ( ( member @ A @ Z5 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ Z5 @ Y5 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ Y5 ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ A6 )
% 12.24/7.96 = X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4856_less__Sup__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [A3: A,S3: set @ A] :
% 12.24/7.96 ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) )
% 12.24/7.96 = ( ? [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ S3 )
% 12.24/7.96 & ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4857_le__Sup__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [X: A,A6: set @ A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) )
% 12.24/7.96 = ( ! [Y6: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y6 @ X )
% 12.24/7.96 => ? [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4858_SUP__eq,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ? [X4: C] :
% 12.24/7.96 ( ( member @ C @ X4 @ B7 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
% 12.24/7.96 => ( ! [J2: C] :
% 12.24/7.96 ( ( member @ C @ J2 @ B7 )
% 12.24/7.96 => ? [X4: B] :
% 12.24/7.96 ( ( member @ B @ X4 @ A6 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4859_less__eq__Sup,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,U: A] :
% 12.24/7.96 ( ! [V4: A] :
% 12.24/7.96 ( ( member @ A @ V4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ V4 ) )
% 12.24/7.96 => ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4860_Sup__subset__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4861_SUP__upper2,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I: B,A6: set @ B,U: A,F: B > A] :
% 12.24/7.96 ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ( ord_less_eq @ A @ U @ ( F @ I ) )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4862_SUP__le__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,U: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
% 12.24/7.96 = ( ! [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4863_SUP__upper,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I: B,A6: set @ B,F: B > A] :
% 12.24/7.96 ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4864_SUP__mono_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [F: B > A,G: B > A,A6: set @ B] :
% 12.24/7.96 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4865_SUP__least,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,F: B > A,U: A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ I4 ) @ U ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4866_SUP__mono,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
% 12.24/7.96 ( ! [N4: B] :
% 12.24/7.96 ( ( member @ B @ N4 @ A6 )
% 12.24/7.96 => ? [X4: C] :
% 12.24/7.96 ( ( member @ C @ X4 @ B7 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4867_SUP__eqI,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,F: B > A,X: A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ I4 ) @ X ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [I3: B] :
% 12.24/7.96 ( ( member @ B @ I3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ I3 ) @ Y5 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ Y5 ) )
% 12.24/7.96 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4868_SUP__lessD,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 12.24/7.96 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
% 12.24/7.96 => ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4869_less__SUP__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [A3: A,F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.96 = ( ? [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4870_le__SUP__iff,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [X: A,F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.96 = ( ! [Y6: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y6 @ X )
% 12.24/7.96 => ? [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ Y6 @ ( F @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4871_SUP__eq__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I6: set @ B,C2: A,F: B > A] :
% 12.24/7.96 ( ( I6
% 12.24/7.96 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 => ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ I6 )
% 12.24/7.96 => ( ord_less_eq @ A @ C2 @ ( F @ I4 ) ) )
% 12.24/7.96 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I6 ) )
% 12.24/7.96 = C2 )
% 12.24/7.96 = ( ! [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 => ( ( F @ X2 )
% 12.24/7.96 = C2 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4872_Sup__inter__less__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4873_SUP__subset__mono,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,B7: set @ B,F: B > A,G: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4874_mono__Sup,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( comple6319245703460814977attice @ B ) )
% 12.24/7.96 => ! [F: A > B,A6: set @ A] :
% 12.24/7.96 ( ( order_mono @ A @ B @ F )
% 12.24/7.96 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4875_card__UNION,axiom,
% 12.24/7.96 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/7.96 ( ( finite_finite2 @ ( set @ A ) @ A6 )
% 12.24/7.96 => ( ! [X3: set @ A] :
% 12.24/7.96 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 12.24/7.96 => ( finite_finite2 @ A @ X3 ) )
% 12.24/7.96 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
% 12.24/7.96 = ( nat2
% 12.24/7.96 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
% 12.24/7.96 @ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
% 12.24/7.96 @ ( collect @ ( set @ ( set @ A ) )
% 12.24/7.96 @ ^ [I8: set @ ( set @ A )] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A6 )
% 12.24/7.96 & ( I8
% 12.24/7.96 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4876_surj__int__encode,axiom,
% 12.24/7.96 ( ( image @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
% 12.24/7.96 = ( top_top @ ( set @ nat ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4877_horner__sum__eq__sum__funpow,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comm_semiring_0 @ A )
% 12.24/7.96 => ( ( groups4207007520872428315er_sum @ B @ A )
% 12.24/7.96 = ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
% 12.24/7.96 ( groups7311177749621191930dd_sum @ nat @ A
% 12.24/7.96 @ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A5 ) @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) )
% 12.24/7.96 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4878_Suc__funpow,axiom,
% 12.24/7.96 ! [N: nat] :
% 12.24/7.96 ( ( compow @ ( nat > nat ) @ N @ suc )
% 12.24/7.96 = ( plus_plus @ nat @ N ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4879_funpow__0,axiom,
% 12.24/7.96 ! [A: $tType,F: A > A,X: A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X )
% 12.24/7.96 = X ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4880_Inf__eq__bot__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( ( complete_Inf_Inf @ A @ A6 )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
% 12.24/7.96 => ? [Y6: A] :
% 12.24/7.96 ( ( member @ A @ Y6 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4881_Inf__atLeastAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4882_cInf__atLeastAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4883_cInf__atLeastLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4884_Inf__atLeastLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4885_surj__fn,axiom,
% 12.24/7.96 ! [A: $tType,F: A > A,N: nat] :
% 12.24/7.96 ( ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) )
% 12.24/7.96 => ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) )
% 12.24/7.96 = ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4886_cInf__greaterThanAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4887_Inf__greaterThanAtMost,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4888_Inf__greaterThanLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [X: A,Y3: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
% 12.24/7.96 = X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4889_cInf__greaterThanLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( dense_linorder @ A ) )
% 12.24/7.96 => ! [Y3: A,X: A] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ X )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
% 12.24/7.96 = Y3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4890_INF__eq__bot__iff,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = ( bot_bot @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
% 12.24/7.96 => ? [Y6: B] :
% 12.24/7.96 ( ( member @ B @ Y6 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ ( F @ Y6 ) @ X2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4891_Compl__UN,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
% 12.24/7.96 ( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
% 12.24/7.96 = ( complete_Inf_Inf @ ( set @ A )
% 12.24/7.96 @ ( image @ B @ ( set @ A )
% 12.24/7.96 @ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
% 12.24/7.96 @ A6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4892_Compl__INT,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
% 12.24/7.96 ( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ ( set @ A )
% 12.24/7.96 @ ( image @ B @ ( set @ A )
% 12.24/7.96 @ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
% 12.24/7.96 @ A6 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4893_Inf__atMostLessThan,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A] :
% 12.24/7.96 ( ( ord_less @ A @ ( top_top @ A ) @ X )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
% 12.24/7.96 = ( bot_bot @ A ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4894_comp__funpow,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,N: nat,F: A > A] :
% 12.24/7.96 ( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
% 12.24/7.96 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4895_Inf__greatest,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,Z3: A] :
% 12.24/7.96 ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4896_le__Inf__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [B2: A,A6: set @ A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A6 ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4897_Inf__lower2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [U: A,A6: set @ A,V: A] :
% 12.24/7.96 ( ( member @ A @ U @ A6 )
% 12.24/7.96 => ( ( ord_less_eq @ A @ U @ V )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4898_Inf__lower,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [X: A,A6: set @ A] :
% 12.24/7.96 ( ( member @ A @ X @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4899_Inf__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A6: set @ A] :
% 12.24/7.96 ( ! [B5: A] :
% 12.24/7.96 ( ( member @ A @ B5 @ B7 )
% 12.24/7.96 => ? [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ A6 )
% 12.24/7.96 & ( ord_less_eq @ A @ X4 @ B5 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4900_Inf__eqI,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,X: A] :
% 12.24/7.96 ( ! [I4: A] :
% 12.24/7.96 ( ( member @ A @ I4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ I4 ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [I3: A] :
% 12.24/7.96 ( ( member @ A @ I3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ I3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ A6 )
% 12.24/7.96 = X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4901_cInf__eq__minimum,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [Z3: A,X8: set @ A] :
% 12.24/7.96 ( ( member @ A @ Z3 @ X8 )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ X8 )
% 12.24/7.96 = Z3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4902_cInf__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 & ( no_top @ A ) )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ A3 @ X3 ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ A3 ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ X8 )
% 12.24/7.96 = A3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4903_Inf__less__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [S3: set @ A,A3: A] :
% 12.24/7.96 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 )
% 12.24/7.96 = ( ? [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ S3 )
% 12.24/7.96 & ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4904_mono__pow,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [F: A > A,N: nat] :
% 12.24/7.96 ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4905_funpow__mult,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,M: nat,F: A > A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F ) )
% 12.24/7.96 = ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4906_int__encode__eq,axiom,
% 12.24/7.96 ! [X: int,Y3: int] :
% 12.24/7.96 ( ( ( nat_int_encode @ X )
% 12.24/7.96 = ( nat_int_encode @ Y3 ) )
% 12.24/7.96 = ( X = Y3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4907_funpow__swap1,axiom,
% 12.24/7.96 ! [A: $tType,F: A > A,N: nat,X: A] :
% 12.24/7.96 ( ( F @ ( compow @ ( A > A ) @ N @ F @ X ) )
% 12.24/7.96 = ( compow @ ( A > A ) @ N @ F @ ( F @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4908_funpow__mod__eq,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,F: A > A,X: A,M: nat] :
% 12.24/7.96 ( ( ( compow @ ( A > A ) @ N @ F @ X )
% 12.24/7.96 = X )
% 12.24/7.96 => ( ( compow @ ( A > A ) @ ( modulo_modulo @ nat @ M @ N ) @ F @ X )
% 12.24/7.96 = ( compow @ ( A > A ) @ M @ F @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4909_Inf__le__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [A6: set @ A,X: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X )
% 12.24/7.96 = ( ! [Y6: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y6 )
% 12.24/7.96 => ? [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4910_INF__eq,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,B7: set @ C,G: C > A,F: B > A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ? [X4: C] :
% 12.24/7.96 ( ( member @ C @ X4 @ B7 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
% 12.24/7.96 => ( ! [J2: C] :
% 12.24/7.96 ( ( member @ C @ J2 @ B7 )
% 12.24/7.96 => ? [X4: B] :
% 12.24/7.96 ( ( member @ B @ X4 @ A6 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4911_cInf__eq__non__empty,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ A3 @ X3 ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [X4: A] :
% 12.24/7.96 ( ( member @ A @ X4 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ A3 ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ X8 )
% 12.24/7.96 = A3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4912_cInf__greatest,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,Z3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ Z3 @ X3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4913_Inf__less__eq,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,U: A] :
% 12.24/7.96 ( ! [V4: A] :
% 12.24/7.96 ( ( member @ A @ V4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ V4 @ U ) )
% 12.24/7.96 => ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ U ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4914_cInf__le__finite,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [X8: set @ A,X: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( member @ A @ X @ X8 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4915_cInf__lessD,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,Z3: A] :
% 12.24/7.96 ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z3 )
% 12.24/7.96 => ? [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 & ( ord_less @ A @ X3 @ Z3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4916_Inf__superset__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A6: set @ A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4917_finite__imp__less__Inf,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,X: A,A3: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( member @ A @ X @ X8 )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ X8 )
% 12.24/7.96 => ( ord_less @ A @ A3 @ X3 ) )
% 12.24/7.96 => ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4918_funpow__times__power,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( monoid_mult @ A )
% 12.24/7.96 => ! [F: A > nat,X: A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ ( F @ X ) @ ( times_times @ A @ X ) )
% 12.24/7.96 = ( times_times @ A @ ( power_power @ A @ X @ ( F @ X ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4919_funpow__Suc__right,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,F: A > A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 12.24/7.96 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4920_funpow_Osimps_I2_J,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,F: A > A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 12.24/7.96 = ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4921_sup__Inf,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [A3: A,B7: set @ A] :
% 12.24/7.96 ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B7 ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( sup_sup @ A @ A3 ) @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4922_Inf__sup__eq__top__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A3: A] :
% 12.24/7.96 ( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
% 12.24/7.96 = ( top_top @ A ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ B7 )
% 12.24/7.96 => ( ( sup_sup @ A @ X2 @ A3 )
% 12.24/7.96 = ( top_top @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4923_funpow__add,axiom,
% 12.24/7.96 ! [A: $tType,M: nat,N: nat,F: A > A] :
% 12.24/7.96 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F )
% 12.24/7.96 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4924_funpow__mono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order @ A )
% 12.24/7.96 => ! [F: A > A,A6: A,B7: A,N: nat] :
% 12.24/7.96 ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ( ord_less_eq @ A @ A6 @ B7 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ A6 ) @ ( compow @ ( A > A ) @ N @ F @ B7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4925_INF__greatest,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,U: A,F: B > A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ ( F @ I4 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4926_le__INF__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [U: A,F: B > A,A6: set @ B] :
% 12.24/7.96 ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.96 = ( ! [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4927_INF__lower2,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I: B,A6: set @ B,F: B > A,U: A] :
% 12.24/7.96 ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ( ord_less_eq @ A @ ( F @ I ) @ U )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4928_INF__mono_H,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [F: B > A,G: B > A,A6: set @ B] :
% 12.24/7.96 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4929_INF__lower,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I: B,A6: set @ B,F: B > A] :
% 12.24/7.96 ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ I ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4930_INF__mono,axiom,
% 12.24/7.96 ! [C: $tType,A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [B7: set @ B,A6: set @ C,F: C > A,G: B > A] :
% 12.24/7.96 ( ! [M6: B] :
% 12.24/7.96 ( ( member @ B @ M6 @ B7 )
% 12.24/7.96 => ? [X4: C] :
% 12.24/7.96 ( ( member @ C @ X4 @ A6 )
% 12.24/7.96 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4931_INF__eqI,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,X: A,F: B > A] :
% 12.24/7.96 ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ X @ ( F @ I4 ) ) )
% 12.24/7.96 => ( ! [Y5: A] :
% 12.24/7.96 ( ! [I3: B] :
% 12.24/7.96 ( ( member @ B @ I3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ ( F @ I3 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/7.96 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.96 = X ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4932_less__INF__D,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [Y3: A,F: B > A,A6: set @ B,I: B] :
% 12.24/7.96 ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.96 => ( ( member @ B @ I @ A6 )
% 12.24/7.96 => ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4933_INF__less__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,A3: A] :
% 12.24/7.96 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
% 12.24/7.96 = ( ? [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4934_INF__sup,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [F: B > A,B7: set @ B,A3: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [B3: B] : ( sup_sup @ A @ ( F @ B3 ) @ A3 )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4935_Inf__sup,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [B7: set @ A,A3: A] :
% 12.24/7.96 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ A @ A
% 12.24/7.96 @ ^ [B3: A] : ( sup_sup @ A @ B3 @ A3 )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4936_sup__INF,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [A3: A,F: B > A,B7: set @ B] :
% 12.24/7.96 ( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [B3: B] : ( sup_sup @ A @ A3 @ ( F @ B3 ) )
% 12.24/7.96 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4937_INF__sup__distrib2,axiom,
% 12.24/7.96 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
% 12.24/7.96 ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [A5: B] :
% 12.24/7.96 ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ C @ A
% 12.24/7.96 @ ^ [B3: C] : ( sup_sup @ A @ ( F @ A5 ) @ ( G @ B3 ) )
% 12.24/7.96 @ B7 ) )
% 12.24/7.96 @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4938_INF__le__iff,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( comple5582772986160207858norder @ A )
% 12.24/7.96 => ! [F: B > A,A6: set @ B,X: A] :
% 12.24/7.96 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ X )
% 12.24/7.96 = ( ! [Y6: A] :
% 12.24/7.96 ( ( ord_less @ A @ X @ Y6 )
% 12.24/7.96 => ? [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ A6 )
% 12.24/7.96 & ( ord_less @ A @ ( F @ X2 ) @ Y6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4939_cINF__greatest,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( condit1219197933456340205attice @ A )
% 12.24/7.96 => ! [A6: set @ B,M: A,F: B > A] :
% 12.24/7.96 ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ A6 )
% 12.24/7.96 => ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4940_INF__eq__iff,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [I6: set @ B,F: B > A,C2: A] :
% 12.24/7.96 ( ( I6
% 12.24/7.96 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 => ( ! [I4: B] :
% 12.24/7.96 ( ( member @ B @ I4 @ I6 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ I4 ) @ C2 ) )
% 12.24/7.96 => ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ I6 ) )
% 12.24/7.96 = C2 )
% 12.24/7.96 = ( ! [X2: B] :
% 12.24/7.96 ( ( member @ B @ X2 @ I6 )
% 12.24/7.96 => ( ( F @ X2 )
% 12.24/7.96 = C2 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4941_finite__less__Inf__iff,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 => ! [X8: set @ A,A3: A] :
% 12.24/7.96 ( ( finite_finite2 @ A @ X8 )
% 12.24/7.96 => ( ( X8
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) )
% 12.24/7.96 = ( ! [X2: A] :
% 12.24/7.96 ( ( member @ A @ X2 @ X8 )
% 12.24/7.96 => ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4942_Inf__le__Sup,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4943_cInf__abs__ge,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( linordered_idom @ A ) )
% 12.24/7.96 => ! [S3: set @ A,A3: A] :
% 12.24/7.96 ( ( S3
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ S3 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4944_uminus__Sup,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple489889107523837845lgebra @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A6 ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4945_uminus__Inf,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple489889107523837845lgebra @ A )
% 12.24/7.96 => ! [A6: set @ A] :
% 12.24/7.96 ( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A6 ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4946_less__eq__Inf__inter,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4947_Kleene__iter__lpfp,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_bot @ A )
% 12.24/7.96 => ! [F: A > A,P5: A,K: nat] :
% 12.24/7.96 ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ( ord_less_eq @ A @ ( F @ P5 ) @ P5 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4948_Kleene__iter__gpfp,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order_top @ A )
% 12.24/7.96 => ! [F: A > A,P5: A,K: nat] :
% 12.24/7.96 ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ( ord_less_eq @ A @ P5 @ ( F @ P5 ) )
% 12.24/7.96 => ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4949_funpow__mono2,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( order @ A )
% 12.24/7.96 => ! [F: A > A,I: nat,J: nat,X: A,Y3: A] :
% 12.24/7.96 ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ( ord_less_eq @ nat @ I @ J )
% 12.24/7.96 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.96 => ( ( ord_less_eq @ A @ X @ ( F @ X ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F @ X ) @ ( compow @ ( A > A ) @ J @ F @ Y3 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4950_INF__superset__mono,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [B7: set @ B,A6: set @ B,F: B > A,G: B > A] :
% 12.24/7.96 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
% 12.24/7.96 => ( ! [X3: B] :
% 12.24/7.96 ( ( member @ B @ X3 @ B7 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4951_uminus__INF,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple489889107523837845lgebra @ A )
% 12.24/7.96 => ! [B7: B > A,A6: set @ B] :
% 12.24/7.96 ( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
% 12.24/7.96 @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4952_uminus__SUP,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple489889107523837845lgebra @ A )
% 12.24/7.96 => ! [B7: B > A,A6: set @ B] :
% 12.24/7.96 ( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
% 12.24/7.96 @ A6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4953_mono__INF,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( comple6319245703460814977attice @ B ) )
% 12.24/7.96 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 12.24/7.96 ( ( order_mono @ A @ B @ F )
% 12.24/7.96 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
% 12.24/7.96 @ ( complete_Inf_Inf @ B
% 12.24/7.96 @ ( image @ C @ B
% 12.24/7.96 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 12.24/7.96 @ I6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4954_mono__Inf,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType] :
% 12.24/7.96 ( ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 & ( comple6319245703460814977attice @ B ) )
% 12.24/7.96 => ! [F: A > B,A6: set @ A] :
% 12.24/7.96 ( ( order_mono @ A @ B @ F )
% 12.24/7.96 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4955_of__nat__def,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ( ( semiring_1_of_nat @ A )
% 12.24/7.96 = ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4956_numeral__add__unfold__funpow,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_numeral @ A )
% 12.24/7.96 => ! [K: num,A3: A] :
% 12.24/7.96 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
% 12.24/7.96 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4957_INF__le__SUP,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType] :
% 12.24/7.96 ( ( comple6319245703460814977attice @ A )
% 12.24/7.96 => ! [A6: set @ B,F: B > A] :
% 12.24/7.96 ( ( A6
% 12.24/7.96 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4958_mono__funpow,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( lattice @ A )
% 12.24/7.96 & ( order_bot @ A ) )
% 12.24/7.96 => ! [Q: A > A] :
% 12.24/7.96 ( ( order_mono @ A @ A @ Q )
% 12.24/7.96 => ( order_mono @ nat @ A
% 12.24/7.96 @ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4959_cInf__asclose,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( condit6923001295902523014norder @ A )
% 12.24/7.96 & ( linordered_idom @ A ) )
% 12.24/7.96 => ! [S3: set @ A,L: A,E: A] :
% 12.24/7.96 ( ( S3
% 12.24/7.96 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.96 => ( ! [X3: A] :
% 12.24/7.96 ( ( member @ A @ X3 @ S3 )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
% 12.24/7.96 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4960_funpow__decreasing,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( lattice @ A )
% 12.24/7.96 & ( order_bot @ A ) )
% 12.24/7.96 => ! [M: nat,N: nat,F: A > A] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.96 => ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4961_funpow__increasing,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ( lattice @ A )
% 12.24/7.96 & ( order_top @ A ) )
% 12.24/7.96 => ! [M: nat,N: nat,F: A > A] :
% 12.24/7.96 ( ( ord_less_eq @ nat @ M @ N )
% 12.24/7.96 => ( ( order_mono @ A @ A @ F )
% 12.24/7.96 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4962_numeral__unfold__funpow,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( semiring_1 @ A )
% 12.24/7.96 => ( ( numeral_numeral @ A )
% 12.24/7.96 = ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4963_SUP__INF,axiom,
% 12.24/7.96 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [P: C > B > A] :
% 12.24/7.96 ( ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [Y6: B] :
% 12.24/7.96 ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ C @ A
% 12.24/7.96 @ ^ [X2: C] : ( P @ X2 @ Y6 )
% 12.24/7.96 @ ( top_top @ ( set @ C ) ) ) )
% 12.24/7.96 @ ( top_top @ ( set @ B ) ) ) )
% 12.24/7.96 = ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ ( B > C ) @ A
% 12.24/7.96 @ ^ [X2: B > C] :
% 12.24/7.96 ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [Y6: B] : ( P @ ( X2 @ Y6 ) @ Y6 )
% 12.24/7.96 @ ( top_top @ ( set @ B ) ) ) )
% 12.24/7.96 @ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4964_INF__SUP,axiom,
% 12.24/7.96 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.96 ( ( comple592849572758109894attice @ A )
% 12.24/7.96 => ! [P: C > B > A] :
% 12.24/7.96 ( ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [Y6: B] :
% 12.24/7.96 ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ C @ A
% 12.24/7.96 @ ^ [X2: C] : ( P @ X2 @ Y6 )
% 12.24/7.96 @ ( top_top @ ( set @ C ) ) ) )
% 12.24/7.96 @ ( top_top @ ( set @ B ) ) ) )
% 12.24/7.96 = ( complete_Sup_Sup @ A
% 12.24/7.96 @ ( image @ ( B > C ) @ A
% 12.24/7.96 @ ^ [F2: B > C] :
% 12.24/7.96 ( complete_Inf_Inf @ A
% 12.24/7.96 @ ( image @ B @ A
% 12.24/7.96 @ ^ [X2: B] : ( P @ ( F2 @ X2 ) @ X2 )
% 12.24/7.96 @ ( top_top @ ( set @ B ) ) ) )
% 12.24/7.96 @ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4965_relpowp__fun__conv,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) )
% 12.24/7.96 = ( ^ [N3: nat,P4: A > A > $o,X2: A,Y6: A] :
% 12.24/7.96 ? [F2: nat > A] :
% 12.24/7.96 ( ( ( F2 @ ( zero_zero @ nat ) )
% 12.24/7.96 = X2 )
% 12.24/7.96 & ( ( F2 @ N3 )
% 12.24/7.96 = Y6 )
% 12.24/7.96 & ! [I5: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ I5 @ N3 )
% 12.24/7.96 => ( P4 @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4966_relpowp__1,axiom,
% 12.24/7.96 ! [A: $tType,P: A > A > $o] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
% 12.24/7.96 = P ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4967_Nat_Ofunpow__code__def,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( funpow @ A )
% 12.24/7.96 = ( compow @ ( A > A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4968_Inf__int__def,axiom,
% 12.24/7.96 ( ( complete_Inf_Inf @ int )
% 12.24/7.96 = ( ^ [X7: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X7 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4969_relpowp__Suc__I2,axiom,
% 12.24/7.96 ! [A: $tType,P: A > A > $o,X: A,Y3: A,N: nat,Z3: A] :
% 12.24/7.96 ( ( P @ X @ Y3 )
% 12.24/7.96 => ( ( compow @ ( A > A > $o ) @ N @ P @ Y3 @ Z3 )
% 12.24/7.96 => ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4970_relpowp__Suc__E2,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 12.24/7.96 => ~ ! [Y5: A] :
% 12.24/7.96 ( ( P @ X @ Y5 )
% 12.24/7.96 => ~ ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4971_relpowp__Suc__D2,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 12.24/7.96 => ? [Y5: A] :
% 12.24/7.96 ( ( P @ X @ Y5 )
% 12.24/7.96 & ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4972_relpowp__Suc__I,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Y3: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y3 )
% 12.24/7.96 => ( ( P @ Y3 @ Z3 )
% 12.24/7.96 => ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4973_relpowp__Suc__E,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
% 12.24/7.96 => ~ ! [Y5: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y5 )
% 12.24/7.96 => ~ ( P @ Y5 @ Z3 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4974_relpowp__0__I,axiom,
% 12.24/7.96 ! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4975_relpowp__0__E,axiom,
% 12.24/7.96 ! [A: $tType,P: A > A > $o,X: A,Y3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y3 )
% 12.24/7.96 => ( X = Y3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4976_relpowp_Osimps_I1_J,axiom,
% 12.24/7.96 ! [A: $tType,R2: A > A > $o] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
% 12.24/7.96 = ( ^ [Y: A,Z2: A] : ( Y = Z2 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4977_relpowp__E,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
% 12.24/7.96 => ( ( ( N
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 => ( X != Z3 ) )
% 12.24/7.96 => ~ ! [Y5: A,M6: nat] :
% 12.24/7.96 ( ( N
% 12.24/7.96 = ( suc @ M6 ) )
% 12.24/7.96 => ( ( compow @ ( A > A > $o ) @ M6 @ P @ X @ Y5 )
% 12.24/7.96 => ~ ( P @ Y5 @ Z3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4978_relpowp__E2,axiom,
% 12.24/7.96 ! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
% 12.24/7.96 ( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
% 12.24/7.96 => ( ( ( N
% 12.24/7.96 = ( zero_zero @ nat ) )
% 12.24/7.96 => ( X != Z3 ) )
% 12.24/7.96 => ~ ! [Y5: A,M6: nat] :
% 12.24/7.96 ( ( N
% 12.24/7.96 = ( suc @ M6 ) )
% 12.24/7.96 => ( ( P @ X @ Y5 )
% 12.24/7.96 => ~ ( compow @ ( A > A > $o ) @ M6 @ P @ Y5 @ Z3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4979_relpowp__bot,axiom,
% 12.24/7.96 ! [A: $tType,N: nat] :
% 12.24/7.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.96 => ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
% 12.24/7.96 = ( bot_bot @ ( A > A > $o ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4980_set__zip,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 12.24/7.96 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 12.24/7.96 = ( collect @ ( product_prod @ A @ B )
% 12.24/7.96 @ ^ [Uu: product_prod @ A @ B] :
% 12.24/7.96 ? [I5: nat] :
% 12.24/7.96 ( ( Uu
% 12.24/7.96 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I5 ) @ ( nth @ B @ Ys2 @ I5 ) ) )
% 12.24/7.96 & ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4981_measure__function__int,axiom,
% 12.24/7.96 fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4982_surj__int__decode,axiom,
% 12.24/7.96 ( ( image @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.96 = ( top_top @ ( set @ int ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4983_int__decode__inverse,axiom,
% 12.24/7.96 ! [N: nat] :
% 12.24/7.96 ( ( nat_int_encode @ ( nat_int_decode @ N ) )
% 12.24/7.96 = N ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4984_int__encode__inverse,axiom,
% 12.24/7.96 ! [X: int] :
% 12.24/7.96 ( ( nat_int_decode @ ( nat_int_encode @ X ) )
% 12.24/7.96 = X ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4985_measure__snd,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,F: A > nat] :
% 12.24/7.96 ( ( fun_is_measure @ A @ F )
% 12.24/7.96 => ( fun_is_measure @ ( product_prod @ B @ A )
% 12.24/7.96 @ ^ [P6: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4986_measure__fst,axiom,
% 12.24/7.96 ! [B: $tType,A: $tType,F: A > nat] :
% 12.24/7.96 ( ( fun_is_measure @ A @ F )
% 12.24/7.96 => ( fun_is_measure @ ( product_prod @ A @ B )
% 12.24/7.96 @ ^ [P6: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4987_lexordp_Omono,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( ord @ A )
% 12.24/7.96 => ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
% 12.24/7.96 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
% 12.24/7.96 ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.96 ( ( X12
% 12.24/7.96 = ( nil @ A ) )
% 12.24/7.96 & ( X24
% 12.24/7.96 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 12.24/7.96 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.96 ( ( X12
% 12.24/7.96 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.96 & ( X24
% 12.24/7.96 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.96 & ( ord_less @ A @ X2 @ Y6 ) )
% 12.24/7.96 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.96 ( ( X12
% 12.24/7.96 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.96 & ( X24
% 12.24/7.96 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.96 & ~ ( ord_less @ A @ X2 @ Y6 )
% 12.24/7.96 & ~ ( ord_less @ A @ Y6 @ X2 )
% 12.24/7.96 & ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4988_is__measure_Osimps,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( fun_is_measure @ A )
% 12.24/7.96 = ( ^ [A5: A > nat] :
% 12.24/7.96 ? [X7: A > nat] :
% 12.24/7.96 ( ^ [Y: A > nat,Z2: A > nat] : ( Y = Z2 )
% 12.24/7.96 @ A5
% 12.24/7.96 @ X7 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4989_is__measure__trivial,axiom,
% 12.24/7.96 ! [A: $tType,F: A > nat] : ( fun_is_measure @ A @ F ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4990_int__decode__eq,axiom,
% 12.24/7.96 ! [X: nat,Y3: nat] :
% 12.24/7.96 ( ( ( nat_int_decode @ X )
% 12.24/7.96 = ( nat_int_decode @ Y3 ) )
% 12.24/7.96 = ( X = Y3 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4991_bot2E,axiom,
% 12.24/7.96 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 12.24/7.96 ~ ( bot_bot @ ( A > B > $o ) @ X @ Y3 ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4992_measure__size,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( size @ A )
% 12.24/7.96 => ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4993_set__conv__nth,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( set2 @ A )
% 12.24/7.96 = ( ^ [Xs3: list @ A] :
% 12.24/7.96 ( collect @ A
% 12.24/7.96 @ ^ [Uu: A] :
% 12.24/7.96 ? [I5: nat] :
% 12.24/7.96 ( ( Uu
% 12.24/7.96 = ( nth @ A @ Xs3 @ I5 ) )
% 12.24/7.96 & ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4994_set__nths,axiom,
% 12.24/7.96 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 12.24/7.96 ( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
% 12.24/7.96 = ( collect @ A
% 12.24/7.96 @ ^ [Uu: A] :
% 12.24/7.96 ? [I5: nat] :
% 12.24/7.96 ( ( Uu
% 12.24/7.96 = ( nth @ A @ Xs @ I5 ) )
% 12.24/7.96 & ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.96 & ( member @ nat @ I5 @ I6 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4995_sup__bot_Osemilattice__neutr__order__axioms,axiom,
% 12.24/7.96 ! [A: $tType] :
% 12.24/7.96 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.96 => ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
% 12.24/7.96 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/7.96 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4996_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 12.24/7.96 ( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
% 12.24/7.96 @ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
% 12.24/7.96 @ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4997_times__int_Oabs__eq,axiom,
% 12.24/7.96 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 12.24/7.96 ( ( times_times @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 12.24/7.96 = ( abs_Integ
% 12.24/7.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.96 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.96 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.96 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) )
% 12.24/7.96 @ Xa3
% 12.24/7.96 @ X ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4998_int_Oabs__induct,axiom,
% 12.24/7.96 ! [P: int > $o,X: int] :
% 12.24/7.96 ( ! [Y5: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y5 ) )
% 12.24/7.96 => ( P @ X ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_4999_semilattice__neutr__order_Oneutr__eq__iff,axiom,
% 12.24/7.96 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/7.96 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.96 => ( ( Z3
% 12.24/7.96 = ( F @ A3 @ B2 ) )
% 12.24/7.96 = ( ( A3 = Z3 )
% 12.24/7.96 & ( B2 = Z3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_5000_semilattice__neutr__order_Oeq__neutr__iff,axiom,
% 12.24/7.96 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/7.96 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.96 => ( ( ( F @ A3 @ B2 )
% 12.24/7.96 = Z3 )
% 12.24/7.96 = ( ( A3 = Z3 )
% 12.24/7.96 & ( B2 = Z3 ) ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_5001_eq__Abs__Integ,axiom,
% 12.24/7.96 ! [Z3: int] :
% 12.24/7.96 ~ ! [X3: nat,Y5: nat] :
% 12.24/7.96 ( Z3
% 12.24/7.96 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y5 ) ) ) ).
% 12.24/7.96
% 12.24/7.96 thf(fact_5002_mono__compose,axiom,
% 12.24/7.96 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( order @ A )
% 12.24/7.97 & ( order @ C ) )
% 12.24/7.97 => ! [Q: A > B > C,F: D > B] :
% 12.24/7.97 ( ( order_mono @ A @ ( B > C ) @ Q )
% 12.24/7.97 => ( order_mono @ A @ ( D > C )
% 12.24/7.97 @ ^ [I5: A,X2: D] : ( Q @ I5 @ ( F @ X2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5003_zero__int__def,axiom,
% 12.24/7.97 ( ( zero_zero @ int )
% 12.24/7.97 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5004_int__def,axiom,
% 12.24/7.97 ( ( semiring_1_of_nat @ int )
% 12.24/7.97 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5005_nat_Oabs__eq,axiom,
% 12.24/7.97 ! [X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( nat2 @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5006_Nats__altdef1,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ( ( semiring_1_Nats @ A )
% 12.24/7.97 = ( collect @ A
% 12.24/7.97 @ ^ [Uu: A] :
% 12.24/7.97 ? [N3: int] :
% 12.24/7.97 ( ( Uu
% 12.24/7.97 = ( ring_1_of_int @ A @ N3 ) )
% 12.24/7.97 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5007_uminus__int_Oabs__eq,axiom,
% 12.24/7.97 ! [X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( abs_Integ
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 )
% 12.24/7.97 @ X ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5008_one__int__def,axiom,
% 12.24/7.97 ( ( one_one @ int )
% 12.24/7.97 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5009_of__int_Oabs__eq,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ! [X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
% 12.24/7.97 @ X ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5010_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 12.24/7.97 ( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
% 12.24/7.97 @ ^ [M2: nat,N3: nat] :
% 12.24/7.97 ( ( dvd_dvd @ nat @ M2 @ N3 )
% 12.24/7.97 & ( M2 != N3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5011_less__int_Oabs__eq,axiom,
% 12.24/7.97 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( ord_less @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
% 12.24/7.97 @ Xa3
% 12.24/7.97 @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5012_less__eq__int_Oabs__eq,axiom,
% 12.24/7.97 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
% 12.24/7.97 @ Xa3
% 12.24/7.97 @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5013_plus__int_Oabs__eq,axiom,
% 12.24/7.97 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( plus_plus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( abs_Integ
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) )
% 12.24/7.97 @ Xa3
% 12.24/7.97 @ X ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5014_minus__int_Oabs__eq,axiom,
% 12.24/7.97 ! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( minus_minus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
% 12.24/7.97 = ( abs_Integ
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) )
% 12.24/7.97 @ Xa3
% 12.24/7.97 @ X ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5015_inf__top_Osemilattice__neutr__order__axioms,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.97 => ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5016_take__bit__numeral__minus__numeral__int,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int )
% 12.24/7.97 @ ^ [Q3: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q3 ) ) )
% 12.24/7.97 @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5017_prod_Oinsert_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [I6: set @ B,P5: B > A,I: B] :
% 12.24/7.97 ( ( finite_finite2 @ B
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I6 )
% 12.24/7.97 & ( ( P5 @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 => ( ( ( member @ B @ I @ I6 )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) )
% 12.24/7.97 & ( ~ ( member @ B @ I @ I6 )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
% 12.24/7.97 = ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5018_sorted__insort__is__snoc,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [Xs: list @ A,A3: A] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.97 => ( ! [X3: A] :
% 12.24/7.97 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ord_less_eq @ A @ X3 @ A3 ) )
% 12.24/7.97 => ( ( linorder_insort_key @ A @ A
% 12.24/7.97 @ ^ [X2: A] : X2
% 12.24/7.97 @ A3
% 12.24/7.97 @ Xs )
% 12.24/7.97 = ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5019_length__insort,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [F: B > A,X: B,Xs: list @ B] :
% 12.24/7.97 ( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
% 12.24/7.97 = ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5020_prod_Oempty_H,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [P5: B > A] :
% 12.24/7.97 ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 12.24/7.97 = ( one_one @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5021_take__bit__numeral__numeral,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5022_prod_Onon__neutral_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [G: B > A,I6: set @ B] :
% 12.24/7.97 ( ( groups1962203154675924110t_prod @ B @ A @ G
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I6 )
% 12.24/7.97 & ( ( G @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5023_insort__key_Osimps_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [F: B > A,X: B,Y3: B,Ys2: list @ B] :
% 12.24/7.97 ( ( ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.97 => ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
% 12.24/7.97 = ( cons @ B @ X @ ( cons @ B @ Y3 @ Ys2 ) ) ) )
% 12.24/7.97 & ( ~ ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.97 => ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
% 12.24/7.97 = ( cons @ B @ Y3 @ ( linorder_insort_key @ B @ A @ F @ X @ Ys2 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5024_sorted__insort,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [X: A,Xs: list @ A] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 12.24/7.97 @ ( linorder_insort_key @ A @ A
% 12.24/7.97 @ ^ [X2: A] : X2
% 12.24/7.97 @ X
% 12.24/7.97 @ Xs ) )
% 12.24/7.97 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5025_prod_Odistrib__triv_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 12.24/7.97 ( ( finite_finite2 @ B @ I6 )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A
% 12.24/7.97 @ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 12.24/7.97 @ I6 )
% 12.24/7.97 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5026_insort__is__Cons,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [Xs: list @ B,F: B > A,A3: B] :
% 12.24/7.97 ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 12.24/7.97 => ( ord_less_eq @ A @ ( F @ A3 ) @ ( F @ X3 ) ) )
% 12.24/7.97 => ( ( linorder_insort_key @ B @ A @ F @ A3 @ Xs )
% 12.24/7.97 = ( cons @ B @ A3 @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5027_sorted__insort__key,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [F: B > A,X: B,Xs: list @ B] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) )
% 12.24/7.97 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5028_prod_Omono__neutral__cong__right_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.97 => ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.97 => ( ( G @ X3 )
% 12.24/7.97 = ( one_one @ A ) ) )
% 12.24/7.97 => ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ S3 )
% 12.24/7.97 => ( ( G @ X3 )
% 12.24/7.97 = ( H2 @ X3 ) ) )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5029_prod_Omono__neutral__cong__left_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.97 => ( ! [I4: B] :
% 12.24/7.97 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.97 => ( ( H2 @ I4 )
% 12.24/7.97 = ( one_one @ A ) ) )
% 12.24/7.97 => ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ S3 )
% 12.24/7.97 => ( ( G @ X3 )
% 12.24/7.97 = ( H2 @ X3 ) ) )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5030_prod_Omono__neutral__right_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.97 => ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.97 => ( ( G @ X3 )
% 12.24/7.97 = ( one_one @ A ) ) )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5031_prod_Omono__neutral__left_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [S3: set @ B,T6: set @ B,G: B > A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
% 12.24/7.97 => ( ! [X3: B] :
% 12.24/7.97 ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
% 12.24/7.97 => ( ( G @ X3 )
% 12.24/7.97 = ( one_one @ A ) ) )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
% 12.24/7.97 = ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5032_prod_Odistrib_H,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ! [I6: set @ B,G: B > A,H2: B > A] :
% 12.24/7.97 ( ( finite_finite2 @ B
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I6 )
% 12.24/7.97 & ( ( G @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ B
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I6 )
% 12.24/7.97 & ( ( H2 @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A
% 12.24/7.97 @ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
% 12.24/7.97 @ I6 )
% 12.24/7.97 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5033_prod_OG__def,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ( ( groups1962203154675924110t_prod @ B @ A )
% 12.24/7.97 = ( ^ [P6: B > A,I8: set @ B] :
% 12.24/7.97 ( if @ A
% 12.24/7.97 @ ( finite_finite2 @ B
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I8 )
% 12.24/7.97 & ( ( P6 @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 @ ( groups7121269368397514597t_prod @ B @ A @ P6
% 12.24/7.97 @ ( collect @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( member @ B @ X2 @ I8 )
% 12.24/7.97 & ( ( P6 @ X2 )
% 12.24/7.97 != ( one_one @ A ) ) ) ) )
% 12.24/7.97 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5034_filter__insort,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [F: B > A,Xs: list @ B,P: B > $o,X: B] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.97 => ( ( P @ X )
% 12.24/7.97 => ( ( filter @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
% 12.24/7.97 = ( linorder_insort_key @ B @ A @ F @ X @ ( filter @ B @ P @ Xs ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5035_and__minus__numerals_I3_J,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5036_and__minus__numerals_I7_J,axiom,
% 12.24/7.97 ! [N: num,M: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5037_and__minus__numerals_I8_J,axiom,
% 12.24/7.97 ! [N: num,M: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5038_and__minus__numerals_I4_J,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5039_int__numeral__not__and__num,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5040_int__numeral__and__not__num,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.97 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5041_insort__key__remove1,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A3: B,Xs: list @ B,F: B > A] :
% 12.24/7.97 ( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
% 12.24/7.97 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.97 => ( ( ( hd @ B
% 12.24/7.97 @ ( filter @ B
% 12.24/7.97 @ ^ [X2: B] :
% 12.24/7.97 ( ( F @ A3 )
% 12.24/7.97 = ( F @ X2 ) )
% 12.24/7.97 @ Xs ) )
% 12.24/7.97 = A3 )
% 12.24/7.97 => ( ( linorder_insort_key @ B @ A @ F @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
% 12.24/7.97 = Xs ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5042_less__eq__int_Orep__eq,axiom,
% 12.24/7.97 ( ( ord_less_eq @ int )
% 12.24/7.97 = ( ^ [X2: int,Xa4: int] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [Y6: nat,Z7: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
% 12.24/7.97 @ ( rep_Integ @ X2 )
% 12.24/7.97 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5043_less__int_Orep__eq,axiom,
% 12.24/7.97 ( ( ord_less @ int )
% 12.24/7.97 = ( ^ [X2: int,Xa4: int] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [Y6: nat,Z7: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
% 12.24/7.97 @ ( rep_Integ @ X2 )
% 12.24/7.97 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5044_sorted__remove1,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [Xs: list @ A,A3: A] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.97 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5045_sorted__map__remove1,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [F: B > A,Xs: list @ B,X: B] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.97 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5046_length__remove1,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.97 ( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 12.24/7.97 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
% 12.24/7.97 & ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 12.24/7.97 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5047_nat_Orep__eq,axiom,
% 12.24/7.97 ( nat2
% 12.24/7.97 = ( ^ [X2: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5048_insort__remove1,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A3: A,Xs: list @ A] :
% 12.24/7.97 ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.97 => ( ( linorder_insort_key @ A @ A
% 12.24/7.97 @ ^ [X2: A] : X2
% 12.24/7.97 @ A3
% 12.24/7.97 @ ( remove1 @ A @ A3 @ Xs ) )
% 12.24/7.97 = Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5049_of__int_Orep__eq,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ( ( ring_1_of_int @ A )
% 12.24/7.97 = ( ^ [X2: int] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
% 12.24/7.97 @ ( rep_Integ @ X2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5050_uminus__int__def,axiom,
% 12.24/7.97 ( ( uminus_uminus @ int )
% 12.24/7.97 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5051_numeral__xor__num,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5052_numeral__and__num,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5053_times__int__def,axiom,
% 12.24/7.97 ( ( times_times @ int )
% 12.24/7.97 = ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5054_minus__int__def,axiom,
% 12.24/7.97 ( ( minus_minus @ int )
% 12.24/7.97 = ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5055_plus__int__def,axiom,
% 12.24/7.97 ( ( plus_plus @ int )
% 12.24/7.97 = ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5056_rp__inv__image__def,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( fun_rp_inv_image @ A @ B )
% 12.24/7.97 = ( 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 ) ) ) )
% 12.24/7.97 @ ^ [R6: set @ ( product_prod @ A @ A ),S7: 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 @ S7 @ F2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5057_ring__1__class_Oof__int__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ( ( ring_1_of_int @ A )
% 12.24/7.97 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5058_card__Min__le__sum,axiom,
% 12.24/7.97 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5059_of__nat__eq__id,axiom,
% 12.24/7.97 ( ( semiring_1_of_nat @ nat )
% 12.24/7.97 = ( id @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5060_id__funpow,axiom,
% 12.24/7.97 ! [A: $tType,N: nat] :
% 12.24/7.97 ( ( compow @ ( A > A ) @ N @ ( id @ A ) )
% 12.24/7.97 = ( id @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5061_rotate0,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( rotate @ A @ ( zero_zero @ nat ) )
% 12.24/7.97 = ( id @ ( list @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5062_group__add__class_Ominus__comp__minus,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( group_add @ A )
% 12.24/7.97 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 12.24/7.97 = ( id @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5063_boolean__algebra__class_Ominus__comp__minus,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( boolea8198339166811842893lgebra @ A )
% 12.24/7.97 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 12.24/7.97 = ( id @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5064_Min__singleton,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [X: A] :
% 12.24/7.97 ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5065_push__bit__0__id,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.97 => ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
% 12.24/7.97 = ( id @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5066_drop__bit__0,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.97 => ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
% 12.24/7.97 = ( id @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5067_Min_Obounded__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5068_Min__gr__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5069_Min__const,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ B,C2: A] :
% 12.24/7.97 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A
% 12.24/7.97 @ ( image @ B @ A
% 12.24/7.97 @ ^ [Uu: B] : C2
% 12.24/7.97 @ A6 ) )
% 12.24/7.97 = C2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5070_Min__insert,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5071_funpow__simps__right_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,F: A > A] :
% 12.24/7.97 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F )
% 12.24/7.97 = ( id @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5072_Min_OcoboundedI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,A3: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A3 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5073_Min__eqI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ! [Y5: A] :
% 12.24/7.97 ( ( member @ A @ Y5 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ Y5 ) )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = X ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5074_Min__le,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5075_Min__in,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5076_Min_Oin__idem,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5077_less__int__def,axiom,
% 12.24/7.97 ( ( ord_less @ int )
% 12.24/7.97 = ( 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 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5078_less__eq__int__def,axiom,
% 12.24/7.97 ( ( ord_less_eq @ int )
% 12.24/7.97 = ( 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 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5079_nat__def,axiom,
% 12.24/7.97 ( nat2
% 12.24/7.97 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5080_Min_OboundedI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ! [A8: A] :
% 12.24/7.97 ( ( member @ A @ A8 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ A8 ) )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5081_Min_OboundedE,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 => ! [A10: A] :
% 12.24/7.97 ( ( member @ A @ A10 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5082_eq__Min__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,M: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( M
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( ( member @ A @ M @ A6 )
% 12.24/7.97 & ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5083_Min__le__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
% 12.24/7.97 = ( ? [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 & ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5084_Min__eq__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,M: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = M )
% 12.24/7.97 = ( ( member @ A @ M @ A6 )
% 12.24/7.97 & ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5085_Min__less__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
% 12.24/7.97 = ( ? [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 & ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5086_Min__insert2,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,A3: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ! [B5: A] :
% 12.24/7.97 ( ( member @ A @ B5 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ A3 @ B5 ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A3 @ A6 ) )
% 12.24/7.97 = A3 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5087_Min__Inf,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( comple5582772986160207858norder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5088_Min_Osubset__imp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5089_Min__antimono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [M7: set @ A,N6: set @ A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
% 12.24/7.97 => ( ( M7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5090_hom__Min__commute,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [H2: A > A,N6: set @ A] :
% 12.24/7.97 ( ! [X3: A,Y5: A] :
% 12.24/7.97 ( ( H2 @ ( ord_min @ A @ X3 @ Y5 ) )
% 12.24/7.97 = ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.97 => ( ( N6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N6 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5091_Min_Osubset,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( B7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.97 => ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5092_Min_Oclosed,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.97 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5093_Min_Oinsert__not__elem,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5094_mono__Min__commute,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( linorder @ A )
% 12.24/7.97 & ( linorder @ B ) )
% 12.24/7.97 => ! [F: A > B,A6: set @ A] :
% 12.24/7.97 ( ( order_mono @ A @ B @ F )
% 12.24/7.97 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( F @ ( lattic643756798350308766er_Min @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5095_Min_Ounion,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.97 => ( ( B7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.97 = ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( lattic643756798350308766er_Min @ A @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5096_Min__add__commute,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( linord4140545234300271783up_add @ A )
% 12.24/7.97 => ! [S3: set @ B,F: B > A,K: A] :
% 12.24/7.97 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.97 => ( ( S3
% 12.24/7.97 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A
% 12.24/7.97 @ ( image @ B @ A
% 12.24/7.97 @ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
% 12.24/7.97 @ S3 ) )
% 12.24/7.97 = ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5097_Min_Oinsert__remove,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = X ) )
% 12.24/7.97 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5098_Min_Oremove,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = X ) )
% 12.24/7.97 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5099_positive__def,axiom,
% 12.24/7.97 ( positive
% 12.24/7.97 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5100_dual__Max,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ( ( lattices_Max @ A
% 12.24/7.97 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5101_dual__max,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ( ( max @ A
% 12.24/7.97 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 12.24/7.97 = ( ord_min @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5102_euclidean__size__nat__def,axiom,
% 12.24/7.97 ( ( euclid6346220572633701492n_size @ nat )
% 12.24/7.97 = ( id @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5103_linorder_OMax_Ocong,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( lattices_Max @ A )
% 12.24/7.97 = ( lattices_Max @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5104_ord_Omax__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( max @ A )
% 12.24/7.97 = ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ B3 @ A5 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5105_ord_Omax_Ocong,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( max @ A )
% 12.24/7.97 = ( max @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5106_dual__min,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ( ( min @ A
% 12.24/7.97 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 12.24/7.97 = ( ord_max @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5107_sum__le__card__Max,axiom,
% 12.24/7.97 ! [A: $tType,A6: set @ A,F: A > nat] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5108_euclidean__size__integer__def,axiom,
% 12.24/7.97 ( ( euclid6346220572633701492n_size @ code_integer )
% 12.24/7.97 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5109_Max__singleton,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [X: A] :
% 12.24/7.97 ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5110_Max__divisors__self__nat,axiom,
% 12.24/7.97 ! [N: nat] :
% 12.24/7.97 ( ( N
% 12.24/7.97 != ( zero_zero @ nat ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ nat
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
% 12.24/7.97 = N ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5111_Max_Obounded__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 12.24/7.97 = ( ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5112_Max__less__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 12.24/7.97 = ( ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5113_Max__const,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ B,C2: A] :
% 12.24/7.97 ( ( finite_finite2 @ B @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A
% 12.24/7.97 @ ( image @ B @ A
% 12.24/7.97 @ ^ [Uu: B] : C2
% 12.24/7.97 @ A6 ) )
% 12.24/7.97 = C2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5114_Max__insert,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5115_minus__Max__eq__Min,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linord5086331880401160121up_add @ A )
% 12.24/7.97 => ! [S3: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.97 => ( ( S3
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
% 12.24/7.97 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5116_minus__Min__eq__Max,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linord5086331880401160121up_add @ A )
% 12.24/7.97 => ! [S3: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.97 => ( ( S3
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5117_ord_Omin_Ocong,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( min @ A )
% 12.24/7.97 = ( min @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5118_ord_Omin__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( min @ A )
% 12.24/7.97 = ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ A5 @ B3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5119_Max_OcoboundedI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,A3: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5120_Max__eq__if,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.97 => ( ! [X3: A] :
% 12.24/7.97 ( ( member @ A @ X3 @ A6 )
% 12.24/7.97 => ? [Xa: A] :
% 12.24/7.97 ( ( member @ A @ Xa @ B7 )
% 12.24/7.97 & ( ord_less_eq @ A @ X3 @ Xa ) ) )
% 12.24/7.97 => ( ! [X3: A] :
% 12.24/7.97 ( ( member @ A @ X3 @ B7 )
% 12.24/7.97 => ? [Xa: A] :
% 12.24/7.97 ( ( member @ A @ Xa @ A6 )
% 12.24/7.97 & ( ord_less_eq @ A @ X3 @ Xa ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5121_Max__eqI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ! [Y5: A] :
% 12.24/7.97 ( ( member @ A @ Y5 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = X ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5122_Max__ge,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5123_Max__in,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5124_Max_Oin__idem,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5125_Max_OboundedI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ! [A8: A] :
% 12.24/7.97 ( ( member @ A @ A8 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ A8 @ X ) )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5126_Max_OboundedE,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
% 12.24/7.97 => ! [A10: A] :
% 12.24/7.97 ( ( member @ A @ A10 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5127_eq__Max__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,M: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( M
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( ( member @ A @ M @ A6 )
% 12.24/7.97 & ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5128_Max__ge__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( ? [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 & ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5129_Max__eq__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,M: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = M )
% 12.24/7.97 = ( ( member @ A @ M @ A6 )
% 12.24/7.97 & ! [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5130_Max__gr__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( ? [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ A6 )
% 12.24/7.97 & ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5131_Max__insert2,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,A3: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ! [B5: A] :
% 12.24/7.97 ( ( member @ A @ B5 @ A6 )
% 12.24/7.97 => ( ord_less_eq @ A @ B5 @ A3 ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A3 @ A6 ) )
% 12.24/7.97 = A3 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5132_Max__Sup,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( comple5582772986160207858norder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5133_Max_Osubset__imp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5134_Max__mono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [M7: set @ A,N6: set @ A] :
% 12.24/7.97 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
% 12.24/7.97 => ( ( M7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5135_hom__Max__commute,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [H2: A > A,N6: set @ A] :
% 12.24/7.97 ( ! [X3: A,Y5: A] :
% 12.24/7.97 ( ( H2 @ ( ord_max @ A @ X3 @ Y5 ) )
% 12.24/7.97 = ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.97 => ( ( N6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N6 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5136_Max_Osubset,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( B7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.97 => ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B7 ) @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5137_Max_Oclosed,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.97 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5138_Max_Oinsert__not__elem,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5139_mono__Max__commute,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( linorder @ A )
% 12.24/7.97 & ( linorder @ B ) )
% 12.24/7.97 => ! [F: A > B,A6: set @ A] :
% 12.24/7.97 ( ( order_mono @ A @ B @ F )
% 12.24/7.97 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( F @ ( lattic643756798349783984er_Max @ A @ A6 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5140_Max_Ounion,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( A6
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.97 => ( ( B7
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.97 = ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5141_Sup__nat__def,axiom,
% 12.24/7.97 ( ( complete_Sup_Sup @ nat )
% 12.24/7.97 = ( ^ [X7: set @ nat] :
% 12.24/7.97 ( if @ nat
% 12.24/7.97 @ ( X7
% 12.24/7.97 = ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.97 @ ( zero_zero @ nat )
% 12.24/7.97 @ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5142_card__le__Suc__Max,axiom,
% 12.24/7.97 ! [S3: set @ nat] :
% 12.24/7.97 ( ( finite_finite2 @ nat @ S3 )
% 12.24/7.97 => ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5143_Max__add__commute,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( linord4140545234300271783up_add @ A )
% 12.24/7.97 => ! [S3: set @ B,F: B > A,K: A] :
% 12.24/7.97 ( ( finite_finite2 @ B @ S3 )
% 12.24/7.97 => ( ( S3
% 12.24/7.97 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A
% 12.24/7.97 @ ( image @ B @ A
% 12.24/7.97 @ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
% 12.24/7.97 @ S3 ) )
% 12.24/7.97 = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5144_divide__nat__def,axiom,
% 12.24/7.97 ( ( divide_divide @ nat )
% 12.24/7.97 = ( ^ [M2: nat,N3: nat] :
% 12.24/7.97 ( if @ nat
% 12.24/7.97 @ ( N3
% 12.24/7.97 = ( zero_zero @ nat ) )
% 12.24/7.97 @ ( zero_zero @ nat )
% 12.24/7.97 @ ( lattic643756798349783984er_Max @ nat
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M2 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5145_gcd__is__Max__divisors__nat,axiom,
% 12.24/7.97 ! [N: nat,M: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.97 => ( ( gcd_gcd @ nat @ M @ N )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ nat
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [D5: nat] :
% 12.24/7.97 ( ( dvd_dvd @ nat @ D5 @ M )
% 12.24/7.97 & ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5146_Max_Oinsert__remove,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = X ) )
% 12.24/7.97 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5147_Max_Oremove,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( member @ A @ X @ A6 )
% 12.24/7.97 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = X ) )
% 12.24/7.97 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.97 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5148_nat__of__integer__def,axiom,
% 12.24/7.97 ( code_nat_of_integer
% 12.24/7.97 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5149_Gcd__eq__Max,axiom,
% 12.24/7.97 ! [M7: set @ nat] :
% 12.24/7.97 ( ( finite_finite2 @ nat @ M7 )
% 12.24/7.97 => ( ( M7
% 12.24/7.97 != ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.97 => ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 12.24/7.97 => ( ( gcd_Gcd @ nat @ M7 )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ nat
% 12.24/7.97 @ ( complete_Inf_Inf @ ( set @ nat )
% 12.24/7.97 @ ( image @ nat @ ( set @ nat )
% 12.24/7.97 @ ^ [M2: nat] :
% 12.24/7.97 ( collect @ nat
% 12.24/7.97 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
% 12.24/7.97 @ M7 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5150_dual__Min,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ( ( lattices_Min @ A
% 12.24/7.97 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5151_mask__integer__def,axiom,
% 12.24/7.97 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 12.24/7.97 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5152_positive_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5153_Max__divisors__self__int,axiom,
% 12.24/7.97 ! [N: int] :
% 12.24/7.97 ( ( N
% 12.24/7.97 != ( zero_zero @ int ) )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ int
% 12.24/7.97 @ ( collect @ int
% 12.24/7.97 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
% 12.24/7.97 = ( abs_abs @ int @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5154_power__transfer,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( ( power @ B )
% 12.24/7.97 & ( power @ A ) )
% 12.24/7.97 => ! [R2: A > B > $o] :
% 12.24/7.97 ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 12.24/7.97 => ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ R2 )
% 12.24/7.97 @ ( power_power @ A )
% 12.24/7.97 @ ( power_power @ B ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5155_linorder_OMin_Ocong,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( lattices_Min @ A )
% 12.24/7.97 = ( lattices_Min @ A ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5156_transfer__rule__numeral,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( ( monoid_add @ B )
% 12.24/7.97 & ( semiring_numeral @ B )
% 12.24/7.97 & ( monoid_add @ A )
% 12.24/7.97 & ( semiring_numeral @ A ) )
% 12.24/7.97 => ! [R2: A > B > $o] :
% 12.24/7.97 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.97 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 12.24/7.97 => ( bNF_rel_fun @ num @ num @ A @ B
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ R2
% 12.24/7.97 @ ( numeral_numeral @ A )
% 12.24/7.97 @ ( numeral_numeral @ B ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5157_transfer__rule__of__int,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( ( ring_1 @ B )
% 12.24/7.97 & ( ring_1 @ A ) )
% 12.24/7.97 => ! [R2: A > B > $o] :
% 12.24/7.97 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.97 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 12.24/7.97 => ( ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
% 12.24/7.97 => ( bNF_rel_fun @ int @ int @ A @ B
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ R2
% 12.24/7.97 @ ( ring_1_of_int @ A )
% 12.24/7.97 @ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5158_gcd__is__Max__divisors__int,axiom,
% 12.24/7.97 ! [N: int,M: int] :
% 12.24/7.97 ( ( N
% 12.24/7.97 != ( zero_zero @ int ) )
% 12.24/7.97 => ( ( gcd_gcd @ int @ M @ N )
% 12.24/7.97 = ( lattic643756798349783984er_Max @ int
% 12.24/7.97 @ ( collect @ int
% 12.24/7.97 @ ^ [D5: int] :
% 12.24/7.97 ( ( dvd_dvd @ int @ D5 @ M )
% 12.24/7.97 & ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5159_transfer__rule__of__nat,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( ( semiring_1 @ B )
% 12.24/7.97 & ( semiring_1 @ A ) )
% 12.24/7.97 => ! [R2: A > B > $o] :
% 12.24/7.97 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.97 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 12.24/7.97 => ( bNF_rel_fun @ nat @ nat @ A @ B
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ R2
% 12.24/7.97 @ ( semiring_1_of_nat @ A )
% 12.24/7.97 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5160_integer__of__nat__def,axiom,
% 12.24/7.97 ( code_integer_of_nat
% 12.24/7.97 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5161_antimono__funpow,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( lattice @ A )
% 12.24/7.97 & ( order_top @ A ) )
% 12.24/7.97 => ! [Q: A > A] :
% 12.24/7.97 ( ( order_mono @ A @ A @ Q )
% 12.24/7.97 => ( order_antimono @ nat @ A
% 12.24/7.97 @ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( top_top @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5162_nat__of__integer__integer__of__nat,axiom,
% 12.24/7.97 ! [N: nat] :
% 12.24/7.97 ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
% 12.24/7.97 = N ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5163_integer__of__nat_Orep__eq,axiom,
% 12.24/7.97 ! [X: nat] :
% 12.24/7.97 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 12.24/7.97 = ( semiring_1_of_nat @ int @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5164_int__of__integer__integer__of__nat,axiom,
% 12.24/7.97 ! [N: nat] :
% 12.24/7.97 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 12.24/7.97 = ( semiring_1_of_nat @ int @ N ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5165_euclidean__size__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ nat @ nat
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( euclid6346220572633701492n_size @ int )
% 12.24/7.97 @ ( euclid6346220572633701492n_size @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5166_euclidean__size__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( euclid6346220572633701492n_size @ nat )
% 12.24/7.97 @ ( euclid6346220572633701492n_size @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5167_dup_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
% 12.24/7.97 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5168_sub_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ int @ int
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
% 12.24/7.97 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5169_less__eq__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( ord_less_eq @ int )
% 12.24/7.97 @ ( ord_less_eq @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5170_less__eq__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( ord_less_eq @ nat )
% 12.24/7.97 @ ( ord_less_eq @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5171_less__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( ord_less @ int )
% 12.24/7.97 @ ( ord_less @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5172_less__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( ord_less @ nat )
% 12.24/7.97 @ ( ord_less @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5173_uminus__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( uminus_uminus @ int )
% 12.24/7.97 @ ( uminus_uminus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5174_times__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( times_times @ nat )
% 12.24/7.97 @ ( times_times @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5175_times__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( times_times @ int )
% 12.24/7.97 @ ( times_times @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5176_natural__of__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ nat @ nat
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ nat2
% 12.24/7.97 @ nat2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5177_Suc_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ suc
% 12.24/7.97 @ suc ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5178_modulo__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( modulo_modulo @ int )
% 12.24/7.97 @ ( modulo_modulo @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5179_modulo__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( modulo_modulo @ nat )
% 12.24/7.97 @ ( modulo_modulo @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5180_sgn__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( sgn_sgn @ int )
% 12.24/7.97 @ ( sgn_sgn @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5181_plus__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( plus_plus @ nat )
% 12.24/7.97 @ ( plus_plus @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5182_plus__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( plus_plus @ int )
% 12.24/7.97 @ ( plus_plus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5183_divide__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( divide_divide @ nat )
% 12.24/7.97 @ ( divide_divide @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5184_divide__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( divide_divide @ int )
% 12.24/7.97 @ ( divide_divide @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5185_abs__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( abs_abs @ int )
% 12.24/7.97 @ ( abs_abs @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5186_minus__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( minus_minus @ int )
% 12.24/7.97 @ ( minus_minus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5187_minus__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( minus_minus @ nat )
% 12.24/7.97 @ ( minus_minus @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5188_integer__of__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ int @ int
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( semiring_1_of_nat @ int )
% 12.24/7.97 @ ( semiring_1_of_nat @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5189_push__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se4730199178511100633sh_bit @ nat )
% 12.24/7.97 @ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5190_push__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se4730199178511100633sh_bit @ int )
% 12.24/7.97 @ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5191_flip__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se8732182000553998342ip_bit @ int )
% 12.24/7.97 @ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5192_flip__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se8732182000553998342ip_bit @ nat )
% 12.24/7.97 @ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5193_take__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se2584673776208193580ke_bit @ nat )
% 12.24/7.97 @ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5194_take__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se2584673776208193580ke_bit @ int )
% 12.24/7.97 @ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5195_and__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5824344872417868541ns_and @ nat )
% 12.24/7.97 @ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5196_and__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5824344872417868541ns_and @ int )
% 12.24/7.97 @ ( bit_se5824344872417868541ns_and @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5197_unset__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se2638667681897837118et_bit @ nat )
% 12.24/7.97 @ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5198_unset__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se2638667681897837118et_bit @ int )
% 12.24/7.97 @ ( bit_se2638667681897837118et_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5199_xor__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5824344971392196577ns_xor @ int )
% 12.24/7.97 @ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5200_xor__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5824344971392196577ns_xor @ nat )
% 12.24/7.97 @ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5201_set__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5668285175392031749et_bit @ int )
% 12.24/7.97 @ ( bit_se5668285175392031749et_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5202_set__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se5668285175392031749et_bit @ nat )
% 12.24/7.97 @ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5203_drop__bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se4197421643247451524op_bit @ nat )
% 12.24/7.97 @ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5204_drop__bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se4197421643247451524op_bit @ int )
% 12.24/7.97 @ ( bit_se4197421643247451524op_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5205_or__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se1065995026697491101ons_or @ nat )
% 12.24/7.97 @ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5206_or__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/7.97 @ ( bit_se1065995026697491101ons_or @ int )
% 12.24/7.97 @ ( bit_se1065995026697491101ons_or @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5207_bit__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 12.24/7.97 @ ( bit_se5641148757651400278ts_bit @ int )
% 12.24/7.97 @ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5208_not__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bit_ri4277139882892585799ns_not @ int )
% 12.24/7.97 @ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5209_bit__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 12.24/7.97 @ ( bit_se5641148757651400278ts_bit @ nat )
% 12.24/7.97 @ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5210_mask__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ int @ int
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bit_se2239418461657761734s_mask @ int )
% 12.24/7.97 @ ( bit_se2239418461657761734s_mask @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5211_mask__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( bit_se2239418461657761734s_mask @ nat )
% 12.24/7.97 @ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5212_division__segment__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ int @ int
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( euclid7384307370059645450egment @ int )
% 12.24/7.97 @ ( euclid7384307370059645450egment @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5213_division__segment__natural_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( euclid7384307370059645450egment @ nat )
% 12.24/7.97 @ ( euclid7384307370059645450egment @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5214_Fract_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ratrel )
% 12.24/7.97 @ ^ [A5: int,B3: int] :
% 12.24/7.97 ( if @ ( product_prod @ int @ int )
% 12.24/7.97 @ ( B3
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ A5 @ B3 ) )
% 12.24/7.97 @ ^ [A5: int,B3: int] :
% 12.24/7.97 ( if @ ( product_prod @ int @ int )
% 12.24/7.97 @ ( B3
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5215_num__of__integer_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ int @ int @ num @ num
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 12.24/7.97 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5216_antimonoD,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( order @ A )
% 12.24/7.97 & ( order @ B ) )
% 12.24/7.97 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.97 ( ( order_antimono @ A @ B @ F )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.97 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5217_antimonoE,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( order @ A )
% 12.24/7.97 & ( order @ B ) )
% 12.24/7.97 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.97 ( ( order_antimono @ A @ B @ F )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.97 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5218_antimonoI,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( order @ A )
% 12.24/7.97 & ( order @ B ) )
% 12.24/7.97 => ! [F: A > B] :
% 12.24/7.97 ( ! [X3: A,Y5: A] :
% 12.24/7.97 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.97 => ( ord_less_eq @ B @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 12.24/7.97 => ( order_antimono @ A @ B @ F ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5219_antimono__def,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( order @ A )
% 12.24/7.97 & ( order @ B ) )
% 12.24/7.97 => ( ( order_antimono @ A @ B )
% 12.24/7.97 = ( ^ [F2: A > B] :
% 12.24/7.97 ! [X2: A,Y6: A] :
% 12.24/7.97 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 12.24/7.97 => ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5220_transfer__rule__of__bool,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( ( zero_neq_one @ B )
% 12.24/7.97 & ( zero_neq_one @ A ) )
% 12.24/7.97 => ! [R2: A > B > $o] :
% 12.24/7.97 ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.97 => ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.97 => ( bNF_rel_fun @ $o @ $o @ A @ B
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 12.24/7.97 @ R2
% 12.24/7.97 @ ( zero_neq_one_of_bool @ A )
% 12.24/7.97 @ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5221_times__rat_Orsp,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5222_plus__rat_Orsp,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5223_integer__of__nat__eq__of__nat,axiom,
% 12.24/7.97 ( code_integer_of_nat
% 12.24/7.97 = ( semiring_1_of_nat @ code_integer ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5224_antimono__iff__le__Suc,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( order @ A )
% 12.24/7.97 => ( ( order_antimono @ nat @ A )
% 12.24/7.97 = ( ^ [F2: nat > A] :
% 12.24/7.97 ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5225_integer__of__nat__0,axiom,
% 12.24/7.97 ( ( code_integer_of_nat @ ( zero_zero @ nat ) )
% 12.24/7.97 = ( zero_zero @ code_integer ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5226_integer__of__nat_Oabs__eq,axiom,
% 12.24/7.97 ( code_integer_of_nat
% 12.24/7.97 = ( ^ [X2: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5227_integer__of__nat__numeral,axiom,
% 12.24/7.97 ! [N: num] :
% 12.24/7.97 ( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
% 12.24/7.97 = ( numeral_numeral @ code_integer @ N ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5228_min__of__antimono,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( linorder @ A )
% 12.24/7.97 & ( linorder @ B ) )
% 12.24/7.97 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.97 ( ( order_antimono @ A @ B @ F )
% 12.24/7.97 => ( ( ord_min @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.97 = ( F @ ( ord_max @ A @ X @ Y3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5229_max__of__antimono,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( linorder @ A )
% 12.24/7.97 & ( linorder @ B ) )
% 12.24/7.97 => ! [F: A > B,X: A,Y3: A] :
% 12.24/7.97 ( ( order_antimono @ A @ B @ F )
% 12.24/7.97 => ( ( ord_max @ B @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.97 = ( F @ ( ord_min @ A @ X @ Y3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5230_integer__of__nat__1,axiom,
% 12.24/7.97 ( ( code_integer_of_nat @ ( one_one @ nat ) )
% 12.24/7.97 = ( one_one @ code_integer ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5231_uminus__rat_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5232_inverse__rat_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] :
% 12.24/7.97 ( if @ ( product_prod @ int @ int )
% 12.24/7.97 @ ( ( product_fst @ int @ int @ X2 )
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] :
% 12.24/7.97 ( if @ ( product_prod @ int @ int )
% 12.24/7.97 @ ( ( product_fst @ int @ int @ X2 )
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5233_plus__rat_Otransfer,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 12.24/7.97 @ ( plus_plus @ rat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5234_inverse__rat_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] :
% 12.24/7.97 ( if @ ( product_prod @ int @ int )
% 12.24/7.97 @ ( ( product_fst @ int @ int @ X2 )
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
% 12.24/7.97 @ ( inverse_inverse @ rat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5235_positive_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
% 12.24/7.97 @ positive ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5236_one__rat_Otransfer,axiom,
% 12.24/7.97 pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5237_zero__rat_Otransfer,axiom,
% 12.24/7.97 pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5238_uminus__rat_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
% 12.24/7.97 @ ( uminus_uminus @ rat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5239_times__rat_Otransfer,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
% 12.24/7.97 @ ( times_times @ rat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5240_times__int_Otransfer,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
% 12.24/7.97 @ ( times_times @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5241_num_Orec__transfer,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,S3: A > B > $o] :
% 12.24/7.97 ( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S3
% 12.24/7.97 @ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
% 12.24/7.97 @ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ A @ B
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ S3 ) ) )
% 12.24/7.97 @ ( rec_num @ A )
% 12.24/7.97 @ ( rec_num @ B ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5242_num__of__integer__def,axiom,
% 12.24/7.97 ( code_num_of_integer
% 12.24/7.97 = ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5243_verit__eq__simplify_I20_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X22: num] :
% 12.24/7.97 ( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
% 12.24/7.97 = ( F22 @ X22 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X22 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5244_verit__eq__simplify_I19_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
% 12.24/7.97 ( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
% 12.24/7.97 = F1 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5245_verit__eq__simplify_I21_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
% 12.24/7.97 ( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
% 12.24/7.97 = ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5246_zero__int_Otransfer,axiom,
% 12.24/7.97 pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5247_int__transfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ pcr_int
% 12.24/7.97 @ ^ [N3: nat] : ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) )
% 12.24/7.97 @ ( semiring_1_of_nat @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5248_uminus__int_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
% 12.24/7.97 @ ( uminus_uminus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5249_nat_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
% 12.24/7.97 @ nat2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5250_one__int_Otransfer,axiom,
% 12.24/7.97 pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5251_of__int_Otransfer,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
% 12.24/7.97 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
% 12.24/7.97 @ ( ring_1_of_int @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5252_less__int_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 12.24/7.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 12.24/7.97 @ ( ord_less @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5253_less__eq__int_Otransfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 12.24/7.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 12.24/7.97 @ ( ord_less_eq @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5254_plus__int_Otransfer,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
% 12.24/7.97 @ ( plus_plus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5255_minus__int_Otransfer,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
% 12.24/7.97 @ ( minus_minus @ int ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5256_num_Ocase__transfer,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,S3: A > B > $o] :
% 12.24/7.97 ( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S3
% 12.24/7.97 @ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ A @ B
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ S3 )
% 12.24/7.97 @ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ A @ B
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ S3 )
% 12.24/7.97 @ ( bNF_rel_fun @ num @ num @ A @ B
% 12.24/7.97 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.97 @ S3 ) ) )
% 12.24/7.97 @ ( case_num @ A )
% 12.24/7.97 @ ( case_num @ B ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5257_times__int_Orsp,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5258_list_Osize__gen_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
% 12.24/7.97 ( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
% 12.24/7.97 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5259_intrel__iff,axiom,
% 12.24/7.97 ! [X: nat,Y3: nat,U: nat,V: nat] :
% 12.24/7.97 ( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ ( product_Pair @ nat @ nat @ U @ V ) )
% 12.24/7.97 = ( ( plus_plus @ nat @ X @ V )
% 12.24/7.97 = ( plus_plus @ nat @ U @ Y3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5260_int_Orel__eq__transfer,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 12.24/7.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ intrel
% 12.24/7.97 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5261_int_Oabs__eq__iff,axiom,
% 12.24/7.97 ! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
% 12.24/7.97 ( ( ( abs_Integ @ X )
% 12.24/7.97 = ( abs_Integ @ Y3 ) )
% 12.24/7.97 = ( intrel @ X @ Y3 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5262_num_Ocase__distrib,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
% 12.24/7.97 ( ( H2 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
% 12.24/7.97 = ( case_num @ B @ ( H2 @ F1 )
% 12.24/7.97 @ ^ [X2: num] : ( H2 @ ( F22 @ X2 ) )
% 12.24/7.97 @ ^ [X2: num] : ( H2 @ ( F32 @ X2 ) )
% 12.24/7.97 @ Num ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5263_verit__eq__simplify_I17_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A,F32: num > A,X22: num] :
% 12.24/7.97 ( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
% 12.24/7.97 = ( F22 @ X22 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5264_verit__eq__simplify_I16_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A,F32: num > A] :
% 12.24/7.97 ( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
% 12.24/7.97 = F1 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5265_zero__int_Orsp,axiom,
% 12.24/7.97 intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5266_verit__eq__simplify_I18_J,axiom,
% 12.24/7.97 ! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
% 12.24/7.97 ( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
% 12.24/7.97 = ( F32 @ X32 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5267_list_Osize__gen_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,X: A > nat] :
% 12.24/7.97 ( ( size_list @ A @ X @ ( nil @ A ) )
% 12.24/7.97 = ( zero_zero @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5268_size__list__estimation,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
% 12.24/7.97 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ( ord_less @ nat @ Y3 @ ( F @ X ) )
% 12.24/7.97 => ( ord_less @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5269_size__list__pointwise,axiom,
% 12.24/7.97 ! [A: $tType,Xs: list @ A,F: A > nat,G: A > nat] :
% 12.24/7.97 ( ! [X3: A] :
% 12.24/7.97 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ord_less_eq @ nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.97 => ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5270_size__list__estimation_H,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
% 12.24/7.97 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.97 => ( ( ord_less_eq @ nat @ Y3 @ ( F @ X ) )
% 12.24/7.97 => ( ord_less_eq @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5271_uminus__int_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5272_nat_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
% 12.24/7.97 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5273_one__int_Orsp,axiom,
% 12.24/7.97 intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5274_of__int_Orsp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ring_1 @ A )
% 12.24/7.97 => ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
% 12.24/7.97 @ ^ [Y: A,Z2: A] : ( Y = Z2 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ A
% 12.24/7.97 @ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5275_intrel__def,axiom,
% 12.24/7.97 ( intrel
% 12.24/7.97 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] :
% 12.24/7.97 ( ( plus_plus @ nat @ X2 @ V5 )
% 12.24/7.97 = ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5276_less__int_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 12.24/7.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5277_less__eq__int_Orsp,axiom,
% 12.24/7.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 12.24/7.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 12.24/7.97 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5278_plus__int_Orsp,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5279_minus__int_Orsp,axiom,
% 12.24/7.97 ( 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 )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 12.24/7.97 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5280_Nitpick_Osize__list__simp_I1_J,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( size_list @ A )
% 12.24/7.97 = ( ^ [F2: A > nat,Xs3: list @ A] :
% 12.24/7.97 ( if @ nat
% 12.24/7.97 @ ( Xs3
% 12.24/7.97 = ( nil @ A ) )
% 12.24/7.97 @ ( zero_zero @ nat )
% 12.24/7.97 @ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5281_execute__make,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 12.24/7.97 @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5282_relpow__finite__bounded1,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 12.24/7.97 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
% 12.24/7.97 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [N3: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/7.97 & ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5283_tl__upt,axiom,
% 12.24/7.97 ! [M: nat,N: nat] :
% 12.24/7.97 ( ( tl @ nat @ ( upt @ M @ N ) )
% 12.24/7.97 = ( upt @ ( suc @ M ) @ N ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5284_relpow__1,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R2 )
% 12.24/7.97 = R2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5285_length__tl,axiom,
% 12.24/7.97 ! [A: $tType,Xs: list @ A] :
% 12.24/7.97 ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
% 12.24/7.97 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5286_finite__relpow,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.97 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5287_tl__replicate,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,X: A] :
% 12.24/7.97 ( ( tl @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.97 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5288_execute__heap,axiom,
% 12.24/7.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
% 12.24/7.97 = ( 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 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5289_execute__guard_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( P @ H2 )
% 12.24/7.97 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5290_Heap__eqI,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
% 12.24/7.97 ( ! [H6: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ F @ H6 )
% 12.24/7.97 = ( heap_Time_execute @ A @ G @ H6 ) )
% 12.24/7.97 => ( F = G ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5291_effectI,axiom,
% 12.24/7.97 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R: A,H: heap_ext @ product_unit,N: nat] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ C2 @ H2 )
% 12.24/7.97 = ( 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 @ H @ N ) ) ) )
% 12.24/7.97 => ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5292_effect__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap_Time_effect @ A )
% 12.24/7.97 = ( ^ [C4: heap_Time_Heap @ A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: A,N3: nat] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ C4 @ H4 )
% 12.24/7.97 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5293_sorted__tl,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [Xs: list @ A] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.97 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5294_relpow__Suc__E,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 12.24/7.97 => ~ ! [Y5: A] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5295_relpow__Suc__I,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Y3: A,N: nat,R2: set @ ( product_prod @ A @ A ),Z3: A] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
% 12.24/7.97 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5296_relpow__Suc__D2,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 12.24/7.97 => ? [Y5: A] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 12.24/7.97 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5297_relpow__Suc__E2,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
% 12.24/7.97 => ~ ! [Y5: A] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 12.24/7.97 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5298_relpow__Suc__I2,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Y3: A,R2: set @ ( product_prod @ A @ A ),Z3: A,N: nat] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
% 12.24/7.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5299_relpow__0__I,axiom,
% 12.24/7.97 ! [A: $tType,X: A,R2: 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 ) @ R2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5300_relpow__0__E,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) )
% 12.24/7.97 => ( X = Y3 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5301_take__tl,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.97 ( ( take @ A @ N @ ( tl @ A @ Xs ) )
% 12.24/7.97 = ( tl @ A @ ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5302_drop__Suc,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.97 ( ( drop @ A @ ( suc @ N ) @ Xs )
% 12.24/7.97 = ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5303_execute__assert_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( P @ X )
% 12.24/7.97 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
% 12.24/7.97 = ( 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 @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5304_execute__tap,axiom,
% 12.24/7.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5305_execute__return,axiom,
% 12.24/7.97 ! [A: $tType,X: A] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
% 12.24/7.97 = ( 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 ) ) )
% 12.24/7.97 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5306_execute__ureturn,axiom,
% 12.24/7.97 ! [A: $tType,X: A] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
% 12.24/7.97 = ( 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 ) ) )
% 12.24/7.97 @ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5307_relpow__E,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 => ( ( ( N
% 12.24/7.97 = ( zero_zero @ nat ) )
% 12.24/7.97 => ( X != Z3 ) )
% 12.24/7.97 => ~ ! [Y5: A,M6: nat] :
% 12.24/7.97 ( ( N
% 12.24/7.97 = ( suc @ M6 ) )
% 12.24/7.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) )
% 12.24/7.97 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5308_relpow__E2,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 => ( ( ( N
% 12.24/7.97 = ( zero_zero @ nat ) )
% 12.24/7.97 => ( X != Z3 ) )
% 12.24/7.97 => ~ ! [Y5: A,M6: nat] :
% 12.24/7.97 ( ( N
% 12.24/7.97 = ( suc @ M6 ) )
% 12.24/7.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
% 12.24/7.97 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5309_tl__take,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.97 ( ( tl @ A @ ( take @ A @ N @ Xs ) )
% 12.24/7.97 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5310_relpow__empty,axiom,
% 12.24/7.97 ! [A: $tType,N: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.97 => ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.97 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5311_relpow__finite__bounded,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
% 12.24/7.97 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5312_Nitpick_Osize__list__simp_I2_J,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( size_size @ ( list @ A ) )
% 12.24/7.97 = ( ^ [Xs3: list @ A] :
% 12.24/7.97 ( if @ nat
% 12.24/7.97 @ ( Xs3
% 12.24/7.97 = ( nil @ A ) )
% 12.24/7.97 @ ( zero_zero @ nat )
% 12.24/7.97 @ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5313_nth__tl,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.97 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
% 12.24/7.97 => ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
% 12.24/7.97 = ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5314_relpow__fun__conv,axiom,
% 12.24/7.97 ! [A: $tType,A3: A,B2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
% 12.24/7.97 = ( ? [F2: nat > A] :
% 12.24/7.97 ( ( ( F2 @ ( zero_zero @ nat ) )
% 12.24/7.97 = A3 )
% 12.24/7.97 & ( ( F2 @ N )
% 12.24/7.97 = B2 )
% 12.24/7.97 & ! [I5: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ I5 @ N )
% 12.24/7.97 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) @ R2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5315_execute__of__list,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [Xs: list @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
% 12.24/7.97 @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5316_execute__new,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [N: nat,X: A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 12.24/7.97 @ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 12.24/7.97 @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5317_take__Suc,axiom,
% 12.24/7.97 ! [A: $tType,Xs: list @ A,N: nat] :
% 12.24/7.97 ( ( Xs
% 12.24/7.97 != ( nil @ A ) )
% 12.24/7.97 => ( ( take @ A @ ( suc @ N ) @ Xs )
% 12.24/7.97 = ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N @ ( tl @ A @ Xs ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5318_ntrancl__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( transitive_ntrancl @ A )
% 12.24/7.97 = ( ^ [N3: nat,R6: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ^ [I5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I5 @ R6 )
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [I5: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I5 )
% 12.24/7.97 & ( ord_less_eq @ nat @ I5 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5319_option_Osize_I4_J,axiom,
% 12.24/7.97 ! [A: $tType,X22: A] :
% 12.24/7.97 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
% 12.24/7.97 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5320_trancl__finite__eq__relpow,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ( transitive_trancl @ A @ R2 )
% 12.24/7.97 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [N3: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/7.97 & ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5321_ntrancl__Zero,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R2 )
% 12.24/7.97 = R2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5322_take__bit__num__simps_I2_J,axiom,
% 12.24/7.97 ! [N: nat] :
% 12.24/7.97 ( ( bit_take_bit_num @ ( suc @ N ) @ one2 )
% 12.24/7.97 = ( some @ num @ one2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5323_take__bit__num__simps_I4_J,axiom,
% 12.24/7.97 ! [N: nat,M: num] :
% 12.24/7.97 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 12.24/7.97 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5324_execute__len,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [A3: array @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H2 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5325_finite__trancl__ntranl,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ( transitive_trancl @ A @ R2 )
% 12.24/7.97 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) @ ( one_one @ nat ) ) @ R2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5326_trancl__set__ntrancl,axiom,
% 12.24/7.97 ! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
% 12.24/7.97 = ( 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 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5327_trancl__power,axiom,
% 12.24/7.97 ! [A: $tType,P5: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R2 ) )
% 12.24/7.97 = ( ? [N3: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/7.97 & ( member @ ( product_prod @ A @ A ) @ P5 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5328_option_Osize__neq,axiom,
% 12.24/7.97 ! [A: $tType,X: option @ A] :
% 12.24/7.97 ( ( size_size @ ( option @ A ) @ X )
% 12.24/7.97 != ( zero_zero @ nat ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5329_option_Osize__gen_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,X: A > nat,X22: A] :
% 12.24/7.97 ( ( size_option @ A @ X @ ( some @ A @ X22 ) )
% 12.24/7.97 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5330_sorted__find__Min,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [Xs: list @ A,P: A > $o] :
% 12.24/7.97 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.97 => ( ? [X4: A] :
% 12.24/7.97 ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
% 12.24/7.97 & ( P @ X4 ) )
% 12.24/7.97 => ( ( find @ A @ P @ Xs )
% 12.24/7.97 = ( some @ A
% 12.24/7.97 @ ( lattic643756798350308766er_Min @ A
% 12.24/7.97 @ ( collect @ A
% 12.24/7.97 @ ^ [X2: A] :
% 12.24/7.97 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 12.24/7.97 & ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5331_timeFrame_Osimps_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,R: A,H2: heap_ext @ product_unit,N2: nat] :
% 12.24/7.97 ( ( 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 @ H2 @ N2 ) ) ) )
% 12.24/7.97 = ( 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 @ H2 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5332_timeFrame__zero,axiom,
% 12.24/7.97 ! [A: $tType,H2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H2 )
% 12.24/7.97 = H2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5333_timeFrame__assoc,axiom,
% 12.24/7.97 ! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
% 12.24/7.97 = ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5334_find__Some__iff2,axiom,
% 12.24/7.97 ! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
% 12.24/7.97 ( ( ( some @ A @ X )
% 12.24/7.97 = ( find @ A @ P @ Xs ) )
% 12.24/7.97 = ( ? [I5: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.97 & ( P @ ( nth @ A @ Xs @ I5 ) )
% 12.24/7.97 & ( X
% 12.24/7.97 = ( nth @ A @ Xs @ I5 ) )
% 12.24/7.97 & ! [J3: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ J3 @ I5 )
% 12.24/7.97 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5335_find__Some__iff,axiom,
% 12.24/7.97 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 12.24/7.97 ( ( ( find @ A @ P @ Xs )
% 12.24/7.97 = ( some @ A @ X ) )
% 12.24/7.97 = ( ? [I5: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.97 & ( P @ ( nth @ A @ Xs @ I5 ) )
% 12.24/7.97 & ( X
% 12.24/7.97 = ( nth @ A @ Xs @ I5 ) )
% 12.24/7.97 & ! [J3: nat] :
% 12.24/7.97 ( ( ord_less @ nat @ J3 @ I5 )
% 12.24/7.97 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5336_tap__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap_Time_tap @ A )
% 12.24/7.97 = ( ^ [F2: ( heap_ext @ product_unit ) > A] :
% 12.24/7.97 ( heap_Time_Heap2 @ A
% 12.24/7.97 @ ^ [H4: 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 @ H4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5337_execute__bind_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 = ( 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 @ H @ N ) ) ) )
% 12.24/7.97 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 12.24/7.97 = ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5338_rtrancl__finite__eq__relpow,axiom,
% 12.24/7.97 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.97 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 12.24/7.97 => ( ( transitive_rtrancl @ A @ R2 )
% 12.24/7.97 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
% 12.24/7.97 @ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
% 12.24/7.97 @ ( collect @ nat
% 12.24/7.97 @ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5339_Heap_Oinject,axiom,
% 12.24/7.97 ! [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 ) ) )] :
% 12.24/7.97 ( ( ( heap_Time_Heap2 @ A @ X )
% 12.24/7.97 = ( heap_Time_Heap2 @ A @ Ya ) )
% 12.24/7.97 = ( X = Ya ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5340_Heap__Time__Monad_Obind__bind,axiom,
% 12.24/7.97 ! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
% 12.24/7.97 = ( heap_Time_bind @ A @ B @ F
% 12.24/7.97 @ ^ [X2: A] : ( heap_Time_bind @ C @ B @ ( G @ X2 ) @ K ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5341_Heap__execute,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A] :
% 12.24/7.97 ( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
% 12.24/7.97 = F ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5342_execute_Osimps,axiom,
% 12.24/7.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F ) )
% 12.24/7.97 = F ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5343_distrib__if__bind,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
% 12.24/7.97 ( ( B2
% 12.24/7.97 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
% 12.24/7.97 = ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
% 12.24/7.97 & ( ~ B2
% 12.24/7.97 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
% 12.24/7.97 = ( heap_Time_bind @ B @ A @ D2 @ F ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5344_Heap_Oexhaust,axiom,
% 12.24/7.97 ! [A: $tType,Y3: heap_Time_Heap @ A] :
% 12.24/7.97 ~ ! [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 12.24/7.97 ( Y3
% 12.24/7.97 != ( heap_Time_Heap2 @ A @ X3 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5345_assert__cong,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F5: A > ( heap_Time_Heap @ B ),X: A] :
% 12.24/7.97 ( ( P = P2 )
% 12.24/7.97 => ( ! [X3: A] :
% 12.24/7.97 ( ( P2 @ X3 )
% 12.24/7.97 => ( ( F @ X3 )
% 12.24/7.97 = ( F5 @ X3 ) ) )
% 12.24/7.97 => ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
% 12.24/7.97 = ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F5 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5346_bind__ureturn,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A] :
% 12.24/7.97 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
% 12.24/7.97 = F ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5347_ureturn__bind,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 12.24/7.97 ( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
% 12.24/7.97 = ( F @ X ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5348_effect__bindE,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,R4: A,N: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 @ H3 @ R4 @ N )
% 12.24/7.97 => ~ ! [H7: heap_ext @ product_unit,R3: B,N1: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ B @ F @ H2 @ H7 @ R3 @ N1 )
% 12.24/7.97 => ! [N22: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ ( G @ R3 ) @ H7 @ H3 @ R4 @ N22 )
% 12.24/7.97 => ( N
% 12.24/7.97 != ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5349_effect__bindI,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,R4: B,N2: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ F @ H2 @ H @ R @ N )
% 12.24/7.97 => ( ( heap_Time_effect @ B @ ( G @ R ) @ H @ H3 @ R4 @ N2 )
% 12.24/7.97 => ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 @ H3 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5350_bind__lift,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
% 12.24/7.97 ( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
% 12.24/7.97 = ( heap_Time_bind @ B @ A @ F
% 12.24/7.97 @ ^ [X2: B] : ( heap_Time_return @ A @ ( G @ X2 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5351_Heap_Osize_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 12.24/7.97 ( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X ) )
% 12.24/7.97 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5352_heap__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap_Time_heap @ A )
% 12.24/7.97 = ( ^ [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 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5353_execute__bind__eq__SomeI,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y3: B,H3: heap_ext @ product_unit,N2: nat] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 = ( 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 @ H @ N ) ) ) )
% 12.24/7.97 => ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H )
% 12.24/7.97 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
% 12.24/7.97 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 12.24/7.97 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5354_bind__return,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A] :
% 12.24/7.97 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
% 12.24/7.97 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 12.24/7.97 @ ^ [Uu: product_unit] : F ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5355_return__bind,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 12.24/7.97 ( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
% 12.24/7.97 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 12.24/7.97 @ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5356_Heap_Osize__gen,axiom,
% 12.24/7.97 ! [A: $tType,Xa3: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 12.24/7.97 ( ( heap_Time_size_Heap @ A @ Xa3 @ ( heap_Time_Heap2 @ A @ X ) )
% 12.24/7.97 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5357_success__bind__executeI,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 = ( 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 @ H @ N ) ) ) )
% 12.24/7.97 => ( ( heap_Time_success @ B @ ( G @ X ) @ H )
% 12.24/7.97 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5358_take__bit__num__def,axiom,
% 12.24/7.97 ( bit_take_bit_num
% 12.24/7.97 = ( ^ [N3: nat,M2: num] :
% 12.24/7.97 ( if @ ( option @ num )
% 12.24/7.97 @ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) )
% 12.24/7.97 = ( zero_zero @ nat ) )
% 12.24/7.97 @ ( none @ num )
% 12.24/7.97 @ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5359_power__int__numeral__neg__numeral,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.97 = ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5360_power__int__1__left,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [N: int] :
% 12.24/7.97 ( ( power_int @ A @ ( one_one @ A ) @ N )
% 12.24/7.97 = ( one_one @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5361_power__int__1__right,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( inverse @ A )
% 12.24/7.97 & ( monoid_mult @ A ) )
% 12.24/7.97 => ! [Y3: A] :
% 12.24/7.97 ( ( power_int @ A @ Y3 @ ( one_one @ int ) )
% 12.24/7.97 = Y3 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5362_power__int__sgn,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,N: int] :
% 12.24/7.97 ( ( sgn_sgn @ A @ ( power_int @ A @ A3 @ N ) )
% 12.24/7.97 = ( power_int @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5363_power__int__mult__distrib__numeral2,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [X: A,W2: num,M: int] :
% 12.24/7.97 ( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5364_power__int__mult__distrib__numeral1,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [W2: num,Y3: A,M: int] :
% 12.24/7.97 ( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) @ M )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5365_power__int__eq__0__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ( power_int @ A @ X @ N )
% 12.24/7.97 = ( zero_zero @ A ) )
% 12.24/7.97 = ( ( X
% 12.24/7.97 = ( zero_zero @ A ) )
% 12.24/7.97 & ( N
% 12.24/7.97 != ( zero_zero @ int ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5366_power__int__0__left,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [M: int] :
% 12.24/7.97 ( ( M
% 12.24/7.97 != ( zero_zero @ int ) )
% 12.24/7.97 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.97 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5367_power__int__0__right,axiom,
% 12.24/7.97 ! [B: $tType] :
% 12.24/7.97 ( ( ( inverse @ B )
% 12.24/7.97 & ( power @ B ) )
% 12.24/7.97 => ! [X: B] :
% 12.24/7.97 ( ( power_int @ B @ X @ ( zero_zero @ int ) )
% 12.24/7.97 = ( one_one @ B ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5368_abs__power__int__minus,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,N: int] :
% 12.24/7.97 ( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
% 12.24/7.97 = ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5369_power__int__of__nat,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( inverse @ A )
% 12.24/7.97 & ( power @ A ) )
% 12.24/7.97 => ! [X: A,N: nat] :
% 12.24/7.97 ( ( power_int @ A @ X @ ( semiring_1_of_nat @ int @ N ) )
% 12.24/7.97 = ( power_power @ A @ X @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5370_power__int__mult__numeral,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: num,N: num] :
% 12.24/7.97 ( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
% 12.24/7.97 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5371_take__bit__num__simps_I1_J,axiom,
% 12.24/7.97 ! [M: num] :
% 12.24/7.97 ( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
% 12.24/7.97 = ( none @ num ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5372_power__int__minus__one__mult__self,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [M: int] :
% 12.24/7.97 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) )
% 12.24/7.97 = ( one_one @ A ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5373_power__int__minus__one__mult__self_H,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [M: int,B2: A] :
% 12.24/7.97 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ B2 ) )
% 12.24/7.97 = B2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5374_power__int__numeral,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( inverse @ A )
% 12.24/7.97 & ( power @ A ) )
% 12.24/7.97 => ! [X: A,N: num] :
% 12.24/7.97 ( ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) )
% 12.24/7.97 = ( power_power @ A @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5375_power__int__add__numeral2,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: num,N: num,B2: A] :
% 12.24/7.97 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5376_power__int__add__numeral,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: num,N: num] :
% 12.24/7.97 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.97 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5377_power__int__minus1__right,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( inverse @ A )
% 12.24/7.97 & ( monoid_mult @ A ) )
% 12.24/7.97 => ! [Y3: A] :
% 12.24/7.97 ( ( power_int @ A @ Y3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 12.24/7.97 = ( inverse_inverse @ A @ Y3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5378_power__int__mono__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,B2: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 12.24/7.97 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) )
% 12.24/7.97 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5379_power__int__minus__left__even,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [N: int,A3: A] :
% 12.24/7.97 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.97 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.97 = ( power_int @ A @ A3 @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5380_power__int__minus__left__odd,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [N: int,A3: A] :
% 12.24/7.97 ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.97 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.97 = ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5381_take__bit__num__simps_I3_J,axiom,
% 12.24/7.97 ! [N: nat,M: num] :
% 12.24/7.97 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 12.24/7.97 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 12.24/7.97 @ ^ [Q3: num] : ( some @ num @ ( bit0 @ Q3 ) )
% 12.24/7.97 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5382_power__int__not__zero,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( N
% 12.24/7.97 = ( zero_zero @ int ) ) )
% 12.24/7.97 => ( ( power_int @ A @ X @ N )
% 12.24/7.97 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5383_zero__le__power__int,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.97 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5384_success__LetI,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( X = T2 )
% 12.24/7.97 => ( ( heap_Time_success @ B @ ( F @ X ) @ H2 )
% 12.24/7.97 => ( heap_Time_success @ B @ ( F @ T2 ) @ H2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5385_power__int__commutes,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
% 12.24/7.97 = ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5386_power__int__mult__distrib,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [X: A,Y3: A,M: int] :
% 12.24/7.97 ( ( power_int @ A @ ( times_times @ A @ X @ Y3 ) @ M )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5387_power__int__mult,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: int,N: int] :
% 12.24/7.97 ( ( power_int @ A @ X @ ( times_times @ int @ M @ N ) )
% 12.24/7.97 = ( power_int @ A @ ( power_int @ A @ X @ M ) @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5388_power__int__inverse,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( power_int @ A @ ( inverse_inverse @ A @ X ) @ N )
% 12.24/7.97 = ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5389_success__ifI,axiom,
% 12.24/7.97 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,E: heap_Time_Heap @ A] :
% 12.24/7.97 ( ( C2
% 12.24/7.97 => ( heap_Time_success @ A @ T2 @ H2 ) )
% 12.24/7.97 => ( ( ~ C2
% 12.24/7.97 => ( heap_Time_success @ A @ E @ H2 ) )
% 12.24/7.97 => ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5390_power__int__divide__distrib,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [X: A,Y3: A,M: int] :
% 12.24/7.97 ( ( power_int @ A @ ( divide_divide @ A @ X @ Y3 ) @ M )
% 12.24/7.97 = ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5391_power__int__abs,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,N: int] :
% 12.24/7.97 ( ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) )
% 12.24/7.97 = ( power_int @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5392_effect__success,axiom,
% 12.24/7.97 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N )
% 12.24/7.97 => ( heap_Time_success @ A @ C2 @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5393_success__effectE,axiom,
% 12.24/7.97 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_success @ A @ C2 @ H2 )
% 12.24/7.97 => ~ ! [R3: A,H7: heap_ext @ product_unit,N4: nat] :
% 12.24/7.97 ~ ( heap_Time_effect @ A @ C2 @ H2 @ H7 @ R3 @ N4 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5394_zero__less__power__int,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.97 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5395_power__int__one__over,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
% 12.24/7.97 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5396_power__int__minus,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( power_int @ A @ X @ ( uminus_uminus @ int @ N ) )
% 12.24/7.97 = ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5397_success__returnI,axiom,
% 12.24/7.97 ! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5398_success__heapI,axiom,
% 12.24/7.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5399_success__guardI,axiom,
% 12.24/7.97 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( P @ H2 )
% 12.24/7.97 => ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5400_success__assertI,axiom,
% 12.24/7.97 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( P @ X )
% 12.24/7.97 => ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5401_success__ureturnI,axiom,
% 12.24/7.97 ! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5402_success__makeI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5403_success__of__listI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5404_success__newI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [N: nat,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5405_success__tapI,axiom,
% 12.24/7.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H2 ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5406_success__lenI,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap @ A )
% 12.24/7.97 => ! [A3: array @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5407_power__int__0__left__If,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [M: int] :
% 12.24/7.97 ( ( ( M
% 12.24/7.97 = ( zero_zero @ int ) )
% 12.24/7.97 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.97 = ( one_one @ A ) ) )
% 12.24/7.97 & ( ( M
% 12.24/7.97 != ( zero_zero @ int ) )
% 12.24/7.97 => ( ( power_int @ A @ ( zero_zero @ A ) @ M )
% 12.24/7.97 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5408_power__int__increasing,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [N: int,N6: int,A3: A] :
% 12.24/7.97 ( ( ord_less_eq @ int @ N @ N6 )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.97 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5409_power__int__strict__increasing,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [N: int,N6: int,A3: A] :
% 12.24/7.97 ( ( ord_less @ int @ N @ N6 )
% 12.24/7.97 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.97 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5410_power__int__minus__one__minus,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [N: int] :
% 12.24/7.97 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
% 12.24/7.97 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5411_power__int__diff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [X: A,M: int,N: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( M != N ) )
% 12.24/7.97 => ( ( power_int @ A @ X @ ( minus_minus @ int @ M @ N ) )
% 12.24/7.97 = ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5412_power__int__minus__one__diff__commute,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [A3: int,B2: int] :
% 12.24/7.97 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
% 12.24/7.97 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5413_success__bind__effectI,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ F @ H2 @ H @ X @ N )
% 12.24/7.97 => ( ( heap_Time_success @ B @ ( G @ X ) @ H )
% 12.24/7.97 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5414_power__int__strict__decreasing,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [N: int,N6: int,A3: A] :
% 12.24/7.97 ( ( ord_less @ int @ N @ N6 )
% 12.24/7.97 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.97 => ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5415_power__int__mono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,Y3: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.97 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.97 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y3 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5416_power__int__strict__antimono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,B2: A,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.97 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
% 12.24/7.97 => ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5417_one__le__power__int,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
% 12.24/7.97 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5418_one__less__power__int,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5419_power__int__add,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: int,N: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( ( plus_plus @ int @ M @ N )
% 12.24/7.97 != ( zero_zero @ int ) ) )
% 12.24/7.97 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N ) )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5420_option_Osize__gen_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,X: A > nat] :
% 12.24/7.97 ( ( size_option @ A @ X @ ( none @ A ) )
% 12.24/7.97 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5421_option_Osize_I3_J,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 12.24/7.97 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5422_power__int__minus__left__distrib,axiom,
% 12.24/7.97 ! [C: $tType,B: $tType,A: $tType] :
% 12.24/7.97 ( ( ( division_ring @ A )
% 12.24/7.97 & ( one @ B )
% 12.24/7.97 & ( uminus @ B ) )
% 12.24/7.97 => ! [X: C,A3: A,N: int] :
% 12.24/7.97 ( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
% 12.24/7.97 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5423_take__bit__num__eq__None__imp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.97 => ! [M: nat,N: num] :
% 12.24/7.97 ( ( ( bit_take_bit_num @ M @ N )
% 12.24/7.97 = ( none @ num ) )
% 12.24/7.97 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5424_and__num__eq__None__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( ( bit_un7362597486090784418nd_num @ M @ N )
% 12.24/7.97 = ( none @ num ) )
% 12.24/7.97 = ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5425_xor__num__eq__None__iff,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.97 => ! [M: num,N: num] :
% 12.24/7.97 ( ( ( bit_un2480387367778600638or_num @ M @ N )
% 12.24/7.97 = ( none @ num ) )
% 12.24/7.97 = ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 12.24/7.97 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5426_power__int__antimono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,B2: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ A3 @ B2 )
% 12.24/7.97 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
% 12.24/7.97 => ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5427_power__int__strict__mono,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [A3: A,B2: A,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ A3 @ B2 )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5428_power__int__le__one,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 12.24/7.97 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 12.24/7.97 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5429_power__int__decreasing,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [N: int,N6: int,A3: A] :
% 12.24/7.97 ( ( ord_less_eq @ int @ N @ N6 )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.97 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.97 => ( ( ( A3
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( N6
% 12.24/7.97 != ( zero_zero @ int ) )
% 12.24/7.97 | ( N
% 12.24/7.97 = ( zero_zero @ int ) ) )
% 12.24/7.97 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5430_power__int__le__imp__le__exp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,M: int,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 12.24/7.97 => ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
% 12.24/7.97 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5431_power__int__le__imp__less__exp,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linordered_field @ A )
% 12.24/7.97 => ! [X: A,M: int,N: int] :
% 12.24/7.97 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 12.24/7.97 => ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
% 12.24/7.97 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 12.24/7.97 => ( ord_less @ int @ M @ N ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5432_power__int__minus__mult,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( field @ A )
% 12.24/7.97 => ! [X: A,N: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( N
% 12.24/7.97 != ( zero_zero @ int ) ) )
% 12.24/7.97 => ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
% 12.24/7.97 = ( power_int @ A @ X @ N ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5433_power__int__add__1_H,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( M
% 12.24/7.97 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 12.24/7.97 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
% 12.24/7.97 = ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5434_power__int__add__1,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [X: A,M: int] :
% 12.24/7.97 ( ( ( X
% 12.24/7.97 != ( zero_zero @ A ) )
% 12.24/7.97 | ( M
% 12.24/7.97 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 12.24/7.97 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
% 12.24/7.97 = ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5435_power__int__minus__left,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( division_ring @ A )
% 12.24/7.97 => ! [N: int,A3: A] :
% 12.24/7.97 ( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.97 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.97 = ( power_int @ A @ A3 @ N ) ) )
% 12.24/7.97 & ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
% 12.24/7.97 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 12.24/7.97 = ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5436_successE,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_success @ A @ F @ H2 )
% 12.24/7.97 => ~ ! [R3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ H7 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5437_and__not__num__eq__None__iff,axiom,
% 12.24/7.97 ! [M: num,N: num] :
% 12.24/7.97 ( ( ( bit_and_not_num @ M @ N )
% 12.24/7.97 = ( none @ num ) )
% 12.24/7.97 = ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 12.24/7.97 = ( zero_zero @ int ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5438_power__int__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( ( inverse @ A )
% 12.24/7.97 & ( power @ A ) )
% 12.24/7.97 => ( ( power_int @ A )
% 12.24/7.97 = ( ^ [X2: A,N3: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) @ ( power_power @ A @ X2 @ ( nat2 @ N3 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5439_execute__bind__success,axiom,
% 12.24/7.97 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( heap_Time_success @ A @ F @ H2 )
% 12.24/7.97 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 12.24/7.97 = ( 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 @ H2 ) ) ) ) @ ( 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 @ H2 ) ) ) ) @ ( 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 @ H2 ) ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5440_effectE,axiom,
% 12.24/7.97 ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.24/7.97 ( ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N )
% 12.24/7.97 => ~ ( ( R
% 12.24/7.97 = ( 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 @ H2 ) ) ) )
% 12.24/7.97 => ( ( H
% 12.24/7.97 = ( 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 @ H2 ) ) ) ) )
% 12.24/7.97 => ( ( N
% 12.24/7.97 = ( 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 @ H2 ) ) ) ) )
% 12.24/7.97 => ~ ( heap_Time_success @ A @ C2 @ H2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5441_in__finite__psubset,axiom,
% 12.24/7.97 ! [A: $tType,A6: set @ A,B7: set @ A] :
% 12.24/7.97 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A6 @ B7 ) @ ( finite_psubset @ A ) )
% 12.24/7.97 = ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.97 & ( finite_finite2 @ A @ B7 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5442_timeFrame_Osimps_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,N: nat] :
% 12.24/7.97 ( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5443_execute__bind_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.97 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
% 12.24/7.97 = ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5444_successI,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.97 => ( heap_Time_success @ A @ F @ H2 ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5445_success__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap_Time_success @ A )
% 12.24/7.97 = ( ^ [F2: heap_Time_Heap @ A,H4: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ F2 @ H4 )
% 12.24/7.97 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5446_execute__guard_I1_J,axiom,
% 12.24/7.97 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ~ ( P @ H2 )
% 12.24/7.97 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5447_execute__assert_I2_J,axiom,
% 12.24/7.97 ! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ~ ( P @ X )
% 12.24/7.97 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5448_finite__psubset__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( finite_psubset @ A )
% 12.24/7.97 = ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 12.24/7.97 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 12.24/7.97 @ ^ [A7: set @ A,B6: set @ A] :
% 12.24/7.97 ( ( ord_less @ ( set @ A ) @ A7 @ B6 )
% 12.24/7.97 & ( finite_finite2 @ A @ B6 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5449_Heap__cases,axiom,
% 12.24/7.97 ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ! [X3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H7 ) ) )
% 12.24/7.97 => ( ( heap_Time_execute @ A @ F @ H2 )
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5450_timeFrame_Ocases,axiom,
% 12.24/7.97 ! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 12.24/7.97 ( ! [N4: nat,R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 12.24/7.97 ( X
% 12.24/7.97 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( 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 @ H6 @ N10 ) ) ) ) )
% 12.24/7.97 => ~ ! [N4: nat] :
% 12.24/7.97 ( X
% 12.24/7.97 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5451_Max_Oinfinite,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ A6 )
% 12.24/7.97 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5452_Min_Oinfinite,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A] :
% 12.24/7.97 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ A6 )
% 12.24/7.97 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5453_guard__def,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( heap_Time_guard @ A )
% 12.24/7.97 = ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( heap_Time_Heap2 @ A
% 12.24/7.97 @ ^ [H4: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H4 ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H4 ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5454_timeFrame_Oelims,axiom,
% 12.24/7.97 ! [A: $tType,X: nat,Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.97 ( ( ( heap_Time_timeFrame @ A @ X @ Xa3 )
% 12.24/7.97 = Y3 )
% 12.24/7.97 => ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 12.24/7.97 ( ( Xa3
% 12.24/7.97 = ( 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 @ H6 @ N10 ) ) ) )
% 12.24/7.97 => ( Y3
% 12.24/7.97 != ( 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 @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) ) )
% 12.24/7.97 => ~ ( ( Xa3
% 12.24/7.97 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.97 => ( Y3
% 12.24/7.97 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5455_in__measure,axiom,
% 12.24/7.97 ! [A: $tType,X: A,Y3: A,F: A > nat] :
% 12.24/7.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measure @ A @ F ) )
% 12.24/7.97 = ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5456_pred__nat__def,axiom,
% 12.24/7.97 ( pred_nat
% 12.24/7.97 = ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.97 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.97 @ ^ [M2: nat,N3: nat] :
% 12.24/7.97 ( N3
% 12.24/7.97 = ( suc @ M2 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5457_Min_Oeq__fold_H,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A )
% 12.24/7.97 = ( ^ [A7: set @ A] :
% 12.24/7.97 ( the2 @ A
% 12.24/7.97 @ ( finite_fold @ A @ ( option @ A )
% 12.24/7.97 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y6 ) )
% 12.24/7.97 @ ( none @ A )
% 12.24/7.97 @ A7 ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5458_sum_Oeq__fold,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_add @ A )
% 12.24/7.97 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 12.24/7.97 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5459_prod_Oeq__fold,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType] :
% 12.24/7.97 ( ( comm_monoid_mult @ A )
% 12.24/7.97 => ( ( groups7121269368397514597t_prod @ B @ A )
% 12.24/7.97 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5460_Max_Oeq__fold,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5461_Min_Oeq__fold,axiom,
% 12.24/7.97 ! [A: $tType] :
% 12.24/7.97 ( ( linorder @ A )
% 12.24/7.97 => ! [A6: set @ A,X: A] :
% 12.24/7.97 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.97 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.97 = ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A6 ) ) ) ) ).
% 12.24/7.97
% 12.24/7.97 thf(fact_5462_execute__bind__case,axiom,
% 12.24/7.97 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit] :
% 12.24/7.97 ( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 )
% 12.24/7.97 = ( 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 ) ) )
% 12.24/7.97 @ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 12.24/7.98 @ ^ [X2: B] :
% 12.24/7.98 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 12.24/7.98 @ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ A @ N3 @ ( heap_Time_execute @ A @ ( G @ X2 ) @ H5 ) ) ) )
% 12.24/7.98 @ ( heap_Time_execute @ B @ F @ H2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5463_less__eq,axiom,
% 12.24/7.98 ! [M: nat,N: nat] :
% 12.24/7.98 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
% 12.24/7.98 = ( ord_less @ nat @ M @ N ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5464_pred__nat__trancl__eq__le,axiom,
% 12.24/7.98 ! [M: nat,N: nat] :
% 12.24/7.98 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
% 12.24/7.98 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5465_Heap__Time__Monad_Obind__def,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( heap_Time_bind @ A @ B )
% 12.24/7.98 = ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
% 12.24/7.98 ( heap_Time_Heap2 @ B
% 12.24/7.98 @ ^ [H4: heap_ext @ product_unit] :
% 12.24/7.98 ( 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 ) ) )
% 12.24/7.98 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 12.24/7.98 @ ^ [R5: A] :
% 12.24/7.98 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 12.24/7.98 @ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ B @ N3 @ ( heap_Time_execute @ B @ ( G2 @ R5 ) @ H5 ) ) ) )
% 12.24/7.98 @ ( heap_Time_execute @ A @ F2 @ H4 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5466_Max_Oeq__fold_H,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ( ( lattic643756798349783984er_Max @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ( the2 @ A
% 12.24/7.98 @ ( finite_fold @ A @ ( option @ A )
% 12.24/7.98 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y6 ) )
% 12.24/7.98 @ ( none @ A )
% 12.24/7.98 @ A7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5467_Sup__fin_Oeq__fold_H,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ( the2 @ A
% 12.24/7.98 @ ( finite_fold @ A @ ( option @ A )
% 12.24/7.98 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y6 ) )
% 12.24/7.98 @ ( none @ A )
% 12.24/7.98 @ A7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5468_Inf__fin_Oeq__fold_H,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ( the2 @ A
% 12.24/7.98 @ ( finite_fold @ A @ ( option @ A )
% 12.24/7.98 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y6 ) )
% 12.24/7.98 @ ( none @ A )
% 12.24/7.98 @ A7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5469_timeFrame_Opelims,axiom,
% 12.24/7.98 ! [A: $tType,X: nat,Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 12.24/7.98 ( ( ( heap_Time_timeFrame @ A @ X @ Xa3 )
% 12.24/7.98 = Y3 )
% 12.24/7.98 => ( ( 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 @ Xa3 ) )
% 12.24/7.98 => ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
% 12.24/7.98 ( ( Xa3
% 12.24/7.98 = ( 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 @ H6 @ N10 ) ) ) )
% 12.24/7.98 => ( ( Y3
% 12.24/7.98 = ( 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 @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) )
% 12.24/7.98 => ~ ( 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 ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) ) ) ) )
% 12.24/7.98 => ~ ( ( Xa3
% 12.24/7.98 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.98 => ( ( Y3
% 12.24/7.98 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/7.98 => ~ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5470_Inf__fin_Osingleton,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = X ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5471_Sup__fin_Osingleton,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = X ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5472_inf__Sup__absorb,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,A3: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.98 => ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 12.24/7.98 = A3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5473_sup__Inf__absorb,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,A3: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.98 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 )
% 12.24/7.98 = A3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5474_Inf__fin_Oinsert,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5475_Sup__fin_Oinsert,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5476_Inf__fin__Min,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ( semilattice_inf @ A )
% 12.24/7.98 & ( linorder @ A ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A )
% 12.24/7.98 = ( lattic643756798350308766er_Min @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5477_Sup__fin__Max,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ( semilattice_sup @ A )
% 12.24/7.98 & ( linorder @ A ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A )
% 12.24/7.98 = ( lattic643756798349783984er_Max @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5478_Inf__fin__le__Sup__fin,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( lattice @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5479_Sup__fin_OcoboundedI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,A3: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5480_Inf__fin_OcoboundedI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,A3: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5481_Inf__fin_Oin__idem,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5482_Sup__fin_Oin__idem,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5483_Sup__fin_Obounded__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
% 12.24/7.98 = ( ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5484_Inf__fin_Obounded__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 12.24/7.98 = ( ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5485_Sup__fin_OboundedI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ! [A8: A] :
% 12.24/7.98 ( ( member @ A @ A8 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ A8 @ X ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5486_Sup__fin_OboundedE,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
% 12.24/7.98 => ! [A10: A] :
% 12.24/7.98 ( ( member @ A @ A10 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5487_Inf__fin_OboundedI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ! [A8: A] :
% 12.24/7.98 ( ( member @ A @ A8 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X @ A8 ) )
% 12.24/7.98 => ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5488_Inf__fin_OboundedE,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 12.24/7.98 => ! [A10: A] :
% 12.24/7.98 ( ( member @ A @ A10 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5489_Sup__fin__Sup,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 12.24/7.98 = ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5490_card_Oeq__fold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( finite_card @ A )
% 12.24/7.98 = ( finite_fold @ A @ nat
% 12.24/7.98 @ ^ [Uu: A] : suc
% 12.24/7.98 @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5491_Inf__fin__Inf,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 12.24/7.98 = ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5492_Inf__fin_Oinfinite,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 12.24/7.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5493_Sup__fin_Oinfinite,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 12.24/7.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5494_Sup__fin_Osubset__imp,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5495_Inf__fin_Osubset__imp,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5496_Inf__fin_Ohom__commute,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [H2: A > A,N6: set @ A] :
% 12.24/7.98 ( ! [X3: A,Y5: A] :
% 12.24/7.98 ( ( H2 @ ( inf_inf @ A @ X3 @ Y5 ) )
% 12.24/7.98 = ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.98 => ( ( N6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
% 12.24/7.98 = ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5497_Sup__fin_Ohom__commute,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [H2: A > A,N6: set @ A] :
% 12.24/7.98 ( ! [X3: A,Y5: A] :
% 12.24/7.98 ( ( H2 @ ( sup_sup @ A @ X3 @ Y5 ) )
% 12.24/7.98 = ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.98 => ( ( N6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
% 12.24/7.98 = ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5498_Inf__fin_Osubset,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.98 => ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5499_Sup__fin_Osubset,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.98 => ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B7 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5500_Inf__fin_Oinsert__not__elem,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5501_Inf__fin_Oclosed,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ! [X3: A,Y5: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.98 => ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5502_Sup__fin_Oclosed,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ! [X3: A,Y5: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.98 => ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5503_Sup__fin_Oinsert__not__elem,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5504_Inf__fin_Ounion,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.98 = ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5505_Sup__fin_Ounion,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.98 = ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5506_Inf__fin_Oeq__fold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5507_Sup__fin_Oeq__fold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5508_inf__Sup1__distrib,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( distrib_lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic5882676163264333800up_fin @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [Uu: A] :
% 12.24/7.98 ? [A5: A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( inf_inf @ A @ X @ A5 ) )
% 12.24/7.98 & ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5509_inf__Sup2__distrib,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( distrib_lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) )
% 12.24/7.98 = ( lattic5882676163264333800up_fin @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [Uu: A] :
% 12.24/7.98 ? [A5: A,B3: A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( inf_inf @ A @ A5 @ B3 ) )
% 12.24/7.98 & ( member @ A @ A5 @ A6 )
% 12.24/7.98 & ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5510_sup__Inf2__distrib,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( distrib_lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.98 => ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) )
% 12.24/7.98 = ( lattic7752659483105999362nf_fin @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [Uu: A] :
% 12.24/7.98 ? [A5: A,B3: A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( sup_sup @ A @ A5 @ B3 ) )
% 12.24/7.98 & ( member @ A @ A5 @ A6 )
% 12.24/7.98 & ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5511_sup__Inf1__distrib,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( distrib_lattice @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
% 12.24/7.98 = ( lattic7752659483105999362nf_fin @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [Uu: A] :
% 12.24/7.98 ? [A5: A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( sup_sup @ A @ X @ A5 ) )
% 12.24/7.98 & ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5512_Inf__fin_Oinsert__remove,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = X ) )
% 12.24/7.98 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5513_Inf__fin_Oremove,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_inf @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 12.24/7.98 = X ) )
% 12.24/7.98 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic7752659483105999362nf_fin @ A @ A6 )
% 12.24/7.98 = ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5514_Sup__fin_Oinsert__remove,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = X ) )
% 12.24/7.98 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
% 12.24/7.98 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5515_Sup__fin_Oremove,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semilattice_sup @ A )
% 12.24/7.98 => ! [A6: set @ A,X: A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( member @ A @ X @ A6 )
% 12.24/7.98 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 12.24/7.98 = X ) )
% 12.24/7.98 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( lattic5882676163264333800up_fin @ A @ A6 )
% 12.24/7.98 = ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5516_raise__def,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( heap_Time_raise @ A )
% 12.24/7.98 = ( ^ [S6: list @ char] :
% 12.24/7.98 ( heap_Time_Heap2 @ A
% 12.24/7.98 @ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5517_Gcd__fin_Oeq__fold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ( ( semiring_gcd_Gcd_fin @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5518_funpow__inj__finite,axiom,
% 12.24/7.98 ! [A: $tType,P5: A > A,X: A] :
% 12.24/7.98 ( ( inj_on @ A @ A @ P5 @ ( top_top @ ( set @ A ) ) )
% 12.24/7.98 => ( ( finite_finite2 @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [Y6: A] :
% 12.24/7.98 ? [N3: nat] :
% 12.24/7.98 ( Y6
% 12.24/7.98 = ( compow @ ( A > A ) @ N3 @ P5 @ X ) ) ) )
% 12.24/7.98 => ~ ! [N4: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
% 12.24/7.98 => ( ( compow @ ( A > A ) @ N4 @ P5 @ X )
% 12.24/7.98 != X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5519_inj__uminus,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [A6: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5520_raise__bind,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,E: list @ char,F: B > ( heap_Time_Heap @ A )] :
% 12.24/7.98 ( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E ) @ F )
% 12.24/7.98 = ( heap_Time_raise @ A @ E ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5521_Gcd__fin_Oempty,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 = ( zero_zero @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5522_Gcd__fin_Oinfinite,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 12.24/7.98 = ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5523_is__unit__Gcd__fin__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) @ ( one_one @ A ) )
% 12.24/7.98 = ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 12.24/7.98 = ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5524_inj__mult__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( idom @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
% 12.24/7.98 = ( A3
% 12.24/7.98 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5525_inj__divide__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( inj_on @ A @ A
% 12.24/7.98 @ ^ [B3: A] : ( divide_divide @ A @ B3 @ A3 )
% 12.24/7.98 @ ( top_top @ ( set @ A ) ) )
% 12.24/7.98 = ( A3
% 12.24/7.98 != ( zero_zero @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5526_inj__fn,axiom,
% 12.24/7.98 ! [A: $tType,F: A > A,N: nat] :
% 12.24/7.98 ( ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.98 => ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5527_subset__image__inj,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,S3: set @ A,F: B > A,T6: set @ B] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F @ T6 ) )
% 12.24/7.98 = ( ? [U5: set @ B] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ B ) @ U5 @ T6 )
% 12.24/7.98 & ( inj_on @ B @ A @ F @ U5 )
% 12.24/7.98 & ( S3
% 12.24/7.98 = ( image @ B @ A @ F @ U5 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5528_inj__list__encode,axiom,
% 12.24/7.98 ! [A6: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5529_inj__prod__encode,axiom,
% 12.24/7.98 ! [A6: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5530_inj__Suc,axiom,
% 12.24/7.98 ! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5531_inj__on__of__nat,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_char_0 @ A )
% 12.24/7.98 => ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5532_inj__int__encode,axiom,
% 12.24/7.98 ! [A6: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5533_inj__int__decode,axiom,
% 12.24/7.98 ! [A6: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5534_linorder__inj__onI,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( order @ A )
% 12.24/7.98 => ! [A6: set @ A,F: A > B] :
% 12.24/7.98 ( ! [X3: A,Y5: A] :
% 12.24/7.98 ( ( ord_less @ A @ X3 @ Y5 )
% 12.24/7.98 => ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 => ( ( member @ A @ Y5 @ A6 )
% 12.24/7.98 => ( ( F @ X3 )
% 12.24/7.98 != ( F @ Y5 ) ) ) ) )
% 12.24/7.98 => ( ! [X3: A,Y5: A] :
% 12.24/7.98 ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 => ( ( member @ A @ Y5 @ A6 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.98 | ( ord_less_eq @ A @ Y5 @ X3 ) ) ) )
% 12.24/7.98 => ( inj_on @ A @ B @ F @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5535_inj__on__mult,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semidom_divide @ A )
% 12.24/7.98 => ! [A3: A,A6: set @ A] :
% 12.24/7.98 ( ( A3
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 => ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5536_linorder__injI,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [F: A > B] :
% 12.24/7.98 ( ! [X3: A,Y5: A] :
% 12.24/7.98 ( ( ord_less @ A @ X3 @ Y5 )
% 12.24/7.98 => ( ( F @ X3 )
% 12.24/7.98 != ( F @ Y5 ) ) )
% 12.24/7.98 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5537_inj__on__strict__subset,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,F: A > B,B7: set @ A,A6: set @ A] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F @ B7 )
% 12.24/7.98 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ ( image @ A @ B @ F @ B7 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5538_inj__of__nat,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_char_0 @ A )
% 12.24/7.98 => ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5539_inj__on__diff__nat,axiom,
% 12.24/7.98 ! [N6: set @ nat,K: nat] :
% 12.24/7.98 ( ! [N4: nat] :
% 12.24/7.98 ( ( member @ nat @ N4 @ N6 )
% 12.24/7.98 => ( ord_less_eq @ nat @ K @ N4 ) )
% 12.24/7.98 => ( inj_on @ nat @ nat
% 12.24/7.98 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
% 12.24/7.98 @ N6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5540_inj__on__set__encode,axiom,
% 12.24/7.98 inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5541_inj__on__iff__surj,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,A6: set @ A,A11: set @ B] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( ? [F2: A > B] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F2 @ A6 )
% 12.24/7.98 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ A11 ) ) )
% 12.24/7.98 = ( ? [G2: B > A] :
% 12.24/7.98 ( ( image @ B @ A @ G2 @ A11 )
% 12.24/7.98 = A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5542_pigeonhole,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
% 12.24/7.98 ( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( finite_card @ B @ A6 ) )
% 12.24/7.98 => ~ ( inj_on @ B @ A @ F @ A6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5543_effect__raiseE,axiom,
% 12.24/7.98 ! [A: $tType,X: list @ char,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
% 12.24/7.98 ~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H2 @ H @ R @ N ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5544_finite__imp__nat__seg__image__inj__on,axiom,
% 12.24/7.98 ! [A: $tType,A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ? [N4: nat,F3: nat > A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 = ( image @ nat @ A @ F3
% 12.24/7.98 @ ( collect @ nat
% 12.24/7.98 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) )
% 12.24/7.98 & ( inj_on @ nat @ A @ F3
% 12.24/7.98 @ ( collect @ nat
% 12.24/7.98 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5545_finite__imp__inj__to__nat__seg,axiom,
% 12.24/7.98 ! [A: $tType,A6: set @ A] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ? [F3: A > nat,N4: nat] :
% 12.24/7.98 ( ( ( image @ A @ nat @ F3 @ A6 )
% 12.24/7.98 = ( collect @ nat
% 12.24/7.98 @ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) )
% 12.24/7.98 & ( inj_on @ A @ nat @ F3 @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5546_inj__image__Compl__subset,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5547_inj__on__nth,axiom,
% 12.24/7.98 ! [A: $tType,Xs: list @ A,I6: set @ nat] :
% 12.24/7.98 ( ( distinct @ A @ Xs )
% 12.24/7.98 => ( ! [X3: nat] :
% 12.24/7.98 ( ( member @ nat @ X3 @ I6 )
% 12.24/7.98 => ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 12.24/7.98 => ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5548_infinite__iff__countable__subset,axiom,
% 12.24/7.98 ! [A: $tType,S3: set @ A] :
% 12.24/7.98 ( ( ~ ( finite_finite2 @ A @ S3 ) )
% 12.24/7.98 = ( ? [F2: nat > A] :
% 12.24/7.98 ( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.98 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5549_infinite__countable__subset,axiom,
% 12.24/7.98 ! [A: $tType,S3: set @ A] :
% 12.24/7.98 ( ~ ( finite_finite2 @ A @ S3 )
% 12.24/7.98 => ? [F3: nat > A] :
% 12.24/7.98 ( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.98 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5550_inj__on__funpow__least,axiom,
% 12.24/7.98 ! [A: $tType,N: nat,F: A > A,S: A] :
% 12.24/7.98 ( ( ( compow @ ( A > A ) @ N @ F @ S )
% 12.24/7.98 = S )
% 12.24/7.98 => ( ! [M6: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M6 )
% 12.24/7.98 => ( ( ord_less @ nat @ M6 @ N )
% 12.24/7.98 => ( ( compow @ ( A > A ) @ M6 @ F @ S )
% 12.24/7.98 != S ) ) )
% 12.24/7.98 => ( inj_on @ nat @ A
% 12.24/7.98 @ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F @ S )
% 12.24/7.98 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5551_inj__on__iff__card__le,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.98 => ( ( ? [F2: A > B] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F2 @ A6 )
% 12.24/7.98 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ B7 ) ) )
% 12.24/7.98 = ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5552_card__inj__on__le,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,F: A > B,A6: set @ A,B7: set @ B] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F @ A6 )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ B7 )
% 12.24/7.98 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.98 => ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5553_card__le__inj,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
% 12.24/7.98 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.98 => ( ( finite_finite2 @ B @ B7 )
% 12.24/7.98 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) )
% 12.24/7.98 => ? [F3: A > B] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A6 ) @ B7 )
% 12.24/7.98 & ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5554_execute__raise,axiom,
% 12.24/7.98 ! [A: $tType,S: list @ char] :
% 12.24/7.98 ( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
% 12.24/7.98 = ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5555_map__sorted__distinct__set__unique,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [F: B > A,Xs: list @ B,Ys2: list @ B] :
% 12.24/7.98 ( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys2 ) ) )
% 12.24/7.98 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.98 => ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.98 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys2 ) )
% 12.24/7.98 => ( ( distinct @ A @ ( map @ B @ A @ F @ Ys2 ) )
% 12.24/7.98 => ( ( ( set2 @ B @ Xs )
% 12.24/7.98 = ( set2 @ B @ Ys2 ) )
% 12.24/7.98 => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5556_Gcd__fin__0__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 12.24/7.98 = ( zero_zero @ A ) )
% 12.24/7.98 = ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.98 & ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5557_greaterThan__Suc,axiom,
% 12.24/7.98 ! [K: nat] :
% 12.24/7.98 ( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
% 12.24/7.98 = ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5558_last__list__update,axiom,
% 12.24/7.98 ! [A: $tType,Xs: list @ A,K: nat,X: A] :
% 12.24/7.98 ( ( Xs
% 12.24/7.98 != ( nil @ A ) )
% 12.24/7.98 => ( ( ( K
% 12.24/7.98 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 12.24/7.98 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 12.24/7.98 = X ) )
% 12.24/7.98 & ( ( K
% 12.24/7.98 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 12.24/7.98 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 12.24/7.98 = ( last @ A @ Xs ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5559_last__upt,axiom,
% 12.24/7.98 ! [I: nat,J: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ I @ J )
% 12.24/7.98 => ( ( last @ nat @ ( upt @ I @ J ) )
% 12.24/7.98 = ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5560_greaterThan__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [I: A,K: A] :
% 12.24/7.98 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 12.24/7.98 = ( ord_less @ A @ K @ I ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5561_greaterThan__subset__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [X: A,Y3: A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y3 ) )
% 12.24/7.98 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5562_last__replicate,axiom,
% 12.24/7.98 ! [A: $tType,N: nat,X: A] :
% 12.24/7.98 ( ( N
% 12.24/7.98 != ( zero_zero @ nat ) )
% 12.24/7.98 => ( ( last @ A @ ( replicate @ A @ N @ X ) )
% 12.24/7.98 = X ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5563_Compl__greaterThan,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [K: A] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 12.24/7.98 = ( set_ord_atMost @ A @ K ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5564_Compl__atMost,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [K: A] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 12.24/7.98 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5565_Sup__greaterThanAtLeast,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 12.24/7.98 => ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
% 12.24/7.98 = ( top_top @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5566_image__uminus__greaterThan,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
% 12.24/7.98 = ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5567_image__uminus__lessThan,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
% 12.24/7.98 = ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5568_last__drop,axiom,
% 12.24/7.98 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.98 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.98 => ( ( last @ A @ ( drop @ A @ N @ Xs ) )
% 12.24/7.98 = ( last @ A @ Xs ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5569_greaterThan__def,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( set_ord_greaterThan @ A )
% 12.24/7.98 = ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5570_ivl__disj__un__one_I5_J,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [L: A,U: A] :
% 12.24/7.98 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 12.24/7.98 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5571_greaterThan__0,axiom,
% 12.24/7.98 ( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
% 12.24/7.98 = ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5572_last__conv__nth,axiom,
% 12.24/7.98 ! [A: $tType,Xs: list @ A] :
% 12.24/7.98 ( ( Xs
% 12.24/7.98 != ( nil @ A ) )
% 12.24/7.98 => ( ( last @ A @ Xs )
% 12.24/7.98 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5573_interval__cases,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit6923001295902523014norder @ A )
% 12.24/7.98 => ! [S3: set @ A] :
% 12.24/7.98 ( ! [A8: A,B5: A,X3: A] :
% 12.24/7.98 ( ( member @ A @ A8 @ S3 )
% 12.24/7.98 => ( ( member @ A @ B5 @ S3 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ A8 @ X3 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ X3 @ B5 )
% 12.24/7.98 => ( member @ A @ X3 @ S3 ) ) ) ) )
% 12.24/7.98 => ? [A8: A,B5: A] :
% 12.24/7.98 ( ( S3
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( top_top @ ( set @ A ) ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_ord_lessThan @ A @ B5 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_ord_atMost @ A @ B5 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_ord_greaterThan @ A @ A8 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_ord_atLeast @ A @ A8 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_or5935395276787703475ssThan @ A @ A8 @ B5 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_or3652927894154168847AtMost @ A @ A8 @ B5 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_or7035219750837199246ssThan @ A @ A8 @ B5 ) )
% 12.24/7.98 | ( S3
% 12.24/7.98 = ( set_or1337092689740270186AtMost @ A @ A8 @ B5 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5574_sum__mult__sum__if__inj,axiom,
% 12.24/7.98 ! [A: $tType,C: $tType,B: $tType] :
% 12.24/7.98 ( ( semiring_0 @ B )
% 12.24/7.98 => ! [F: A > B,G: C > B,A6: set @ A,B7: set @ C] :
% 12.24/7.98 ( ( inj_on @ ( product_prod @ A @ C ) @ B
% 12.24/7.98 @ ( product_case_prod @ A @ C @ B
% 12.24/7.98 @ ^ [A5: A,B3: C] : ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 12.24/7.98 @ ( product_Sigma @ A @ C @ A6
% 12.24/7.98 @ ^ [Uu: A] : B7 ) )
% 12.24/7.98 => ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
% 12.24/7.98 = ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
% 12.24/7.98 @ ( collect @ B
% 12.24/7.98 @ ^ [Uu: B] :
% 12.24/7.98 ? [A5: A,B3: C] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 12.24/7.98 & ( member @ A @ A5 @ A6 )
% 12.24/7.98 & ( member @ C @ B3 @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5575_arg__min__inj__eq,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( order @ B )
% 12.24/7.98 => ! [F: A > B,P: A > $o,A3: A] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F @ ( collect @ A @ P ) )
% 12.24/7.98 => ( ( P @ A3 )
% 12.24/7.98 => ( ! [Y5: A] :
% 12.24/7.98 ( ( P @ Y5 )
% 12.24/7.98 => ( ord_less_eq @ B @ ( F @ A3 ) @ ( F @ Y5 ) ) )
% 12.24/7.98 => ( ( lattices_ord_arg_min @ A @ B @ F @ P )
% 12.24/7.98 = A3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5576_atLeast__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [I: A,K: A] :
% 12.24/7.98 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 12.24/7.98 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5577_atLeast__0,axiom,
% 12.24/7.98 ( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
% 12.24/7.98 = ( top_top @ ( set @ nat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5578_atLeast__subset__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [X: A,Y3: A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y3 ) )
% 12.24/7.98 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5579_Compl__atLeast,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [K: A] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 12.24/7.98 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5580_Compl__lessThan,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [K: A] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 12.24/7.98 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5581_Icc__subset__Ici__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [L: A,H2: A,L4: A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) )
% 12.24/7.98 = ( ~ ( ord_less_eq @ A @ L @ H2 )
% 12.24/7.98 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5582_image__uminus__atMost,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
% 12.24/7.98 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5583_image__uminus__atLeast,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
% 12.24/7.98 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5584_arg__min__nat__lemma,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 12.24/7.98 ( ( P @ K )
% 12.24/7.98 => ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
% 12.24/7.98 & ! [Y4: A] :
% 12.24/7.98 ( ( P @ Y4 )
% 12.24/7.98 => ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y4 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5585_arg__min__nat__le,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 12.24/7.98 ( ( P @ X )
% 12.24/7.98 => ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5586_atLeast__def,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( set_ord_atLeast @ A )
% 12.24/7.98 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5587_arg__min__equality,axiom,
% 12.24/7.98 ! [A: $tType,C: $tType] :
% 12.24/7.98 ( ( order @ A )
% 12.24/7.98 => ! [P: C > $o,K: C,F: C > A] :
% 12.24/7.98 ( ( P @ K )
% 12.24/7.98 => ( ! [X3: C] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( F @ K ) @ ( F @ X3 ) ) )
% 12.24/7.98 => ( ( F @ ( lattices_ord_arg_min @ C @ A @ F @ P ) )
% 12.24/7.98 = ( F @ K ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5588_arg__minI,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( ord @ B )
% 12.24/7.98 => ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
% 12.24/7.98 ( ( P @ X )
% 12.24/7.98 => ( ! [Y5: A] :
% 12.24/7.98 ( ( P @ Y5 )
% 12.24/7.98 => ~ ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X ) ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 => ( ! [Y4: A] :
% 12.24/7.98 ( ( P @ Y4 )
% 12.24/7.98 => ~ ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 12.24/7.98 => ( Q @ X3 ) ) )
% 12.24/7.98 => ( Q @ ( lattices_ord_arg_min @ A @ B @ F @ P ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5589_atLeast__Suc__greaterThan,axiom,
% 12.24/7.98 ! [K: nat] :
% 12.24/7.98 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 12.24/7.98 = ( set_ord_greaterThan @ nat @ K ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5590_Ex__inj__on__UNION__Sigma,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,A6: B > ( set @ A ),I6: set @ B] :
% 12.24/7.98 ? [F3: A > ( product_prod @ B @ A )] :
% 12.24/7.98 ( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) )
% 12.24/7.98 & ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) ) @ ( product_Sigma @ B @ A @ I6 @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5591_ivl__disj__un__one_I8_J,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [L: A,U: A] :
% 12.24/7.98 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 12.24/7.98 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5592_Ici__subset__Ioi__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 12.24/7.98 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5593_card__cartesian__product,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
% 12.24/7.98 ( ( finite_card @ ( product_prod @ A @ B )
% 12.24/7.98 @ ( product_Sigma @ A @ B @ A6
% 12.24/7.98 @ ^ [Uu: A] : B7 ) )
% 12.24/7.98 = ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5594_ivl__disj__un__one_I7_J,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [L: A,U: A] :
% 12.24/7.98 ( ( ord_less_eq @ A @ L @ U )
% 12.24/7.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 12.24/7.98 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5595_ivl__disj__un__one_I6_J,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [L: A,U: A] :
% 12.24/7.98 ( ( ord_less @ A @ L @ U )
% 12.24/7.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 12.24/7.98 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5596_lists__length__Suc__eq,axiom,
% 12.24/7.98 ! [A: $tType,A6: set @ A,N: nat] :
% 12.24/7.98 ( ( collect @ ( list @ A )
% 12.24/7.98 @ ^ [Xs3: list @ A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 12.24/7.98 & ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.98 = ( suc @ N ) ) ) )
% 12.24/7.98 = ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
% 12.24/7.98 @ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
% 12.24/7.98 @ ^ [Xs3: list @ A,N3: A] : ( cons @ A @ N3 @ Xs3 ) )
% 12.24/7.98 @ ( product_Sigma @ ( list @ A ) @ A
% 12.24/7.98 @ ( collect @ ( list @ A )
% 12.24/7.98 @ ^ [Xs3: list @ A] :
% 12.24/7.98 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
% 12.24/7.98 & ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.98 = N ) ) )
% 12.24/7.98 @ ^ [Uu: list @ A] : A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5597_arg__min__on__def,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( lattic7623131987881927897min_on @ B @ A )
% 12.24/7.98 = ( ^ [F2: B > A,S7: set @ B] :
% 12.24/7.98 ( lattices_ord_arg_min @ B @ A @ F2
% 12.24/7.98 @ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5598_atLeast__Suc,axiom,
% 12.24/7.98 ! [K: nat] :
% 12.24/7.98 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 12.24/7.98 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5599_Compl__Times__UNIV1,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,A6: set @ B] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 12.24/7.98 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 12.24/7.98 @ ^ [Uu: A] : A6 ) )
% 12.24/7.98 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 12.24/7.98 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5600_Compl__Times__UNIV2,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,A6: set @ A] :
% 12.24/7.98 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 12.24/7.98 @ ( product_Sigma @ A @ B @ A6
% 12.24/7.98 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 12.24/7.98 = ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A6 )
% 12.24/7.98 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5601_mult__inj__if__coprime__nat,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,F: A > nat,A6: set @ A,G: B > nat,B7: set @ B] :
% 12.24/7.98 ( ( inj_on @ A @ nat @ F @ A6 )
% 12.24/7.98 => ( ( inj_on @ B @ nat @ G @ B7 )
% 12.24/7.98 => ( ! [A8: A,B5: B] :
% 12.24/7.98 ( ( member @ A @ A8 @ A6 )
% 12.24/7.98 => ( ( member @ B @ B5 @ B7 )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ nat @ ( F @ A8 ) @ ( G @ B5 ) ) ) )
% 12.24/7.98 => ( inj_on @ ( product_prod @ A @ B ) @ nat
% 12.24/7.98 @ ( product_case_prod @ A @ B @ nat
% 12.24/7.98 @ ^ [A5: A,B3: B] : ( times_times @ nat @ ( F @ A5 ) @ ( G @ B3 ) ) )
% 12.24/7.98 @ ( product_Sigma @ A @ B @ A6
% 12.24/7.98 @ ^ [Uu: A] : B7 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5602_coprime__mult__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.98 = ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5603_coprime__mult__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.98 = ( ( algebr8660921524188924756oprime @ A @ C2 @ A3 )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5604_coprime__minus__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5605_coprime__minus__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5606_coprime__self,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
% 12.24/7.98 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5607_coprime__mod__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( euclid3725896446679973847miring @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( A3
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B2 @ A3 ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5608_coprime__mod__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( euclid3725896446679973847miring @ A )
% 12.24/7.98 => ! [B2: A,A3: A] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5609_coprime__power__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A,N: nat] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B2 @ N ) )
% 12.24/7.98 = ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 | ( N
% 12.24/7.98 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5610_coprime__power__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,N: nat,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
% 12.24/7.98 = ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 | ( N
% 12.24/7.98 = ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5611_coprime__imp__gcd__eq__1,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.98 = ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5612_coprime__0__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.98 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5613_coprime__0__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.98 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5614_coprime__mult__self__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.98 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5615_coprime__mult__self__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.24/7.98 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5616_is__unit__gcd,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5617_arg__min__natI,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,K: A,M: A > nat] :
% 12.24/7.98 ( ( P @ K )
% 12.24/7.98 => ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5618_coprime__add__one__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5619_coprime__add__one__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5620_coprime__diff__one__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ring_gcd @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5621_coprime__doff__one__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ring_gcd @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5622_coprime__dvd__mult__right__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 12.24/7.98 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5623_coprime__dvd__mult__left__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.98 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5624_divides__mult,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ B2 @ C2 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5625_coprime__Suc__0__left,axiom,
% 12.24/7.98 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5626_coprime__Suc__0__right,axiom,
% 12.24/7.98 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5627_coprime__1__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5628_coprime__1__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5629_coprime__Suc__left__nat,axiom,
% 12.24/7.98 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5630_coprime__Suc__right__nat,axiom,
% 12.24/7.98 ! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5631_coprime__commute,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A )
% 12.24/7.98 = ( ^ [B3: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5632_coprime__divisors,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A,D2: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ B2 @ D2 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5633_coprime__crossproduct__nat,axiom,
% 12.24/7.98 ! [A3: nat,D2: nat,B2: nat,C2: nat] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ nat @ A3 @ D2 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ nat @ B2 @ C2 )
% 12.24/7.98 => ( ( ( times_times @ nat @ A3 @ C2 )
% 12.24/7.98 = ( times_times @ nat @ B2 @ D2 ) )
% 12.24/7.98 = ( ( A3 = B2 )
% 12.24/7.98 & ( C2 = D2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5634_is__unit__right__imp__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [B2: A,A3: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5635_is__unit__left__imp__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5636_coprime__common__divisor,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5637_coprime__absorb__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [Y3: A,X: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ Y3 @ X )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
% 12.24/7.98 = ( dvd_dvd @ A @ Y3 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5638_coprime__imp__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [C2: A,D2: A,A3: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 12.24/7.98 => ( ! [E2: A] :
% 12.24/7.98 ( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
% 12.24/7.98 => ( ( dvd_dvd @ A @ E2 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ E2 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ E2 @ C2 ) ) ) )
% 12.24/7.98 => ( ! [E2: A] :
% 12.24/7.98 ( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
% 12.24/7.98 => ( ( dvd_dvd @ A @ E2 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ E2 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ E2 @ D2 ) ) ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5639_coprime__absorb__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [X: A,Y3: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ X @ Y3 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
% 12.24/7.98 = ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5640_not__coprimeI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ C2 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 12.24/7.98 => ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 12.24/7.98 => ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5641_not__coprimeE,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.98 => ~ ! [C3: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ C3 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C3 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5642_coprime__def,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A )
% 12.24/7.98 = ( ^ [A5: A,B3: A] :
% 12.24/7.98 ! [C4: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ C4 @ A5 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C4 @ B3 )
% 12.24/7.98 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5643_coprimeI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( algebraic_semidom @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ! [C3: A] :
% 12.24/7.98 ( ( dvd_dvd @ A @ C3 @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ A @ C3 @ B2 )
% 12.24/7.98 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5644_coprime__common__divisor__nat,axiom,
% 12.24/7.98 ! [A3: nat,B2: nat,X: nat] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
% 12.24/7.98 => ( ( dvd_dvd @ nat @ X @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ nat @ X @ B2 )
% 12.24/7.98 => ( X
% 12.24/7.98 = ( one_one @ nat ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5645_gcd__eq__1__imp__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.98 = ( one_one @ A ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5646_coprime__iff__gcd__eq__1,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A )
% 12.24/7.98 = ( ^ [A5: A,B3: A] :
% 12.24/7.98 ( ( gcd_gcd @ A @ A5 @ B3 )
% 12.24/7.98 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5647_gcd__mult__right__right__cancel,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.98 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5648_gcd__mult__right__left__cancel,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
% 12.24/7.98 => ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 12.24/7.98 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5649_gcd__mult__left__right__cancel,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 12.24/7.98 => ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
% 12.24/7.98 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5650_gcd__mult__left__left__cancel,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [B2: A,C2: A,A3: A] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 12.24/7.98 => ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 )
% 12.24/7.98 = ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5651_mult__mod__cancel__right,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ( euclid8851590272496341667cancel @ A )
% 12.24/7.98 & ( semiring_gcd @ A ) )
% 12.24/7.98 => ! [A3: A,N: A,M: A,B2: A] :
% 12.24/7.98 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M )
% 12.24/7.98 = ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M ) )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ M @ N )
% 12.24/7.98 => ( ( modulo_modulo @ A @ A3 @ M )
% 12.24/7.98 = ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5652_mult__mod__cancel__left,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ( euclid8851590272496341667cancel @ A )
% 12.24/7.98 & ( semiring_gcd @ A ) )
% 12.24/7.98 => ! [N: A,A3: A,M: A,B2: A] :
% 12.24/7.98 ( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M )
% 12.24/7.98 = ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M ) )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ A @ M @ N )
% 12.24/7.98 => ( ( modulo_modulo @ A @ A3 @ M )
% 12.24/7.98 = ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5653_invertible__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( euclid8851590272496341667cancel @ A )
% 12.24/7.98 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.98 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.98 = ( one_one @ A ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5654_gcd__coprime__exists,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 => ? [A15: A,B8: A] :
% 12.24/7.98 ( ( A3
% 12.24/7.98 = ( times_times @ A @ A15 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 12.24/7.98 & ( B2
% 12.24/7.98 = ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ A @ A15 @ B8 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5655_gcd__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A,A4: A,B4: A] :
% 12.24/7.98 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 => ( ( A3
% 12.24/7.98 = ( times_times @ A @ A4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 12.24/7.98 => ( ( B2
% 12.24/7.98 = ( times_times @ A @ B4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ A4 @ B4 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5656_div__gcd__coprime,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( semiring_gcd @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( ( A3
% 12.24/7.98 != ( zero_zero @ A ) )
% 12.24/7.98 | ( B2
% 12.24/7.98 != ( zero_zero @ A ) ) )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5657_coprime__diff__one__right__nat,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5658_coprime__diff__one__left__nat,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.98 => ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5659_set__encode__vimage__Suc,axiom,
% 12.24/7.98 ! [A6: set @ nat] :
% 12.24/7.98 ( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A6 ) )
% 12.24/7.98 = ( divide_divide @ nat @ ( nat_set_encode @ A6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5660_floor__rat__def,axiom,
% 12.24/7.98 ( ( archim6421214686448440834_floor @ rat )
% 12.24/7.98 = ( ^ [X2: rat] :
% 12.24/7.98 ( the @ int
% 12.24/7.98 @ ^ [Z7: int] :
% 12.24/7.98 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z7 ) @ X2 )
% 12.24/7.98 & ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5661_sorted__insort__insert__key,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [F: B > A,Xs: list @ B,X: B] :
% 12.24/7.98 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
% 12.24/7.98 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5662_coprime__abs__left__iff,axiom,
% 12.24/7.98 ! [K: int,L: int] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ int @ ( abs_abs @ int @ K ) @ L )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5663_coprime__abs__right__iff,axiom,
% 12.24/7.98 ! [K: int,L: int] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ int @ K @ ( abs_abs @ int @ L ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5664_coprime__int__iff,axiom,
% 12.24/7.98 ! [M: nat,N: nat] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ nat @ M @ N ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5665_normalize__stable,axiom,
% 12.24/7.98 ! [Q4: int,P5: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ int @ P5 @ Q4 )
% 12.24/7.98 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
% 12.24/7.98 = ( product_Pair @ int @ int @ P5 @ Q4 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5666_coprime__nat__abs__left__iff,axiom,
% 12.24/7.98 ! [K: int,N: nat] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5667_coprime__nat__abs__right__iff,axiom,
% 12.24/7.98 ! [N: nat,K: int] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 12.24/7.98 = ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5668_vimage__if,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,C2: B,A6: set @ B,D2: B,B7: set @ A] :
% 12.24/7.98 ( ( ( member @ B @ C2 @ A6 )
% 12.24/7.98 => ( ( ( member @ B @ D2 @ A6 )
% 12.24/7.98 => ( ( vimage @ A @ B
% 12.24/7.98 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 12.24/7.98 @ A6 )
% 12.24/7.98 = ( top_top @ ( set @ A ) ) ) )
% 12.24/7.98 & ( ~ ( member @ B @ D2 @ A6 )
% 12.24/7.98 => ( ( vimage @ A @ B
% 12.24/7.98 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 12.24/7.98 @ A6 )
% 12.24/7.98 = B7 ) ) ) )
% 12.24/7.98 & ( ~ ( member @ B @ C2 @ A6 )
% 12.24/7.98 => ( ( ( member @ B @ D2 @ A6 )
% 12.24/7.98 => ( ( vimage @ A @ B
% 12.24/7.98 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 12.24/7.98 @ A6 )
% 12.24/7.98 = ( uminus_uminus @ ( set @ A ) @ B7 ) ) )
% 12.24/7.98 & ( ~ ( member @ B @ D2 @ A6 )
% 12.24/7.98 => ( ( vimage @ A @ B
% 12.24/7.98 @ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
% 12.24/7.98 @ A6 )
% 12.24/7.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5669_vimage__Compl,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 12.24/7.98 ( ( vimage @ A @ B @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) )
% 12.24/7.98 = ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5670_coprime__crossproduct__int,axiom,
% 12.24/7.98 ! [A3: int,D2: int,B2: int,C2: int] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ int @ A3 @ D2 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ int @ B2 @ C2 )
% 12.24/7.98 => ( ( ( times_times @ int @ ( abs_abs @ int @ A3 ) @ ( abs_abs @ int @ C2 ) )
% 12.24/7.98 = ( times_times @ int @ ( abs_abs @ int @ B2 ) @ ( abs_abs @ int @ D2 ) ) )
% 12.24/7.98 = ( ( ( abs_abs @ int @ A3 )
% 12.24/7.98 = ( abs_abs @ int @ B2 ) )
% 12.24/7.98 & ( ( abs_abs @ int @ C2 )
% 12.24/7.98 = ( abs_abs @ int @ D2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5671_finite__vimage__Suc__iff,axiom,
% 12.24/7.98 ! [F4: set @ nat] :
% 12.24/7.98 ( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F4 ) )
% 12.24/7.98 = ( finite_finite2 @ nat @ F4 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5672_Nitpick_OThe__psimp,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,X: A] :
% 12.24/7.98 ( ( P
% 12.24/7.98 = ( ^ [Y: A,Z2: A] : ( Y = Z2 )
% 12.24/7.98 @ X ) )
% 12.24/7.98 => ( ( the @ A @ P )
% 12.24/7.98 = X ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5673_vimage__Suc__insert__Suc,axiom,
% 12.24/7.98 ! [N: nat,A6: set @ nat] :
% 12.24/7.98 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A6 ) )
% 12.24/7.98 = ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5674_vimage__Suc__insert__0,axiom,
% 12.24/7.98 ! [A6: set @ nat] :
% 12.24/7.98 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A6 ) )
% 12.24/7.98 = ( vimage @ nat @ nat @ suc @ A6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5675_coprime__common__divisor__int,axiom,
% 12.24/7.98 ! [A3: int,B2: int,X: int] :
% 12.24/7.98 ( ( algebr8660921524188924756oprime @ int @ A3 @ B2 )
% 12.24/7.98 => ( ( dvd_dvd @ int @ X @ A3 )
% 12.24/7.98 => ( ( dvd_dvd @ int @ X @ B2 )
% 12.24/7.98 => ( ( abs_abs @ int @ X )
% 12.24/7.98 = ( one_one @ int ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5676_sorted__insort__insert,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [Xs: list @ A,X: A] :
% 12.24/7.98 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.98 => ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 12.24/7.98 @ ( linord329482645794927042rt_key @ A @ A
% 12.24/7.98 @ ^ [X2: A] : X2
% 12.24/7.98 @ X
% 12.24/7.98 @ Xs ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5677_card__vimage__inj__on__le,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,F: A > B,D4: set @ A,A6: set @ B] :
% 12.24/7.98 ( ( inj_on @ A @ B @ F @ D4 )
% 12.24/7.98 => ( ( finite_finite2 @ B @ A6 )
% 12.24/7.98 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ D4 ) ) @ ( finite_card @ B @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5678_set__decode__div__2,axiom,
% 12.24/7.98 ! [X: nat] :
% 12.24/7.98 ( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 12.24/7.98 = ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5679_old_Orec__nat__def,axiom,
% 12.24/7.98 ! [T: $tType] :
% 12.24/7.98 ( ( rec_nat @ T )
% 12.24/7.98 = ( ^ [F12: T,F23: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5680_rat__floor__lemma,axiom,
% 12.24/7.98 ! [A3: int,B2: int] :
% 12.24/7.98 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 & ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5681_sgn__rat,axiom,
% 12.24/7.98 ! [A3: int,B2: int] :
% 12.24/7.98 ( ( sgn_sgn @ rat @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ring_1_of_int @ rat @ ( times_times @ int @ ( sgn_sgn @ int @ A3 ) @ ( sgn_sgn @ int @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5682_minus__rat__cancel,axiom,
% 12.24/7.98 ! [A3: int,B2: int] :
% 12.24/7.98 ( ( fract @ ( uminus_uminus @ int @ A3 ) @ ( uminus_uminus @ int @ B2 ) )
% 12.24/7.98 = ( fract @ A3 @ B2 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5683_minus__rat,axiom,
% 12.24/7.98 ! [A3: int,B2: int] :
% 12.24/7.98 ( ( uminus_uminus @ rat @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( fract @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5684_divide__rat,axiom,
% 12.24/7.98 ! [A3: int,B2: int,C2: int,D2: int] :
% 12.24/7.98 ( ( divide_divide @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( fract @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ C2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5685_mult__rat,axiom,
% 12.24/7.98 ! [A3: int,B2: int,C2: int,D2: int] :
% 12.24/7.98 ( ( times_times @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( fract @ ( times_times @ int @ A3 @ C2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5686_less__rat,axiom,
% 12.24/7.98 ! [B2: int,D2: int,A3: int,C2: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( D2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5687_le__rat,axiom,
% 12.24/7.98 ! [B2: int,D2: int,A3: int,C2: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( D2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5688_add__rat,axiom,
% 12.24/7.98 ! [B2: int,D2: int,A3: int,C2: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( D2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5689_diff__rat,axiom,
% 12.24/7.98 ! [B2: int,D2: int,A3: int,C2: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( D2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( minus_minus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( fract @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5690_Rat__cases,axiom,
% 12.24/7.98 ! [Q4: rat] :
% 12.24/7.98 ~ ! [A8: int,B5: int] :
% 12.24/7.98 ( ( Q4
% 12.24/7.98 = ( fract @ A8 @ B5 ) )
% 12.24/7.98 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 12.24/7.98 => ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5691_Rat__induct,axiom,
% 12.24/7.98 ! [P: rat > $o,Q4: rat] :
% 12.24/7.98 ( ! [A8: int,B5: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
% 12.24/7.98 => ( P @ ( fract @ A8 @ B5 ) ) ) )
% 12.24/7.98 => ( P @ Q4 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5692_Fract__of__nat__eq,axiom,
% 12.24/7.98 ! [K: nat] :
% 12.24/7.98 ( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
% 12.24/7.98 = ( semiring_1_of_nat @ rat @ K ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5693_eq__rat_I3_J,axiom,
% 12.24/7.98 ! [A3: int,C2: int] :
% 12.24/7.98 ( ( fract @ ( zero_zero @ int ) @ A3 )
% 12.24/7.98 = ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5694_eq__rat_I1_J,axiom,
% 12.24/7.98 ! [B2: int,D2: int,A3: int,C2: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( D2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( ( fract @ A3 @ B2 )
% 12.24/7.98 = ( fract @ C2 @ D2 ) )
% 12.24/7.98 = ( ( times_times @ int @ A3 @ D2 )
% 12.24/7.98 = ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5695_mult__rat__cancel,axiom,
% 12.24/7.98 ! [C2: int,A3: int,B2: int] :
% 12.24/7.98 ( ( C2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( fract @ ( times_times @ int @ C2 @ A3 ) @ ( times_times @ int @ C2 @ B2 ) )
% 12.24/7.98 = ( fract @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5696_eq__rat_I2_J,axiom,
% 12.24/7.98 ! [A3: int] :
% 12.24/7.98 ( ( fract @ A3 @ ( zero_zero @ int ) )
% 12.24/7.98 = ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5697_Rat__induct__pos,axiom,
% 12.24/7.98 ! [P: rat > $o,Q4: rat] :
% 12.24/7.98 ( ! [A8: int,B5: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 12.24/7.98 => ( P @ ( fract @ A8 @ B5 ) ) )
% 12.24/7.98 => ( P @ Q4 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5698_rat__number__collapse_I6_J,axiom,
% 12.24/7.98 ! [K: int] :
% 12.24/7.98 ( ( fract @ K @ ( zero_zero @ int ) )
% 12.24/7.98 = ( zero_zero @ rat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5699_rat__number__collapse_I1_J,axiom,
% 12.24/7.98 ! [K: int] :
% 12.24/7.98 ( ( fract @ ( zero_zero @ int ) @ K )
% 12.24/7.98 = ( zero_zero @ rat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5700_One__rat__def,axiom,
% 12.24/7.98 ( ( one_one @ rat )
% 12.24/7.98 = ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5701_Fract__of__int__eq,axiom,
% 12.24/7.98 ! [K: int] :
% 12.24/7.98 ( ( fract @ K @ ( one_one @ int ) )
% 12.24/7.98 = ( ring_1_of_int @ rat @ K ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5702_quotient__of__def,axiom,
% 12.24/7.98 ( quotient_of
% 12.24/7.98 = ( ^ [X2: rat] :
% 12.24/7.98 ( the @ ( product_prod @ int @ int )
% 12.24/7.98 @ ^ [Pair: product_prod @ int @ int] :
% 12.24/7.98 ( ( X2
% 12.24/7.98 = ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
% 12.24/7.98 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5703_Rat__cases__nonzero,axiom,
% 12.24/7.98 ! [Q4: rat] :
% 12.24/7.98 ( ! [A8: int,B5: int] :
% 12.24/7.98 ( ( Q4
% 12.24/7.98 = ( fract @ A8 @ B5 ) )
% 12.24/7.98 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 12.24/7.98 => ( ( A8
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) )
% 12.24/7.98 => ( Q4
% 12.24/7.98 = ( zero_zero @ rat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5704_Zero__rat__def,axiom,
% 12.24/7.98 ( ( zero_zero @ rat )
% 12.24/7.98 = ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5705_rat__number__expand_I3_J,axiom,
% 12.24/7.98 ( ( numeral_numeral @ rat )
% 12.24/7.98 = ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5706_rat__number__collapse_I3_J,axiom,
% 12.24/7.98 ! [W2: num] :
% 12.24/7.98 ( ( fract @ ( numeral_numeral @ int @ W2 ) @ ( one_one @ int ) )
% 12.24/7.98 = ( numeral_numeral @ rat @ W2 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5707_quotient__of__unique,axiom,
% 12.24/7.98 ! [R: rat] :
% 12.24/7.98 ? [X3: product_prod @ int @ int] :
% 12.24/7.98 ( ( R
% 12.24/7.98 = ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
% 12.24/7.98 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
% 12.24/7.98 & ! [Y4: product_prod @ int @ int] :
% 12.24/7.98 ( ( ( R
% 12.24/7.98 = ( fract @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
% 12.24/7.98 & ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y4 ) )
% 12.24/7.98 & ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
% 12.24/7.98 => ( Y4 = X3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5708_Fract_Oabs__eq,axiom,
% 12.24/7.98 ( fract
% 12.24/7.98 = ( ^ [Xa4: int,X2: int] :
% 12.24/7.98 ( abs_Rat
% 12.24/7.98 @ ( if @ ( product_prod @ int @ int )
% 12.24/7.98 @ ( X2
% 12.24/7.98 = ( zero_zero @ int ) )
% 12.24/7.98 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.98 @ ( product_Pair @ int @ int @ Xa4 @ X2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5709_positive__rat,axiom,
% 12.24/7.98 ! [A3: int,B2: int] :
% 12.24/7.98 ( ( positive @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5710_Fract__less__zero__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
% 12.24/7.98 = ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5711_zero__less__Fract__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5712_one__less__Fract__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ord_less @ int @ B2 @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5713_Fract__less__one__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
% 12.24/7.98 = ( ord_less @ int @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5714_rat__number__collapse_I5_J,axiom,
% 12.24/7.98 ( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
% 12.24/7.98 = ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5715_Fract_Otransfer,axiom,
% 12.24/7.98 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
% 12.24/7.98 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.98 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
% 12.24/7.98 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.98 @ pcr_rat )
% 12.24/7.98 @ ^ [A5: int,B3: int] :
% 12.24/7.98 ( if @ ( product_prod @ int @ int )
% 12.24/7.98 @ ( B3
% 12.24/7.98 = ( zero_zero @ int ) )
% 12.24/7.98 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/7.98 @ ( product_Pair @ int @ int @ A5 @ B3 ) )
% 12.24/7.98 @ fract ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5716_Fract__add__one,axiom,
% 12.24/7.98 ! [N: int,M: int] :
% 12.24/7.98 ( ( N
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
% 12.24/7.98 = ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5717_Fract__le__zero__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
% 12.24/7.98 = ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5718_zero__le__Fract__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5719_one__le__Fract__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5720_Fract__le__one__iff,axiom,
% 12.24/7.98 ! [B2: int,A3: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 12.24/7.98 => ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
% 12.24/7.98 = ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5721_rat__number__expand_I5_J,axiom,
% 12.24/7.98 ! [K: num] :
% 12.24/7.98 ( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
% 12.24/7.98 = ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5722_rat__number__collapse_I4_J,axiom,
% 12.24/7.98 ! [W2: num] :
% 12.24/7.98 ( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( one_one @ int ) )
% 12.24/7.98 = ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5723_Rats__cases_H,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [X: A] :
% 12.24/7.98 ( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
% 12.24/7.98 => ~ ! [A8: int,B5: int] :
% 12.24/7.98 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
% 12.24/7.98 => ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
% 12.24/7.98 => ( X
% 12.24/7.98 != ( divide_divide @ A @ ( ring_1_of_int @ A @ A8 ) @ ( ring_1_of_int @ A @ B5 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5724_horner__sum__transfer,axiom,
% 12.24/7.98 ! [C: $tType,A: $tType,B: $tType,D: $tType] :
% 12.24/7.98 ( ( ( comm_semiring_0 @ B )
% 12.24/7.98 & ( comm_semiring_0 @ A ) )
% 12.24/7.98 => ! [A6: A > B > $o,B7: C > D > $o] :
% 12.24/7.98 ( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.98 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 12.24/7.98 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 12.24/7.98 => ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B7 @ A6 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A6 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B7 ) @ A6 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5725_of__rat__neg__numeral__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [W2: num] :
% 12.24/7.98 ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) )
% 12.24/7.98 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5726_Rats__minus__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: A] :
% 12.24/7.98 ( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( field_char_0_Rats @ A ) )
% 12.24/7.98 = ( member @ A @ A3 @ ( field_char_0_Rats @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5727_of__rat__0,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
% 12.24/7.98 = ( zero_zero @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5728_of__rat__eq__0__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat] :
% 12.24/7.98 ( ( ( field_char_0_of_rat @ A @ A3 )
% 12.24/7.98 = ( zero_zero @ A ) )
% 12.24/7.98 = ( A3
% 12.24/7.98 = ( zero_zero @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5729_zero__eq__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat] :
% 12.24/7.98 ( ( ( zero_zero @ A )
% 12.24/7.98 = ( field_char_0_of_rat @ A @ A3 ) )
% 12.24/7.98 = ( ( zero_zero @ rat )
% 12.24/7.98 = A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5730_one__eq__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat] :
% 12.24/7.98 ( ( ( one_one @ A )
% 12.24/7.98 = ( field_char_0_of_rat @ A @ A3 ) )
% 12.24/7.98 = ( ( one_one @ rat )
% 12.24/7.98 = A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5731_of__rat__eq__1__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat] :
% 12.24/7.98 ( ( ( field_char_0_of_rat @ A @ A3 )
% 12.24/7.98 = ( one_one @ A ) )
% 12.24/7.98 = ( A3
% 12.24/7.98 = ( one_one @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5732_of__rat__1,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
% 12.24/7.98 = ( one_one @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5733_of__rat__of__nat__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [N: nat] :
% 12.24/7.98 ( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
% 12.24/7.98 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5734_of__rat__less__0__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
% 12.24/7.98 = ( ord_less @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5735_zero__less__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 12.24/7.98 = ( ord_less @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5736_of__rat__le__0__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
% 12.24/7.98 = ( ord_less_eq @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5737_zero__le__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 12.24/7.98 = ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5738_of__rat__neg__one,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 12.24/7.98 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5739_of__rat__less__1__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 12.24/7.98 = ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5740_one__less__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 12.24/7.98 = ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5741_of__rat__le__1__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 12.24/7.98 = ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5742_one__le__of__rat__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat] :
% 12.24/7.98 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 12.24/7.98 = ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5743_Rats__mult,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: A,B2: A] :
% 12.24/7.98 ( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
% 12.24/7.98 => ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
% 12.24/7.98 => ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5744_Rats__0,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5745_Rats__1,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5746_Rats__of__nat,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5747_of__rat__minus,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat] :
% 12.24/7.98 ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ A3 ) )
% 12.24/7.98 = ( uminus_uminus @ A @ ( field_char_0_of_rat @ A @ A3 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5748_of__rat__less,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat,S: rat] :
% 12.24/7.98 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
% 12.24/7.98 = ( ord_less @ rat @ R @ S ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5749_of__rat__less__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linordered_field @ A )
% 12.24/7.98 => ! [R: rat,S: rat] :
% 12.24/7.98 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
% 12.24/7.98 = ( ord_less_eq @ rat @ R @ S ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5750_of__rat__mult,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [A3: rat,B2: rat] :
% 12.24/7.98 ( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
% 12.24/7.98 = ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5751_list__all2__conv__all__nth,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( list_all2 @ A @ B )
% 12.24/7.98 = ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
% 12.24/7.98 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 12.24/7.98 = ( size_size @ ( list @ B ) @ Ys3 ) )
% 12.24/7.98 & ! [I5: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.98 => ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ B @ Ys3 @ I5 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5752_list__all2__all__nthI,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
% 12.24/7.98 ( ( ( size_size @ ( list @ A ) @ A3 )
% 12.24/7.98 = ( size_size @ ( list @ B ) @ B2 ) )
% 12.24/7.98 => ( ! [N4: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ A3 ) )
% 12.24/7.98 => ( P @ ( nth @ A @ A3 @ N4 ) @ ( nth @ B @ B2 @ N4 ) ) )
% 12.24/7.98 => ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5753_list__all2__nthD2,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
% 12.24/7.98 ( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
% 12.24/7.98 => ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.98 => ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5754_list__all2__nthD,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
% 12.24/7.98 ( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
% 12.24/7.98 => ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.98 => ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5755_of__rat__rat,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( field_char_0 @ A )
% 12.24/7.98 => ! [B2: int,A3: int] :
% 12.24/7.98 ( ( B2
% 12.24/7.98 != ( zero_zero @ int ) )
% 12.24/7.98 => ( ( field_char_0_of_rat @ A @ ( fract @ A3 @ B2 ) )
% 12.24/7.98 = ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5756_sum__list__transfer,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( ( monoid_add @ B )
% 12.24/7.98 & ( monoid_add @ A ) )
% 12.24/7.98 => ! [A6: A > B > $o] :
% 12.24/7.98 ( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 12.24/7.98 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 12.24/7.98 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5757_prod__list__transfer,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( ( monoid_mult @ B )
% 12.24/7.98 & ( monoid_mult @ A ) )
% 12.24/7.98 => ! [A6: A > B > $o] :
% 12.24/7.98 ( ( A6 @ ( one_one @ A ) @ ( one_one @ B ) )
% 12.24/7.98 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 12.24/7.98 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5758_bot_Oordering__top__axioms,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( order_bot @ A )
% 12.24/7.98 => ( ordering_top @ A
% 12.24/7.98 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/7.98 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 )
% 12.24/7.98 @ ( bot_bot @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5759_dropWhile__nth,axiom,
% 12.24/7.98 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 12.24/7.98 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 12.24/7.98 => ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
% 12.24/7.98 = ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5760_prod__list_OCons,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( monoid_mult @ A )
% 12.24/7.98 => ! [X: A,Xs: list @ A] :
% 12.24/7.98 ( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs ) )
% 12.24/7.98 = ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5761_prod__list_ONil,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( monoid_mult @ A )
% 12.24/7.98 => ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
% 12.24/7.98 = ( one_one @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5762_prod__list_Oappend,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( monoid_mult @ A )
% 12.24/7.98 => ! [Xs: list @ A,Ys2: list @ A] :
% 12.24/7.98 ( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs @ Ys2 ) )
% 12.24/7.98 = ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ ( groups5270119922927024881d_list @ A @ Ys2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5763_ordering__top_Oextremum__uniqueI,axiom,
% 12.24/7.98 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 12.24/7.98 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/7.98 => ( ( Less_eq @ Top @ A3 )
% 12.24/7.98 => ( A3 = Top ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5764_ordering__top_Onot__eq__extremum,axiom,
% 12.24/7.98 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 12.24/7.98 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/7.98 => ( ( A3 != Top )
% 12.24/7.98 = ( Less @ A3 @ Top ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5765_ordering__top_Oextremum__unique,axiom,
% 12.24/7.98 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 12.24/7.98 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/7.98 => ( ( Less_eq @ Top @ A3 )
% 12.24/7.98 = ( A3 = Top ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5766_ordering__top_Oextremum__strict,axiom,
% 12.24/7.98 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 12.24/7.98 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/7.98 => ~ ( Less @ Top @ A3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5767_ordering__top_Oextremum,axiom,
% 12.24/7.98 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
% 12.24/7.98 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/7.98 => ( Less_eq @ A3 @ Top ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5768_length__dropWhile__le,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5769_sorted__dropWhile,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [Xs: list @ A,P: A > $o] :
% 12.24/7.98 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.98 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5770_prod__list__zero__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ( semiring_1 @ A )
% 12.24/7.98 & ( semiri3467727345109120633visors @ A ) )
% 12.24/7.98 => ! [Xs: list @ A] :
% 12.24/7.98 ( ( ( groups5270119922927024881d_list @ A @ Xs )
% 12.24/7.98 = ( zero_zero @ A ) )
% 12.24/7.98 = ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5771_gcd__nat_Oordering__top__axioms,axiom,
% 12.24/7.98 ( ordering_top @ nat @ ( dvd_dvd @ nat )
% 12.24/7.98 @ ^ [M2: nat,N3: nat] :
% 12.24/7.98 ( ( dvd_dvd @ nat @ M2 @ N3 )
% 12.24/7.98 & ( M2 != N3 ) )
% 12.24/7.98 @ ( zero_zero @ nat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5772_prod__list_Oeq__foldr,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( monoid_mult @ A )
% 12.24/7.98 => ( ( groups5270119922927024881d_list @ A )
% 12.24/7.98 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5773_top_Oordering__top__axioms,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( order_top @ A )
% 12.24/7.98 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5774_bot__nat__0_Oordering__top__axioms,axiom,
% 12.24/7.98 ( ordering_top @ nat
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 )
% 12.24/7.98 @ ( zero_zero @ nat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5775_mlex__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( mlex_prod @ A )
% 12.24/7.98 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 12.24/7.98 ( collect @ ( product_prod @ A @ A )
% 12.24/7.98 @ ( product_case_prod @ A @ A @ $o
% 12.24/7.98 @ ^ [X2: A,Y6: A] :
% 12.24/7.98 ( ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
% 12.24/7.98 | ( ( ord_less_eq @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
% 12.24/7.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y6 ) @ R6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5776_arg__min__SOME__Min,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( linorder @ B )
% 12.24/7.98 => ! [S3: set @ A,F: A > B] :
% 12.24/7.98 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.98 => ( ( lattic7623131987881927897min_on @ A @ B @ F @ S3 )
% 12.24/7.98 = ( fChoice @ A
% 12.24/7.98 @ ^ [Y6: A] :
% 12.24/7.98 ( ( member @ A @ Y6 @ S3 )
% 12.24/7.98 & ( ( F @ Y6 )
% 12.24/7.98 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ S3 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5777_prod__decode__triangle__add,axiom,
% 12.24/7.98 ! [K: nat,M: nat] :
% 12.24/7.98 ( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) )
% 12.24/7.98 = ( nat_prod_decode_aux @ K @ M ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5778_prod__decode__eq,axiom,
% 12.24/7.98 ! [X: nat,Y3: nat] :
% 12.24/7.98 ( ( ( nat_prod_decode @ X )
% 12.24/7.98 = ( nat_prod_decode @ Y3 ) )
% 12.24/7.98 = ( X = Y3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5779_some__equality,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A3: A] :
% 12.24/7.98 ( ( P @ A3 )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 => ( X3 = A3 ) )
% 12.24/7.98 => ( ( fChoice @ A @ P )
% 12.24/7.98 = A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5780_some__eq__trivial,axiom,
% 12.24/7.98 ! [A: $tType,X: A] :
% 12.24/7.98 ( ( fChoice @ A
% 12.24/7.98 @ ^ [Y6: A] : ( Y6 = X ) )
% 12.24/7.98 = X ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5781_some__sym__eq__trivial,axiom,
% 12.24/7.98 ! [A: $tType,X: A] :
% 12.24/7.98 ( ( fChoice @ A
% 12.24/7.98 @ ( ^ [Y: A,Z2: A] : ( Y = Z2 )
% 12.24/7.98 @ X ) )
% 12.24/7.98 = X ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5782_Eps__case__prod__eq,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType,X: A,Y3: B] :
% 12.24/7.98 ( ( fChoice @ ( product_prod @ A @ B )
% 12.24/7.98 @ ( product_case_prod @ A @ B @ $o
% 12.24/7.98 @ ^ [X9: A,Y8: B] :
% 12.24/7.98 ( ( X = X9 )
% 12.24/7.98 & ( Y3 = Y8 ) ) ) )
% 12.24/7.98 = ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5783_prod__decode__inverse,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
% 12.24/7.98 = N ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5784_prod__encode__inverse,axiom,
% 12.24/7.98 ! [X: product_prod @ nat @ nat] :
% 12.24/7.98 ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
% 12.24/7.98 = X ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5785_inj__prod__decode,axiom,
% 12.24/7.98 ! [A6: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5786_someI2,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
% 12.24/7.98 ( ( P @ A3 )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 => ( Q @ X3 ) )
% 12.24/7.98 => ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5787_someI__ex,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o] :
% 12.24/7.98 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.98 => ( P @ ( fChoice @ A @ P ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5788_someI2__ex,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,Q: A > $o] :
% 12.24/7.98 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 => ( Q @ X3 ) )
% 12.24/7.98 => ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5789_someI2__bex,axiom,
% 12.24/7.98 ! [A: $tType,A6: set @ A,P: A > $o,Q: A > $o] :
% 12.24/7.98 ( ? [X4: A] :
% 12.24/7.98 ( ( member @ A @ X4 @ A6 )
% 12.24/7.98 & ( P @ X4 ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 & ( P @ X3 ) )
% 12.24/7.98 => ( Q @ X3 ) )
% 12.24/7.98 => ( Q
% 12.24/7.98 @ ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A6 )
% 12.24/7.98 & ( P @ X2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5790_some__eq__ex,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o] :
% 12.24/7.98 ( ( P @ ( fChoice @ A @ P ) )
% 12.24/7.98 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5791_some1__equality,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A3: A] :
% 12.24/7.98 ( ? [X4: A] :
% 12.24/7.98 ( ( P @ X4 )
% 12.24/7.98 & ! [Y5: A] :
% 12.24/7.98 ( ( P @ Y5 )
% 12.24/7.98 => ( Y5 = X4 ) ) )
% 12.24/7.98 => ( ( P @ A3 )
% 12.24/7.98 => ( ( fChoice @ A @ P )
% 12.24/7.98 = A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5792_verit__sko__ex_H,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A6: $o] :
% 12.24/7.98 ( ( ( P @ ( fChoice @ A @ P ) )
% 12.24/7.98 = A6 )
% 12.24/7.98 => ( ( ? [X7: A] : ( P @ X7 ) )
% 12.24/7.98 = A6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5793_verit__sko__forall,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ^ [P3: A > $o] :
% 12.24/7.98 ! [X6: A] : ( P3 @ X6 ) )
% 12.24/7.98 = ( ^ [P4: A > $o] :
% 12.24/7.98 ( P4
% 12.24/7.98 @ ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ~ ( P4 @ X2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5794_verit__sko__forall_H,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A6: $o] :
% 12.24/7.98 ( ( ( P
% 12.24/7.98 @ ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ~ ( P @ X2 ) ) )
% 12.24/7.98 = A6 )
% 12.24/7.98 => ( ( ! [X7: A] : ( P @ X7 ) )
% 12.24/7.98 = A6 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5795_verit__sko__forall_H_H,axiom,
% 12.24/7.98 ! [A: $tType,B7: A,A6: A,P: A > $o] :
% 12.24/7.98 ( ( B7 = A6 )
% 12.24/7.98 => ( ( ( fChoice @ A @ P )
% 12.24/7.98 = A6 )
% 12.24/7.98 = ( ( fChoice @ A @ P )
% 12.24/7.98 = B7 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5796_verit__sko__ex__indirect,axiom,
% 12.24/7.98 ! [A: $tType,X: A,P: A > $o] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( fChoice @ A @ P ) )
% 12.24/7.98 => ( ( ? [X7: A] : ( P @ X7 ) )
% 12.24/7.98 = ( P @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5797_verit__sko__ex__indirect2,axiom,
% 12.24/7.98 ! [A: $tType,X: A,P: A > $o,P2: A > $o] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( fChoice @ A @ P ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 = ( P2 @ X3 ) )
% 12.24/7.98 => ( ( ? [X7: A] : ( P2 @ X7 ) )
% 12.24/7.98 = ( P @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5798_verit__sko__forall__indirect,axiom,
% 12.24/7.98 ! [A: $tType,X: A,P: A > $o] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ~ ( P @ X2 ) ) )
% 12.24/7.98 => ( ( ! [X7: A] : ( P @ X7 ) )
% 12.24/7.98 = ( P @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5799_verit__sko__forall__indirect2,axiom,
% 12.24/7.98 ! [A: $tType,X: A,P: A > $o,P2: A > $o] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ~ ( P @ X2 ) ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 = ( P2 @ X3 ) )
% 12.24/7.98 => ( ( ! [X7: A] : ( P2 @ X7 ) )
% 12.24/7.98 = ( P @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5800_Nitpick_OEps__psimp,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,X: A,Y3: A] :
% 12.24/7.98 ( ( P @ X )
% 12.24/7.98 => ( ~ ( P @ Y3 )
% 12.24/7.98 => ( ( ( fChoice @ A @ P )
% 12.24/7.98 = Y3 )
% 12.24/7.98 => ( ( fChoice @ A @ P )
% 12.24/7.98 = X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5801_someI,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,X: A] :
% 12.24/7.98 ( ( P @ X )
% 12.24/7.98 => ( P @ ( fChoice @ A @ P ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5802_Eps__cong,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,Q: A > $o] :
% 12.24/7.98 ( ! [X3: A] :
% 12.24/7.98 ( ( P @ X3 )
% 12.24/7.98 = ( Q @ X3 ) )
% 12.24/7.98 => ( ( fChoice @ A @ P )
% 12.24/7.98 = ( fChoice @ A @ Q ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5803_tfl__some,axiom,
% 12.24/7.98 ! [A: $tType,P7: A > $o,X4: A] :
% 12.24/7.98 ( ( P7 @ X4 )
% 12.24/7.98 => ( P7 @ ( fChoice @ A @ P7 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5804_some__eq__imp,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,A3: A,B2: A] :
% 12.24/7.98 ( ( ( fChoice @ A @ P )
% 12.24/7.98 = A3 )
% 12.24/7.98 => ( ( P @ B2 )
% 12.24/7.98 => ( P @ A3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5805_some__in__eq,axiom,
% 12.24/7.98 ! [A: $tType,A6: set @ A] :
% 12.24/7.98 ( ( member @ A
% 12.24/7.98 @ ( fChoice @ A
% 12.24/7.98 @ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
% 12.24/7.98 @ A6 )
% 12.24/7.98 = ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5806_exE__some,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,C2: A] :
% 12.24/7.98 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.98 => ( ( C2
% 12.24/7.98 = ( fChoice @ A @ P ) )
% 12.24/7.98 => ( P @ C2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5807_split__paired__Eps,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( fChoice @ ( product_prod @ A @ B ) )
% 12.24/7.98 = ( ^ [P4: ( product_prod @ A @ B ) > $o] :
% 12.24/7.98 ( fChoice @ ( product_prod @ A @ B )
% 12.24/7.98 @ ( product_case_prod @ A @ B @ $o
% 12.24/7.98 @ ^ [A5: A,B3: B] : ( P4 @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5808_prod__decode__def,axiom,
% 12.24/7.98 ( nat_prod_decode
% 12.24/7.98 = ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5809_Eps__case__prod,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,P: A > B > $o] :
% 12.24/7.98 ( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
% 12.24/7.98 = ( fChoice @ ( product_prod @ A @ B )
% 12.24/7.98 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5810_surj__prod__decode,axiom,
% 12.24/7.98 ( ( image @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.98 = ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5811_mlex__less,axiom,
% 12.24/7.98 ! [A: $tType,F: A > nat,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.98 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5812_mlex__iff,axiom,
% 12.24/7.98 ! [A: $tType,X: A,Y3: A,F: A > nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) )
% 12.24/7.98 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.98 | ( ( ( F @ X )
% 12.24/7.98 = ( F @ Y3 ) )
% 12.24/7.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5813_mlex__leq,axiom,
% 12.24/7.98 ! [A: $tType,F: A > nat,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.98 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
% 12.24/7.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
% 12.24/7.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5814_list__decode_Oelims,axiom,
% 12.24/7.98 ! [X: nat,Y3: list @ nat] :
% 12.24/7.98 ( ( ( nat_list_decode @ X )
% 12.24/7.98 = Y3 )
% 12.24/7.98 => ( ( ( X
% 12.24/7.98 = ( zero_zero @ nat ) )
% 12.24/7.98 => ( Y3
% 12.24/7.98 != ( nil @ nat ) ) )
% 12.24/7.98 => ~ ! [N4: nat] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( suc @ N4 ) )
% 12.24/7.98 => ( Y3
% 12.24/7.98 != ( product_case_prod @ nat @ nat @ ( list @ nat )
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 12.24/7.98 @ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5815_arg__min__def,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( lattices_ord_arg_min @ B @ A )
% 12.24/7.98 = ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5816_list__decode_Osimps_I2_J,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( nat_list_decode @ ( suc @ N ) )
% 12.24/7.98 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 12.24/7.98 @ ( nat_prod_decode @ N ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5817_list__encode__inverse,axiom,
% 12.24/7.98 ! [X: list @ nat] :
% 12.24/7.98 ( ( nat_list_decode @ ( nat_list_encode @ X ) )
% 12.24/7.98 = X ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5818_list__decode__inverse,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( nat_list_encode @ ( nat_list_decode @ N ) )
% 12.24/7.98 = N ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5819_list__decode__eq,axiom,
% 12.24/7.98 ! [X: nat,Y3: nat] :
% 12.24/7.98 ( ( ( nat_list_decode @ X )
% 12.24/7.98 = ( nat_list_decode @ Y3 ) )
% 12.24/7.98 = ( X = Y3 ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5820_inj__list__decode,axiom,
% 12.24/7.98 ! [A6: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A6 ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5821_is__arg__min__antimono,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( order @ B )
% 12.24/7.98 => ! [F: A > B,P: A > $o,X: A,Y3: A] :
% 12.24/7.98 ( ( lattic501386751177426532rg_min @ A @ B @ F @ P @ X )
% 12.24/7.98 => ( ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) )
% 12.24/7.98 => ( ( P @ Y3 )
% 12.24/7.98 => ( lattic501386751177426532rg_min @ A @ B @ F @ P @ Y3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5822_is__arg__min__linorder,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( linorder @ B )
% 12.24/7.98 => ( ( lattic501386751177426532rg_min @ A @ B )
% 12.24/7.98 = ( ^ [F2: A > B,P4: A > $o,X2: A] :
% 12.24/7.98 ( ( P4 @ X2 )
% 12.24/7.98 & ! [Y6: A] :
% 12.24/7.98 ( ( P4 @ Y6 )
% 12.24/7.98 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5823_is__arg__min__def,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( lattic501386751177426532rg_min @ B @ A )
% 12.24/7.98 = ( ^ [F2: B > A,P4: B > $o,X2: B] :
% 12.24/7.98 ( ( P4 @ X2 )
% 12.24/7.98 & ~ ? [Y6: B] :
% 12.24/7.98 ( ( P4 @ Y6 )
% 12.24/7.98 & ( ord_less @ A @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5824_list__decode_Osimps_I1_J,axiom,
% 12.24/7.98 ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 12.24/7.98 = ( nil @ nat ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5825_is__arg__min__arg__min__nat,axiom,
% 12.24/7.98 ! [A: $tType,P: A > $o,X: A,M: A > nat] :
% 12.24/7.98 ( ( P @ X )
% 12.24/7.98 => ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5826_surj__list__decode,axiom,
% 12.24/7.98 ( ( image @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
% 12.24/7.98 = ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5827_ex__is__arg__min__if__finite,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( order @ B )
% 12.24/7.98 => ! [S3: set @ A,F: A > B] :
% 12.24/7.98 ( ( finite_finite2 @ A @ S3 )
% 12.24/7.98 => ( ( S3
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ? [X_12: A] :
% 12.24/7.98 ( lattic501386751177426532rg_min @ A @ B @ F
% 12.24/7.98 @ ^ [X2: A] : ( member @ A @ X2 @ S3 )
% 12.24/7.98 @ X_12 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5828_list__decode_Opelims,axiom,
% 12.24/7.98 ! [X: nat,Y3: list @ nat] :
% 12.24/7.98 ( ( ( nat_list_decode @ X )
% 12.24/7.98 = Y3 )
% 12.24/7.98 => ( ( accp @ nat @ nat_list_decode_rel @ X )
% 12.24/7.98 => ( ( ( X
% 12.24/7.98 = ( zero_zero @ nat ) )
% 12.24/7.98 => ( ( Y3
% 12.24/7.98 = ( nil @ nat ) )
% 12.24/7.98 => ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
% 12.24/7.98 => ~ ! [N4: nat] :
% 12.24/7.98 ( ( X
% 12.24/7.98 = ( suc @ N4 ) )
% 12.24/7.98 => ( ( Y3
% 12.24/7.98 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 12.24/7.98 @ ( nat_prod_decode @ N4 ) ) )
% 12.24/7.98 => ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5829_mono__cSUP,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.98 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 & ( condit1219197933456340205attice @ B ) )
% 12.24/7.98 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 12.24/7.98 ( ( order_mono @ A @ B @ F )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A6 @ I6 ) )
% 12.24/7.98 => ( ( I6
% 12.24/7.98 != ( bot_bot @ ( set @ C ) ) )
% 12.24/7.98 => ( ord_less_eq @ B
% 12.24/7.98 @ ( complete_Sup_Sup @ B
% 12.24/7.98 @ ( image @ C @ B
% 12.24/7.98 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 12.24/7.98 @ I6 ) )
% 12.24/7.98 @ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5830_mono__cSup,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 & ( condit1219197933456340205attice @ B ) )
% 12.24/7.98 => ! [F: A > B,A6: set @ A] :
% 12.24/7.98 ( ( order_mono @ A @ B @ F )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5831_bdd__above_OI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ A,M7: A] :
% 12.24/7.98 ( ! [X3: A] :
% 12.24/7.98 ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X3 @ M7 ) )
% 12.24/7.98 => ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5832_bdd__above_OI2,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ B,F: B > A,M7: A] :
% 12.24/7.98 ( ! [X3: B] :
% 12.24/7.98 ( ( member @ B @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
% 12.24/7.98 => ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5833_bdd__above_OE,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( condit941137186595557371_above @ A @ A6 )
% 12.24/7.98 => ~ ! [M8: A] :
% 12.24/7.98 ~ ! [X4: A] :
% 12.24/7.98 ( ( member @ A @ X4 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ X4 @ M8 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5834_bdd__above_Ounfold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ? [M9: A] :
% 12.24/7.98 ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A7 )
% 12.24/7.98 => ( ord_less_eq @ A @ X2 @ M9 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5835_cSup__upper2,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [X: A,X8: set @ A,Y3: A] :
% 12.24/7.98 ( ( member @ A @ X @ X8 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ Y3 @ X )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ X8 )
% 12.24/7.98 => ( ord_less_eq @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5836_cSup__upper,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [X: A,X8: set @ A] :
% 12.24/7.98 ( ( member @ A @ X @ X8 )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ X8 )
% 12.24/7.98 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5837_list__decode_Opsimps_I1_J,axiom,
% 12.24/7.98 ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 12.24/7.98 => ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 12.24/7.98 = ( nil @ nat ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5838_list__decode_Opinduct,axiom,
% 12.24/7.98 ! [A0: nat,P: nat > $o] :
% 12.24/7.98 ( ( accp @ nat @ nat_list_decode_rel @ A0 )
% 12.24/7.98 => ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 12.24/7.98 => ( P @ ( zero_zero @ nat ) ) )
% 12.24/7.98 => ( ! [N4: nat] :
% 12.24/7.98 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) )
% 12.24/7.98 => ( ! [X4: nat,Y4: nat] :
% 12.24/7.98 ( ( ( product_Pair @ nat @ nat @ X4 @ Y4 )
% 12.24/7.98 = ( nat_prod_decode @ N4 ) )
% 12.24/7.98 => ( P @ Y4 ) )
% 12.24/7.98 => ( P @ ( suc @ N4 ) ) ) )
% 12.24/7.98 => ( P @ A0 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5839_cSUP__upper,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [X: B,A6: set @ B,F: B > A] :
% 12.24/7.98 ( ( member @ B @ X @ A6 )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( F @ X ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5840_cSUP__upper2,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [F: B > A,A6: set @ B,X: B,U: A] :
% 12.24/7.98 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( member @ B @ X @ A6 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ U @ ( F @ X ) )
% 12.24/7.98 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5841_cSup__le__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [S3: set @ A,A3: A] :
% 12.24/7.98 ( ( S3
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ S3 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A3 )
% 12.24/7.98 = ( ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ S3 )
% 12.24/7.98 => ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5842_cSup__mono,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [B7: set @ A,A6: set @ A] :
% 12.24/7.98 ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ A6 )
% 12.24/7.98 => ( ! [B5: A] :
% 12.24/7.98 ( ( member @ A @ B5 @ B7 )
% 12.24/7.98 => ? [X4: A] :
% 12.24/7.98 ( ( member @ A @ X4 @ A6 )
% 12.24/7.98 & ( ord_less_eq @ A @ B5 @ X4 ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B7 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5843_less__cSup__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit6923001295902523014norder @ A )
% 12.24/7.98 => ! [X8: set @ A,Y3: A] :
% 12.24/7.98 ( ( X8
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ X8 )
% 12.24/7.98 => ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
% 12.24/7.98 = ( ? [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ X8 )
% 12.24/7.98 & ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5844_cSUP__lessD,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 12.24/7.98 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
% 12.24/7.98 => ( ( member @ B @ I @ A6 )
% 12.24/7.98 => ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5845_cSUP__le__iff,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ B,F: B > A,U: A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
% 12.24/7.98 = ( ! [X2: B] :
% 12.24/7.98 ( ( member @ B @ X2 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5846_cSUP__mono,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ B,G: C > A,B7: set @ C,F: B > A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B7 ) )
% 12.24/7.98 => ( ! [N4: B] :
% 12.24/7.98 ( ( member @ B @ N4 @ A6 )
% 12.24/7.98 => ? [X4: C] :
% 12.24/7.98 ( ( member @ C @ X4 @ B7 )
% 12.24/7.98 & ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5847_cSup__subset__mono,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ B7 )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5848_less__cSUP__iff,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit6923001295902523014norder @ A )
% 12.24/7.98 => ! [A6: set @ B,F: B > A,A3: A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.98 = ( ? [X2: B] :
% 12.24/7.98 ( ( member @ B @ X2 @ A6 )
% 12.24/7.98 & ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5849_list__decode_Opsimps_I2_J,axiom,
% 12.24/7.98 ! [N: nat] :
% 12.24/7.98 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
% 12.24/7.98 => ( ( nat_list_decode @ ( suc @ N ) )
% 12.24/7.98 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 12.24/7.98 @ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
% 12.24/7.98 @ ( nat_prod_decode @ N ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5850_cSUP__subset__mono,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B7 ) )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 12.24/7.98 => ( ! [X3: B] :
% 12.24/7.98 ( ( member @ B @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5851_cSup__inter__less__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( condit941137186595557371_above @ A @ A6 )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ B7 )
% 12.24/7.98 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5852_division__segment__integer__def,axiom,
% 12.24/7.98 ( ( euclid7384307370059645450egment @ code_integer )
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5853_mono__cInf,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 & ( condit1219197933456340205attice @ B ) )
% 12.24/7.98 => ! [F: A > B,A6: set @ A] :
% 12.24/7.98 ( ( order_mono @ A @ B @ F )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ A6 )
% 12.24/7.98 => ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5854_mono__cINF,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/7.98 ( ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 & ( condit1219197933456340205attice @ B ) )
% 12.24/7.98 => ! [F: A > B,A6: C > A,I6: set @ C] :
% 12.24/7.98 ( ( order_mono @ A @ B @ F )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A6 @ I6 ) )
% 12.24/7.98 => ( ( I6
% 12.24/7.98 != ( bot_bot @ ( set @ C ) ) )
% 12.24/7.98 => ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
% 12.24/7.98 @ ( complete_Inf_Inf @ B
% 12.24/7.98 @ ( image @ C @ B
% 12.24/7.98 @ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
% 12.24/7.98 @ I6 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5855_bdd__belowI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ A,M: A] :
% 12.24/7.98 ( ! [X3: A] :
% 12.24/7.98 ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ M @ X3 ) )
% 12.24/7.98 => ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5856_bdd__below_OI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ A,M7: A] :
% 12.24/7.98 ( ! [X3: A] :
% 12.24/7.98 ( ( member @ A @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ M7 @ X3 ) )
% 12.24/7.98 => ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5857_bdd__below__uminus,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X8: set @ A] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
% 12.24/7.98 = ( condit941137186595557371_above @ A @ X8 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5858_bdd__above__uminus,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ordered_ab_group_add @ A )
% 12.24/7.98 => ! [X8: set @ A] :
% 12.24/7.98 ( ( condit941137186595557371_above @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
% 12.24/7.98 = ( condit1013018076250108175_below @ A @ X8 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5859_bdd__below_OI2,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ B,M7: A,F: B > A] :
% 12.24/7.98 ( ! [X3: B] :
% 12.24/7.98 ( ( member @ B @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ M7 @ ( F @ X3 ) ) )
% 12.24/7.98 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5860_bdd__belowI2,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ B,M: A,F: B > A] :
% 12.24/7.98 ( ! [X3: B] :
% 12.24/7.98 ( ( member @ B @ X3 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
% 12.24/7.98 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5861_cInf__lower2,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [X: A,X8: set @ A,Y3: A] :
% 12.24/7.98 ( ( member @ A @ X @ X8 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ X @ Y3 )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ X8 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5862_cInf__lower,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [X: A,X8: set @ A] :
% 12.24/7.98 ( ( member @ A @ X @ X8 )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ X8 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5863_bdd__below_Ounfold,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ? [M9: A] :
% 12.24/7.98 ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A7 )
% 12.24/7.98 => ( ord_less_eq @ A @ M9 @ X2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5864_bdd__below_OE,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ A6 )
% 12.24/7.98 => ~ ! [M8: A] :
% 12.24/7.98 ~ ! [X4: A] :
% 12.24/7.98 ( ( member @ A @ X4 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ M8 @ X4 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5865_cINF__lower2,axiom,
% 12.24/7.98 ! [B: $tType,A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [F: B > A,A6: set @ B,X: B,U: A] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( member @ B @ X @ A6 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ ( F @ X ) @ U )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5866_cINF__lower,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [F: B > A,A6: set @ B,X: B] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( member @ B @ X @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ X ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5867_le__cInf__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [S3: set @ A,A3: A] :
% 12.24/7.98 ( ( S3
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ S3 )
% 12.24/7.98 => ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S3 ) )
% 12.24/7.98 = ( ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ S3 )
% 12.24/7.98 => ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5868_cInf__mono,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [B7: set @ A,A6: set @ A] :
% 12.24/7.98 ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ A6 )
% 12.24/7.98 => ( ! [B5: A] :
% 12.24/7.98 ( ( member @ A @ B5 @ B7 )
% 12.24/7.98 => ? [X4: A] :
% 12.24/7.98 ( ( member @ A @ X4 @ A6 )
% 12.24/7.98 & ( ord_less_eq @ A @ X4 @ B5 ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5869_cInf__less__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit6923001295902523014norder @ A )
% 12.24/7.98 => ! [X8: set @ A,Y3: A] :
% 12.24/7.98 ( ( X8
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ X8 )
% 12.24/7.98 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 )
% 12.24/7.98 = ( ? [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ X8 )
% 12.24/7.98 & ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5870_less__cINF__D,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [F: B > A,A6: set @ B,Y3: A,I: B] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.98 => ( ( member @ B @ I @ A6 )
% 12.24/7.98 => ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5871_cINF__mono,axiom,
% 12.24/7.98 ! [C: $tType,A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [B7: set @ B,F: C > A,A6: set @ C,G: B > A] :
% 12.24/7.98 ( ( B7
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F @ A6 ) )
% 12.24/7.98 => ( ! [M6: B] :
% 12.24/7.98 ( ( member @ B @ M6 @ B7 )
% 12.24/7.98 => ? [X4: C] :
% 12.24/7.98 ( ( member @ C @ X4 @ A6 )
% 12.24/7.98 & ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5872_le__cINF__iff,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ B,F: B > A,U: A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
% 12.24/7.98 = ( ! [X2: B] :
% 12.24/7.98 ( ( member @ B @ X2 @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5873_cInf__superset__mono,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ B7 )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B7 ) @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5874_cINF__less__iff,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit6923001295902523014norder @ A )
% 12.24/7.98 => ! [A6: set @ B,F: B > A,A3: A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.98 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
% 12.24/7.98 = ( ? [X2: B] :
% 12.24/7.98 ( ( member @ B @ X2 @ A6 )
% 12.24/7.98 & ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5875_uminus__integer__def,axiom,
% 12.24/7.98 ( ( uminus_uminus @ code_integer )
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5876_abs__integer__def,axiom,
% 12.24/7.98 ( ( abs_abs @ code_integer )
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5877_sgn__integer__def,axiom,
% 12.24/7.98 ( ( sgn_sgn @ code_integer )
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5878_cINF__superset__mono,axiom,
% 12.24/7.98 ! [A: $tType,B: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ B ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B7 ) )
% 12.24/7.98 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
% 12.24/7.98 => ( ! [X3: B] :
% 12.24/7.98 ( ( member @ B @ X3 @ B7 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5879_less__eq__cInf__inter,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ A,B7: set @ A] :
% 12.24/7.98 ( ( condit1013018076250108175_below @ A @ A6 )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ B7 )
% 12.24/7.98 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5880_cInf__le__cSup,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [A6: set @ A] :
% 12.24/7.98 ( ( A6
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ A6 )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ A6 )
% 12.24/7.98 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5881_not__integer__def,axiom,
% 12.24/7.98 ( ( bit_ri4277139882892585799ns_not @ code_integer )
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5882_Code__Numeral_Odup__def,axiom,
% 12.24/7.98 ( code_dup
% 12.24/7.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
% 12.24/7.98 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5883_flip__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se8732182000553998342ip_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5884_set__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se5668285175392031749et_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5885_Code__Numeral_Odup__code_I1_J,axiom,
% 12.24/7.98 ( ( code_dup @ ( zero_zero @ code_integer ) )
% 12.24/7.98 = ( zero_zero @ code_integer ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5886_dup_Orep__eq,axiom,
% 12.24/7.98 ! [X: code_integer] :
% 12.24/7.98 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 12.24/7.98 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5887_dup_Oabs__eq,axiom,
% 12.24/7.98 ! [X: int] :
% 12.24/7.98 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 12.24/7.98 = ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5888_take__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se2584673776208193580ke_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5889_drop__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se4197421643247451524op_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5890_push__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se4730199178511100633sh_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5891_unset__bit__integer__def,axiom,
% 12.24/7.98 ( ( bit_se2638667681897837118et_bit @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5892_Code__Numeral_Osub__code_I9_J,axiom,
% 12.24/7.98 ! [M: num,N: num] :
% 12.24/7.98 ( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/7.98 = ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5893_Code__Numeral_Osub__code_I8_J,axiom,
% 12.24/7.98 ! [M: num,N: num] :
% 12.24/7.98 ( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/7.98 = ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5894_xor__integer__def,axiom,
% 12.24/7.98 ( ( bit_se5824344971392196577ns_xor @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5895_Code__Numeral_Osub__code_I1_J,axiom,
% 12.24/7.98 ( ( code_sub @ one2 @ one2 )
% 12.24/7.98 = ( zero_zero @ code_integer ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5896_Code__Numeral_Osub__code_I6_J,axiom,
% 12.24/7.98 ! [M: num,N: num] :
% 12.24/7.98 ( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/7.98 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5897_Code__Numeral_Osub__code_I7_J,axiom,
% 12.24/7.98 ! [M: num,N: num] :
% 12.24/7.98 ( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/7.98 = ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5898_sub_Orep__eq,axiom,
% 12.24/7.98 ! [X: num,Xa3: num] :
% 12.24/7.98 ( ( code_int_of_integer @ ( code_sub @ X @ Xa3 ) )
% 12.24/7.98 = ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa3 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5899_sub_Oabs__eq,axiom,
% 12.24/7.98 ( code_sub
% 12.24/7.98 = ( ^ [Xa4: num,X2: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5900_plus__integer__def,axiom,
% 12.24/7.98 ( ( plus_plus @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5901_times__integer__def,axiom,
% 12.24/7.98 ( ( times_times @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5902_minus__integer__def,axiom,
% 12.24/7.98 ( ( minus_minus @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5903_divide__integer__def,axiom,
% 12.24/7.98 ( ( divide_divide @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5904_modulo__integer__def,axiom,
% 12.24/7.98 ( ( modulo_modulo @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5905_and__integer__def,axiom,
% 12.24/7.98 ( ( bit_se5824344872417868541ns_and @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5906_or__integer__def,axiom,
% 12.24/7.98 ( ( bit_se1065995026697491101ons_or @ code_integer )
% 12.24/7.98 = ( 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 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5907_Sup__int__def,axiom,
% 12.24/7.98 ( ( complete_Sup_Sup @ int )
% 12.24/7.98 = ( ^ [X7: set @ int] :
% 12.24/7.98 ( the @ int
% 12.24/7.98 @ ^ [X2: int] :
% 12.24/7.98 ( ( member @ int @ X2 @ X7 )
% 12.24/7.98 & ! [Y6: int] :
% 12.24/7.98 ( ( member @ int @ Y6 @ X7 )
% 12.24/7.98 => ( ord_less_eq @ int @ Y6 @ X2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5908_lexordp__conv__lexord,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ( ( ord_lexordp @ A )
% 12.24/7.98 = ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5909_sorted__wrt__iff__nth__Suc__transp,axiom,
% 12.24/7.98 ! [A: $tType,P: A > A > $o,Xs: list @ A] :
% 12.24/7.98 ( ( transp @ A @ P )
% 12.24/7.98 => ( ( sorted_wrt @ A @ P @ Xs )
% 12.24/7.98 = ( ! [I5: nat] :
% 12.24/7.98 ( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/7.98 => ( P @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5910_lexordp__simps_I3_J,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
% 12.24/7.98 ( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
% 12.24/7.98 = ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.98 | ( ~ ( ord_less @ A @ Y3 @ X )
% 12.24/7.98 & ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5911_lexordp__irreflexive,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [Xs: list @ A] :
% 12.24/7.98 ( ! [X3: A] :
% 12.24/7.98 ~ ( ord_less @ A @ X3 @ X3 )
% 12.24/7.98 => ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5912_lexordp_OCons,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.98 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.98 => ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5913_lexordp_OCons__eq,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.98 ( ~ ( ord_less @ A @ X @ Y3 )
% 12.24/7.98 => ( ~ ( ord_less @ A @ Y3 @ X )
% 12.24/7.98 => ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 12.24/7.98 => ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5914_lexordp__append__leftD,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
% 12.24/7.98 ( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
% 12.24/7.98 => ( ! [A8: A] :
% 12.24/7.98 ~ ( ord_less @ A @ A8 @ A8 )
% 12.24/7.98 => ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5915_Sup__eq__Inf,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ( ( complete_Sup_Sup @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ( complete_Inf_Inf @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [B3: A] :
% 12.24/7.98 ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A7 )
% 12.24/7.98 => ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5916_Inf__eq__Sup,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ( ( complete_Inf_Inf @ A )
% 12.24/7.98 = ( ^ [A7: set @ A] :
% 12.24/7.98 ( complete_Sup_Sup @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [B3: A] :
% 12.24/7.98 ! [X2: A] :
% 12.24/7.98 ( ( member @ A @ X2 @ A7 )
% 12.24/7.98 => ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5917_lexordp__induct,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 12.24/7.98 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 12.24/7.98 => ( ! [Y5: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y5 @ Ys4 ) )
% 12.24/7.98 => ( ! [X3: A,Xs2: list @ A,Y5: A,Ys4: list @ A] :
% 12.24/7.98 ( ( ord_less @ A @ X3 @ Y5 )
% 12.24/7.98 => ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y5 @ Ys4 ) ) )
% 12.24/7.98 => ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
% 12.24/7.98 ( ( ord_lexordp @ A @ Xs2 @ Ys4 )
% 12.24/7.98 => ( ( P @ Xs2 @ Ys4 )
% 12.24/7.98 => ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys4 ) ) ) )
% 12.24/7.98 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5918_lexordp__cases,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ! [Xs: list @ A,Ys2: list @ A] :
% 12.24/7.98 ( ( ord_lexordp @ A @ Xs @ Ys2 )
% 12.24/7.98 => ( ( ( Xs
% 12.24/7.98 = ( nil @ A ) )
% 12.24/7.98 => ! [Y5: A,Ys5: list @ A] :
% 12.24/7.98 ( Ys2
% 12.24/7.98 != ( cons @ A @ Y5 @ Ys5 ) ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ? [Xs4: list @ A] :
% 12.24/7.98 ( Xs
% 12.24/7.98 = ( cons @ A @ X3 @ Xs4 ) )
% 12.24/7.98 => ! [Y5: A] :
% 12.24/7.98 ( ? [Ys5: list @ A] :
% 12.24/7.98 ( Ys2
% 12.24/7.98 = ( cons @ A @ Y5 @ Ys5 ) )
% 12.24/7.98 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 12.24/7.98 => ~ ! [X3: A,Xs4: list @ A] :
% 12.24/7.98 ( ( Xs
% 12.24/7.98 = ( cons @ A @ X3 @ Xs4 ) )
% 12.24/7.98 => ! [Ys5: list @ A] :
% 12.24/7.98 ( ( Ys2
% 12.24/7.98 = ( cons @ A @ X3 @ Ys5 ) )
% 12.24/7.98 => ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5919_lexordp_Osimps,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( ord_lexordp @ A )
% 12.24/7.98 = ( ^ [A12: list @ A,A23: list @ A] :
% 12.24/7.98 ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.98 ( ( A12
% 12.24/7.98 = ( nil @ A ) )
% 12.24/7.98 & ( A23
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 12.24/7.98 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.98 ( ( A12
% 12.24/7.98 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.98 & ( A23
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.98 & ( ord_less @ A @ X2 @ Y6 ) )
% 12.24/7.98 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.98 ( ( A12
% 12.24/7.98 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.98 & ( A23
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.98 & ~ ( ord_less @ A @ X2 @ Y6 )
% 12.24/7.98 & ~ ( ord_less @ A @ Y6 @ X2 )
% 12.24/7.98 & ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5920_lexordp_Ocases,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [A1: list @ A,A22: list @ A] :
% 12.24/7.98 ( ( ord_lexordp @ A @ A1 @ A22 )
% 12.24/7.98 => ( ( ( A1
% 12.24/7.98 = ( nil @ A ) )
% 12.24/7.98 => ! [Y5: A,Ys4: list @ A] :
% 12.24/7.98 ( A22
% 12.24/7.98 != ( cons @ A @ Y5 @ Ys4 ) ) )
% 12.24/7.98 => ( ! [X3: A] :
% 12.24/7.98 ( ? [Xs2: list @ A] :
% 12.24/7.98 ( A1
% 12.24/7.98 = ( cons @ A @ X3 @ Xs2 ) )
% 12.24/7.98 => ! [Y5: A] :
% 12.24/7.98 ( ? [Ys4: list @ A] :
% 12.24/7.98 ( A22
% 12.24/7.98 = ( cons @ A @ Y5 @ Ys4 ) )
% 12.24/7.98 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 12.24/7.98 => ~ ! [X3: A,Y5: A,Xs2: list @ A] :
% 12.24/7.98 ( ( A1
% 12.24/7.98 = ( cons @ A @ X3 @ Xs2 ) )
% 12.24/7.98 => ! [Ys4: list @ A] :
% 12.24/7.98 ( ( A22
% 12.24/7.98 = ( cons @ A @ Y5 @ Ys4 ) )
% 12.24/7.98 => ( ~ ( ord_less @ A @ X3 @ Y5 )
% 12.24/7.98 => ( ~ ( ord_less @ A @ Y5 @ X3 )
% 12.24/7.98 => ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5921_lexordp__append__left__rightI,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ! [X: A,Y3: A,Us: list @ A,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.98 ( ( ord_less @ A @ X @ Y3 )
% 12.24/7.98 => ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5922_lexordp__iff,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( linorder @ A )
% 12.24/7.98 => ( ( ord_lexordp @ A )
% 12.24/7.98 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.98 ( ? [X2: A,Vs2: list @ A] :
% 12.24/7.98 ( Ys3
% 12.24/7.98 = ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Vs2 ) ) )
% 12.24/7.98 | ? [Us2: list @ A,A5: A,B3: A,Vs2: list @ A,Ws: list @ A] :
% 12.24/7.98 ( ( ord_less @ A @ A5 @ B3 )
% 12.24/7.98 & ( Xs3
% 12.24/7.98 = ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
% 12.24/7.98 & ( Ys3
% 12.24/7.98 = ( append @ A @ Us2 @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5923_cInf__cSup,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [S3: set @ A] :
% 12.24/7.98 ( ( S3
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit1013018076250108175_below @ A @ S3 )
% 12.24/7.98 => ( ( complete_Inf_Inf @ A @ S3 )
% 12.24/7.98 = ( complete_Sup_Sup @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ! [Y6: A] :
% 12.24/7.98 ( ( member @ A @ Y6 @ S3 )
% 12.24/7.98 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5924_cSup__cInf,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( condit1219197933456340205attice @ A )
% 12.24/7.98 => ! [S3: set @ A] :
% 12.24/7.98 ( ( S3
% 12.24/7.98 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.98 => ( ( condit941137186595557371_above @ A @ S3 )
% 12.24/7.98 => ( ( complete_Sup_Sup @ A @ S3 )
% 12.24/7.98 = ( complete_Inf_Inf @ A
% 12.24/7.98 @ ( collect @ A
% 12.24/7.98 @ ^ [X2: A] :
% 12.24/7.98 ! [Y6: A] :
% 12.24/7.98 ( ( member @ A @ Y6 @ S3 )
% 12.24/7.98 => ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5925_transp__gr,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ( transp @ A
% 12.24/7.98 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5926_transp__ge,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( preorder @ A )
% 12.24/7.98 => ( transp @ A
% 12.24/7.98 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5927_ord__class_Olexordp__def,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( ord @ A )
% 12.24/7.98 => ( ( ord_lexordp @ A )
% 12.24/7.98 = ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
% 12.24/7.98 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
% 12.24/7.98 ( ? [Y6: A,Ys3: list @ A] :
% 12.24/7.98 ( ( X12
% 12.24/7.98 = ( nil @ A ) )
% 12.24/7.98 & ( X24
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) ) )
% 12.24/7.98 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.98 ( ( X12
% 12.24/7.98 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.98 & ( X24
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.98 & ( ord_less @ A @ X2 @ Y6 ) )
% 12.24/7.98 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/7.98 ( ( X12
% 12.24/7.98 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/7.98 & ( X24
% 12.24/7.98 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/7.98 & ~ ( ord_less @ A @ X2 @ Y6 )
% 12.24/7.98 & ~ ( ord_less @ A @ Y6 @ X2 )
% 12.24/7.98 & ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5928_Sup__Inf,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple592849572758109894attice @ A )
% 12.24/7.98 => ! [A6: set @ ( set @ A )] :
% 12.24/7.98 ( ( complete_Sup_Sup @ A @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A6 ) )
% 12.24/7.98 = ( complete_Inf_Inf @ A
% 12.24/7.98 @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
% 12.24/7.98 @ ( collect @ ( set @ A )
% 12.24/7.98 @ ^ [Uu: set @ A] :
% 12.24/7.98 ? [F2: ( set @ A ) > A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 12.24/7.98 & ! [X2: set @ A] :
% 12.24/7.98 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 12.24/7.98 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5929_Inf__Sup__le,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple592849572758109894attice @ A )
% 12.24/7.98 => ! [A6: set @ ( set @ A )] :
% 12.24/7.98 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
% 12.24/7.98 @ ( complete_Sup_Sup @ A
% 12.24/7.98 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 12.24/7.98 @ ( collect @ ( set @ A )
% 12.24/7.98 @ ^ [Uu: set @ A] :
% 12.24/7.98 ? [F2: ( set @ A ) > A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 12.24/7.98 & ! [X2: set @ A] :
% 12.24/7.98 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 12.24/7.98 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5930_Sup__Inf__le,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( comple6319245703460814977attice @ A )
% 12.24/7.98 => ! [A6: set @ ( set @ A )] :
% 12.24/7.98 ( ord_less_eq @ A
% 12.24/7.98 @ ( complete_Sup_Sup @ A
% 12.24/7.98 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 12.24/7.98 @ ( collect @ ( set @ A )
% 12.24/7.98 @ ^ [Uu: set @ A] :
% 12.24/7.98 ? [F2: ( set @ A ) > A] :
% 12.24/7.98 ( ( Uu
% 12.24/7.98 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 12.24/7.98 & ! [X2: set @ A] :
% 12.24/7.98 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 12.24/7.98 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) )
% 12.24/7.98 @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) ) ) ) ).
% 12.24/7.98
% 12.24/7.98 thf(fact_5931_finite__Inf__Sup,axiom,
% 12.24/7.98 ! [A: $tType] :
% 12.24/7.98 ( ( finite8700451911770168679attice @ A )
% 12.24/7.98 => ! [A6: set @ ( set @ A )] :
% 12.24/7.98 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
% 12.24/7.99 @ ( complete_Sup_Sup @ A
% 12.24/7.99 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
% 12.24/7.99 @ ( collect @ ( set @ A )
% 12.24/7.99 @ ^ [Uu: set @ A] :
% 12.24/7.99 ? [F2: ( set @ A ) > A] :
% 12.24/7.99 ( ( Uu
% 12.24/7.99 = ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
% 12.24/7.99 & ! [X2: set @ A] :
% 12.24/7.99 ( ( member @ ( set @ A ) @ X2 @ A6 )
% 12.24/7.99 => ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5932_SUP__INF__set,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType] :
% 12.24/7.99 ( ( comple592849572758109894attice @ A )
% 12.24/7.99 => ! [G: B > A,A6: set @ ( set @ B )] :
% 12.24/7.99 ( ( complete_Sup_Sup @ A
% 12.24/7.99 @ ( image @ ( set @ B ) @ A
% 12.24/7.99 @ ^ [X2: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ X2 ) )
% 12.24/7.99 @ A6 ) )
% 12.24/7.99 = ( complete_Inf_Inf @ A
% 12.24/7.99 @ ( image @ ( set @ B ) @ A
% 12.24/7.99 @ ^ [X2: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ X2 ) )
% 12.24/7.99 @ ( collect @ ( set @ B )
% 12.24/7.99 @ ^ [Uu: set @ B] :
% 12.24/7.99 ? [F2: ( set @ B ) > B] :
% 12.24/7.99 ( ( Uu
% 12.24/7.99 = ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
% 12.24/7.99 & ! [X2: set @ B] :
% 12.24/7.99 ( ( member @ ( set @ B ) @ X2 @ A6 )
% 12.24/7.99 => ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5933_INF__SUP__set,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType] :
% 12.24/7.99 ( ( comple592849572758109894attice @ A )
% 12.24/7.99 => ! [G: B > A,A6: set @ ( set @ B )] :
% 12.24/7.99 ( ( complete_Inf_Inf @ A
% 12.24/7.99 @ ( image @ ( set @ B ) @ A
% 12.24/7.99 @ ^ [B6: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) )
% 12.24/7.99 @ A6 ) )
% 12.24/7.99 = ( complete_Sup_Sup @ A
% 12.24/7.99 @ ( image @ ( set @ B ) @ A
% 12.24/7.99 @ ^ [B6: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) )
% 12.24/7.99 @ ( collect @ ( set @ B )
% 12.24/7.99 @ ^ [Uu: set @ B] :
% 12.24/7.99 ? [F2: ( set @ B ) > B] :
% 12.24/7.99 ( ( Uu
% 12.24/7.99 = ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
% 12.24/7.99 & ! [X2: set @ B] :
% 12.24/7.99 ( ( member @ ( set @ B ) @ X2 @ A6 )
% 12.24/7.99 => ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5934_lfp__funpow,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,N: nat] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
% 12.24/7.99 = ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5935_lfp__Kleene__iter,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,K: nat] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( bot_bot @ A ) )
% 12.24/7.99 = ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) )
% 12.24/7.99 => ( ( complete_lattice_lfp @ A @ F )
% 12.24/7.99 = ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5936_transp__le,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( preorder @ A )
% 12.24/7.99 => ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5937_transp__less,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( preorder @ A )
% 12.24/7.99 => ( transp @ A @ ( ord_less @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5938_lfp__ordinal__induct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,P: A > $o] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ! [S4: A] :
% 12.24/7.99 ( ( P @ S4 )
% 12.24/7.99 => ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F ) )
% 12.24/7.99 => ( P @ ( F @ S4 ) ) ) )
% 12.24/7.99 => ( ! [M8: set @ A] :
% 12.24/7.99 ( ! [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ M8 )
% 12.24/7.99 => ( P @ X4 ) )
% 12.24/7.99 => ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
% 12.24/7.99 => ( P @ ( complete_lattice_lfp @ A @ F ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5939_def__lfp__induct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [A6: A,F: A > A,P: A] :
% 12.24/7.99 ( ( A6
% 12.24/7.99 = ( complete_lattice_lfp @ A @ F ) )
% 12.24/7.99 => ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ A6 @ P ) ) @ P )
% 12.24/7.99 => ( ord_less_eq @ A @ A6 @ P ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5940_lfp__induct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,P: A] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) @ P )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5941_le__rel__bool__arg__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( ord @ A )
% 12.24/7.99 => ( ( ord_less_eq @ ( $o > A ) )
% 12.24/7.99 = ( ^ [X7: $o > A,Y9: $o > A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y9 @ $false ) )
% 12.24/7.99 & ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5942_lfp__mono,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,G: A > A] :
% 12.24/7.99 ( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5943_lfp__lowerbound,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,A6: A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ ( F @ A6 ) @ A6 )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ A6 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5944_lfp__greatest,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,A6: A] :
% 12.24/7.99 ( ! [U2: A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ ( F @ U2 ) @ U2 )
% 12.24/7.99 => ( ord_less_eq @ A @ A6 @ U2 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ A6 @ ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5945_lfp__lfp,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A > A] :
% 12.24/7.99 ( ! [X3: A,Y5: A,W: A,Z4: A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.99 => ( ( ord_less_eq @ A @ W @ Z4 )
% 12.24/7.99 => ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
% 12.24/7.99 => ( ( complete_lattice_lfp @ A
% 12.24/7.99 @ ^ [X2: A] : ( complete_lattice_lfp @ A @ ( F @ X2 ) ) )
% 12.24/7.99 = ( complete_lattice_lfp @ A
% 12.24/7.99 @ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5946_lfp__eqI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F4: A > A,X: A] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F4 )
% 12.24/7.99 => ( ( ( F4 @ X )
% 12.24/7.99 = X )
% 12.24/7.99 => ( ! [Z4: A] :
% 12.24/7.99 ( ( ( F4 @ Z4 )
% 12.24/7.99 = Z4 )
% 12.24/7.99 => ( ord_less_eq @ A @ X @ Z4 ) )
% 12.24/7.99 => ( ( complete_lattice_lfp @ A @ F4 )
% 12.24/7.99 = X ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5947_lfp__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ( ( complete_lattice_lfp @ A )
% 12.24/7.99 = ( ^ [F2: A > A] :
% 12.24/7.99 ( complete_Inf_Inf @ A
% 12.24/7.99 @ ( collect @ A
% 12.24/7.99 @ ^ [U3: A] : ( ord_less_eq @ A @ ( F2 @ U3 ) @ U3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5948_char__of__take__bit__eq,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.99 => ! [N: nat,M: A] :
% 12.24/7.99 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
% 12.24/7.99 => ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
% 12.24/7.99 = ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5949_num__of__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.99 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 12.24/7.99 @ code_num_of_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5950_Max_Osemilattice__order__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
% 12.24/7.99 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/7.99 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5951_char__of__nat,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.99 => ! [N: nat] :
% 12.24/7.99 ( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
% 12.24/7.99 = ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5952_integer_Oid__abs__transfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ int @ int @ code_integer
% 12.24/7.99 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.99 @ code_pcr_integer
% 12.24/7.99 @ ^ [X2: int] : X2
% 12.24/7.99 @ code_integer_of_int ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5953_integer_Orep__transfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.99 @ ^ [X2: int] : X2
% 12.24/7.99 @ code_int_of_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5954_zero__integer_Otransfer,axiom,
% 12.24/7.99 code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5955_less__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.99 @ ( ord_less @ int )
% 12.24/7.99 @ ( ord_less @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5956_plus__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5957_uminus__integer_Otransfer,axiom,
% 12.24/7.99 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5958_times__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5959_one__integer_Otransfer,axiom,
% 12.24/7.99 code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5960_less__eq__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.99 @ ( ord_less_eq @ int )
% 12.24/7.99 @ ( ord_less_eq @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5961_minus__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5962_abs__integer_Otransfer,axiom,
% 12.24/7.99 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5963_integer__of__nat_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ code_pcr_integer
% 12.24/7.99 @ ( semiring_1_of_nat @ int )
% 12.24/7.99 @ code_integer_of_nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5964_divide__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5965_nat__of__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ nat2
% 12.24/7.99 @ code_nat_of_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5966_modulo__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5967_sgn__integer_Otransfer,axiom,
% 12.24/7.99 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5968_dup_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
% 12.24/7.99 @ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
% 12.24/7.99 @ code_dup ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5969_take__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se2584673776208193580ke_bit @ int )
% 12.24/7.99 @ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5970_not__integer_Otransfer,axiom,
% 12.24/7.99 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5971_and__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5972_bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ int )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5973_or__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5974_xor__integer_Otransfer,axiom,
% 12.24/7.99 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 ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5975_push__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se4730199178511100633sh_bit @ int )
% 12.24/7.99 @ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5976_drop__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se4197421643247451524op_bit @ int )
% 12.24/7.99 @ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5977_mask__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ code_pcr_integer
% 12.24/7.99 @ ( bit_se2239418461657761734s_mask @ int )
% 12.24/7.99 @ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5978_unset__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se2638667681897837118et_bit @ int )
% 12.24/7.99 @ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5979_set__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se5668285175392031749et_bit @ int )
% 12.24/7.99 @ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5980_flip__bit__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 12.24/7.99 @ ( bit_se8732182000553998342ip_bit @ int )
% 12.24/7.99 @ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5981_division__segment__integer_Otransfer,axiom,
% 12.24/7.99 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5982_euclidean__size__integer_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 12.24/7.99 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/7.99 @ ( euclid6346220572633701492n_size @ int )
% 12.24/7.99 @ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5983_sub_Otransfer,axiom,
% 12.24/7.99 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
% 12.24/7.99 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.99 @ ( bNF_rel_fun @ num @ num @ int @ code_integer
% 12.24/7.99 @ ^ [Y: num,Z2: num] : ( Y = Z2 )
% 12.24/7.99 @ code_pcr_integer )
% 12.24/7.99 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
% 12.24/7.99 @ code_sub ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5984_Inf__fin_Osemilattice__order__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_inf @ A )
% 12.24/7.99 => ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5985_Min_Osemilattice__order__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5986_UNIV__char__of__nat,axiom,
% 12.24/7.99 ( ( top_top @ ( set @ char ) )
% 12.24/7.99 = ( image @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5987_inj__on__char__of__nat,axiom,
% 12.24/7.99 inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5988_Sup__fin_Osemilattice__order__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_sup @ A )
% 12.24/7.99 => ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
% 12.24/7.99 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/7.99 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5989_char__of__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_un5681908812861735899ations @ A )
% 12.24/7.99 => ( ( unique5772411509450598832har_of @ A )
% 12.24/7.99 = ( ^ [N3: A] :
% 12.24/7.99 ( char2
% 12.24/7.99 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
% 12.24/7.99 @ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5990_range__nat__of__char,axiom,
% 12.24/7.99 ( ( image @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
% 12.24/7.99 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5991_Gcd__fin__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( semiring_gcd_Gcd_fin @ A )
% 12.24/7.99 = ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5992_char_Osize_I2_J,axiom,
% 12.24/7.99 ! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 12.24/7.99 ( ( size_size @ char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 12.24/7.99 = ( zero_zero @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5993_nat__of__char__less__256,axiom,
% 12.24/7.99 ! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5994_assert__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( heap_Time_assert @ A )
% 12.24/7.99 = ( ^ [P4: A > $o,X2: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X2 ) @ ( heap_Time_return @ A @ X2 ) @ ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5995_char_Osize__gen,axiom,
% 12.24/7.99 ! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
% 12.24/7.99 ( ( size_char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
% 12.24/7.99 = ( zero_zero @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5996_n__lists_Osimps_I2_J,axiom,
% 12.24/7.99 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/7.99 ( ( n_lists @ A @ ( suc @ N ) @ Xs )
% 12.24/7.99 = ( concat @ ( list @ A )
% 12.24/7.99 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 12.24/7.99 @ ^ [Ys3: list @ A] :
% 12.24/7.99 ( map @ A @ ( list @ A )
% 12.24/7.99 @ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
% 12.24/7.99 @ Xs )
% 12.24/7.99 @ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5997_comm__append__is__replicate,axiom,
% 12.24/7.99 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 12.24/7.99 ( ( Xs
% 12.24/7.99 != ( nil @ A ) )
% 12.24/7.99 => ( ( Ys2
% 12.24/7.99 != ( nil @ A ) )
% 12.24/7.99 => ( ( ( append @ A @ Xs @ Ys2 )
% 12.24/7.99 = ( append @ A @ Ys2 @ Xs ) )
% 12.24/7.99 => ? [N4: nat,Zs2: list @ A] :
% 12.24/7.99 ( ( ord_less @ nat @ ( one_one @ nat ) @ N4 )
% 12.24/7.99 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N4 @ Zs2 ) )
% 12.24/7.99 = ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5998_strict__sorted__equal__Uniq,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( uniq @ ( list @ A )
% 12.24/7.99 @ ^ [Xs3: list @ A] :
% 12.24/7.99 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
% 12.24/7.99 & ( ( set2 @ A @ Xs3 )
% 12.24/7.99 = A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_5999_integer__of__natural__of__nat,axiom,
% 12.24/7.99 ! [N: nat] :
% 12.24/7.99 ( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 12.24/7.99 = ( semiring_1_of_nat @ code_integer @ N ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6000_integer__of__natural__of__integer,axiom,
% 12.24/7.99 ! [K: code_integer] :
% 12.24/7.99 ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
% 12.24/7.99 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6001_natural__of__integer__of__natural,axiom,
% 12.24/7.99 ! [N: code_natural] :
% 12.24/7.99 ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
% 12.24/7.99 = N ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6002_card__def,axiom,
% 12.24/7.99 ! [B: $tType] :
% 12.24/7.99 ( ( finite_card @ B )
% 12.24/7.99 = ( finite_folding_F @ B @ nat
% 12.24/7.99 @ ^ [Uu: B] : suc
% 12.24/7.99 @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6003_length__code,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( size_size @ ( list @ A ) )
% 12.24/7.99 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6004_gen__length__code_I2_J,axiom,
% 12.24/7.99 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 12.24/7.99 ( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
% 12.24/7.99 = ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6005_less__than__iff,axiom,
% 12.24/7.99 ! [X: nat,Y3: nat] :
% 12.24/7.99 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ less_than )
% 12.24/7.99 = ( ord_less @ nat @ X @ Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6006_Code__Numeral_Osub__code_I4_J,axiom,
% 12.24/7.99 ! [N: num] :
% 12.24/7.99 ( ( code_sub @ one2 @ ( bit0 @ N ) )
% 12.24/7.99 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6007_Code__Numeral_Odup__code_I3_J,axiom,
% 12.24/7.99 ! [N: num] :
% 12.24/7.99 ( ( code_dup @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6008_less__integer__code_I7_J,axiom,
% 12.24/7.99 ! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6009_less__integer__code_I3_J,axiom,
% 12.24/7.99 ! [L: num] :
% 12.24/7.99 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6010_plus__integer__code_I6_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6011_less__eq__integer__code_I7_J,axiom,
% 12.24/7.99 ! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6012_less__eq__integer__code_I3_J,axiom,
% 12.24/7.99 ! [L: num] :
% 12.24/7.99 ~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6013_less__integer__code_I9_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 12.24/7.99 = ( ord_less @ num @ L @ K ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6014_less__eq__integer__code_I9_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 12.24/7.99 = ( ord_less_eq @ num @ L @ K ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6015_minus__integer__code_I6_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_sub @ N @ M ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6016_divmod__abs__code_I2_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 12.24/7.99 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6017_Code__Numeral_Osub__code_I5_J,axiom,
% 12.24/7.99 ! [N: num] :
% 12.24/7.99 ( ( code_sub @ one2 @ ( bit1 @ N ) )
% 12.24/7.99 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6018_pair__less__def,axiom,
% 12.24/7.99 ( fun_pair_less
% 12.24/7.99 = ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6019_Gcd__int__set__eq__fold,axiom,
% 12.24/7.99 ! [Xs: list @ int] :
% 12.24/7.99 ( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
% 12.24/7.99 = ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6020_Gcd__set__eq__fold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
% 12.24/7.99 = ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6021_Gcd__fin_Oset__eq__fold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
% 12.24/7.99 = ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6022_mono__transfer,axiom,
% 12.24/7.99 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 12.24/7.99 ( ( ( order @ B )
% 12.24/7.99 & ( order @ D )
% 12.24/7.99 & ( order @ C )
% 12.24/7.99 & ( order @ A ) )
% 12.24/7.99 => ! [A6: A > B > $o,B7: C > D > $o] :
% 12.24/7.99 ( ( bi_total @ A @ B @ A6 )
% 12.24/7.99 => ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A6
% 12.24/7.99 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
% 12.24/7.99 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.99 @ ( ord_less_eq @ A )
% 12.24/7.99 @ ( ord_less_eq @ B ) )
% 12.24/7.99 => ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B7
% 12.24/7.99 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B7
% 12.24/7.99 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/7.99 @ ( ord_less_eq @ C )
% 12.24/7.99 @ ( ord_less_eq @ D ) )
% 12.24/7.99 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B7 )
% 12.24/7.99 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 )
% 12.24/7.99 @ ( order_mono @ A @ C )
% 12.24/7.99 @ ( order_mono @ B @ D ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6023_binomial__def,axiom,
% 12.24/7.99 ( binomial
% 12.24/7.99 = ( ^ [N3: nat,K3: nat] :
% 12.24/7.99 ( finite_card @ ( set @ nat )
% 12.24/7.99 @ ( collect @ ( set @ nat )
% 12.24/7.99 @ ^ [K6: set @ nat] :
% 12.24/7.99 ( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 12.24/7.99 & ( ( finite_card @ nat @ K6 )
% 12.24/7.99 = K3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6024_Pow__Compl,axiom,
% 12.24/7.99 ! [A: $tType,A6: set @ A] :
% 12.24/7.99 ( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/7.99 = ( collect @ ( set @ A )
% 12.24/7.99 @ ^ [Uu: set @ A] :
% 12.24/7.99 ? [B6: set @ A] :
% 12.24/7.99 ( ( Uu
% 12.24/7.99 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 12.24/7.99 & ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6025_int_Obi__total,axiom,
% 12.24/7.99 bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6026_integer_Obi__total,axiom,
% 12.24/7.99 bi_total @ int @ code_integer @ code_pcr_integer ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6027_Gcd__nat__set__eq__fold,axiom,
% 12.24/7.99 ! [Xs: list @ nat] :
% 12.24/7.99 ( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
% 12.24/7.99 = ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6028_Heap__lub__empty,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
% 12.24/7.99 = ( heap_Time_Heap2 @ A
% 12.24/7.99 @ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6029_gfp__Kleene__iter,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,K: nat] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( top_top @ A ) )
% 12.24/7.99 = ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) )
% 12.24/7.99 => ( ( complete_lattice_gfp @ A @ F )
% 12.24/7.99 = ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6030_gfp__gfp,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A > A] :
% 12.24/7.99 ( ! [X3: A,Y5: A,W: A,Z4: A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ X3 @ Y5 )
% 12.24/7.99 => ( ( ord_less_eq @ A @ W @ Z4 )
% 12.24/7.99 => ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
% 12.24/7.99 => ( ( complete_lattice_gfp @ A
% 12.24/7.99 @ ^ [X2: A] : ( complete_lattice_gfp @ A @ ( F @ X2 ) ) )
% 12.24/7.99 = ( complete_lattice_gfp @ A
% 12.24/7.99 @ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6031_gfp__mono,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,G: A > A] :
% 12.24/7.99 ( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ ( complete_lattice_gfp @ A @ G ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6032_gfp__upperbound,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [X8: A,F: A > A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ X8 @ ( F @ X8 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6033_gfp__least,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,X8: A] :
% 12.24/7.99 ( ! [U2: A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ U2 @ ( F @ U2 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ U2 @ X8 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ X8 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6034_gfp__eqI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F4: A > A,X: A] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F4 )
% 12.24/7.99 => ( ( ( F4 @ X )
% 12.24/7.99 = X )
% 12.24/7.99 => ( ! [Z4: A] :
% 12.24/7.99 ( ( ( F4 @ Z4 )
% 12.24/7.99 = Z4 )
% 12.24/7.99 => ( ord_less_eq @ A @ Z4 @ X ) )
% 12.24/7.99 => ( ( complete_lattice_gfp @ A @ F4 )
% 12.24/7.99 = X ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6035_gfp__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ( ( complete_lattice_gfp @ A )
% 12.24/7.99 = ( ^ [F2: A > A] :
% 12.24/7.99 ( complete_Sup_Sup @ A
% 12.24/7.99 @ ( collect @ A
% 12.24/7.99 @ ^ [U3: A] : ( ord_less_eq @ A @ U3 @ ( F2 @ U3 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6036_coinduct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,X8: A] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
% 12.24/7.99 => ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6037_def__coinduct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [A6: A,F: A > A,X8: A] :
% 12.24/7.99 ( ( A6
% 12.24/7.99 = ( complete_lattice_gfp @ A @ F ) )
% 12.24/7.99 => ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ A6 ) ) )
% 12.24/7.99 => ( ord_less_eq @ A @ X8 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6038_coinduct__lemma,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [X8: A,F: A > A] :
% 12.24/7.99 ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
% 12.24/7.99 => ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6039_gfp__ordinal__induct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,P: A > $o] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ! [S4: A] :
% 12.24/7.99 ( ( P @ S4 )
% 12.24/7.99 => ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ S4 )
% 12.24/7.99 => ( P @ ( F @ S4 ) ) ) )
% 12.24/7.99 => ( ! [M8: set @ A] :
% 12.24/7.99 ( ! [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ M8 )
% 12.24/7.99 => ( P @ X4 ) )
% 12.24/7.99 => ( P @ ( complete_Inf_Inf @ A @ M8 ) ) )
% 12.24/7.99 => ( P @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6040_gfp__funpow,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A,N: nat] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
% 12.24/7.99 = ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6041_lfp__le__gfp,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple6319245703460814977attice @ A )
% 12.24/7.99 => ! [F: A > A] :
% 12.24/7.99 ( ( order_mono @ A @ A @ F )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6042_Code__Numeral_Osub__code_I2_J,axiom,
% 12.24/7.99 ! [M: num] :
% 12.24/7.99 ( ( code_sub @ ( bit0 @ M ) @ one2 )
% 12.24/7.99 = ( code_Pos @ ( bitM @ M ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6043_is__singleton__altdef,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( is_singleton @ A )
% 12.24/7.99 = ( ^ [A7: set @ A] :
% 12.24/7.99 ( ( finite_card @ A @ A7 )
% 12.24/7.99 = ( one_one @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6044_Code__Numeral_OPos__def,axiom,
% 12.24/7.99 ( code_Pos
% 12.24/7.99 = ( numeral_numeral @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6045_Pos__fold_I2_J,axiom,
% 12.24/7.99 ! [K: num] :
% 12.24/7.99 ( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
% 12.24/7.99 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6046_Pos__fold_I1_J,axiom,
% 12.24/7.99 ( ( numeral_numeral @ code_integer @ one2 )
% 12.24/7.99 = ( code_Pos @ one2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6047_Pos__fold_I3_J,axiom,
% 12.24/7.99 ! [K: num] :
% 12.24/7.99 ( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
% 12.24/7.99 = ( code_Pos @ ( bit1 @ K ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6048_one__integer__code,axiom,
% 12.24/7.99 ( ( one_one @ code_integer )
% 12.24/7.99 = ( code_Pos @ one2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6049_Code__Numeral_Odup__code_I2_J,axiom,
% 12.24/7.99 ! [N: num] :
% 12.24/7.99 ( ( code_dup @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6050_less__integer__code_I4_J,axiom,
% 12.24/7.99 ! [K: num] :
% 12.24/7.99 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6051_less__integer__code_I2_J,axiom,
% 12.24/7.99 ! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6052_plus__integer__code_I3_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6053_less__eq__integer__code_I4_J,axiom,
% 12.24/7.99 ! [K: num] :
% 12.24/7.99 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6054_less__eq__integer__code_I2_J,axiom,
% 12.24/7.99 ! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6055_Code__Numeral_ONeg__def,axiom,
% 12.24/7.99 ( code_Neg
% 12.24/7.99 = ( ^ [N3: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6056_uminus__integer__code_I2_J,axiom,
% 12.24/7.99 ! [M: num] :
% 12.24/7.99 ( ( uminus_uminus @ code_integer @ ( code_Pos @ M ) )
% 12.24/7.99 = ( code_Neg @ M ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6057_uminus__integer__code_I3_J,axiom,
% 12.24/7.99 ! [M: num] :
% 12.24/7.99 ( ( uminus_uminus @ code_integer @ ( code_Neg @ M ) )
% 12.24/7.99 = ( code_Pos @ M ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6058_less__integer__code_I8_J,axiom,
% 12.24/7.99 ! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6059_less__integer__code_I6_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6060_times__integer__code_I3_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6061_less__eq__integer__code_I8_J,axiom,
% 12.24/7.99 ! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6062_less__eq__integer__code_I6_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6063_less__integer__code_I5_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 12.24/7.99 = ( ord_less @ num @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6064_less__eq__integer__code_I5_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 12.24/7.99 = ( ord_less_eq @ num @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6065_minus__integer__code_I3_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_sub @ M @ N ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6066_divmod__abs__code_I1_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 12.24/7.99 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6067_minus__integer__code_I5_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6068_minus__integer__code_I4_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6069_times__integer__code_I6_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6070_times__integer__code_I5_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6071_times__integer__code_I4_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6072_plus__integer__code_I4_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
% 12.24/7.99 = ( code_sub @ M @ N ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6073_plus__integer__code_I5_J,axiom,
% 12.24/7.99 ! [M: num,N: num] :
% 12.24/7.99 ( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
% 12.24/7.99 = ( code_sub @ N @ M ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6074_divmod__abs__code_I3_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
% 12.24/7.99 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6075_divmod__abs__code_I4_J,axiom,
% 12.24/7.99 ! [K: num,L: num] :
% 12.24/7.99 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
% 12.24/7.99 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6076_Code__Numeral_Osub__code_I3_J,axiom,
% 12.24/7.99 ! [M: num] :
% 12.24/7.99 ( ( code_sub @ ( bit1 @ M ) @ one2 )
% 12.24/7.99 = ( code_Pos @ ( bit0 @ M ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6077_prod__list__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( monoid_mult @ A )
% 12.24/7.99 => ( ( groups5270119922927024881d_list @ A )
% 12.24/7.99 = ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6078_semilattice__order__set_Osubset__imp,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.99 => ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6079_semilattice__set_OF_Ocong,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( lattic1715443433743089157tice_F @ A )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6080_Inf__fin__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_inf @ A )
% 12.24/7.99 => ( ( lattic7752659483105999362nf_fin @ A )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6081_Sup__fin__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_sup @ A )
% 12.24/7.99 => ( ( lattic5882676163264333800up_fin @ A )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6082_Max__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( ( lattic643756798349783984er_Max @ A )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ ( ord_max @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6083_Min__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( ( lattic643756798350308766er_Min @ A )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ ( ord_min @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6084_semilattice__order__set_OcoboundedI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.99 => ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6085_semilattice__order__set_OboundedE,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 12.24/7.99 => ! [A10: A] :
% 12.24/7.99 ( ( member @ A @ A10 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ A10 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6086_semilattice__order__set_OboundedI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [A8: A] :
% 12.24/7.99 ( ( member @ A @ A8 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ A8 ) )
% 12.24/7.99 => ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6087_semilattice__order__set_Obounded__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 12.24/7.99 = ( ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6088_sum__list__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( monoid_add @ A )
% 12.24/7.99 => ( ( groups8242544230860333062m_list @ A )
% 12.24/7.99 = ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6089_semilattice__set_Oeq__fold_H,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 12.24/7.99 = ( the2 @ A
% 12.24/7.99 @ ( finite_fold @ A @ ( option @ A )
% 12.24/7.99 @ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F @ X2 ) @ Y6 ) )
% 12.24/7.99 @ ( none @ A )
% 12.24/7.99 @ A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6090_semilattice__set_Oinsert__remove,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = X ) )
% 12.24/7.99 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6091_Min_Osemilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( lattic149705377957585745ce_set @ A @ ( ord_min @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6092_Max_Osemilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( lattic149705377957585745ce_set @ A @ ( ord_max @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6093_Inf__fin_Osemilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_inf @ A )
% 12.24/7.99 => ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6094_Sup__fin_Osemilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semilattice_sup @ A )
% 12.24/7.99 => ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6095_semilattice__order__set_Oaxioms_I2_J,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/7.99 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/7.99 => ( lattic149705377957585745ce_set @ A @ F ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6096_semilattice__set_Oin__idem,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6097_semilattice__set_Osingleton,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 = X ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6098_semilattice__set_Ohom__commute,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,H2: A > A,N6: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ! [X3: A,Y5: A] :
% 12.24/7.99 ( ( H2 @ ( F @ X3 @ Y5 ) )
% 12.24/7.99 = ( F @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
% 12.24/7.99 => ( ( finite_finite2 @ A @ N6 )
% 12.24/7.99 => ( ( N6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F @ N6 ) )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ F @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6099_semilattice__set_Osubset,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( B7
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.99 => ( ( F @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
% 12.24/7.99 = ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6100_semilattice__set_Oinsert__not__elem,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ~ ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6101_semilattice__set_Oinsert,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6102_semilattice__set_Oclosed,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.99 => ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6103_semilattice__set_Ounion,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.99 => ( ( B7
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.99 = ( F @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6104_semilattice__set_Oeq__fold,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( finite_fold @ A @ A @ F @ X @ A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6105_semilattice__set_Oinfinite,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 12.24/7.99 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6106_semilattice__set_Oremove,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 = ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 12.24/7.99 = X ) )
% 12.24/7.99 & ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
% 12.24/7.99 = ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6107_Least__mono,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( ( order @ A )
% 12.24/7.99 & ( order @ B ) )
% 12.24/7.99 => ! [F: A > B,S3: set @ A] :
% 12.24/7.99 ( ( order_mono @ A @ B @ F )
% 12.24/7.99 => ( ? [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ S3 )
% 12.24/7.99 & ! [Xa2: A] :
% 12.24/7.99 ( ( member @ A @ Xa2 @ S3 )
% 12.24/7.99 => ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
% 12.24/7.99 => ( ( ord_Least @ B
% 12.24/7.99 @ ^ [Y6: B] : ( member @ B @ Y6 @ ( image @ A @ B @ F @ S3 ) ) )
% 12.24/7.99 = ( F
% 12.24/7.99 @ ( ord_Least @ A
% 12.24/7.99 @ ^ [X2: A] : ( member @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6108_cr__int__def,axiom,
% 12.24/7.99 ( cr_int
% 12.24/7.99 = ( ^ [X2: product_prod @ nat @ nat] :
% 12.24/7.99 ( ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/7.99 @ ( abs_Integ @ X2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6109_Least__eq__0,axiom,
% 12.24/7.99 ! [P: nat > $o] :
% 12.24/7.99 ( ( P @ ( zero_zero @ nat ) )
% 12.24/7.99 => ( ( ord_Least @ nat @ P )
% 12.24/7.99 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6110_Least__le,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,K: A] :
% 12.24/7.99 ( ( P @ K )
% 12.24/7.99 => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6111_Least1I,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( order @ A )
% 12.24/7.99 => ! [P: A > $o] :
% 12.24/7.99 ( ? [X4: A] :
% 12.24/7.99 ( ( P @ X4 )
% 12.24/7.99 & ! [Y5: A] :
% 12.24/7.99 ( ( P @ Y5 )
% 12.24/7.99 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 12.24/7.99 & ! [Y5: A] :
% 12.24/7.99 ( ( ( P @ Y5 )
% 12.24/7.99 & ! [Ya2: A] :
% 12.24/7.99 ( ( P @ Ya2 )
% 12.24/7.99 => ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
% 12.24/7.99 => ( Y5 = X4 ) ) )
% 12.24/7.99 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6112_Least1__le,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( order @ A )
% 12.24/7.99 => ! [P: A > $o,Z3: A] :
% 12.24/7.99 ( ? [X4: A] :
% 12.24/7.99 ( ( P @ X4 )
% 12.24/7.99 & ! [Y5: A] :
% 12.24/7.99 ( ( P @ Y5 )
% 12.24/7.99 => ( ord_less_eq @ A @ X4 @ Y5 ) )
% 12.24/7.99 & ! [Y5: A] :
% 12.24/7.99 ( ( ( P @ Y5 )
% 12.24/7.99 & ! [Ya2: A] :
% 12.24/7.99 ( ( P @ Ya2 )
% 12.24/7.99 => ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
% 12.24/7.99 => ( Y5 = X4 ) ) )
% 12.24/7.99 => ( ( P @ Z3 )
% 12.24/7.99 => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6113_LeastI2__order,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( order @ A )
% 12.24/7.99 => ! [P: A > $o,X: A,Q: A > $o] :
% 12.24/7.99 ( ( P @ X )
% 12.24/7.99 => ( ! [Y5: A] :
% 12.24/7.99 ( ( P @ Y5 )
% 12.24/7.99 => ( ord_less_eq @ A @ X @ Y5 ) )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 => ( ! [Y4: A] :
% 12.24/7.99 ( ( P @ Y4 )
% 12.24/7.99 => ( ord_less_eq @ A @ X3 @ Y4 ) )
% 12.24/7.99 => ( Q @ X3 ) ) )
% 12.24/7.99 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6114_Least__equality,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( order @ A )
% 12.24/7.99 => ! [P: A > $o,X: A] :
% 12.24/7.99 ( ( P @ X )
% 12.24/7.99 => ( ! [Y5: A] :
% 12.24/7.99 ( ( P @ Y5 )
% 12.24/7.99 => ( ord_less_eq @ A @ X @ Y5 ) )
% 12.24/7.99 => ( ( ord_Least @ A @ P )
% 12.24/7.99 = X ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6115_LeastI2__wellorder,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,A3: A,Q: A > $o] :
% 12.24/7.99 ( ( P @ A3 )
% 12.24/7.99 => ( ! [A8: A] :
% 12.24/7.99 ( ( P @ A8 )
% 12.24/7.99 => ( ! [B11: A] :
% 12.24/7.99 ( ( P @ B11 )
% 12.24/7.99 => ( ord_less_eq @ A @ A8 @ B11 ) )
% 12.24/7.99 => ( Q @ A8 ) ) )
% 12.24/7.99 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6116_LeastI2__wellorder__ex,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,Q: A > $o] :
% 12.24/7.99 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.99 => ( ! [A8: A] :
% 12.24/7.99 ( ( P @ A8 )
% 12.24/7.99 => ( ! [B11: A] :
% 12.24/7.99 ( ( P @ B11 )
% 12.24/7.99 => ( ord_less_eq @ A @ A8 @ B11 ) )
% 12.24/7.99 => ( Q @ A8 ) ) )
% 12.24/7.99 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6117_not__less__Least,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [K: A,P: A > $o] :
% 12.24/7.99 ( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
% 12.24/7.99 => ~ ( P @ K ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6118_LeastI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,K: A] :
% 12.24/7.99 ( ( P @ K )
% 12.24/7.99 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6119_LeastI2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,A3: A,Q: A > $o] :
% 12.24/7.99 ( ( P @ A3 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 => ( Q @ X3 ) )
% 12.24/7.99 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6120_LeastI__ex,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o] :
% 12.24/7.99 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.99 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6121_LeastI2__ex,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ! [P: A > $o,Q: A > $o] :
% 12.24/7.99 ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 => ( Q @ X3 ) )
% 12.24/7.99 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6122_Least__Suc2,axiom,
% 12.24/7.99 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 12.24/7.99 ( ( P @ N )
% 12.24/7.99 => ( ( Q @ M )
% 12.24/7.99 => ( ~ ( P @ ( zero_zero @ nat ) )
% 12.24/7.99 => ( ! [K2: nat] :
% 12.24/7.99 ( ( P @ ( suc @ K2 ) )
% 12.24/7.99 = ( Q @ K2 ) )
% 12.24/7.99 => ( ( ord_Least @ nat @ P )
% 12.24/7.99 = ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6123_Least__Suc,axiom,
% 12.24/7.99 ! [P: nat > $o,N: nat] :
% 12.24/7.99 ( ( P @ N )
% 12.24/7.99 => ( ~ ( P @ ( zero_zero @ nat ) )
% 12.24/7.99 => ( ( ord_Least @ nat @ P )
% 12.24/7.99 = ( suc
% 12.24/7.99 @ ( ord_Least @ nat
% 12.24/7.99 @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6124_Least__Min,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ! [P: A > $o] :
% 12.24/7.99 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 12.24/7.99 => ( ? [X_1: A] : ( P @ X_1 )
% 12.24/7.99 => ( ( ord_Least @ A @ P )
% 12.24/7.99 = ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6125_int_Opcr__cr__eq,axiom,
% 12.24/7.99 pcr_int = cr_int ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6126_Quotient__int,axiom,
% 12.24/7.99 quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6127_list__ex__length,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( list_ex @ A )
% 12.24/7.99 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 12.24/7.99 ? [N3: nat] :
% 12.24/7.99 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/7.99 & ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6128_map__of__zip__nth,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
% 12.24/7.99 ( ( ( size_size @ ( list @ A ) @ Xs )
% 12.24/7.99 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.99 => ( ( distinct @ A @ Xs )
% 12.24/7.99 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.99 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
% 12.24/7.99 = ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6129_and_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.99 => ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6130_semilattice__neutr__order_Oaxioms_I1_J,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/7.99 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6131_sup__bot_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.99 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6132_inf__top_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.99 => ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6133_gcd__nat_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6134_or_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.99 => ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6135_max__nat_Osemilattice__neutr__axioms,axiom,
% 12.24/7.99 semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6136_map__upds__list__update2__drop,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys2: list @ B,Y3: B] :
% 12.24/7.99 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
% 12.24/7.99 => ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys2 @ I @ Y3 ) )
% 12.24/7.99 = ( map_upds @ A @ B @ M @ Xs @ Ys2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6137_natLess__def,axiom,
% 12.24/7.99 ( bNF_Ca8459412986667044542atLess
% 12.24/7.99 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6138_relChain__def,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( ord @ B )
% 12.24/7.99 => ( ( bNF_Ca3754400796208372196lChain @ A @ B )
% 12.24/7.99 = ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
% 12.24/7.99 ! [I5: A,J3: A] :
% 12.24/7.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I5 @ J3 ) @ R5 )
% 12.24/7.99 => ( ord_less_eq @ B @ ( As @ I5 ) @ ( As @ J3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6139_map__upds__append1,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M: A > ( option @ B ),X: A] :
% 12.24/7.99 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 12.24/7.99 => ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
% 12.24/7.99 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6140_restrict__upd__same,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y3: B] :
% 12.24/7.99 ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y3 ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.99 = ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6141_restrict__complement__singleton__eq,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 12.24/7.99 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.99 = ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6142_Restr__natLeq,axiom,
% 12.24/7.99 ! [N: nat] :
% 12.24/7.99 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
% 12.24/7.99 @ ( product_Sigma @ nat @ nat
% 12.24/7.99 @ ( collect @ nat
% 12.24/7.99 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
% 12.24/7.99 @ ^ [Uu: nat] :
% 12.24/7.99 ( collect @ nat
% 12.24/7.99 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) )
% 12.24/7.99 = ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.99 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.99 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.99 ( ( ord_less @ nat @ X2 @ N )
% 12.24/7.99 & ( ord_less @ nat @ Y6 @ N )
% 12.24/7.99 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6143_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
% 12.24/7.99 @ ^ [X2: A] : X2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6144_natLeq__def,axiom,
% 12.24/7.99 ( bNF_Ca8665028551170535155natLeq
% 12.24/7.99 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6145_Restr__natLeq2,axiom,
% 12.24/7.99 ! [N: nat] :
% 12.24/7.99 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
% 12.24/7.99 @ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
% 12.24/7.99 @ ^ [Uu: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 12.24/7.99 = ( collect @ ( product_prod @ nat @ nat )
% 12.24/7.99 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/7.99 @ ^ [X2: nat,Y6: nat] :
% 12.24/7.99 ( ( ord_less @ nat @ X2 @ N )
% 12.24/7.99 & ( ord_less @ nat @ Y6 @ N )
% 12.24/7.99 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6146_fun__cong__unused__0,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,C: $tType] :
% 12.24/7.99 ( ( zero @ B )
% 12.24/7.99 => ! [F: ( A > B ) > C,G: C] :
% 12.24/7.99 ( ( F
% 12.24/7.99 = ( ^ [X2: A > B] : G ) )
% 12.24/7.99 => ( ( F
% 12.24/7.99 @ ^ [X2: A] : ( zero_zero @ B ) )
% 12.24/7.99 = G ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6147_natLeq__underS__less,axiom,
% 12.24/7.99 ! [N: nat] :
% 12.24/7.99 ( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
% 12.24/7.99 = ( collect @ nat
% 12.24/7.99 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6148_add_Ogroup__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( group_add @ A )
% 12.24/7.99 => ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6149_inv__o__cancel,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 12.24/7.99 = ( id @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6150_inv__into__f__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ X ) )
% 12.24/7.99 = X ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6151_inv__identity,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
% 12.24/7.99 @ ^ [A5: A] : A5 )
% 12.24/7.99 = ( ^ [A5: A] : A5 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6152_inv__id,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
% 12.24/7.99 = ( id @ A ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6153_inv__into__image__cancel,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,S3: set @ A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ A6 )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ A6 )
% 12.24/7.99 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ ( image @ A @ B @ F @ S3 ) )
% 12.24/7.99 = S3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6154_o__inv__o__cancel,axiom,
% 12.24/7.99 ! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
% 12.24/7.99 = G ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6155_inv__into__def,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( hilbert_inv_into @ A @ B )
% 12.24/7.99 = ( ^ [A7: set @ A,F2: A > B,X2: B] :
% 12.24/7.99 ( fChoice @ A
% 12.24/7.99 @ ^ [Y6: A] :
% 12.24/7.99 ( ( member @ A @ Y6 @ A7 )
% 12.24/7.99 & ( ( F2 @ Y6 )
% 12.24/7.99 = X2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6156_inv__into__def2,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( hilbert_inv_into @ A @ B )
% 12.24/7.99 = ( ^ [A7: set @ A,F2: A > B,X2: B] :
% 12.24/7.99 ( fChoice @ A
% 12.24/7.99 @ ^ [Y6: A] :
% 12.24/7.99 ( ( member @ A @ Y6 @ A7 )
% 12.24/7.99 & ( ( F2 @ Y6 )
% 12.24/7.99 = X2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6157_group_Oleft__cancel,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( ( F @ A3 @ B2 )
% 12.24/7.99 = ( F @ A3 @ C2 ) )
% 12.24/7.99 = ( B2 = C2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6158_group_Oleft__inverse,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( F @ ( Inverse @ A3 ) @ A3 )
% 12.24/7.99 = Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6159_group_Oright__cancel,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,B2: A,A3: A,C2: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( ( F @ B2 @ A3 )
% 12.24/7.99 = ( F @ C2 @ A3 ) )
% 12.24/7.99 = ( B2 = C2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6160_group_Oright__inverse,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( F @ A3 @ ( Inverse @ A3 ) )
% 12.24/7.99 = Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6161_group_Oinverse__unique,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( ( F @ A3 @ B2 )
% 12.24/7.99 = Z3 )
% 12.24/7.99 => ( ( Inverse @ A3 )
% 12.24/7.99 = B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6162_group_Oinverse__inverse,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( Inverse @ ( Inverse @ A3 ) )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6163_group_Oinverse__neutral,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( Inverse @ Z3 )
% 12.24/7.99 = Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6164_group_Ogroup__left__neutral,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( F @ Z3 @ A3 )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6165_group_Oinverse__distrib__swap,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/7.99 => ( ( Inverse @ ( F @ A3 @ B2 ) )
% 12.24/7.99 = ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6166_inv__into__injective,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,A6: set @ A,F: A > B,X: B,Y3: B] :
% 12.24/7.99 ( ( ( hilbert_inv_into @ A @ B @ A6 @ F @ X )
% 12.24/7.99 = ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 ) )
% 12.24/7.99 => ( ( member @ B @ X @ ( image @ A @ B @ F @ A6 ) )
% 12.24/7.99 => ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A6 ) )
% 12.24/7.99 => ( X = Y3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6167_inv__into__into,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,X: A,F: B > A,A6: set @ B] :
% 12.24/7.99 ( ( member @ A @ X @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.99 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F @ X ) @ A6 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6168_f__inv__into__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,Y3: A,F: B > A,A6: set @ B] :
% 12.24/7.99 ( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.99 => ( ( F @ ( hilbert_inv_into @ B @ A @ A6 @ F @ Y3 ) )
% 12.24/7.99 = Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6169_surj__f__inv__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A,Y3: A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ Y3 ) )
% 12.24/7.99 = Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6170_surj__iff__all,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( ! [X2: A] :
% 12.24/7.99 ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ X2 ) )
% 12.24/7.99 = X2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6171_image__f__inv__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ A6 ) )
% 12.24/7.99 = A6 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6172_surj__imp__inv__eq,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [X3: B] :
% 12.24/7.99 ( ( G @ ( F @ X3 ) )
% 12.24/7.99 = X3 )
% 12.24/7.99 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 12.24/7.99 = G ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6173_inv__into__f__eq,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A,Y3: B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( ( F @ X )
% 12.24/7.99 = Y3 )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 )
% 12.24/7.99 = X ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6174_inv__f__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 12.24/7.99 = X ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6175_inv__f__eq,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,X: A,Y3: B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( ( F @ X )
% 12.24/7.99 = Y3 )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y3 )
% 12.24/7.99 = X ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6176_inj__imp__inv__eq,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,F: A > B,G: B > A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [X3: B] :
% 12.24/7.99 ( ( F @ ( G @ X3 ) )
% 12.24/7.99 = X3 )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
% 12.24/7.99 = G ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6177_inv__equality,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType,G: B > A,F: A > B] :
% 12.24/7.99 ( ! [X3: A] :
% 12.24/7.99 ( ( G @ ( F @ X3 ) )
% 12.24/7.99 = X3 )
% 12.24/7.99 => ( ! [Y5: B] :
% 12.24/7.99 ( ( F @ ( G @ Y5 ) )
% 12.24/7.99 = Y5 )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
% 12.24/7.99 = G ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6178_image__inv__into__cancel,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A,A6: set @ B,A11: set @ A,B13: set @ A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ A6 )
% 12.24/7.99 = A11 )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ B13 @ A11 )
% 12.24/7.99 => ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B13 ) )
% 12.24/7.99 = B13 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6179_inv__def,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A] :
% 12.24/7.99 ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 12.24/7.99 = ( ^ [Y6: A] :
% 12.24/7.99 ( fChoice @ B
% 12.24/7.99 @ ^ [X2: B] :
% 12.24/7.99 ( ( F @ X2 )
% 12.24/7.99 = Y6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6180_inj__on__inv__into,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,B7: set @ A,F: B > A,A6: set @ B] :
% 12.24/7.99 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A6 ) )
% 12.24/7.99 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B7 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6181_inj__transfer,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,P: A > $o,X: A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [Y5: B] :
% 12.24/7.99 ( ( member @ B @ Y5 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 12.24/7.99 => ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y5 ) ) )
% 12.24/7.99 => ( P @ X ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6182_image__inv__f__f,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ A6 ) )
% 12.24/7.99 = A6 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6183_inj__imp__surj__inv,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6184_surj__imp__inj__inv,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6185_inv__into__comp,axiom,
% 12.24/7.99 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A6: set @ C,X: B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) )
% 12.24/7.99 => ( ( inj_on @ C @ A @ G @ A6 )
% 12.24/7.99 => ( ( member @ B @ X @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) ) )
% 12.24/7.99 => ( ( hilbert_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 12.24/7.99 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A6 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image @ C @ A @ G @ A6 ) @ F ) @ X ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6186_inv__unique__comp,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 12.24/7.99 ( ( ( comp @ B @ A @ A @ F @ G )
% 12.24/7.99 = ( id @ A ) )
% 12.24/7.99 => ( ( ( comp @ A @ B @ B @ G @ F )
% 12.24/7.99 = ( id @ B ) )
% 12.24/7.99 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 12.24/7.99 = G ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6187_surj__iff,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: B > A] :
% 12.24/7.99 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
% 12.24/7.99 = ( id @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6188_inj__iff,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/7.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 12.24/7.99 = ( id @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6189_bijection_Oinv__comp__left,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 12.24/7.99 = ( id @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6190_bijection_Oinv__comp__right,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 12.24/7.99 = ( id @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6191_bijection_Oeq__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( ( F @ A3 )
% 12.24/7.99 = ( F @ B2 ) )
% 12.24/7.99 = ( A3 = B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6192_bijection_OeqI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( ( F @ A3 )
% 12.24/7.99 = ( F @ B2 ) )
% 12.24/7.99 => ( A3 = B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6193_bijection_Osurj,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6194_bijection_Oinj,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6195_bijection_Oinv__right__eq__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,B2: A,A3: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( B2
% 12.24/7.99 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
% 12.24/7.99 = ( ( F @ B2 )
% 12.24/7.99 = A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6196_bijection_Oinv__left__eq__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 12.24/7.99 = B2 )
% 12.24/7.99 = ( ( F @ B2 )
% 12.24/7.99 = A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6197_bijection_Oeq__inv__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 12.24/7.99 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
% 12.24/7.99 = ( A3 = B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6198_bijection_Oinv__right,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6199_bijection_Oinv__left,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ ( F @ A3 ) )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6200_bijection_Oeq__invI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A,A3: A,B2: A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
% 12.24/7.99 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
% 12.24/7.99 => ( A3 = B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6201_bijection_Osurj__inv,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6202_bijection_Oinj__inv,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A] :
% 12.24/7.99 ( ( hilbert_bijection @ A @ F )
% 12.24/7.99 => ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6203_iteratesp_Omono,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A] :
% 12.24/7.99 ( order_mono @ ( A > $o ) @ ( A > $o )
% 12.24/7.99 @ ^ [P6: A > $o,X2: A] :
% 12.24/7.99 ( ? [Y6: A] :
% 12.24/7.99 ( ( X2
% 12.24/7.99 = ( F @ Y6 ) )
% 12.24/7.99 & ( P6 @ Y6 ) )
% 12.24/7.99 | ? [M9: set @ A] :
% 12.24/7.99 ( ( X2
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M9 ) )
% 12.24/7.99 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 12.24/7.99 & ! [Y6: A] :
% 12.24/7.99 ( ( member @ A @ Y6 @ M9 )
% 12.24/7.99 => ( P6 @ Y6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6204_finite__subset__wf,axiom,
% 12.24/7.99 ! [A: $tType,A6: set @ A] :
% 12.24/7.99 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( wf @ ( set @ A )
% 12.24/7.99 @ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 12.24/7.99 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 12.24/7.99 @ ^ [X7: set @ A,Y9: set @ A] :
% 12.24/7.99 ( ( ord_less @ ( set @ A ) @ X7 @ Y9 )
% 12.24/7.99 & ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6205_wf,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wellorder @ A )
% 12.24/7.99 => ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6206_ccpo__Sup__least,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [A6: set @ A,Z3: A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( member @ A @ X3 @ A6 )
% 12.24/7.99 => ( ord_less_eq @ A @ X3 @ Z3 ) )
% 12.24/7.99 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6207_ccpo__Sup__upper,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [A6: set @ A,X: A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6208_wf__iff__no__infinite__down__chain,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( wf @ A )
% 12.24/7.99 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ~ ? [F2: nat > A] :
% 12.24/7.99 ! [I5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I5 ) ) @ ( F2 @ I5 ) ) @ R5 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6209_wf__no__infinite__down__chainE,axiom,
% 12.24/7.99 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 12.24/7.99 ( ( wf @ A @ R )
% 12.24/7.99 => ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K2 ) ) @ ( F @ K2 ) ) @ R ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6210_chain__singleton,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6211_wf__if__measure,axiom,
% 12.24/7.99 ! [A: $tType,P: A > $o,F: A > nat,G: A > A] :
% 12.24/7.99 ( ! [X3: A] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 => ( ord_less @ nat @ ( F @ ( G @ X3 ) ) @ ( F @ X3 ) ) )
% 12.24/7.99 => ( wf @ A
% 12.24/7.99 @ ( collect @ ( product_prod @ A @ A )
% 12.24/7.99 @ ( product_case_prod @ A @ A @ $o
% 12.24/7.99 @ ^ [Y6: A,X2: A] :
% 12.24/7.99 ( ( P @ X2 )
% 12.24/7.99 & ( Y6
% 12.24/7.99 = ( G @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6212_wf__bounded__measure,axiom,
% 12.24/7.99 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
% 12.24/7.99 ( ! [A8: A,B5: A] :
% 12.24/7.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
% 12.24/7.99 => ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
% 12.24/7.99 & ( ord_less_eq @ nat @ ( F @ B5 ) @ ( Ub @ A8 ) )
% 12.24/7.99 & ( ord_less @ nat @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
% 12.24/7.99 => ( wf @ A @ R ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6213_wf__linord__ex__has__least,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
% 12.24/7.99 ( ( wf @ A @ R )
% 12.24/7.99 => ( ! [X3: A,Y5: A] :
% 12.24/7.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y5 ) @ ( transitive_trancl @ A @ R ) )
% 12.24/7.99 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
% 12.24/7.99 => ( ( P @ K )
% 12.24/7.99 => ? [X3: B] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 & ! [Y4: B] :
% 12.24/7.99 ( ( P @ Y4 )
% 12.24/7.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y4 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6214_reduction__pair__lemma,axiom,
% 12.24/7.99 ! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( fun_reduction_pair @ A @ P )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
% 12.24/7.99 => ( ( wf @ A @ S3 )
% 12.24/7.99 => ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6215_wf__bounded__set,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
% 12.24/7.99 ( ! [A8: A,B5: A] :
% 12.24/7.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
% 12.24/7.99 => ( ( finite_finite2 @ B @ ( Ub @ A8 ) )
% 12.24/7.99 & ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
% 12.24/7.99 & ( ord_less_eq @ ( set @ B ) @ ( F @ B5 ) @ ( Ub @ A8 ) )
% 12.24/7.99 & ( ord_less @ ( set @ B ) @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
% 12.24/7.99 => ( wf @ A @ R ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6216_in__chain__finite,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( member @ A @ ( complete_Sup_Sup @ A @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6217_iteratesp__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ( ( comple7512665784863727008ratesp @ A )
% 12.24/7.99 = ( ^ [F2: A > A] :
% 12.24/7.99 ( complete_lattice_lfp @ ( A > $o )
% 12.24/7.99 @ ^ [P6: A > $o,X2: A] :
% 12.24/7.99 ( ? [Y6: A] :
% 12.24/7.99 ( ( X2
% 12.24/7.99 = ( F2 @ Y6 ) )
% 12.24/7.99 & ( P6 @ Y6 ) )
% 12.24/7.99 | ? [M9: set @ A] :
% 12.24/7.99 ( ( X2
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M9 ) )
% 12.24/7.99 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 12.24/7.99 & ! [Y6: A] :
% 12.24/7.99 ( ( member @ A @ Y6 @ M9 )
% 12.24/7.99 => ( P6 @ Y6 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6218_wf__int__ge__less__than2,axiom,
% 12.24/7.99 ! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6219_wf__int__ge__less__than,axiom,
% 12.24/7.99 ! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6220_wf__pair__less,axiom,
% 12.24/7.99 wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6221_wf__less,axiom,
% 12.24/7.99 wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6222_iteratesp_Osimps,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ( ( comple7512665784863727008ratesp @ A )
% 12.24/7.99 = ( ^ [F2: A > A,A5: A] :
% 12.24/7.99 ( ? [X2: A] :
% 12.24/7.99 ( ( A5
% 12.24/7.99 = ( F2 @ X2 ) )
% 12.24/7.99 & ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) )
% 12.24/7.99 | ? [M9: set @ A] :
% 12.24/7.99 ( ( A5
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M9 ) )
% 12.24/7.99 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 12.24/7.99 & ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ M9 )
% 12.24/7.99 => ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6223_iteratesp_Ocases,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A,A3: A] :
% 12.24/7.99 ( ( comple7512665784863727008ratesp @ A @ F @ A3 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( F @ X3 ) )
% 12.24/7.99 => ~ ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
% 12.24/7.99 => ~ ! [M8: set @ A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M8 ) )
% 12.24/7.99 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
% 12.24/7.99 => ~ ! [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ M8 )
% 12.24/7.99 => ( comple7512665784863727008ratesp @ A @ F @ X4 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6224_iteratesp_OSup,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [M7: set @ A,F: A > A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( member @ A @ X3 @ M7 )
% 12.24/7.99 => ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
% 12.24/7.99 => ( comple7512665784863727008ratesp @ A @ F @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6225_reduction__pair__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( fun_reduction_pair @ A )
% 12.24/7.99 = ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
% 12.24/7.99 ( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
% 12.24/7.99 & ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6226_admissible__chfin,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [P: A > $o] :
% 12.24/7.99 ( ! [S4: set @ A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S4 )
% 12.24/7.99 => ( finite_finite2 @ A @ S4 ) )
% 12.24/7.99 => ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6227_reduction__pairI,axiom,
% 12.24/7.99 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( wf @ A @ R2 )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 12.24/7.99 => ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6228_wf__no__loop,axiom,
% 12.24/7.99 ! [B: $tType,R2: set @ ( product_prod @ B @ B )] :
% 12.24/7.99 ( ( ( relcomp @ B @ B @ B @ R2 @ R2 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
% 12.24/7.99 => ( wf @ B @ R2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6229_union__comp__emptyL,axiom,
% 12.24/7.99 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.99 => ( ( ( relcomp @ A @ A @ A @ B7 @ C6 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.99 => ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) @ C6 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6230_union__comp__emptyR,axiom,
% 12.24/7.99 ! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( ( relcomp @ A @ A @ A @ A6 @ B7 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.99 => ( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 12.24/7.99 => ( ( relcomp @ A @ A @ A @ A6 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B7 @ C6 ) )
% 12.24/7.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6231_relpow_Osimps_I2_J,axiom,
% 12.24/7.99 ! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 )
% 12.24/7.99 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) @ R2 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6232_admissible__disj,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [P: A > $o,Q: A > $o] :
% 12.24/7.99 ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
% 12.24/7.99 => ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
% 12.24/7.99 => ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
% 12.24/7.99 @ ^ [X2: A] :
% 12.24/7.99 ( ( P @ X2 )
% 12.24/7.99 | ( Q @ X2 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6233_max__ext__compat,axiom,
% 12.24/7.99 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 12.24/7.99 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6234_min__ext__compat,axiom,
% 12.24/7.99 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
% 12.24/7.99 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6235_ntrancl__Suc,axiom,
% 12.24/7.99 ! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( transitive_ntrancl @ A @ ( suc @ N ) @ R2 )
% 12.24/7.99 = ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6236_and_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.99 => ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6237_monoid_Oright__neutral,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 12.24/7.99 ( ( monoid @ A @ F @ Z3 )
% 12.24/7.99 => ( ( F @ A3 @ Z3 )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6238_monoid_Oleft__neutral,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 12.24/7.99 ( ( monoid @ A @ F @ Z3 )
% 12.24/7.99 => ( ( F @ Z3 @ A3 )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6239_relpow_Osimps_I1_J,axiom,
% 12.24/7.99 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 12.24/7.99 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 )
% 12.24/7.99 = ( id2 @ A ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6240_add_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( monoid_add @ A )
% 12.24/7.99 => ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6241_mult_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( monoid_mult @ A )
% 12.24/7.99 => ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6242_sup__bot_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.99 => ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6243_inf__top_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.99 => ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6244_gcd__nat_Omonoid__axioms,axiom,
% 12.24/7.99 monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6245_or_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.99 => ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6246_xor_Omonoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.99 => ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6247_max__nat_Omonoid__axioms,axiom,
% 12.24/7.99 monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6248_pair__leq__def,axiom,
% 12.24/7.99 ( fun_pair_leq
% 12.24/7.99 = ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6249_cr__integer__def,axiom,
% 12.24/7.99 ( code_cr_integer
% 12.24/7.99 = ( ^ [X2: int,Y6: code_integer] :
% 12.24/7.99 ( X2
% 12.24/7.99 = ( code_int_of_integer @ Y6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6250_bit__integer__def,axiom,
% 12.24/7.99 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 12.24/7.99 = ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6251_integer_Opcr__cr__eq,axiom,
% 12.24/7.99 code_pcr_integer = code_cr_integer ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6252_arg__min__list_Osimps_I2_J,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( linorder @ B )
% 12.24/7.99 => ! [F: A > B,X: A,Y3: A,Zs: list @ A] :
% 12.24/7.99 ( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
% 12.24/7.99 = ( if @ A @ ( ord_less_eq @ B @ ( F @ X ) @ ( F @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6253_iterates_OSup,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [M7: set @ A,F: A > A] :
% 12.24/7.99 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( member @ A @ X3 @ M7 )
% 12.24/7.99 => ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 12.24/7.99 => ( member @ A @ ( complete_Sup_Sup @ A @ M7 ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6254_iterates_Osimps,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [A3: A,F: A > A] :
% 12.24/7.99 ( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
% 12.24/7.99 = ( ? [X2: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( F @ X2 ) )
% 12.24/7.99 & ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 12.24/7.99 | ? [M9: set @ A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M9 ) )
% 12.24/7.99 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
% 12.24/7.99 & ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ M9 )
% 12.24/7.99 => ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6255_iterates_Ocases,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [A3: A,F: A > A] :
% 12.24/7.99 ( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( F @ X3 ) )
% 12.24/7.99 => ~ ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
% 12.24/7.99 => ~ ! [M8: set @ A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 = ( complete_Sup_Sup @ A @ M8 ) )
% 12.24/7.99 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
% 12.24/7.99 => ~ ! [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ M8 )
% 12.24/7.99 => ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6256_arg__min__list_Oelims,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( linorder @ B )
% 12.24/7.99 => ! [X: A > B,Xa3: list @ A,Y3: A] :
% 12.24/7.99 ( ( ( arg_min_list @ A @ B @ X @ Xa3 )
% 12.24/7.99 = Y3 )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( Xa3
% 12.24/7.99 = ( cons @ A @ X3 @ ( nil @ A ) ) )
% 12.24/7.99 => ( Y3 != X3 ) )
% 12.24/7.99 => ( ! [X3: A,Y5: A,Zs2: list @ A] :
% 12.24/7.99 ( ( Xa3
% 12.24/7.99 = ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
% 12.24/7.99 => ( Y3
% 12.24/7.99 != ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) )
% 12.24/7.99 => ~ ( ( Xa3
% 12.24/7.99 = ( nil @ A ) )
% 12.24/7.99 => ( Y3
% 12.24/7.99 != ( undefined @ A ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6257_chain__iterates,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A] :
% 12.24/7.99 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6258_iterates__le__f,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [X: A,F: A > A] :
% 12.24/7.99 ( ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F ) )
% 12.24/7.99 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( ord_less_eq @ A @ X @ ( F @ X ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6259_fixp__induct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [P: A > $o,F: A > A] :
% 12.24/7.99 ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
% 12.24/7.99 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( P @ X3 )
% 12.24/7.99 => ( P @ ( F @ X3 ) ) )
% 12.24/7.99 => ( P @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6260_iterates__fixp,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A] :
% 12.24/7.99 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( member @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6261_fixp__unfold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A] :
% 12.24/7.99 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( ( comple115746919287870866o_fixp @ A @ F )
% 12.24/7.99 = ( F @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6262_fixp__lowerbound,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comple9053668089753744459l_ccpo @ A )
% 12.24/7.99 => ! [F: A > A,Z3: A] :
% 12.24/7.99 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 12.24/7.99 => ( ( ord_less_eq @ A @ ( F @ Z3 ) @ Z3 )
% 12.24/7.99 => ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ Z3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6263_arg__min__list_Opelims,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType] :
% 12.24/7.99 ( ( linorder @ B )
% 12.24/7.99 => ! [X: A > B,Xa3: list @ A,Y3: A] :
% 12.24/7.99 ( ( ( arg_min_list @ A @ B @ X @ Xa3 )
% 12.24/7.99 = Y3 )
% 12.24/7.99 => ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa3 ) )
% 12.24/7.99 => ( ! [X3: A] :
% 12.24/7.99 ( ( Xa3
% 12.24/7.99 = ( cons @ A @ X3 @ ( nil @ A ) ) )
% 12.24/7.99 => ( ( Y3 = X3 )
% 12.24/7.99 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
% 12.24/7.99 => ( ! [X3: A,Y5: A,Zs2: list @ A] :
% 12.24/7.99 ( ( Xa3
% 12.24/7.99 = ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
% 12.24/7.99 => ( ( Y3
% 12.24/7.99 = ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) )
% 12.24/7.99 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) ) )
% 12.24/7.99 => ~ ( ( Xa3
% 12.24/7.99 = ( nil @ A ) )
% 12.24/7.99 => ( ( Y3
% 12.24/7.99 = ( undefined @ A ) )
% 12.24/7.99 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6264_sorted__sort__key,axiom,
% 12.24/7.99 ! [A: $tType,B: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ! [F: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6265_sorted__sort,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( sorted_wrt @ A @ ( ord_less_eq @ A )
% 12.24/7.99 @ ( linorder_sort_key @ A @ A
% 12.24/7.99 @ ^ [X2: A] : X2
% 12.24/7.99 @ Xs ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6266_sorted__sort__id,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( linorder @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/7.99 => ( ( linorder_sort_key @ A @ A
% 12.24/7.99 @ ^ [X2: A] : X2
% 12.24/7.99 @ Xs )
% 12.24/7.99 = Xs ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6267_length__removeAll__less,axiom,
% 12.24/7.99 ! [A: $tType,X: A,Xs: list @ A] :
% 12.24/7.99 ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 12.24/7.99 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6268_card__Plus__conv__if,axiom,
% 12.24/7.99 ! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
% 12.24/7.99 ( ( ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 & ( finite_finite2 @ B @ B7 ) )
% 12.24/7.99 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
% 12.24/7.99 = ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) )
% 12.24/7.99 & ( ~ ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 & ( finite_finite2 @ B @ B7 ) )
% 12.24/7.99 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
% 12.24/7.99 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6269_length__removeAll__less__eq,axiom,
% 12.24/7.99 ! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6270_and_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_ri3973907225187159222ations @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6271_lcm__altdef__int,axiom,
% 12.24/7.99 ( ( gcd_lcm @ int )
% 12.24/7.99 = ( ^ [A5: int,B3: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A5 ) @ ( abs_abs @ int @ B3 ) ) @ ( gcd_gcd @ int @ A5 @ B3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6272_lcm_Obottom__left__bottom,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6273_lcm_Obottom__right__bottom,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6274_lcm__neg1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( ring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 12.24/7.99 = ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6275_lcm__neg2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( ring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 12.24/7.99 = ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6276_lcm__0__iff__int,axiom,
% 12.24/7.99 ! [M: int,N: int] :
% 12.24/7.99 ( ( ( gcd_lcm @ int @ M @ N )
% 12.24/7.99 = ( zero_zero @ int ) )
% 12.24/7.99 = ( ( M
% 12.24/7.99 = ( zero_zero @ int ) )
% 12.24/7.99 | ( N
% 12.24/7.99 = ( zero_zero @ int ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6277_lcm__neg__numeral__1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( ring_gcd @ A )
% 12.24/7.99 => ! [N: num,A3: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
% 12.24/7.99 = ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6278_lcm__neg__numeral__2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( ring_gcd @ A )
% 12.24/7.99 => ! [A3: A,N: num] :
% 12.24/7.99 ( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 12.24/7.99 = ( gcd_lcm @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6279_lcm__eq__1__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 12.24/7.99 = ( one_one @ A ) )
% 12.24/7.99 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6280_lcm__1__iff__int,axiom,
% 12.24/7.99 ! [M: int,N: int] :
% 12.24/7.99 ( ( ( gcd_lcm @ int @ M @ N )
% 12.24/7.99 = ( one_one @ int ) )
% 12.24/7.99 = ( ( ( M
% 12.24/7.99 = ( one_one @ int ) )
% 12.24/7.99 | ( M
% 12.24/7.99 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 12.24/7.99 & ( ( N
% 12.24/7.99 = ( one_one @ int ) )
% 12.24/7.99 | ( N
% 12.24/7.99 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6281_lcm__ge__0__int,axiom,
% 12.24/7.99 ! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6282_semilattice__neutr_Oaxioms_I2_J,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/7.99 ( ( semilattice_neutr @ A @ F @ Z3 )
% 12.24/7.99 => ( comm_monoid @ A @ F @ Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6283_lcm__neg1__int,axiom,
% 12.24/7.99 ! [X: int,Y3: int] :
% 12.24/7.99 ( ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
% 12.24/7.99 = ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6284_lcm__neg2__int,axiom,
% 12.24/7.99 ! [X: int,Y3: int] :
% 12.24/7.99 ( ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
% 12.24/7.99 = ( gcd_lcm @ int @ X @ Y3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6285_lcm__pos__int,axiom,
% 12.24/7.99 ! [M: int,N: int] :
% 12.24/7.99 ( ( M
% 12.24/7.99 != ( zero_zero @ int ) )
% 12.24/7.99 => ( ( N
% 12.24/7.99 != ( zero_zero @ int ) )
% 12.24/7.99 => ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ M @ N ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6286_comm__monoid_Ocomm__neutral,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A3: A] :
% 12.24/7.99 ( ( comm_monoid @ A @ F @ Z3 )
% 12.24/7.99 => ( ( F @ A3 @ Z3 )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6287_zero__eq__lcm__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( zero_zero @ A )
% 12.24/7.99 = ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 | ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6288_lcm__eq__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 | ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6289_add_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comm_monoid_add @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6290_mult_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( comm_monoid_mult @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6291_lcm__mult__unit1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 12.24/7.99 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6292_lcm__mult__unit2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 12.24/7.99 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6293_lcm__div__unit1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 12.24/7.99 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6294_lcm__div__unit2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 12.24/7.99 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6295_sup__bot_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4967611905675639751up_bot @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6296_inf__top_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bounde4346867609351753570nf_top @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6297_or_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6298_gcd__nat_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6299_xor_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( bit_se359711467146920520ations @ A )
% 12.24/7.99 => ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6300_lcm__cases__int,axiom,
% 12.24/7.99 ! [X: int,Y3: int,P: int > $o] :
% 12.24/7.99 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.99 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.99 => ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) )
% 12.24/7.99 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 12.24/7.99 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 12.24/7.99 => ( P @ ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 12.24/7.99 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 12.24/7.99 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
% 12.24/7.99 => ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
% 12.24/7.99 => ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
% 12.24/7.99 => ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
% 12.24/7.99 => ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
% 12.24/7.99 => ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6301_max__nat_Ocomm__monoid__axioms,axiom,
% 12.24/7.99 comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6302_lcm__unique__int,axiom,
% 12.24/7.99 ! [D2: int,A3: int,B2: int] :
% 12.24/7.99 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
% 12.24/7.99 & ( dvd_dvd @ int @ A3 @ D2 )
% 12.24/7.99 & ( dvd_dvd @ int @ B2 @ D2 )
% 12.24/7.99 & ! [E3: int] :
% 12.24/7.99 ( ( ( dvd_dvd @ int @ A3 @ E3 )
% 12.24/7.99 & ( dvd_dvd @ int @ B2 @ E3 ) )
% 12.24/7.99 => ( dvd_dvd @ int @ D2 @ E3 ) ) )
% 12.24/7.99 = ( D2
% 12.24/7.99 = ( gcd_lcm @ int @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6303_prod__gcd__lcm__int,axiom,
% 12.24/7.99 ! [M: int,N: int] :
% 12.24/7.99 ( ( times_times @ int @ ( abs_abs @ int @ M ) @ ( abs_abs @ int @ N ) )
% 12.24/7.99 = ( times_times @ int @ ( gcd_gcd @ int @ M @ N ) @ ( gcd_lcm @ int @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6304_Lcm__int__set__eq__fold,axiom,
% 12.24/7.99 ! [Xs: list @ int] :
% 12.24/7.99 ( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
% 12.24/7.99 = ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6305_Lcm__fin_Oeq__fold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( semiring_gcd_Lcm_fin @ A )
% 12.24/7.99 = ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6306_lcm__0__iff__nat,axiom,
% 12.24/7.99 ! [M: nat,N: nat] :
% 12.24/7.99 ( ( ( gcd_lcm @ nat @ M @ N )
% 12.24/7.99 = ( zero_zero @ nat ) )
% 12.24/7.99 = ( ( M
% 12.24/7.99 = ( zero_zero @ nat ) )
% 12.24/7.99 | ( N
% 12.24/7.99 = ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6307_lcm__1__iff__nat,axiom,
% 12.24/7.99 ! [M: nat,N: nat] :
% 12.24/7.99 ( ( ( gcd_lcm @ nat @ M @ N )
% 12.24/7.99 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.99 = ( ( M
% 12.24/7.99 = ( suc @ ( zero_zero @ nat ) ) )
% 12.24/7.99 & ( N
% 12.24/7.99 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6308_lcm__int__int__eq,axiom,
% 12.24/7.99 ! [M: nat,N: nat] :
% 12.24/7.99 ( ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 12.24/7.99 = ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6309_Lcm__UNIV,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6310_Lcm__empty,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6311_Lcm__1__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( one_one @ A ) )
% 12.24/7.99 = ( ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ A6 )
% 12.24/7.99 => ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6312_Lcm__fin_Oinfinite,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6313_Lcm__fin_Oempty,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6314_is__unit__Lcm__fin__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) @ ( one_one @ A ) )
% 12.24/7.99 = ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6315_lcm__nat__abs__right__eq,axiom,
% 12.24/7.99 ! [N: nat,K: int] :
% 12.24/7.99 ( ( gcd_lcm @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
% 12.24/7.99 = ( nat2 @ ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6316_lcm__nat__abs__left__eq,axiom,
% 12.24/7.99 ! [K: int,N: nat] :
% 12.24/7.99 ( ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
% 12.24/7.99 = ( nat2 @ ( gcd_lcm @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6317_prod__gcd__lcm__nat,axiom,
% 12.24/7.99 ( ( times_times @ nat )
% 12.24/7.99 = ( ^ [M2: nat,N3: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M2 @ N3 ) @ ( gcd_lcm @ nat @ M2 @ N3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6318_lcm__pos__nat,axiom,
% 12.24/7.99 ! [M: nat,N: nat] :
% 12.24/7.99 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 12.24/7.99 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/7.99 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M @ N ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6319_Lcm__no__multiple,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ! [M6: A] :
% 12.24/7.99 ( ( M6
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ? [X4: A] :
% 12.24/7.99 ( ( member @ A @ X4 @ A6 )
% 12.24/7.99 & ~ ( dvd_dvd @ A @ X4 @ M6 ) ) )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6320_Lcm__0__iff_H,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( ~ ? [L3: A] :
% 12.24/7.99 ( ( L3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 & ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ A6 )
% 12.24/7.99 => ( dvd_dvd @ A @ X2 @ L3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6321_Lcm__eq__0__I,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( member @ A @ ( zero_zero @ A ) @ A6 )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6322_Lcm__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6323_Lcm__int__greater__eq__0,axiom,
% 12.24/7.99 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Lcm @ int @ K5 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6324_Lcm__fin__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6325_lcm__nat__def,axiom,
% 12.24/7.99 ( ( gcd_lcm @ nat )
% 12.24/7.99 = ( ^ [X2: nat,Y6: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y6 ) @ ( gcd_gcd @ nat @ X2 @ Y6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6326_Lcm__fin__1__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 12.24/7.99 = ( one_one @ A ) )
% 12.24/7.99 = ( ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ A6 )
% 12.24/7.99 => ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
% 12.24/7.99 & ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6327_Lcm__no__units,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ( ( gcd_Lcm @ A )
% 12.24/7.99 = ( ^ [A7: set @ A] :
% 12.24/7.99 ( gcd_Lcm @ A
% 12.24/7.99 @ ( minus_minus @ ( set @ A ) @ A7
% 12.24/7.99 @ ( collect @ A
% 12.24/7.99 @ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6328_lcm__int__def,axiom,
% 12.24/7.99 ( ( gcd_lcm @ int )
% 12.24/7.99 = ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6329_Lcm__set__eq__fold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
% 12.24/7.99 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6330_Lcm__fin_Oset__eq__fold,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [Xs: list @ A] :
% 12.24/7.99 ( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
% 12.24/7.99 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6331_Lcm__fin__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( semiring_gcd_Lcm_fin @ A )
% 12.24/7.99 = ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6332_Lcm__int__def,axiom,
% 12.24/7.99 ( ( gcd_Lcm @ int )
% 12.24/7.99 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6333_Lcm__int__eq,axiom,
% 12.24/7.99 ! [N6: set @ nat] :
% 12.24/7.99 ( ( gcd_Lcm @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
% 12.24/7.99 = ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N6 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6334_Lcm__eq__0__I__nat,axiom,
% 12.24/7.99 ! [A6: set @ nat] :
% 12.24/7.99 ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
% 12.24/7.99 => ( ( gcd_Lcm @ nat @ A6 )
% 12.24/7.99 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6335_Lcm__0__iff__nat,axiom,
% 12.24/7.99 ! [A6: set @ nat] :
% 12.24/7.99 ( ( finite_finite2 @ nat @ A6 )
% 12.24/7.99 => ( ( ( gcd_Lcm @ nat @ A6 )
% 12.24/7.99 = ( zero_zero @ nat ) )
% 12.24/7.99 = ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6336_Lcm__nat__infinite,axiom,
% 12.24/7.99 ! [M7: set @ nat] :
% 12.24/7.99 ( ~ ( finite_finite2 @ nat @ M7 )
% 12.24/7.99 => ( ( gcd_Lcm @ nat @ M7 )
% 12.24/7.99 = ( zero_zero @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6337_Lcm__nat__empty,axiom,
% 12.24/7.99 ( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.99 = ( one_one @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6338_Lcm__nat__set__eq__fold,axiom,
% 12.24/7.99 ! [Xs: list @ nat] :
% 12.24/7.99 ( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
% 12.24/7.99 = ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6339_Lcm__eq__Max__nat,axiom,
% 12.24/7.99 ! [M7: set @ nat] :
% 12.24/7.99 ( ( finite_finite2 @ nat @ M7 )
% 12.24/7.99 => ( ( M7
% 12.24/7.99 != ( bot_bot @ ( set @ nat ) ) )
% 12.24/7.99 => ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
% 12.24/7.99 => ( ! [M6: nat,N4: nat] :
% 12.24/7.99 ( ( member @ nat @ M6 @ M7 )
% 12.24/7.99 => ( ( member @ nat @ N4 @ M7 )
% 12.24/7.99 => ( member @ nat @ ( gcd_lcm @ nat @ M6 @ N4 ) @ M7 ) ) )
% 12.24/7.99 => ( ( gcd_Lcm @ nat @ M7 )
% 12.24/7.99 = ( lattic643756798349783984er_Max @ nat @ M7 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6340_Lcm__nat__def,axiom,
% 12.24/7.99 ( ( gcd_Lcm @ nat )
% 12.24/7.99 = ( ^ [M9: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M9 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M9 ) @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6341_Lcm__coprime_H,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( finite_card @ A @ A6 )
% 12.24/7.99 != ( zero_zero @ nat ) )
% 12.24/7.99 => ( ! [A8: A,B5: A] :
% 12.24/7.99 ( ( member @ A @ A8 @ A6 )
% 12.24/7.99 => ( ( member @ A @ B5 @ A6 )
% 12.24/7.99 => ( ( A8 != B5 )
% 12.24/7.99 => ( algebr8660921524188924756oprime @ A @ A8 @ B5 ) ) ) )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A
% 12.24/7.99 @ ( groups7121269368397514597t_prod @ A @ A
% 12.24/7.99 @ ^ [X2: A] : X2
% 12.24/7.99 @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6342_normalize__idem,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6343_lcm_Onormalize__bottom,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6344_normalize__eq__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( A3
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6345_normalize__0,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6346_normalize__mult__normalize__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6347_normalize__mult__normalize__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6348_gcd_Onormalize__bottom,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6349_normalize__1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6350_dvd__normalize__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6351_normalize__dvd__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
% 12.24/7.99 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6352_coprime__normalize__left__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
% 12.24/7.99 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6353_coprime__normalize__right__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6354_gcd_Otop__left__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_gcd @ A @ ( zero_zero @ A ) @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6355_gcd_Otop__right__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_gcd @ A @ A3 @ ( zero_zero @ A ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6356_lcm_Otop__right__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6357_lcm_Otop__left__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6358_normalize__mult__unit__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [B2: A,A3: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6359_normalize__mult__unit__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6360_lcm__mult__gcd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6361_gcd__mult__lcm,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6362_lcm__mult__distrib_H,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6363_lcm__mult__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6364_lcm__mult__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6365_normalize__nat__def,axiom,
% 12.24/7.99 ( ( normal6383669964737779283malize @ nat )
% 12.24/7.99 = ( id @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6366_gcd__mult__distrib_H,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6367_gcd__mult__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,C2: A,B2: A] :
% 12.24/7.99 ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6368_gcd__mult__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6369_dvd__normalize__div,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal6328177297339901930cative @ A )
% 12.24/7.99 => ! [B2: A,A3: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6370_normalize__idem__imp__is__unit__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = A3 )
% 12.24/7.99 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 = ( A3
% 12.24/7.99 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6371_is__unit__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6372_normalize__1__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( one_one @ A ) )
% 12.24/7.99 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6373_associated__unit,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6374_normalize__int__def,axiom,
% 12.24/7.99 ( ( normal6383669964737779283malize @ int )
% 12.24/7.99 = ( abs_abs @ int ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6375_semilattice__neutr__set_OF_Ocong,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( lattic5214292709420241887eutr_F @ A )
% 12.24/7.99 = ( lattic5214292709420241887eutr_F @ A ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6376_associated__iff__dvd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 = ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.24/7.99 & ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6377_associated__eqI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.24/7.99 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.24/7.99 => ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = A3 )
% 12.24/7.99 => ( ( ( normal6383669964737779283malize @ A @ B2 )
% 12.24/7.99 = B2 )
% 12.24/7.99 => ( A3 = B2 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6378_associatedD2,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6379_associatedD1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 => ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6380_associatedI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ B2 )
% 12.24/7.99 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6381_normalize__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal6328177297339901930cative @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6382_coprime__crossproduct,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,D2: A,B2: A,C2: A] :
% 12.24/7.99 ( ( algebr8660921524188924756oprime @ A @ A3 @ D2 )
% 12.24/7.99 => ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
% 12.24/7.99 => ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
% 12.24/7.99 = ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D2 ) ) )
% 12.24/7.99 = ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 & ( ( normal6383669964737779283malize @ A @ C2 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ D2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6383_lcm__gcd__prod,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6384_Gcd__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [C2: A,A6: set @ A] :
% 12.24/7.99 ( ( gcd_Gcd @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6385_lcm__coprime,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.99 => ( ( gcd_lcm @ A @ A3 @ B2 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6386_Lcm__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A,C2: A] :
% 12.24/7.99 ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( gcd_Lcm @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6387_lcm__gcd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( ( gcd_lcm @ A )
% 12.24/7.99 = ( ^ [A5: A,B3: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B3 ) @ ( gcd_gcd @ A @ A5 @ B3 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6388_Lcm__fin__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A,B2: A] :
% 12.24/7.99 ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ( semiring_gcd_Lcm_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6389_Gcd__fin__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A,B2: A] :
% 12.24/7.99 ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( semiring_gcd_Gcd_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6390_gcd__lcm,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( B2
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6391_semilattice__neutr__set_Oinsert__remove,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6392_semilattice__neutr__set_Oremove,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 12.24/7.99 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6393_semilattice__neutr__set_Ointro,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/7.99 ( ( semilattice_neutr @ A @ F @ Z3 )
% 12.24/7.99 => ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6394_semilattice__neutr__set_Oaxioms,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6395_semilattice__neutr__set__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A )
% 12.24/7.99 = ( semilattice_neutr @ A ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6396_semilattice__neutr__set_Oempty,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 = Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6397_semilattice__neutr__set_Oin__idem,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ X @ A6 )
% 12.24/7.99 => ( ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 12.24/7.99 = ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6398_semilattice__neutr__set_Oinfinite,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 12.24/7.99 = Z3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6399_semilattice__neutr__set_Oeq__fold,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
% 12.24/7.99 = ( finite_fold @ A @ A @ F @ Z3 @ A6 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6400_semilattice__neutr__set_Osubset,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/7.99 => ( ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 12.24/7.99 = ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6401_semilattice__neutr__set_Oinsert,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
% 12.24/7.99 = ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6402_semilattice__neutr__set_Ounion,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.99 => ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
% 12.24/7.99 = ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6403_semilattice__neutr__set_Oclosed,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
% 12.24/7.99 ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( A6
% 12.24/7.99 != ( bot_bot @ ( set @ A ) ) )
% 12.24/7.99 => ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 12.24/7.99 => ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6404_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6405_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6406_semilattice__order__neutr__set_Osubset__imp,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 12.24/7.99 => ( ( finite_finite2 @ A @ B7 )
% 12.24/7.99 => ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6407_lcm_Obounded__quasi__semilattice__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6408_semilattice__order__neutr__set_Oaxioms_I2_J,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6409_semilattice__order__neutr__set_Oaxioms_I1_J,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6410_semilattice__order__neutr__set_OboundedE,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 12.24/7.99 => ! [A10: A] :
% 12.24/7.99 ( ( member @ A @ A10 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ A10 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6411_semilattice__order__neutr__set_OboundedI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ! [A8: A] :
% 12.24/7.99 ( ( member @ A @ A8 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ A8 ) )
% 12.24/7.99 => ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6412_semilattice__order__neutr__set_OcoboundedI,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( member @ A @ A3 @ A6 )
% 12.24/7.99 => ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6413_semilattice__order__neutr__set_Obounded__iff,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( finite_finite2 @ A @ A6 )
% 12.24/7.99 => ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
% 12.24/7.99 = ( ! [X2: A] :
% 12.24/7.99 ( ( member @ A @ X2 @ A6 )
% 12.24/7.99 => ( Less_eq @ X @ X2 ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6414_semilattice__order__neutr__set__def,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( lattic3600114342068043075tr_set @ A )
% 12.24/7.99 = ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/7.99 ( ( semila1105856199041335345_order @ A @ F2 @ Z7 @ Less_eq2 @ Less2 )
% 12.24/7.99 & ( lattic5652469242046573047tr_set @ A @ F2 @ Z7 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6415_semilattice__order__neutr__set_Ointro,axiom,
% 12.24/7.99 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/7.99 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/7.99 => ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
% 12.24/7.99 => ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6416_gcd_Obounded__quasi__semilattice__axioms,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6417_normalize__div,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
% 12.24/7.99 = ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6418_normalize__unit__factor,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6419_unit__factor__simps_I1_J,axiom,
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
% 12.24/7.99 = ( zero_zero @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6420_unit__factor__idem,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 12.24/7.99 = ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6421_unit__factor__simps_I2_J,axiom,
% 12.24/7.99 ! [N: nat] :
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
% 12.24/7.99 = ( one_one @ nat ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6422_unit__factor__eq__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( unit_f5069060285200089521factor @ A @ A3 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( A3
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6423_unit__factor__0,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semido2269285787275462019factor @ A )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
% 12.24/7.99 = ( zero_zero @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6424_unit__factor__1,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6425_normalize__mult__unit__factor,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6426_unit__factor__mult__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
% 12.24/7.99 = A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6427_div__unit__factor,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6428_div__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
% 12.24/7.99 = ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6429_inv__unit__factor__eq__0__iff,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 = ( A3
% 12.24/7.99 = ( zero_zero @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6430_unit__factor__mult__unit__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semido2269285787275462019factor @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
% 12.24/7.99 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6431_unit__factor__mult__unit__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semido2269285787275462019factor @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6432_mult__one__div__unit__factor,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
% 12.24/7.99 = ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6433_unit__factor__lcm,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 | ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 & ( ~ ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 | ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6434_unit__factor__normalize,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6435_normalize__unit__factor__eqI,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = ( normal6383669964737779283malize @ A @ B2 ) )
% 12.24/7.99 => ( ( ( unit_f5069060285200089521factor @ A @ A3 )
% 12.24/7.99 = ( unit_f5069060285200089521factor @ A @ B2 ) )
% 12.24/7.99 => ( A3 = B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6436_lcm__mult__distrib,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [K: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ K @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6437_mult__lcm__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.99 = ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6438_mult__lcm__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6439_unit__factor__nat__def,axiom,
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ nat )
% 12.24/7.99 = ( ^ [N3: nat] :
% 12.24/7.99 ( if @ nat
% 12.24/7.99 @ ( N3
% 12.24/7.99 = ( zero_zero @ nat ) )
% 12.24/7.99 @ ( zero_zero @ nat )
% 12.24/7.99 @ ( one_one @ nat ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6440_dvd__unit__factor__div,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal6328177297339901930cative @ A )
% 12.24/7.99 => ! [B2: A,A3: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ B2 @ A3 )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6441_unit__factor__self,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6442_unit__factor__int__def,axiom,
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ int )
% 12.24/7.99 = ( sgn_sgn @ int ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6443_unit__factor__mult,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal6328177297339901930cative @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6444_unit__factor__dvd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ B2 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6445_is__unit__unit__factor,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semido2269285787275462019factor @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 12.24/7.99 = A3 ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6446_gcd__mult__distrib,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [K: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ K @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6447_mult__gcd__right,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [A3: A,B2: A,C2: A] :
% 12.24/7.99 ( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 )
% 12.24/7.99 = ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6448_mult__gcd__left,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [C2: A,A3: A,B2: A] :
% 12.24/7.99 ( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6449_unit__factor__is__unit,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semido2269285787275462019factor @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( A3
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6450_unit__factor__gcd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A3: A,B2: A] :
% 12.24/7.99 ( ( ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 & ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 & ( ~ ( ( A3
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 & ( B2
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6451_coprime__crossproduct_H,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiri6843258321239162965malize @ A )
% 12.24/7.99 => ! [B2: A,D2: A,A3: A,C2: A] :
% 12.24/7.99 ( ( B2
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( ( unit_f5069060285200089521factor @ A @ B2 )
% 12.24/7.99 = ( unit_f5069060285200089521factor @ A @ D2 ) )
% 12.24/7.99 => ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 12.24/7.99 => ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
% 12.24/7.99 => ( ( ( times_times @ A @ A3 @ D2 )
% 12.24/7.99 = ( times_times @ A @ B2 @ C2 ) )
% 12.24/7.99 = ( ( A3 = C2 )
% 12.24/7.99 & ( B2 = D2 ) ) ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6452_unit__factor__Lcm,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 & ( ( ( gcd_Lcm @ A @ A6 )
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6453_unit__factor__Gcd,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_Gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( ( ( gcd_Gcd @ A @ A6 )
% 12.24/7.99 = ( zero_zero @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
% 12.24/7.99 = ( zero_zero @ A ) ) )
% 12.24/7.99 & ( ( ( gcd_Gcd @ A @ A6 )
% 12.24/7.99 != ( zero_zero @ A ) )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
% 12.24/7.99 = ( one_one @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6454_normalize__idem__imp__unit__factor__eq,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( normal8620421768224518004emidom @ A )
% 12.24/7.99 => ! [A3: A] :
% 12.24/7.99 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 12.24/7.99 = A3 )
% 12.24/7.99 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 12.24/7.99 = ( zero_neq_one_of_bool @ A
% 12.24/7.99 @ ( A3
% 12.24/7.99 != ( zero_zero @ A ) ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6455_unit__factor__Lcm__fin,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) )
% 12.24/7.99 = ( zero_neq_one_of_bool @ A
% 12.24/7.99 @ ( ( semiring_gcd_Lcm_fin @ A @ A6 )
% 12.24/7.99 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6456_unit__factor__Gcd__fin,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( semiring_gcd @ A )
% 12.24/7.99 => ! [A6: set @ A] :
% 12.24/7.99 ( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) )
% 12.24/7.99 = ( zero_neq_one_of_bool @ A
% 12.24/7.99 @ ( ( semiring_gcd_Gcd_fin @ A @ A6 )
% 12.24/7.99 != ( zero_zero @ A ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6457_times__num__def,axiom,
% 12.24/7.99 ( ( times_times @ num )
% 12.24/7.99 = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6458_nat__of__num__pos,axiom,
% 12.24/7.99 ! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6459_nat__of__num_Osimps_I2_J,axiom,
% 12.24/7.99 ! [X: num] :
% 12.24/7.99 ( ( nat_of_num @ ( bit0 @ X ) )
% 12.24/7.99 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 12.24/7.99
% 12.24/7.99 thf(fact_6460_is__num__normalize_I5_J,axiom,
% 12.24/7.99 ! [A: $tType] :
% 12.24/7.99 ( ( neg_numeral @ A )
% 12.24/7.99 => ! [X: A] :
% 12.24/8.00 ( ( neg_numeral_is_num @ A @ X )
% 12.24/8.00 => ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6461_is__num__numeral,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ! [K: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6462_nat__of__num__neq__0,axiom,
% 12.24/8.00 ! [X: num] :
% 12.24/8.00 ( ( nat_of_num @ X )
% 12.24/8.00 != ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6463_num__eq__iff,axiom,
% 12.24/8.00 ( ( ^ [Y: num,Z2: num] : ( Y = Z2 ) )
% 12.24/8.00 = ( ^ [X2: num,Y6: num] :
% 12.24/8.00 ( ( nat_of_num @ X2 )
% 12.24/8.00 = ( nat_of_num @ Y6 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6464_nat__of__num__numeral,axiom,
% 12.24/8.00 ( nat_of_num
% 12.24/8.00 = ( numeral_numeral @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6465_nat__of__num__inverse,axiom,
% 12.24/8.00 ! [X: num] :
% 12.24/8.00 ( ( num_of_nat @ ( nat_of_num @ X ) )
% 12.24/8.00 = X ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6466_is__num__normalize_I4_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6467_is__num__add__left__commute,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ! [X: A,Y3: A,Z3: A] :
% 12.24/8.00 ( ( neg_numeral_is_num @ A @ X )
% 12.24/8.00 => ( ( neg_numeral_is_num @ A @ Y3 )
% 12.24/8.00 => ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y3 @ Z3 ) )
% 12.24/8.00 = ( plus_plus @ A @ Y3 @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6468_is__num__add__commute,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ! [X: A,Y3: A] :
% 12.24/8.00 ( ( neg_numeral_is_num @ A @ X )
% 12.24/8.00 => ( ( neg_numeral_is_num @ A @ Y3 )
% 12.24/8.00 => ( ( plus_plus @ A @ X @ Y3 )
% 12.24/8.00 = ( plus_plus @ A @ Y3 @ X ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6469_is__num__normalize_I6_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ! [X: A,Y3: A] :
% 12.24/8.00 ( ( neg_numeral_is_num @ A @ X )
% 12.24/8.00 => ( ( neg_numeral_is_num @ A @ Y3 )
% 12.24/8.00 => ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6470_nat__of__num__code_I1_J,axiom,
% 12.24/8.00 ( ( nat_of_num @ one2 )
% 12.24/8.00 = ( one_one @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6471_nat__of__num__add,axiom,
% 12.24/8.00 ! [X: num,Y3: num] :
% 12.24/8.00 ( ( nat_of_num @ ( plus_plus @ num @ X @ Y3 ) )
% 12.24/8.00 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6472_nat__of__num__mult,axiom,
% 12.24/8.00 ! [X: num,Y3: num] :
% 12.24/8.00 ( ( nat_of_num @ ( times_times @ num @ X @ Y3 ) )
% 12.24/8.00 = ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6473_nat__of__num__inc,axiom,
% 12.24/8.00 ! [X: num] :
% 12.24/8.00 ( ( nat_of_num @ ( inc @ X ) )
% 12.24/8.00 = ( suc @ ( nat_of_num @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6474_less__num__def,axiom,
% 12.24/8.00 ( ( ord_less @ num )
% 12.24/8.00 = ( ^ [M2: num,N3: num] : ( ord_less @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6475_less__eq__num__def,axiom,
% 12.24/8.00 ( ( ord_less_eq @ num )
% 12.24/8.00 = ( ^ [M2: num,N3: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6476_nat__of__num__code_I2_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( nat_of_num @ ( bit0 @ N ) )
% 12.24/8.00 = ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6477_nat__of__num__sqr,axiom,
% 12.24/8.00 ! [X: num] :
% 12.24/8.00 ( ( nat_of_num @ ( sqr @ X ) )
% 12.24/8.00 = ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6478_nat__of__num_Osimps_I1_J,axiom,
% 12.24/8.00 ( ( nat_of_num @ one2 )
% 12.24/8.00 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6479_nat__of__num_Osimps_I3_J,axiom,
% 12.24/8.00 ! [X: num] :
% 12.24/8.00 ( ( nat_of_num @ ( bit1 @ X ) )
% 12.24/8.00 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6480_num__of__nat__inverse,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 12.24/8.00 => ( ( nat_of_num @ ( num_of_nat @ N ) )
% 12.24/8.00 = N ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6481_plus__num__def,axiom,
% 12.24/8.00 ( ( plus_plus @ num )
% 12.24/8.00 = ( ^ [M2: num,N3: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6482_nat__of__num__code_I3_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( nat_of_num @ ( bit1 @ N ) )
% 12.24/8.00 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6483_is__num_Ocases,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ! [A3: A] :
% 12.24/8.00 ( ( neg_numeral_is_num @ A @ A3 )
% 12.24/8.00 => ( ( A3
% 12.24/8.00 != ( one_one @ A ) )
% 12.24/8.00 => ( ! [X3: A] :
% 12.24/8.00 ( ( A3
% 12.24/8.00 = ( uminus_uminus @ A @ X3 ) )
% 12.24/8.00 => ~ ( neg_numeral_is_num @ A @ X3 ) )
% 12.24/8.00 => ~ ! [X3: A,Y5: A] :
% 12.24/8.00 ( ( A3
% 12.24/8.00 = ( plus_plus @ A @ X3 @ Y5 ) )
% 12.24/8.00 => ( ( neg_numeral_is_num @ A @ X3 )
% 12.24/8.00 => ~ ( neg_numeral_is_num @ A @ Y5 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6484_is__num_Osimps,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( neg_numeral @ A )
% 12.24/8.00 => ( ( neg_numeral_is_num @ A )
% 12.24/8.00 = ( ^ [A5: A] :
% 12.24/8.00 ( ( A5
% 12.24/8.00 = ( one_one @ A ) )
% 12.24/8.00 | ? [X2: A] :
% 12.24/8.00 ( ( A5
% 12.24/8.00 = ( uminus_uminus @ A @ X2 ) )
% 12.24/8.00 & ( neg_numeral_is_num @ A @ X2 ) )
% 12.24/8.00 | ? [X2: A,Y6: A] :
% 12.24/8.00 ( ( A5
% 12.24/8.00 = ( plus_plus @ A @ X2 @ Y6 ) )
% 12.24/8.00 & ( neg_numeral_is_num @ A @ X2 )
% 12.24/8.00 & ( neg_numeral_is_num @ A @ Y6 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6485_arg__max__nat__lemma,axiom,
% 12.24/8.00 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 12.24/8.00 ( ( P @ K )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 12.24/8.00 => ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) )
% 12.24/8.00 & ! [Y4: A] :
% 12.24/8.00 ( ( P @ Y4 )
% 12.24/8.00 => ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6486_arg__max__nat__le,axiom,
% 12.24/8.00 ! [A: $tType,P: A > $o,X: A,F: A > nat,B2: nat] :
% 12.24/8.00 ( ( P @ X )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 12.24/8.00 => ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6487_arg__max__natI,axiom,
% 12.24/8.00 ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
% 12.24/8.00 ( ( P @ K )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
% 12.24/8.00 => ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6488_arg__max__equality,axiom,
% 12.24/8.00 ! [A: $tType,C: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ! [P: C > $o,K: C,F: C > A] :
% 12.24/8.00 ( ( P @ K )
% 12.24/8.00 => ( ! [X3: C] :
% 12.24/8.00 ( ( P @ X3 )
% 12.24/8.00 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ K ) ) )
% 12.24/8.00 => ( ( F @ ( lattices_ord_arg_max @ C @ A @ F @ P ) )
% 12.24/8.00 = ( F @ K ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6489_arg__maxI,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( ord @ B )
% 12.24/8.00 => ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
% 12.24/8.00 ( ( P @ X )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ~ ( ord_less @ B @ ( F @ X ) @ ( F @ Y5 ) ) )
% 12.24/8.00 => ( ! [X3: A] :
% 12.24/8.00 ( ( P @ X3 )
% 12.24/8.00 => ( ! [Y4: A] :
% 12.24/8.00 ( ( P @ Y4 )
% 12.24/8.00 => ~ ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 12.24/8.00 => ( Q @ X3 ) ) )
% 12.24/8.00 => ( Q @ ( lattices_ord_arg_max @ A @ B @ F @ P ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6490_arg__max__on__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ( ( lattic1883929316492267755max_on @ B @ A )
% 12.24/8.00 = ( ^ [F2: B > A,S7: set @ B] :
% 12.24/8.00 ( lattices_ord_arg_max @ B @ A @ F2
% 12.24/8.00 @ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6491_arg__max__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ( ( lattices_ord_arg_max @ B @ A )
% 12.24/8.00 = ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751176901750rg_max @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6492_is__arg__max__linorder,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( linorder @ B )
% 12.24/8.00 => ( ( lattic501386751176901750rg_max @ A @ B )
% 12.24/8.00 = ( ^ [F2: A > B,P4: A > $o,X2: A] :
% 12.24/8.00 ( ( P4 @ X2 )
% 12.24/8.00 & ! [Y6: A] :
% 12.24/8.00 ( ( P4 @ Y6 )
% 12.24/8.00 => ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6493_is__arg__max__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ( ( lattic501386751176901750rg_max @ B @ A )
% 12.24/8.00 = ( ^ [F2: B > A,P4: B > $o,X2: B] :
% 12.24/8.00 ( ( P4 @ X2 )
% 12.24/8.00 & ~ ? [Y6: B] :
% 12.24/8.00 ( ( P4 @ Y6 )
% 12.24/8.00 & ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6494_int__of__integer__of__natural,axiom,
% 12.24/8.00 ! [N: code_natural] :
% 12.24/8.00 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 12.24/8.00 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6495_integer__of__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural] :
% 12.24/8.00 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 12.24/8.00 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6496_zero__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
% 12.24/8.00 = ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6497_plus__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6498_nat__of__natural__numeral,axiom,
% 12.24/8.00 ! [K: num] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
% 12.24/8.00 = ( numeral_numeral @ nat @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6499_minus__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6500_times__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6501_one__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( code_nat_of_natural @ ( one_one @ code_natural ) )
% 12.24/8.00 = ( one_one @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6502_modulo__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6503_divide__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6504_nat__of__natural__min,axiom,
% 12.24/8.00 ! [K: code_natural,L: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
% 12.24/8.00 = ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6505_nat__of__natural__max,axiom,
% 12.24/8.00 ! [K: code_natural,L: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
% 12.24/8.00 = ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6506_division__segment__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
% 12.24/8.00 = ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6507_euclidean__size__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( euclid6346220572633701492n_size @ code_natural )
% 12.24/8.00 = ( ^ [X2: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6508_of__nat__of__natural,axiom,
% 12.24/8.00 ! [N: code_natural] :
% 12.24/8.00 ( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6509_nat__of__natural__of__nat,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6510_nat__of__natural__induct,axiom,
% 12.24/8.00 ! [Y3: nat,P: nat > $o] :
% 12.24/8.00 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ( ! [X3: code_natural] : ( P @ ( code_nat_of_natural @ X3 ) )
% 12.24/8.00 => ( P @ Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6511_nat__of__natural__cases,axiom,
% 12.24/8.00 ! [Y3: nat] :
% 12.24/8.00 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ~ ! [X3: code_natural] :
% 12.24/8.00 ( Y3
% 12.24/8.00 != ( code_nat_of_natural @ X3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6512_nat__of__natural,axiom,
% 12.24/8.00 ! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6513_set__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6514_unset__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6515_flip__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6516_drop__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6517_bit__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 12.24/8.00 = ( ^ [X2: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6518_take__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6519_or__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6520_xor__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6521_and__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6522_mask__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
% 12.24/8.00 = ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6523_push__bit__natural_Orep__eq,axiom,
% 12.24/8.00 ! [X: nat,Xa3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 = ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6524_natural__eq__iff,axiom,
% 12.24/8.00 ( ( ^ [Y: code_natural,Z2: code_natural] : ( Y = Z2 ) )
% 12.24/8.00 = ( ^ [M2: code_natural,N3: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ M2 )
% 12.24/8.00 = ( code_nat_of_natural @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6525_natural__eqI,axiom,
% 12.24/8.00 ! [M: code_natural,N: code_natural] :
% 12.24/8.00 ( ( ( code_nat_of_natural @ M )
% 12.24/8.00 = ( code_nat_of_natural @ N ) )
% 12.24/8.00 => ( M = N ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6526_nat__of__natural__inject,axiom,
% 12.24/8.00 ! [X: code_natural,Y3: code_natural] :
% 12.24/8.00 ( ( ( code_nat_of_natural @ X )
% 12.24/8.00 = ( code_nat_of_natural @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6527_less__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( ord_less @ code_natural )
% 12.24/8.00 = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6528_less__eq__natural_Orep__eq,axiom,
% 12.24/8.00 ( ( ord_less_eq @ code_natural )
% 12.24/8.00 = ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6529_natural__of__integer_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_integer] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
% 12.24/8.00 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6530_natural__decr,axiom,
% 12.24/8.00 ! [N: code_natural] :
% 12.24/8.00 ( ( N
% 12.24/8.00 != ( zero_zero @ code_natural ) )
% 12.24/8.00 => ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6531_Nat_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_integer] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( code_Nat @ X ) )
% 12.24/8.00 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6532_Code__Numeral_Osize__nat__def,axiom,
% 12.24/8.00 code_size_nat = code_nat_of_natural ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6533_natural__zero__minus__one,axiom,
% 12.24/8.00 ( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
% 12.24/8.00 = ( zero_zero @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6534_log_Osimps,axiom,
% 12.24/8.00 ( log
% 12.24/8.00 = ( ^ [B3: code_natural,I5: code_natural] :
% 12.24/8.00 ( if @ code_natural
% 12.24/8.00 @ ( ( ord_less_eq @ code_natural @ B3 @ ( one_one @ code_natural ) )
% 12.24/8.00 | ( ord_less @ code_natural @ I5 @ B3 ) )
% 12.24/8.00 @ ( one_one @ code_natural )
% 12.24/8.00 @ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B3 @ ( divide_divide @ code_natural @ I5 @ B3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6535_log_Oelims,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
% 12.24/8.00 ( ( ( log @ X @ Xa3 )
% 12.24/8.00 = Y3 )
% 12.24/8.00 => ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 12.24/8.00 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 = ( one_one @ code_natural ) ) )
% 12.24/8.00 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 12.24/8.00 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6536_minus__shift__def,axiom,
% 12.24/8.00 ( minus_shift
% 12.24/8.00 = ( ^ [R5: code_natural,K3: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K3 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R5 @ K3 ) @ L3 ) @ ( minus_minus @ code_natural @ K3 @ L3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6537_natural_Osimps_I4_J,axiom,
% 12.24/8.00 ! [T: $tType,F1: T,F22: code_natural > T] :
% 12.24/8.00 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 12.24/8.00 = F1 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6538_range,axiom,
% 12.24/8.00 ! [K: code_natural,S: product_prod @ code_natural @ code_natural] :
% 12.24/8.00 ( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K )
% 12.24/8.00 => ( ord_less @ code_natural @ ( product_fst @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( range @ K @ S ) ) @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6539_select__weight__member,axiom,
% 12.24/8.00 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A ),S: product_prod @ code_natural @ code_natural] :
% 12.24/8.00 ( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
% 12.24/8.00 => ( member @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select_weight @ A @ Xs @ S ) ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6540_select__weight__drop__zero,axiom,
% 12.24/8.00 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
% 12.24/8.00 ( ( select_weight @ A
% 12.24/8.00 @ ( filter @ ( product_prod @ code_natural @ A )
% 12.24/8.00 @ ( product_case_prod @ code_natural @ A @ $o
% 12.24/8.00 @ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
% 12.24/8.00 @ Xs ) )
% 12.24/8.00 = ( select_weight @ A @ Xs ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6541_pick__member,axiom,
% 12.24/8.00 ! [A: $tType,I: code_natural,Xs: list @ ( product_prod @ code_natural @ A )] :
% 12.24/8.00 ( ( ord_less @ code_natural @ I @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
% 12.24/8.00 => ( member @ A @ ( pick @ A @ Xs @ I ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6542_pick_Osimps,axiom,
% 12.24/8.00 ! [A: $tType,I: code_natural,X: product_prod @ code_natural @ A,Xs: list @ ( product_prod @ code_natural @ A )] :
% 12.24/8.00 ( ( ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
% 12.24/8.00 => ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
% 12.24/8.00 = ( product_snd @ code_natural @ A @ X ) ) )
% 12.24/8.00 & ( ~ ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
% 12.24/8.00 => ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
% 12.24/8.00 = ( pick @ A @ Xs @ ( minus_minus @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6543_pick__drop__zero,axiom,
% 12.24/8.00 ! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
% 12.24/8.00 ( ( pick @ A
% 12.24/8.00 @ ( filter @ ( product_prod @ code_natural @ A )
% 12.24/8.00 @ ( product_case_prod @ code_natural @ A @ $o
% 12.24/8.00 @ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
% 12.24/8.00 @ Xs ) )
% 12.24/8.00 = ( pick @ A @ Xs ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6544_log_Opelims,axiom,
% 12.24/8.00 ! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
% 12.24/8.00 ( ( ( log @ X @ Xa3 )
% 12.24/8.00 = Y3 )
% 12.24/8.00 => ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) )
% 12.24/8.00 => ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 12.24/8.00 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 = ( one_one @ code_natural ) ) )
% 12.24/8.00 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 12.24/8.00 | ( ord_less @ code_natural @ Xa3 @ X ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) )
% 12.24/8.00 => ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6545_bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 12.24/8.00 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6546_pick__same,axiom,
% 12.24/8.00 ! [A: $tType,L: nat,Xs: list @ A] :
% 12.24/8.00 ( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 12.24/8.00 => ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
% 12.24/8.00 = ( nth @ A @ Xs @ L ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6547_euclidean__size__natural__def,axiom,
% 12.24/8.00 ( ( euclid6346220572633701492n_size @ code_natural )
% 12.24/8.00 = ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6548_natural__of__nat__of__natural__inverse,axiom,
% 12.24/8.00 ! [N: code_natural] :
% 12.24/8.00 ( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6549_nat__of__natural__of__nat__inverse,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6550_nat__of__natural__inverse,axiom,
% 12.24/8.00 ! [X: code_natural] :
% 12.24/8.00 ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
% 12.24/8.00 = X ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6551_natural__of__nat__of__nat,axiom,
% 12.24/8.00 ( code_natural_of_nat
% 12.24/8.00 = ( semiring_1_of_nat @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6552_natural__of__nat__inverse,axiom,
% 12.24/8.00 ! [Y3: nat] :
% 12.24/8.00 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y3 ) )
% 12.24/8.00 = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6553_natural__of__nat__cases,axiom,
% 12.24/8.00 ! [X: code_natural] :
% 12.24/8.00 ~ ! [Y5: nat] :
% 12.24/8.00 ( ( X
% 12.24/8.00 = ( code_natural_of_nat @ Y5 ) )
% 12.24/8.00 => ~ ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6554_natural__of__nat__induct,axiom,
% 12.24/8.00 ! [P: code_natural > $o,X: code_natural] :
% 12.24/8.00 ( ! [Y5: nat] :
% 12.24/8.00 ( ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ( P @ ( code_natural_of_nat @ Y5 ) ) )
% 12.24/8.00 => ( P @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6555_natural__of__nat__inject,axiom,
% 12.24/8.00 ! [X: nat,Y3: nat] :
% 12.24/8.00 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 => ( ( ( code_natural_of_nat @ X )
% 12.24/8.00 = ( code_natural_of_nat @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6556_minus__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( minus_minus @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6557_one__natural__def,axiom,
% 12.24/8.00 ( ( one_one @ code_natural )
% 12.24/8.00 = ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6558_divide__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( divide_divide @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6559_plus__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( plus_plus @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6560_zero__natural__def,axiom,
% 12.24/8.00 ( ( zero_zero @ code_natural )
% 12.24/8.00 = ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6561_less__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( ord_less @ nat @ Xa3 @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6562_less__eq__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( ord_less_eq @ nat @ Xa3 @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6563_times__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( times_times @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6564_modulo__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6565_take__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6566_xor__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6567_or__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6568_and__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6569_bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6570_push__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6571_mask__natural_Oabs__eq,axiom,
% 12.24/8.00 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 12.24/8.00 = ( ^ [X2: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6572_drop__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se4197421643247451524op_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6573_euclidean__size__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6574_flip__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6575_unset__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se2638667681897837118et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6576_division__segment__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6577_set__bit__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [Xa3: nat,X: nat] :
% 12.24/8.00 ( ( bit_se5668285175392031749et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa3 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6578_integer__of__natural_Oabs__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6579_natural__of__integer_Oabs__eq,axiom,
% 12.24/8.00 ! [X: int] :
% 12.24/8.00 ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6580_Nat_Oabs__eq,axiom,
% 12.24/8.00 ! [X: int] :
% 12.24/8.00 ( ( code_Nat @ ( code_integer_of_int @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6581_mask__natural__def,axiom,
% 12.24/8.00 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 12.24/8.00 = ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6582_case__natural__def,axiom,
% 12.24/8.00 ! [T: $tType] :
% 12.24/8.00 ( ( code_case_natural @ T )
% 12.24/8.00 = ( ^ [F12: T,F23: code_natural > T] :
% 12.24/8.00 ( code_rec_natural @ T @ F12
% 12.24/8.00 @ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6583_natural_Osimps_I6_J,axiom,
% 12.24/8.00 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 12.24/8.00 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 12.24/8.00 = F1 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6584_rec__natural__def,axiom,
% 12.24/8.00 ! [T: $tType] :
% 12.24/8.00 ( ( code_rec_natural @ T )
% 12.24/8.00 = ( ^ [F12: T,F23: code_natural > T > T,X2: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6585_Nat__def,axiom,
% 12.24/8.00 ( code_Nat
% 12.24/8.00 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6586_natural__of__integer__def,axiom,
% 12.24/8.00 ( code_n4118661773612635043nteger
% 12.24/8.00 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6587_division__segment__natural__def,axiom,
% 12.24/8.00 ( ( euclid7384307370059645450egment @ code_natural )
% 12.24/8.00 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6588_integer__of__natural__def,axiom,
% 12.24/8.00 ( code_i5400310926305786745atural
% 12.24/8.00 = ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6589_set__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se5668285175392031749et_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6590_unset__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se2638667681897837118et_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6591_take__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se2584673776208193580ke_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6592_push__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se4730199178511100633sh_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6593_drop__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se4197421643247451524op_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6594_flip__bit__natural__def,axiom,
% 12.24/8.00 ( ( bit_se8732182000553998342ip_bit @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6595_and__natural__def,axiom,
% 12.24/8.00 ( ( bit_se5824344872417868541ns_and @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6596_xor__natural__def,axiom,
% 12.24/8.00 ( ( bit_se5824344971392196577ns_xor @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6597_plus__natural__def,axiom,
% 12.24/8.00 ( ( plus_plus @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6598_minus__natural__def,axiom,
% 12.24/8.00 ( ( minus_minus @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6599_divide__natural__def,axiom,
% 12.24/8.00 ( ( divide_divide @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6600_times__natural__def,axiom,
% 12.24/8.00 ( ( times_times @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6601_modulo__natural__def,axiom,
% 12.24/8.00 ( ( modulo_modulo @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6602_or__natural__def,axiom,
% 12.24/8.00 ( ( bit_se1065995026697491101ons_or @ code_natural )
% 12.24/8.00 = ( 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 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6603_Code__Numeral_OSuc__def,axiom,
% 12.24/8.00 ( code_Suc
% 12.24/8.00 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6604_cr__natural__def,axiom,
% 12.24/8.00 ( code_cr_natural
% 12.24/8.00 = ( ^ [X2: nat,Y6: code_natural] :
% 12.24/8.00 ( X2
% 12.24/8.00 = ( code_nat_of_natural @ Y6 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6605_natural_Oinject,axiom,
% 12.24/8.00 ! [Natural: code_natural,Natural2: code_natural] :
% 12.24/8.00 ( ( ( code_Suc @ Natural )
% 12.24/8.00 = ( code_Suc @ Natural2 ) )
% 12.24/8.00 = ( Natural = Natural2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6606_natural_Osimps_I7_J,axiom,
% 12.24/8.00 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 12.24/8.00 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 12.24/8.00 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6607_natural_Osimps_I5_J,axiom,
% 12.24/8.00 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 12.24/8.00 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 12.24/8.00 = ( F22 @ Natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6608_Suc_Orep__eq,axiom,
% 12.24/8.00 ! [X: code_natural] :
% 12.24/8.00 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 12.24/8.00 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6609_natural_Oexhaust,axiom,
% 12.24/8.00 ! [Y3: code_natural] :
% 12.24/8.00 ( ( Y3
% 12.24/8.00 != ( zero_zero @ code_natural ) )
% 12.24/8.00 => ~ ! [Natural3: code_natural] :
% 12.24/8.00 ( Y3
% 12.24/8.00 != ( code_Suc @ Natural3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6610_natural_Odistinct_I1_J,axiom,
% 12.24/8.00 ! [Natural2: code_natural] :
% 12.24/8.00 ( ( zero_zero @ code_natural )
% 12.24/8.00 != ( code_Suc @ Natural2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6611_natural_Odistinct_I2_J,axiom,
% 12.24/8.00 ! [Natural4: code_natural] :
% 12.24/8.00 ( ( code_Suc @ Natural4 )
% 12.24/8.00 != ( zero_zero @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6612_Suc_Oabs__eq,axiom,
% 12.24/8.00 ! [X: nat] :
% 12.24/8.00 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 12.24/8.00 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6613_Suc__natural__minus__one,axiom,
% 12.24/8.00 ! [N: code_natural] :
% 12.24/8.00 ( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6614_Nat_Otransfer,axiom,
% 12.24/8.00 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6615_fold__atLeastAtMost__nat_Opsimps,axiom,
% 12.24/8.00 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 12.24/8.00 ( ( 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 ) ) ) )
% 12.24/8.00 => ( ( ( ord_less @ nat @ B2 @ A3 )
% 12.24/8.00 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 12.24/8.00 = Acc2 ) )
% 12.24/8.00 & ( ~ ( ord_less @ nat @ B2 @ A3 )
% 12.24/8.00 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 12.24/8.00 = ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6616_Suc_Otransfer,axiom,
% 12.24/8.00 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6617_natural_Opcr__cr__eq,axiom,
% 12.24/8.00 code_pcr_natural = code_cr_natural ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6618_natural_Oid__abs__transfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ code_pcr_natural
% 12.24/8.00 @ ^ [X2: nat] : X2
% 12.24/8.00 @ code_natural_of_nat ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6619_less__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 12.24/8.00 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/8.00 @ ( ord_less @ nat )
% 12.24/8.00 @ ( ord_less @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6620_plus__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6621_minus__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6622_zero__natural_Otransfer,axiom,
% 12.24/8.00 code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6623_one__natural_Otransfer,axiom,
% 12.24/8.00 code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6624_natural_Obi__total,axiom,
% 12.24/8.00 bi_total @ nat @ code_natural @ code_pcr_natural ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6625_natural_Orep__transfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ^ [X2: nat] : X2
% 12.24/8.00 @ code_nat_of_natural ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6626_less__eq__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 12.24/8.00 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/8.00 @ ( ord_less_eq @ nat )
% 12.24/8.00 @ ( ord_less_eq @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6627_divide__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6628_times__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6629_modulo__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6630_fold__atLeastAtMost__nat_Opinduct,axiom,
% 12.24/8.00 ! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
% 12.24/8.00 ( ( 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 ) ) ) )
% 12.24/8.00 => ( ! [F3: nat > A > A,A8: nat,B5: nat,Acc3: A] :
% 12.24/8.00 ( ( 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 ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A8 @ ( product_Pair @ nat @ A @ B5 @ Acc3 ) ) ) )
% 12.24/8.00 => ( ( ~ ( ord_less @ nat @ B5 @ A8 )
% 12.24/8.00 => ( P @ F3 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B5 @ ( F3 @ A8 @ Acc3 ) ) )
% 12.24/8.00 => ( P @ F3 @ A8 @ B5 @ Acc3 ) ) )
% 12.24/8.00 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6631_take__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se2584673776208193580ke_bit @ nat )
% 12.24/8.00 @ ( bit_se2584673776208193580ke_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6632_or__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6633_xor__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6634_and__natural_Otransfer,axiom,
% 12.24/8.00 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 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6635_bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
% 12.24/8.00 @ ^ [Y: nat > $o,Z2: nat > $o] : ( Y = Z2 )
% 12.24/8.00 @ ( bit_se5641148757651400278ts_bit @ nat )
% 12.24/8.00 @ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6636_mask__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ code_pcr_natural
% 12.24/8.00 @ ( bit_se2239418461657761734s_mask @ nat )
% 12.24/8.00 @ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6637_push__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se4730199178511100633sh_bit @ nat )
% 12.24/8.00 @ ( bit_se4730199178511100633sh_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6638_drop__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se4197421643247451524op_bit @ nat )
% 12.24/8.00 @ ( bit_se4197421643247451524op_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6639_set__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se5668285175392031749et_bit @ nat )
% 12.24/8.00 @ ( bit_se5668285175392031749et_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6640_division__segment__natural_Otransfer,axiom,
% 12.24/8.00 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6641_unset__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se2638667681897837118et_bit @ nat )
% 12.24/8.00 @ ( bit_se2638667681897837118et_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6642_flip__bit__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 12.24/8.00 @ ( bit_se8732182000553998342ip_bit @ nat )
% 12.24/8.00 @ ( bit_se8732182000553998342ip_bit @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6643_euclidean__size__natural_Otransfer,axiom,
% 12.24/8.00 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ ( euclid6346220572633701492n_size @ nat )
% 12.24/8.00 @ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6644_integer__of__natural_Otransfer,axiom,
% 12.24/8.00 bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6645_natural__of__integer_Otransfer,axiom,
% 12.24/8.00 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6646_fold__atLeastAtMost__nat_Opelims,axiom,
% 12.24/8.00 ! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
% 12.24/8.00 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
% 12.24/8.00 = Y3 )
% 12.24/8.00 => ( ( 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) )
% 12.24/8.00 => ~ ( ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
% 12.24/8.00 => ( Y3 = Xc ) )
% 12.24/8.00 & ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
% 12.24/8.00 => ( Y3
% 12.24/8.00 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) )
% 12.24/8.00 => ~ ( 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6647_prod_Ocomm__monoid__list__set__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid_mult @ A )
% 12.24/8.00 => ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6648_prod_Oreindex__bij__betw__not__neutral,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/8.00 ( ( comm_monoid_mult @ A )
% 12.24/8.00 => ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
% 12.24/8.00 ( ( finite_finite2 @ B @ S5 )
% 12.24/8.00 => ( ( finite_finite2 @ C @ T5 )
% 12.24/8.00 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/8.00 => ( ! [A8: B] :
% 12.24/8.00 ( ( member @ B @ A8 @ S5 )
% 12.24/8.00 => ( ( G @ ( H2 @ A8 ) )
% 12.24/8.00 = ( one_one @ A ) ) )
% 12.24/8.00 => ( ! [B5: C] :
% 12.24/8.00 ( ( member @ C @ B5 @ T5 )
% 12.24/8.00 => ( ( G @ B5 )
% 12.24/8.00 = ( one_one @ A ) ) )
% 12.24/8.00 => ( ( groups7121269368397514597t_prod @ B @ A
% 12.24/8.00 @ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
% 12.24/8.00 @ S3 )
% 12.24/8.00 = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6649_bij__betw__of__nat,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semiring_char_0 @ A )
% 12.24/8.00 => ! [N6: set @ nat,A6: set @ A] :
% 12.24/8.00 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A6 )
% 12.24/8.00 = ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
% 12.24/8.00 = A6 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6650_bij__betw__Suc,axiom,
% 12.24/8.00 ! [M7: set @ nat,N6: set @ nat] :
% 12.24/8.00 ( ( bij_betw @ nat @ nat @ suc @ M7 @ N6 )
% 12.24/8.00 = ( ( image @ nat @ nat @ suc @ M7 )
% 12.24/8.00 = N6 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6651_bijection__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( hilbert_bijection @ A )
% 12.24/8.00 = ( ^ [F2: A > A] : ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6652_bijection_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( hilbert_bijection @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6653_bijection_Obij,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A] :
% 12.24/8.00 ( ( hilbert_bijection @ A @ F )
% 12.24/8.00 => ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6654_bij__betw__funpow,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A,S3: set @ A,N: nat] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ S3 @ S3 )
% 12.24/8.00 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ S3 @ S3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6655_bij__fn,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A,N: nat] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6656_bij__uminus,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ab_group_add @ A )
% 12.24/8.00 => ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6657_bij__imp__bij__inv,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6658_bij__inv__eq__iff,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,P5: A > B,X: A,Y3: B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ P5 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( X
% 12.24/8.00 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P5 @ Y3 ) )
% 12.24/8.00 = ( ( P5 @ X )
% 12.24/8.00 = Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6659_inv__inv__eq,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) )
% 12.24/8.00 = F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6660_bij__betw__inv__into__right,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,F: A > B,A6: set @ A,A11: set @ B,A4: B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 12.24/8.00 => ( ( member @ B @ A4 @ A11 )
% 12.24/8.00 => ( ( F @ ( hilbert_inv_into @ A @ B @ A6 @ F @ A4 ) )
% 12.24/8.00 = A4 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6661_bij__betw__inv__into__left,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 12.24/8.00 => ( ( member @ A @ A3 @ A6 )
% 12.24/8.00 => ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ A3 ) )
% 12.24/8.00 = A3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6662_inv__into__inv__into__eq,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 12.24/8.00 => ( ( member @ A @ A3 @ A6 )
% 12.24/8.00 => ( ( hilbert_inv_into @ B @ A @ A11 @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ A3 )
% 12.24/8.00 = ( F @ A3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6663_bij__betw__inv__into,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,B7: set @ B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ A6 @ B7 )
% 12.24/8.00 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B7 @ A6 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6664_bij__betw__inv__into__subset,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,B7: set @ A,B13: set @ B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ A6 @ A11 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
% 12.24/8.00 => ( ( ( image @ A @ B @ F @ B7 )
% 12.24/8.00 = B13 )
% 12.24/8.00 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B13 @ B7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6665_bij__image__Compl__eq,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 12.24/8.00 = ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6666_o__inv__distrib,axiom,
% 12.24/8.00 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > A] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F @ G ) )
% 12.24/8.00 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6667_inv__fn,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A,N: nat] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 12.24/8.00 = ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6668_mono__inv,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( ( linorder @ A )
% 12.24/8.00 & ( linorder @ B ) )
% 12.24/8.00 => ! [F: A > B] :
% 12.24/8.00 ( ( order_mono @ A @ B @ F )
% 12.24/8.00 => ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6669_bijection_Obij__inv,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A] :
% 12.24/8.00 ( ( hilbert_bijection @ A @ F )
% 12.24/8.00 => ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6670_bij__image__Collect__eq,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,F: A > B,P: A > $o] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( image @ A @ B @ F @ ( collect @ A @ P ) )
% 12.24/8.00 = ( collect @ B
% 12.24/8.00 @ ^ [Y6: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y6 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6671_bij__int__decode,axiom,
% 12.24/8.00 bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6672_bij__int__encode,axiom,
% 12.24/8.00 bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6673_infinite__imp__bij__betw2,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ A,A3: A] :
% 12.24/8.00 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/8.00 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6674_infinite__imp__bij__betw,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ A,A3: A] :
% 12.24/8.00 ( ~ ( finite_finite2 @ A @ A6 )
% 12.24/8.00 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6675_ex__bij__betw__finite__nat,axiom,
% 12.24/8.00 ! [A: $tType,M7: set @ A] :
% 12.24/8.00 ( ( finite_finite2 @ A @ M7 )
% 12.24/8.00 => ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6676_ex__bij__betw__nat__finite,axiom,
% 12.24/8.00 ! [A: $tType,M7: set @ A] :
% 12.24/8.00 ( ( finite_finite2 @ A @ M7 )
% 12.24/8.00 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6677_mono__bij__Inf,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( ( comple5582772986160207858norder @ A )
% 12.24/8.00 & ( comple5582772986160207858norder @ B ) )
% 12.24/8.00 => ! [F: A > B,A6: set @ A] :
% 12.24/8.00 ( ( order_mono @ A @ B @ F )
% 12.24/8.00 => ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( F @ ( complete_Inf_Inf @ A @ A6 ) )
% 12.24/8.00 = ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6678_bij__prod__decode,axiom,
% 12.24/8.00 bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6679_inj__imp__bij__betw__inv,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B,M7: set @ A] :
% 12.24/8.00 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ M7 ) @ M7 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6680_ex__bij__betw__nat__finite__1,axiom,
% 12.24/8.00 ! [A: $tType,M7: set @ A] :
% 12.24/8.00 ( ( finite_finite2 @ A @ M7 )
% 12.24/8.00 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6681_bij__list__decode,axiom,
% 12.24/8.00 bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6682_bij__list__encode,axiom,
% 12.24/8.00 bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6683_bij__prod__encode,axiom,
% 12.24/8.00 bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6684_bij__vimage__eq__inv__image,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 12.24/8.00 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 12.24/8.00 => ( ( vimage @ A @ B @ F @ A6 )
% 12.24/8.00 = ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ A6 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6685_fn__o__inv__fn__is__id,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A,N: nat] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
% 12.24/8.00 = ( ^ [X2: A] : X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6686_inv__fn__o__fn__is__id,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A,N: nat] :
% 12.24/8.00 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 12.24/8.00 = ( ^ [X2: A] : X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6687_sum_Oreindex__bij__betw__not__neutral,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/8.00 ( ( comm_monoid_add @ A )
% 12.24/8.00 => ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
% 12.24/8.00 ( ( finite_finite2 @ B @ S5 )
% 12.24/8.00 => ( ( finite_finite2 @ C @ T5 )
% 12.24/8.00 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 12.24/8.00 => ( ! [A8: B] :
% 12.24/8.00 ( ( member @ B @ A8 @ S5 )
% 12.24/8.00 => ( ( G @ ( H2 @ A8 ) )
% 12.24/8.00 = ( zero_zero @ A ) ) )
% 12.24/8.00 => ( ! [B5: C] :
% 12.24/8.00 ( ( member @ C @ B5 @ T5 )
% 12.24/8.00 => ( ( G @ B5 )
% 12.24/8.00 = ( zero_zero @ A ) ) )
% 12.24/8.00 => ( ( groups7311177749621191930dd_sum @ B @ A
% 12.24/8.00 @ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
% 12.24/8.00 @ S3 )
% 12.24/8.00 = ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6688_sum_Ocomm__monoid__list__set__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid_add @ A )
% 12.24/8.00 => ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6689_prod_H__def,axiom,
% 12.24/8.00 ! [C: $tType,A: $tType] :
% 12.24/8.00 ( ( comm_monoid_mult @ A )
% 12.24/8.00 => ( ( groups1962203154675924110t_prod @ C @ A )
% 12.24/8.00 = ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6690_natLeq__on__wo__rel,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( bNF_Wellorder_wo_rel @ nat
% 12.24/8.00 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.00 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.00 @ ^ [X2: nat,Y6: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ X2 @ N )
% 12.24/8.00 & ( ord_less @ nat @ Y6 @ N )
% 12.24/8.00 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6691_sum_H__def,axiom,
% 12.24/8.00 ! [C: $tType,A: $tType] :
% 12.24/8.00 ( ( comm_monoid_add @ A )
% 12.24/8.00 => ( ( groups1027152243600224163dd_sum @ C @ A )
% 12.24/8.00 = ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6692_Heap_Osize__gen__o__map,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 12.24/8.00 ( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
% 12.24/8.00 = ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6693_max_Osemilattice__order__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semilattice_order @ A @ ( ord_max @ A )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6694_Heap_Oinj__map,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,F: A > B] :
% 12.24/8.00 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 12.24/8.00 => ( inj_on @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Time_Heap @ A ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6695_semilattice__order__set_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6696_Heap_Omap__comp,axiom,
% 12.24/8.00 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
% 12.24/8.00 ( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 12.24/8.00 = ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6697_semilattice__neutr__order_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
% 12.24/8.00 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6698_Heap_Omap__ident,axiom,
% 12.24/8.00 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 12.24/8.00 ( ( heap_Time_map_Heap @ A @ A
% 12.24/8.00 @ ^ [X2: A] : X2
% 12.24/8.00 @ T2 )
% 12.24/8.00 = T2 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6699_semilattice__order_Omono,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A,D2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ C2 )
% 12.24/8.00 => ( ( Less_eq @ B2 @ D2 )
% 12.24/8.00 => ( Less_eq @ ( F @ A3 @ B2 ) @ ( F @ C2 @ D2 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6700_semilattice__order_OorderE,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( A3
% 12.24/8.00 = ( F @ A3 @ B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6701_semilattice__order_OorderI,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( A3
% 12.24/8.00 = ( F @ A3 @ B2 ) )
% 12.24/8.00 => ( Less_eq @ A3 @ B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6702_semilattice__order_Oabsorb1,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( ( F @ A3 @ B2 )
% 12.24/8.00 = A3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6703_semilattice__order_Oabsorb2,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ B2 @ A3 )
% 12.24/8.00 => ( ( F @ A3 @ B2 )
% 12.24/8.00 = B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6704_semilattice__order_Oabsorb3,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ B2 )
% 12.24/8.00 => ( ( F @ A3 @ B2 )
% 12.24/8.00 = A3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6705_semilattice__order_Oabsorb4,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ B2 @ A3 )
% 12.24/8.00 => ( ( F @ A3 @ B2 )
% 12.24/8.00 = B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6706_semilattice__order_OboundedE,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
% 12.24/8.00 => ~ ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ~ ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6707_semilattice__order_OboundedI,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( ( Less_eq @ A3 @ C2 )
% 12.24/8.00 => ( Less_eq @ A3 @ ( F @ B2 @ C2 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6708_semilattice__order_Oorder__iff,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 = ( A3
% 12.24/8.00 = ( F @ A3 @ B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6709_semilattice__order_Ocobounded1,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( Less_eq @ ( F @ A3 @ B2 ) @ A3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6710_semilattice__order_Ocobounded2,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( Less_eq @ ( F @ A3 @ B2 ) @ B2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6711_semilattice__order_Oabsorb__iff1,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 = ( ( F @ A3 @ B2 )
% 12.24/8.00 = A3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6712_semilattice__order_Oabsorb__iff2,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ B2 @ A3 )
% 12.24/8.00 = ( ( F @ A3 @ B2 )
% 12.24/8.00 = B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6713_semilattice__order_Obounded__iff,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
% 12.24/8.00 = ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 & ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6714_semilattice__order_OcoboundedI1,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ C2 )
% 12.24/8.00 => ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6715_semilattice__order_OcoboundedI2,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ B2 @ C2 )
% 12.24/8.00 => ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6716_semilattice__order_Ostrict__boundedE,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ ( F @ B2 @ C2 ) )
% 12.24/8.00 => ~ ( ( Less @ A3 @ B2 )
% 12.24/8.00 => ~ ( Less @ A3 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6717_semilattice__order_Ostrict__order__iff,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ B2 )
% 12.24/8.00 = ( ( A3
% 12.24/8.00 = ( F @ A3 @ B2 ) )
% 12.24/8.00 & ( A3 != B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6718_semilattice__order_Ostrict__coboundedI1,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ C2 )
% 12.24/8.00 => ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6719_semilattice__order_Ostrict__coboundedI2,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ B2 @ C2 )
% 12.24/8.00 => ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6720_Heap_Omap__id,axiom,
% 12.24/8.00 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 12.24/8.00 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
% 12.24/8.00 = T2 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6721_Heap_Omap__id0,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
% 12.24/8.00 = ( id @ ( heap_Time_Heap @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6722_semilattice__order__set__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( lattic4895041142388067077er_set @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
% 12.24/8.00 & ( lattic149705377957585745ce_set @ A @ F2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6723_semilattice__order__set_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/8.00 => ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6724_semilattice__neutr__order__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semila1105856199041335345_order @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ( semilattice_neutr @ A @ F2 @ Z7 )
% 12.24/8.00 & ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6725_semilattice__neutr__order_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semilattice_neutr @ A @ F @ Z3 )
% 12.24/8.00 => ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6726_inf_Osemilattice__order__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_inf @ A )
% 12.24/8.00 => ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6727_min_Osemilattice__order__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6728_sup_Osemilattice__order__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_sup @ A )
% 12.24/8.00 => ( semilattice_order @ A @ ( sup_sup @ A )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6729_lexordp__eq_Osimps,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ( ( ord_lexordp_eq @ A )
% 12.24/8.00 = ( ^ [A12: list @ A,A23: list @ A] :
% 12.24/8.00 ( ? [Ys3: list @ A] :
% 12.24/8.00 ( ( A12
% 12.24/8.00 = ( nil @ A ) )
% 12.24/8.00 & ( A23 = Ys3 ) )
% 12.24/8.00 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/8.00 ( ( A12
% 12.24/8.00 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/8.00 & ( A23
% 12.24/8.00 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/8.00 & ( ord_less @ A @ X2 @ Y6 ) )
% 12.24/8.00 | ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
% 12.24/8.00 ( ( A12
% 12.24/8.00 = ( cons @ A @ X2 @ Xs3 ) )
% 12.24/8.00 & ( A23
% 12.24/8.00 = ( cons @ A @ Y6 @ Ys3 ) )
% 12.24/8.00 & ~ ( ord_less @ A @ X2 @ Y6 )
% 12.24/8.00 & ~ ( ord_less @ A @ Y6 @ X2 )
% 12.24/8.00 & ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6730_lexordp__eq__simps_I4_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
% 12.24/8.00 ( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
% 12.24/8.00 = ( ( ord_less @ A @ X @ Y3 )
% 12.24/8.00 | ( ~ ( ord_less @ A @ Y3 @ X )
% 12.24/8.00 & ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6731_lexordp__eq_OCons,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 12.24/8.00 ( ( ord_less @ A @ X @ Y3 )
% 12.24/8.00 => ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6732_lexordp__eq_OCons__eq,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
% 12.24/8.00 ( ~ ( ord_less @ A @ X @ Y3 )
% 12.24/8.00 => ( ~ ( ord_less @ A @ Y3 @ X )
% 12.24/8.00 => ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
% 12.24/8.00 => ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6733_lexordp__eq_Ocases,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ! [A1: list @ A,A22: list @ A] :
% 12.24/8.00 ( ( ord_lexordp_eq @ A @ A1 @ A22 )
% 12.24/8.00 => ( ( A1
% 12.24/8.00 != ( nil @ A ) )
% 12.24/8.00 => ( ! [X3: A] :
% 12.24/8.00 ( ? [Xs2: list @ A] :
% 12.24/8.00 ( A1
% 12.24/8.00 = ( cons @ A @ X3 @ Xs2 ) )
% 12.24/8.00 => ! [Y5: A] :
% 12.24/8.00 ( ? [Ys4: list @ A] :
% 12.24/8.00 ( A22
% 12.24/8.00 = ( cons @ A @ Y5 @ Ys4 ) )
% 12.24/8.00 => ~ ( ord_less @ A @ X3 @ Y5 ) ) )
% 12.24/8.00 => ~ ! [X3: A,Y5: A,Xs2: list @ A] :
% 12.24/8.00 ( ( A1
% 12.24/8.00 = ( cons @ A @ X3 @ Xs2 ) )
% 12.24/8.00 => ! [Ys4: list @ A] :
% 12.24/8.00 ( ( A22
% 12.24/8.00 = ( cons @ A @ Y5 @ Ys4 ) )
% 12.24/8.00 => ( ~ ( ord_less @ A @ X3 @ Y5 )
% 12.24/8.00 => ( ~ ( ord_less @ A @ Y5 @ X3 )
% 12.24/8.00 => ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6734_bdd__below__primitive__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( ( condit1013018076250108175_below @ A )
% 12.24/8.00 = ( condit16957441358409770ng_bdd @ A
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6735_Quotient__natural,axiom,
% 12.24/8.00 ( quotient @ nat @ code_natural
% 12.24/8.00 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 )
% 12.24/8.00 @ code_natural_of_nat
% 12.24/8.00 @ code_nat_of_natural
% 12.24/8.00 @ code_cr_natural ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6736_bdd__above__primitive__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( ( condit941137186595557371_above @ A )
% 12.24/8.00 = ( condit16957441358409770ng_bdd @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6737_card_Ofolding__on__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
% 12.24/8.00 @ ^ [Uu: A] : suc ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6738_bdd__below_Opreordering__bdd__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( condit622319405099724424ng_bdd @ A
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6739_bdd__above_Opreordering__bdd__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6740_Quotient__integer,axiom,
% 12.24/8.00 ( quotient @ int @ code_integer
% 12.24/8.00 @ ^ [Y: int,Z2: int] : ( Y = Z2 )
% 12.24/8.00 @ code_integer_of_int
% 12.24/8.00 @ code_int_of_integer
% 12.24/8.00 @ code_cr_integer ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6741_type__definition__integer,axiom,
% 12.24/8.00 type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6742_finite__subset__Union__chain,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ A,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 12.24/8.00 ( ( finite_finite2 @ A @ A6 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
% 12.24/8.00 => ( ( B12
% 12.24/8.00 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 12.24/8.00 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 12.24/8.00 => ~ ! [B9: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ B9 @ B12 )
% 12.24/8.00 => ~ ( ord_less_eq @ ( set @ A ) @ A6 @ B9 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6743_wait__def,axiom,
% 12.24/8.00 ( heap_Time_wait
% 12.24/8.00 = ( ^ [N3: nat] :
% 12.24/8.00 ( heap_Time_Heap2 @ product_unit
% 12.24/8.00 @ ^ [H4: 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 @ H4 @ N3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6744_subset_Ochain__empty,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6745_subset__chain__def,axiom,
% 12.24/8.00 ! [A: $tType,A16: set @ ( set @ A ),C7: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C7 )
% 12.24/8.00 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C7 @ A16 )
% 12.24/8.00 & ! [X2: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X2 @ C7 )
% 12.24/8.00 => ! [Y6: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Y6 @ C7 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y6 )
% 12.24/8.00 | ( ord_less_eq @ ( set @ A ) @ Y6 @ X2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6746_subset__Zorn,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/8.00 ( ! [C8: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
% 12.24/8.00 => ? [X4: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X4 @ A6 )
% 12.24/8.00 & ! [Xa2: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Xa2 @ C8 )
% 12.24/8.00 => ( ord_less_eq @ ( set @ A ) @ Xa2 @ X4 ) ) ) )
% 12.24/8.00 => ? [X3: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 12.24/8.00 & ! [Xa: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Xa @ A6 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 12.24/8.00 => ( Xa = X3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6747_subset__chain__insert,axiom,
% 12.24/8.00 ! [A: $tType,A16: set @ ( set @ A ),B7: set @ A,B12: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ B7 @ B12 ) )
% 12.24/8.00 = ( ( member @ ( set @ A ) @ B7 @ A16 )
% 12.24/8.00 & ! [X2: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X2 @ B12 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ X2 @ B7 )
% 12.24/8.00 | ( ord_less_eq @ ( set @ A ) @ B7 @ X2 ) ) )
% 12.24/8.00 & ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6748_subset__Zorn_H,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/8.00 ( ! [C8: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
% 12.24/8.00 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) @ A6 ) )
% 12.24/8.00 => ? [X3: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X3 @ A6 )
% 12.24/8.00 & ! [Xa: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Xa @ A6 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 12.24/8.00 => ( Xa = X3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6749_subset_Ochain__extend,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),Z3: set @ A] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ( member @ ( set @ A ) @ Z3 @ A6 )
% 12.24/8.00 => ( ! [X3: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X3 @ C6 )
% 12.24/8.00 => ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ X3
% 12.24/8.00 @ Z3 ) )
% 12.24/8.00 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C6 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6750_subset_Ochain__total,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X: set @ A,Y3: set @ A] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ( member @ ( set @ A ) @ X @ C6 )
% 12.24/8.00 => ( ( member @ ( set @ A ) @ Y3 @ C6 )
% 12.24/8.00 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ X
% 12.24/8.00 @ Y3 )
% 12.24/8.00 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ Y3
% 12.24/8.00 @ X ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6751_subset_OchainI,axiom,
% 12.24/8.00 ! [A: $tType,C6: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.00 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
% 12.24/8.00 => ( ! [X3: set @ A,Y5: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X3 @ C6 )
% 12.24/8.00 => ( ( member @ ( set @ A ) @ Y5 @ C6 )
% 12.24/8.00 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ X3
% 12.24/8.00 @ Y5 )
% 12.24/8.00 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ Y5
% 12.24/8.00 @ X3 ) ) ) )
% 12.24/8.00 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6752_subset_Ochain__def,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 = ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
% 12.24/8.00 & ! [X2: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X2 @ C6 )
% 12.24/8.00 => ! [Y6: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Y6 @ C6 )
% 12.24/8.00 => ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ X2
% 12.24/8.00 @ Y6 )
% 12.24/8.00 | ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ^ [Y: set @ A,Z2: set @ A] : ( Y = Z2 )
% 12.24/8.00 @ Y6
% 12.24/8.00 @ X2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6753_uminus__unit__def,axiom,
% 12.24/8.00 ( ( uminus_uminus @ product_unit )
% 12.24/8.00 = ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6754_subset__Zorn__nonempty,axiom,
% 12.24/8.00 ! [A: $tType,A16: set @ ( set @ A )] :
% 12.24/8.00 ( ( A16
% 12.24/8.00 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 12.24/8.00 => ( ! [C9: set @ ( set @ A )] :
% 12.24/8.00 ( ( C9
% 12.24/8.00 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 12.24/8.00 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C9 )
% 12.24/8.00 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) @ A16 ) ) )
% 12.24/8.00 => ? [X3: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ X3 @ A16 )
% 12.24/8.00 & ! [Xa: set @ A] :
% 12.24/8.00 ( ( member @ ( set @ A ) @ Xa @ A16 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
% 12.24/8.00 => ( Xa = X3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6755_dependent__wellorder__choice,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( wellorder @ A )
% 12.24/8.00 => ! [P: ( A > B ) > A > B > $o] :
% 12.24/8.00 ( ! [R3: B,F3: A > B,G5: A > B,X3: A] :
% 12.24/8.00 ( ! [Y4: A] :
% 12.24/8.00 ( ( ord_less @ A @ Y4 @ X3 )
% 12.24/8.00 => ( ( F3 @ Y4 )
% 12.24/8.00 = ( G5 @ Y4 ) ) )
% 12.24/8.00 => ( ( P @ F3 @ X3 @ R3 )
% 12.24/8.00 = ( P @ G5 @ X3 @ R3 ) ) )
% 12.24/8.00 => ( ! [X3: A,F3: A > B] :
% 12.24/8.00 ( ! [Y4: A] :
% 12.24/8.00 ( ( ord_less @ A @ Y4 @ X3 )
% 12.24/8.00 => ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
% 12.24/8.00 => ? [X_1: B] : ( P @ F3 @ X3 @ X_1 ) )
% 12.24/8.00 => ? [F3: A > B] :
% 12.24/8.00 ! [X4: A] : ( P @ F3 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6756_Union__in__chain,axiom,
% 12.24/8.00 ! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 12.24/8.00 ( ( finite_finite2 @ ( set @ A ) @ B12 )
% 12.24/8.00 => ( ( B12
% 12.24/8.00 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 12.24/8.00 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 12.24/8.00 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6757_Inter__in__chain,axiom,
% 12.24/8.00 ! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
% 12.24/8.00 ( ( finite_finite2 @ ( set @ A ) @ B12 )
% 12.24/8.00 => ( ( B12
% 12.24/8.00 != ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 12.24/8.00 => ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
% 12.24/8.00 => ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6758_chain__subset__alt__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( chain_subset @ A )
% 12.24/8.00 = ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6759_chains__alt__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( chains @ A )
% 12.24/8.00 = ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6760_sup2CI,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
% 12.24/8.00 ( ( ~ ( B7 @ X @ Y3 )
% 12.24/8.00 => ( A6 @ X @ Y3 ) )
% 12.24/8.00 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6761_sup2I2,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
% 12.24/8.00 ( ( B7 @ X @ Y3 )
% 12.24/8.00 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6762_sup2I1,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,A6: A > B > $o,X: A,Y3: B,B7: A > B > $o] :
% 12.24/8.00 ( ( A6 @ X @ Y3 )
% 12.24/8.00 => ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6763_sup2E,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,A6: A > B > $o,B7: A > B > $o,X: A,Y3: B] :
% 12.24/8.00 ( ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 )
% 12.24/8.00 => ( ~ ( A6 @ X @ Y3 )
% 12.24/8.00 => ( B7 @ X @ Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6764_pred__on_Onot__maxchain__Some,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ A,P: A > A > $o,C6: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A6 @ P @ C6 )
% 12.24/8.00 => ( ~ ( pred_maxchain @ A @ A6 @ P @ C6 )
% 12.24/8.00 => ( ( pred_chain @ A @ A6 @ P
% 12.24/8.00 @ ( fChoice @ ( set @ A )
% 12.24/8.00 @ ^ [D7: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A6 @ P @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ C6
% 12.24/8.00 @ ( fChoice @ ( set @ A )
% 12.24/8.00 @ ^ [D7: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A6 @ P @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6765_subset__maxchain__max,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X8: set @ A] :
% 12.24/8.00 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ( member @ ( set @ A ) @ X8 @ A6 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ X8 )
% 12.24/8.00 => ( ( complete_Sup_Sup @ ( set @ A ) @ C6 )
% 12.24/8.00 = X8 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6766_subset_OHausdorff,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/8.00 ? [X_12: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X_12 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6767_subset_Omaxchain__def,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 = ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 & ~ ? [S7: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ S7 )
% 12.24/8.00 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ S7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6768_subset_Omaxchain__imp__chain,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6769_subset_Onot__maxchain__Some,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) )
% 12.24/8.00 @ ( fChoice @ ( set @ ( set @ A ) )
% 12.24/8.00 @ ^ [D7: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) )
% 12.24/8.00 & ( ord_less @ ( set @ ( set @ A ) ) @ C6
% 12.24/8.00 @ ( fChoice @ ( set @ ( set @ A ) )
% 12.24/8.00 @ ^ [D7: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6770_pred__on_Omaxchain__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( pred_maxchain @ A )
% 12.24/8.00 = ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A7 @ P4 @ C10 )
% 12.24/8.00 & ~ ? [S7: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A7 @ P4 @ S7 )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ C10 @ S7 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6771_pred__on_Osuc__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( pred_suc @ A )
% 12.24/8.00 = ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
% 12.24/8.00 ( if @ ( set @ A )
% 12.24/8.00 @ ( ~ ( pred_chain @ A @ A7 @ P4 @ C10 )
% 12.24/8.00 | ( pred_maxchain @ A @ A7 @ P4 @ C10 ) )
% 12.24/8.00 @ C10
% 12.24/8.00 @ ( fChoice @ ( set @ A )
% 12.24/8.00 @ ^ [D7: set @ A] :
% 12.24/8.00 ( ( pred_chain @ A @ A7 @ P4 @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ C10 @ D7 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6772_subset_Osuc__def,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 | ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
% 12.24/8.00 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 = C6 ) )
% 12.24/8.00 & ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 | ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
% 12.24/8.00 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 = ( fChoice @ ( set @ ( set @ A ) )
% 12.24/8.00 @ ^ [D7: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
% 12.24/8.00 & ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6773_subset_Onot__chain__suc,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 12.24/8.00 ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 = X8 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6774_subset_Omaxchain__suc,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 = X8 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6775_subset_Ochain__suc,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6776_subset_Osubset__suc,axiom,
% 12.24/8.00 ! [A: $tType,X8: set @ ( set @ A ),Y7: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.00 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 12.24/8.00 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6777_subset_Osuc__subset,axiom,
% 12.24/8.00 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6778_subset_Osuc__in__carrier,axiom,
% 12.24/8.00 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.00 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 )
% 12.24/8.00 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6779_subset_Ochain__sucD,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.00 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 )
% 12.24/8.00 & ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6780_subset_Osuc__not__equals,axiom,
% 12.24/8.00 ! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
% 12.24/8.00 ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 => ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
% 12.24/8.00 != C6 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6781_alloc__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( heap @ A )
% 12.24/8.00 => ( ( array_alloc @ A )
% 12.24/8.00 = ( ^ [Xs3: list @ A,H4: heap_ext @ product_unit] :
% 12.24/8.00 ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H4 ) )
% 12.24/8.00 @ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H4 ) ) @ Xs3
% 12.24/8.00 @ ( lim_update @ product_unit
% 12.24/8.00 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H4 ) @ ( one_one @ nat ) )
% 12.24/8.00 @ H4 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6782_Code__Numeral_Osub__def,axiom,
% 12.24/8.00 ( code_sub
% 12.24/8.00 = ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
% 12.24/8.00 @ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6783_present__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( heap @ A )
% 12.24/8.00 => ( ( array_present @ A )
% 12.24/8.00 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A5 ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6784_array_Osize_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,X: nat] :
% 12.24/8.00 ( ( size_size @ ( array @ A ) @ ( array2 @ A @ X ) )
% 12.24/8.00 = ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6785_array_Osize__gen,axiom,
% 12.24/8.00 ! [A: $tType,Xa3: A > nat,X: nat] :
% 12.24/8.00 ( ( size_array @ A @ Xa3 @ ( array2 @ A @ X ) )
% 12.24/8.00 = ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6786_noteq__def,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( ( heap @ A )
% 12.24/8.00 & ( heap @ B ) )
% 12.24/8.00 => ( ( array_noteq @ A @ B )
% 12.24/8.00 = ( ^ [R5: array @ A,S6: array @ B] :
% 12.24/8.00 ( ( ( typerep_typerep @ A @ ( type2 @ A ) )
% 12.24/8.00 != ( typerep_typerep @ B @ ( type2 @ B ) ) )
% 12.24/8.00 | ( ( addr_of_array @ A @ R5 )
% 12.24/8.00 != ( addr_of_array @ B @ S6 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6787_get__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( heap @ A )
% 12.24/8.00 => ( ( array_get @ A )
% 12.24/8.00 = ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H4 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A5 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6788_Heap_Oempty__def,axiom,
% 12.24/8.00 ( empty
% 12.24/8.00 = ( heap_ext2 @ product_unit
% 12.24/8.00 @ ^ [Uu: typerep,Uv: nat] : ( nil @ nat )
% 12.24/8.00 @ ^ [Uu: typerep,Uv: nat] : ( zero_zero @ nat )
% 12.24/8.00 @ ( zero_zero @ nat )
% 12.24/8.00 @ product_Unity ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6789_coset__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( coset @ A )
% 12.24/8.00 = ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6790_compl__coset,axiom,
% 12.24/8.00 ! [A: $tType,Xs: list @ A] :
% 12.24/8.00 ( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
% 12.24/8.00 = ( set2 @ A @ Xs ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6791_Nitpick_Otranclp__unfold,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( transitive_tranclp @ A )
% 12.24/8.00 = ( ^ [R5: A > A > $o,A5: A,B3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6792_Field__natLeq__on,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( ( field2 @ nat
% 12.24/8.00 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.00 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.00 @ ^ [X2: nat,Y6: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ X2 @ N )
% 12.24/8.00 & ( ord_less @ nat @ Y6 @ N )
% 12.24/8.00 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
% 12.24/8.00 = ( collect @ nat
% 12.24/8.00 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6793_less__nat__rel,axiom,
% 12.24/8.00 ( ( ord_less @ nat )
% 12.24/8.00 = ( transitive_tranclp @ nat
% 12.24/8.00 @ ^ [M2: nat,N3: nat] :
% 12.24/8.00 ( N3
% 12.24/8.00 = ( suc @ M2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6794_tranclp__power,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( transitive_tranclp @ A )
% 12.24/8.00 = ( ^ [P4: A > A > $o,X2: A,Y6: A] :
% 12.24/8.00 ? [N3: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 12.24/8.00 & ( compow @ ( A > A > $o ) @ N3 @ P4 @ X2 @ Y6 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6795_underS__Field3,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 12.24/8.00 ( ( ( field2 @ A @ R )
% 12.24/8.00 != ( bot_bot @ ( set @ A ) ) )
% 12.24/8.00 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6796_underS__Field2,axiom,
% 12.24/8.00 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 12.24/8.00 ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 12.24/8.00 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6797_natLeq__on__Well__order,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( order_well_order_on @ nat
% 12.24/8.00 @ ( field2 @ nat
% 12.24/8.00 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.00 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.00 @ ^ [X2: nat,Y6: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ X2 @ N )
% 12.24/8.00 & ( ord_less @ nat @ Y6 @ N )
% 12.24/8.00 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
% 12.24/8.00 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.00 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.00 @ ^ [X2: nat,Y6: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ X2 @ N )
% 12.24/8.00 & ( ord_less @ nat @ Y6 @ N )
% 12.24/8.00 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6798_Quotient3__int,axiom,
% 12.24/8.00 quotient3 @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6799_natLeq__on__well__order__on,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( order_well_order_on @ nat
% 12.24/8.00 @ ( collect @ nat
% 12.24/8.00 @ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
% 12.24/8.00 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.00 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.00 @ ^ [X2: nat,Y6: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ X2 @ N )
% 12.24/8.00 & ( ord_less @ nat @ Y6 @ N )
% 12.24/8.00 & ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6800_embedS__Field,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 12.24/8.00 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6801_list__all__length,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( list_all @ A )
% 12.24/8.00 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 12.24/8.00 ! [N3: nat] :
% 12.24/8.00 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 12.24/8.00 => ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6802_ofilter__subset__embedS,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ A6 )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ B7 )
% 12.24/8.00 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/8.00 = ( bNF_Wellorder_embedS @ A @ A
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A6
% 12.24/8.00 @ ^ [Uu: A] : A6 ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ B7
% 12.24/8.00 @ ^ [Uu: A] : B7 ) )
% 12.24/8.00 @ ( id @ A ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6803_embedS__iff,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 12.24/8.00 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 12.24/8.00 = ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6804_ofilterIncl__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( bNF_We413866401316099525erIncl @ A )
% 12.24/8.00 = ( ^ [R5: set @ ( product_prod @ A @ A )] :
% 12.24/8.00 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 12.24/8.00 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 12.24/8.00 @ ^ [A7: set @ A,B6: set @ A] :
% 12.24/8.00 ( ( order_ofilter @ A @ R5 @ A7 )
% 12.24/8.00 & ( A7
% 12.24/8.00 != ( field2 @ A @ R5 ) )
% 12.24/8.00 & ( order_ofilter @ A @ R5 @ B6 )
% 12.24/8.00 & ( B6
% 12.24/8.00 != ( field2 @ A @ R5 ) )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6805_ofilter__subset__embedS__iso,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ A6 )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ B7 )
% 12.24/8.00 => ( ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/8.00 = ( bNF_Wellorder_embedS @ A @ A
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A6
% 12.24/8.00 @ ^ [Uu: A] : A6 ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ B7
% 12.24/8.00 @ ^ [Uu: A] : B7 ) )
% 12.24/8.00 @ ( id @ A ) ) )
% 12.24/8.00 & ( ( A6 = B7 )
% 12.24/8.00 = ( bNF_Wellorder_iso @ A @ A
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A6
% 12.24/8.00 @ ^ [Uu: A] : A6 ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ B7
% 12.24/8.00 @ ^ [Uu: A] : B7 ) )
% 12.24/8.00 @ ( id @ A ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6806_ofilter__ordLess,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ A6 )
% 12.24/8.00 => ( ( ord_less @ ( set @ A ) @ A6 @ ( field2 @ A @ R ) )
% 12.24/8.00 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 12.24/8.00 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A6
% 12.24/8.00 @ ^ [Uu: A] : A6 ) )
% 12.24/8.00 @ R )
% 12.24/8.00 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6807_ofilter__subset__ordLess,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ A6 )
% 12.24/8.00 => ( ( order_ofilter @ A @ R @ B7 )
% 12.24/8.00 => ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 12.24/8.00 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 12.24/8.00 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A6
% 12.24/8.00 @ ^ [Uu: A] : A6 ) )
% 12.24/8.00 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 12.24/8.00 @ ( product_Sigma @ A @ A @ B7
% 12.24/8.00 @ ^ [Uu: A] : B7 ) ) )
% 12.24/8.00 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6808_bsqr__ofilter,axiom,
% 12.24/8.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),D4: set @ ( product_prod @ A @ A )] :
% 12.24/8.00 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 12.24/8.00 => ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R ) @ D4 )
% 12.24/8.00 => ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D4
% 12.24/8.00 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
% 12.24/8.00 @ ^ [Uu: A] : ( field2 @ A @ R ) ) )
% 12.24/8.00 => ( ~ ? [A8: A] :
% 12.24/8.00 ( ( field2 @ A @ R )
% 12.24/8.00 = ( order_under @ A @ R @ A8 ) )
% 12.24/8.00 => ? [A9: set @ A] :
% 12.24/8.00 ( ( order_ofilter @ A @ R @ A9 )
% 12.24/8.00 & ( ord_less @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 12.24/8.00 & ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D4
% 12.24/8.00 @ ( product_Sigma @ A @ A @ A9
% 12.24/8.00 @ ^ [Uu: A] : A9 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6809_internalize__ordLess,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 12.24/8.00 ( ( member @ ( 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 ) )
% 12.24/8.00 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 12.24/8.00 ( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 12.24/8.00 & ( member @ ( 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 ) )
% 12.24/8.00 & ( 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 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6810_of__int__code_I1_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ring_1 @ A )
% 12.24/8.00 => ! [K: num] :
% 12.24/8.00 ( ( ring_1_of_int @ A @ ( neg @ K ) )
% 12.24/8.00 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6811_type__definition__natural,axiom,
% 12.24/8.00 type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6812_less__int__code_I3_J,axiom,
% 12.24/8.00 ! [L: num] :
% 12.24/8.00 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6813_less__int__code_I7_J,axiom,
% 12.24/8.00 ! [K: num] : ( ord_less @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6814_less__eq__int__code_I7_J,axiom,
% 12.24/8.00 ! [K: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6815_less__eq__int__code_I3_J,axiom,
% 12.24/8.00 ! [L: num] :
% 12.24/8.00 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6816_nat__code_I1_J,axiom,
% 12.24/8.00 ! [K: num] :
% 12.24/8.00 ( ( nat2 @ ( neg @ K ) )
% 12.24/8.00 = ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6817_plus__int__code_I6_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( plus_plus @ int @ ( neg @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6818_less__eq__int__code_I9_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ( ( ord_less_eq @ int @ ( neg @ K ) @ ( neg @ L ) )
% 12.24/8.00 = ( ord_less_eq @ num @ L @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6819_less__int__code_I9_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ( ( ord_less @ int @ ( neg @ K ) @ ( neg @ L ) )
% 12.24/8.00 = ( ord_less @ num @ L @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6820_Int_Osub__code_I4_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( sub @ one2 @ ( bit0 @ N ) )
% 12.24/8.00 = ( neg @ ( bitM @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6821_Int_Osub__code_I5_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( sub @ one2 @ ( bit1 @ N ) )
% 12.24/8.00 = ( neg @ ( bit0 @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6822_Int_Osub__code_I1_J,axiom,
% 12.24/8.00 ( ( sub @ one2 @ one2 )
% 12.24/8.00 = ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6823_Int_Osub__def,axiom,
% 12.24/8.00 ( sub
% 12.24/8.00 = ( ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6824_minus__int__code_I6_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( minus_minus @ int @ ( neg @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( sub @ N @ M ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6825_Int_Osub__code_I9_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 12.24/8.00 = ( minus_minus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6826_Int_Osub__code_I8_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 12.24/8.00 = ( plus_plus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6827_Int_Odup__code_I1_J,axiom,
% 12.24/8.00 ( ( dup @ ( zero_zero @ int ) )
% 12.24/8.00 = ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6828_Int_Odup__def,axiom,
% 12.24/8.00 ( dup
% 12.24/8.00 = ( ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6829_Int_Odup__code_I3_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( dup @ ( neg @ N ) )
% 12.24/8.00 = ( neg @ ( bit0 @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6830_Int_Osub__code_I6_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 12.24/8.00 = ( dup @ ( sub @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6831_Int_Osub__code_I7_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 12.24/8.00 = ( dup @ ( sub @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6832_Int_Osub__code_I2_J,axiom,
% 12.24/8.00 ! [M: num] :
% 12.24/8.00 ( ( sub @ ( bit0 @ M ) @ one2 )
% 12.24/8.00 = ( pos @ ( bitM @ M ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6833_Int_Osub__code_I3_J,axiom,
% 12.24/8.00 ! [M: num] :
% 12.24/8.00 ( ( sub @ ( bit1 @ M ) @ one2 )
% 12.24/8.00 = ( pos @ ( bit0 @ M ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6834_less__int__code_I8_J,axiom,
% 12.24/8.00 ! [K: num,L: num] : ( ord_less @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6835_less__int__code_I6_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ~ ( ord_less @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6836_less__eq__int__code_I6_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ~ ( ord_less_eq @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6837_less__eq__int__code_I8_J,axiom,
% 12.24/8.00 ! [K: num,L: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6838_Int_ONeg__def,axiom,
% 12.24/8.00 ( neg
% 12.24/8.00 = ( ^ [N3: num] : ( uminus_uminus @ int @ ( pos @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6839_uminus__int__code_I2_J,axiom,
% 12.24/8.00 ! [M: num] :
% 12.24/8.00 ( ( uminus_uminus @ int @ ( pos @ M ) )
% 12.24/8.00 = ( neg @ M ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6840_uminus__int__code_I3_J,axiom,
% 12.24/8.00 ! [M: num] :
% 12.24/8.00 ( ( uminus_uminus @ int @ ( neg @ M ) )
% 12.24/8.00 = ( pos @ M ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6841_nat__code_I3_J,axiom,
% 12.24/8.00 ! [K: num] :
% 12.24/8.00 ( ( nat2 @ ( pos @ K ) )
% 12.24/8.00 = ( nat_of_num @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6842_times__int__code_I3_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( times_times @ int @ ( pos @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6843_Int_OPos__def,axiom,
% 12.24/8.00 ( pos
% 12.24/8.00 = ( numeral_numeral @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6844_less__eq__int__code_I2_J,axiom,
% 12.24/8.00 ! [L: num] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6845_less__eq__int__code_I4_J,axiom,
% 12.24/8.00 ! [K: num] :
% 12.24/8.00 ~ ( ord_less_eq @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6846_of__int__code_I3_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ring_1 @ A )
% 12.24/8.00 => ! [K: num] :
% 12.24/8.00 ( ( ring_1_of_int @ A @ ( pos @ K ) )
% 12.24/8.00 = ( numeral_numeral @ A @ K ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6847_one__int__code,axiom,
% 12.24/8.00 ( ( one_one @ int )
% 12.24/8.00 = ( pos @ one2 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6848_plus__int__code_I3_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( plus_plus @ int @ ( pos @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6849_less__int__code_I4_J,axiom,
% 12.24/8.00 ! [K: num] :
% 12.24/8.00 ~ ( ord_less @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6850_less__int__code_I2_J,axiom,
% 12.24/8.00 ! [L: num] : ( ord_less @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6851_less__eq__int__code_I5_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ( ( ord_less_eq @ int @ ( pos @ K ) @ ( pos @ L ) )
% 12.24/8.00 = ( ord_less_eq @ num @ K @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6852_less__int__code_I5_J,axiom,
% 12.24/8.00 ! [K: num,L: num] :
% 12.24/8.00 ( ( ord_less @ int @ ( pos @ K ) @ ( pos @ L ) )
% 12.24/8.00 = ( ord_less @ num @ K @ L ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6853_minus__int__code_I3_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( minus_minus @ int @ ( pos @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( sub @ M @ N ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6854_Int_Odup__code_I2_J,axiom,
% 12.24/8.00 ! [N: num] :
% 12.24/8.00 ( ( dup @ ( pos @ N ) )
% 12.24/8.00 = ( pos @ ( bit0 @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6855_minus__int__code_I4_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( minus_minus @ int @ ( pos @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6856_minus__int__code_I5_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( minus_minus @ int @ ( neg @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6857_times__int__code_I4_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( times_times @ int @ ( pos @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6858_times__int__code_I5_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( times_times @ int @ ( neg @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6859_times__int__code_I6_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( times_times @ int @ ( neg @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6860_plus__int__code_I5_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( plus_plus @ int @ ( neg @ M ) @ ( pos @ N ) )
% 12.24/8.00 = ( sub @ N @ M ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6861_plus__int__code_I4_J,axiom,
% 12.24/8.00 ! [M: num,N: num] :
% 12.24/8.00 ( ( plus_plus @ int @ ( pos @ M ) @ ( neg @ N ) )
% 12.24/8.00 = ( sub @ M @ N ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6862_prod__list_Ocomm__monoid__list__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid_mult @ A )
% 12.24/8.00 => ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6863_Least__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ord @ A )
% 12.24/8.00 => ( ( ord_Least @ A )
% 12.24/8.00 = ( ^ [P4: A > $o] :
% 12.24/8.00 ( the @ A
% 12.24/8.00 @ ^ [X2: A] :
% 12.24/8.00 ( ( P4 @ X2 )
% 12.24/8.00 & ! [Y6: A] :
% 12.24/8.00 ( ( P4 @ Y6 )
% 12.24/8.00 => ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6864_Ball__def__raw,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ball @ A )
% 12.24/8.00 = ( ^ [A7: set @ A,P4: A > $o] :
% 12.24/8.00 ! [X2: A] :
% 12.24/8.00 ( ( member @ A @ X2 @ A7 )
% 12.24/8.00 => ( P4 @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6865_sum__list_Ocomm__monoid__list__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid_add @ A )
% 12.24/8.00 => ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6866_Greatest__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ( ( order_Greatest @ A )
% 12.24/8.00 = ( ^ [P4: A > $o] :
% 12.24/8.00 ( the @ A
% 12.24/8.00 @ ^ [X2: A] :
% 12.24/8.00 ( ( P4 @ X2 )
% 12.24/8.00 & ! [Y6: A] :
% 12.24/8.00 ( ( P4 @ Y6 )
% 12.24/8.00 => ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6867_ord_OLeast__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( least @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
% 12.24/8.00 ( the @ A
% 12.24/8.00 @ ^ [X2: A] :
% 12.24/8.00 ( ( P4 @ X2 )
% 12.24/8.00 & ! [Y6: A] :
% 12.24/8.00 ( ( P4 @ Y6 )
% 12.24/8.00 => ( Less_eq2 @ X2 @ Y6 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6868_ord_OLeast_Ocong,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( least @ A )
% 12.24/8.00 = ( least @ A ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6869_GreatestI__nat,axiom,
% 12.24/8.00 ! [P: nat > $o,K: nat,B2: nat] :
% 12.24/8.00 ( ( P @ K )
% 12.24/8.00 => ( ! [Y5: nat] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 12.24/8.00 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6870_Greatest__le__nat,axiom,
% 12.24/8.00 ! [P: nat > $o,K: nat,B2: nat] :
% 12.24/8.00 ( ( P @ K )
% 12.24/8.00 => ( ! [Y5: nat] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 12.24/8.00 => ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6871_GreatestI__ex__nat,axiom,
% 12.24/8.00 ! [P: nat > $o,B2: nat] :
% 12.24/8.00 ( ? [X_1: nat] : ( P @ X_1 )
% 12.24/8.00 => ( ! [Y5: nat] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less_eq @ nat @ Y5 @ B2 ) )
% 12.24/8.00 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6872_GreatestI2__order,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ! [P: A > $o,X: A,Q: A > $o] :
% 12.24/8.00 ( ( P @ X )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/8.00 => ( ! [X3: A] :
% 12.24/8.00 ( ( P @ X3 )
% 12.24/8.00 => ( ! [Y4: A] :
% 12.24/8.00 ( ( P @ Y4 )
% 12.24/8.00 => ( ord_less_eq @ A @ Y4 @ X3 ) )
% 12.24/8.00 => ( Q @ X3 ) ) )
% 12.24/8.00 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6873_Greatest__equality,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ! [P: A > $o,X: A] :
% 12.24/8.00 ( ( P @ X )
% 12.24/8.00 => ( ! [Y5: A] :
% 12.24/8.00 ( ( P @ Y5 )
% 12.24/8.00 => ( ord_less_eq @ A @ Y5 @ X ) )
% 12.24/8.00 => ( ( order_Greatest @ A @ P )
% 12.24/8.00 = X ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6874_sum__encode__def,axiom,
% 12.24/8.00 ( nat_sum_encode
% 12.24/8.00 = ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 12.24/8.00 @ ^ [B3: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6875_disjnt__ge__max,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ! [Y7: set @ A,X8: set @ A] :
% 12.24/8.00 ( ( finite_finite2 @ A @ Y7 )
% 12.24/8.00 => ( ! [X3: A] :
% 12.24/8.00 ( ( member @ A @ X3 @ X8 )
% 12.24/8.00 => ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y7 ) @ X3 ) )
% 12.24/8.00 => ( disjnt @ A @ X8 @ Y7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6876_inj__sum__encode,axiom,
% 12.24/8.00 ! [A6: set @ ( sum_sum @ nat @ nat )] : ( inj_on @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ A6 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6877_sum__encode__eq,axiom,
% 12.24/8.00 ! [X: sum_sum @ nat @ nat,Y3: sum_sum @ nat @ nat] :
% 12.24/8.00 ( ( ( nat_sum_encode @ X )
% 12.24/8.00 = ( nat_sum_encode @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6878_bij__sum__encode,axiom,
% 12.24/8.00 bij_betw @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6879_surj__sum__encode,axiom,
% 12.24/8.00 ( ( image @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
% 12.24/8.00 = ( top_top @ ( set @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6880_int__encode__def,axiom,
% 12.24/8.00 ( nat_int_encode
% 12.24/8.00 = ( ^ [I5: int] : ( nat_sum_encode @ ( if @ ( sum_sum @ nat @ nat ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I5 ) @ ( sum_Inl @ nat @ nat @ ( nat2 @ I5 ) ) @ ( sum_Inr @ nat @ nat @ ( nat2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ I5 ) @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6881_ordering__top_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 12.24/8.00 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/8.00 => ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6882_prod_OPlus,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,C: $tType] :
% 12.24/8.00 ( ( comm_monoid_mult @ A )
% 12.24/8.00 => ! [A6: set @ B,B7: set @ C,G: ( sum_sum @ B @ C ) > A] :
% 12.24/8.00 ( ( finite_finite2 @ B @ A6 )
% 12.24/8.00 => ( ( finite_finite2 @ C @ B7 )
% 12.24/8.00 => ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A6 @ B7 ) )
% 12.24/8.00 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B7 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6883_ordering__top__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ordering_top_axioms @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o,Top2: A] :
% 12.24/8.00 ! [A5: A] : ( Less_eq2 @ A5 @ Top2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6884_ordering__top__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Top: A] :
% 12.24/8.00 ( ! [A8: A] : ( Less_eq @ A8 @ Top )
% 12.24/8.00 => ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6885_ndepth__Push__Node__aux,axiom,
% 12.24/8.00 ! [A: $tType,I: nat,F: nat > ( sum_sum @ A @ nat ),K: nat] :
% 12.24/8.00 ( ( ( case_nat @ ( sum_sum @ A @ nat ) @ ( sum_Inr @ nat @ A @ ( suc @ I ) ) @ F @ K )
% 12.24/8.00 = ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) )
% 12.24/8.00 => ( ord_less_eq @ nat
% 12.24/8.00 @ ( suc
% 12.24/8.00 @ ( ord_Least @ nat
% 12.24/8.00 @ ^ [X2: nat] :
% 12.24/8.00 ( ( F @ X2 )
% 12.24/8.00 = ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) )
% 12.24/8.00 @ K ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6886_le__sum__encode__Inl,axiom,
% 12.24/8.00 ! [X: nat,Y3: nat] :
% 12.24/8.00 ( ( ord_less_eq @ nat @ X @ Y3 )
% 12.24/8.00 => ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inl @ nat @ nat @ Y3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6887_le__sum__encode__Inr,axiom,
% 12.24/8.00 ! [X: nat,Y3: nat] :
% 12.24/8.00 ( ( ord_less_eq @ nat @ X @ Y3 )
% 12.24/8.00 => ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inr @ nat @ nat @ Y3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6888_sum_Osize_I3_J,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,X13: A] :
% 12.24/8.00 ( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X13 ) )
% 12.24/8.00 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6889_sum_Osize_I4_J,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,X22: B] :
% 12.24/8.00 ( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X22 ) )
% 12.24/8.00 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6890_sum_Osize__neq,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
% 12.24/8.00 ( ( size_size @ ( sum_sum @ A @ B ) @ X )
% 12.24/8.00 != ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6891_prod_Osize__neq,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,X: product_prod @ A @ B] :
% 12.24/8.00 ( ( size_size @ ( product_prod @ A @ B ) @ X )
% 12.24/8.00 != ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6892_sum_Osize__gen_I1_J,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,Xa3: A > nat,X: B > nat,X13: A] :
% 12.24/8.00 ( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inl @ A @ B @ X13 ) )
% 12.24/8.00 = ( plus_plus @ nat @ ( Xa3 @ X13 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6893_sum_Osize__gen_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,Xa3: A > nat,X: B > nat,X22: B] :
% 12.24/8.00 ( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inr @ B @ A @ X22 ) )
% 12.24/8.00 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6894_sum__decode__def,axiom,
% 12.24/8.00 ( nat_sum_decode
% 12.24/8.00 = ( ^ [N3: nat] : ( if @ ( sum_sum @ nat @ nat ) @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( sum_Inl @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( sum_Inr @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6895_sum__decode__inverse,axiom,
% 12.24/8.00 ! [N: nat] :
% 12.24/8.00 ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
% 12.24/8.00 = N ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6896_sum__encode__inverse,axiom,
% 12.24/8.00 ! [X: sum_sum @ nat @ nat] :
% 12.24/8.00 ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
% 12.24/8.00 = X ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6897_sum__decode__eq,axiom,
% 12.24/8.00 ! [X: nat,Y3: nat] :
% 12.24/8.00 ( ( ( nat_sum_decode @ X )
% 12.24/8.00 = ( nat_sum_decode @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6898_inj__sum__decode,axiom,
% 12.24/8.00 ! [A6: set @ nat] : ( inj_on @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ A6 ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6899_surj__sum__decode,axiom,
% 12.24/8.00 ( ( image @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
% 12.24/8.00 = ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6900_bij__sum__decode,axiom,
% 12.24/8.00 bij_betw @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6901_int__decode__def,axiom,
% 12.24/8.00 ( nat_int_decode
% 12.24/8.00 = ( ^ [N3: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
% 12.24/8.00 @ ^ [B3: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B3 ) ) @ ( one_one @ int ) )
% 12.24/8.00 @ ( nat_sum_decode @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6902_Node__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( old_Node @ B @ A )
% 12.24/8.00 = ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
% 12.24/8.00 @ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
% 12.24/8.00 ? [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat,K3: nat] :
% 12.24/8.00 ( ( P6
% 12.24/8.00 = ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X2 ) )
% 12.24/8.00 & ( ( F2 @ K3 )
% 12.24/8.00 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6903_Node__K0__I,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
% 12.24/8.00 ( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
% 12.24/8.00 @ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
% 12.24/8.00 @ ^ [K3: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
% 12.24/8.00 @ A3 )
% 12.24/8.00 @ ( old_Node @ A @ B ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6904_Push__neq__K0,axiom,
% 12.24/8.00 ! [A: $tType,K: nat,F: nat > ( sum_sum @ A @ nat )] :
% 12.24/8.00 ( ( old_Push @ A @ ( sum_Inr @ nat @ A @ ( suc @ K ) ) @ F )
% 12.24/8.00 != ( ^ [Z7: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6905_ndepth__def,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( old_ndepth @ A @ B )
% 12.24/8.00 = ( ^ [N3: old_node @ A @ B] :
% 12.24/8.00 ( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
% 12.24/8.00 @ ^ [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat] :
% 12.24/8.00 ( ord_Least @ nat
% 12.24/8.00 @ ^ [K3: nat] :
% 12.24/8.00 ( ( F2 @ K3 )
% 12.24/8.00 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
% 12.24/8.00 @ ( old_Rep_Node @ A @ B @ N3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6906_ndepth__K0,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
% 12.24/8.00 ( ( old_ndepth @ A @ B
% 12.24/8.00 @ ( old_Abs_Node @ B @ A
% 12.24/8.00 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 12.24/8.00 @ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 12.24/8.00 @ X ) ) )
% 12.24/8.00 = ( zero_zero @ nat ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6907_ndepth__Push__Node,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,I: nat,N: old_node @ A @ B] :
% 12.24/8.00 ( ( old_ndepth @ A @ B @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ I ) ) @ N ) )
% 12.24/8.00 = ( suc @ ( old_ndepth @ A @ B @ N ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6908_Atom__def,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( old_Atom @ A @ B )
% 12.24/8.00 = ( ^ [X2: sum_sum @ A @ nat] :
% 12.24/8.00 ( insert @ ( old_node @ A @ B )
% 12.24/8.00 @ ( old_Abs_Node @ B @ A
% 12.24/8.00 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 12.24/8.00 @ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 12.24/8.00 @ X2 ) )
% 12.24/8.00 @ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6909_Scons__def,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType] :
% 12.24/8.00 ( ( old_Scons @ A @ B )
% 12.24/8.00 = ( ^ [M9: set @ ( old_node @ A @ B ),N7: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6910_ntrunc__0,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M7 )
% 12.24/8.00 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6911_ntrunc__Atom,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,K: nat,A3: sum_sum @ A @ nat] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Atom @ A @ B @ A3 ) )
% 12.24/8.00 = ( old_Atom @ A @ B @ A3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6912_ntrunc__Scons,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B ),N6: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Scons @ A @ B @ M7 @ N6 ) )
% 12.24/8.00 = ( old_Scons @ A @ B @ ( old_ntrunc @ A @ B @ K @ M7 ) @ ( old_ntrunc @ A @ B @ K @ N6 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6913_ntrunc__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B )
% 12.24/8.00 = ( ^ [K3: nat,N7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( collect @ ( old_node @ A @ B )
% 12.24/8.00 @ ^ [N3: old_node @ A @ B] :
% 12.24/8.00 ( ( member @ ( old_node @ A @ B ) @ N3 @ N7 )
% 12.24/8.00 & ( ord_less @ nat @ ( old_ndepth @ A @ B @ N3 ) @ K3 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6914_ntrunc__one__In0,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M7 ) )
% 12.24/8.00 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6915_ntrunc__one__In1,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M7 ) )
% 12.24/8.00 = ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6916_ntrunc__In0,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In0 @ A @ B @ M7 ) )
% 12.24/8.00 = ( old_In0 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6917_ntrunc__In1,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In1 @ A @ B @ M7 ) )
% 12.24/8.00 = ( old_In1 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6918_ntrunc__Leaf,axiom,
% 12.24/8.00 ! [B: $tType,A: $tType,K: nat,A3: A] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Leaf @ A @ B @ A3 ) )
% 12.24/8.00 = ( old_Leaf @ A @ B @ A3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6919_In1__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( old_In1 @ A @ B )
% 12.24/8.00 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6920_ntrunc__Numb,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType,K: nat,I: nat] :
% 12.24/8.00 ( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Numb @ A @ B @ I ) )
% 12.24/8.00 = ( old_Numb @ A @ B @ I ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6921_In0__def,axiom,
% 12.24/8.00 ! [A: $tType,B: $tType] :
% 12.24/8.00 ( ( old_In0 @ A @ B )
% 12.24/8.00 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6922_ordering__top_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( ordering_top_axioms @ A @ Less_eq @ Top )
% 12.24/8.00 => ( ordering_top @ A @ Less_eq @ Less @ Top ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6923_ordering__top__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ordering_top @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,Top2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 12.24/8.00 & ( ordering_top_axioms @ A @ Less_eq2 @ Top2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6924_ordering__top_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
% 12.24/8.00 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 12.24/8.00 => ( ordering @ A @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6925_ordering__strictI,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less_eq @ A8 @ B5 )
% 12.24/8.00 = ( ( Less @ A8 @ B5 )
% 12.24/8.00 | ( A8 = B5 ) ) )
% 12.24/8.00 => ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less @ A8 @ B5 )
% 12.24/8.00 => ~ ( Less @ B5 @ A8 ) )
% 12.24/8.00 => ( ! [A8: A] :
% 12.24/8.00 ~ ( Less @ A8 @ A8 )
% 12.24/8.00 => ( ! [A8: A,B5: A,C3: A] :
% 12.24/8.00 ( ( Less @ A8 @ B5 )
% 12.24/8.00 => ( ( Less @ B5 @ C3 )
% 12.24/8.00 => ( Less @ A8 @ C3 ) ) )
% 12.24/8.00 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6926_ordering_Onot__eq__order__implies__strict,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( A3 != B2 )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( Less @ A3 @ B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6927_ordering_Ostrict__implies__not__eq,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ B2 )
% 12.24/8.00 => ( A3 != B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6928_ordering_Ostrict__iff__order,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less @ A3 @ B2 )
% 12.24/8.00 = ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 & ( A3 != B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6929_ordering_Oorder__iff__strict,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 = ( ( Less @ A3 @ B2 )
% 12.24/8.00 | ( A3 = B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6930_ordering_Oantisym,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( ( Less_eq @ B2 @ A3 )
% 12.24/8.00 => ( A3 = B2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6931_ordering_Oeq__iff,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ( A3 = B2 )
% 12.24/8.00 = ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 & ( Less_eq @ B2 @ A3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6932_ordering__dualI,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( ordering @ A
% 12.24/8.00 @ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
% 12.24/8.00 @ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
% 12.24/8.00 => ( ordering @ A @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6933_order_Oordering__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6934_dual__order_Oordering__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( order @ A )
% 12.24/8.00 => ( ordering @ A
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6935_nth__item_Oelims,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
% 12.24/8.00 ( ( ( nth_item @ A @ X )
% 12.24/8.00 = Y3 )
% 12.24/8.00 => ( ( ( X
% 12.24/8.00 = ( zero_zero @ nat ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 != ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
% 12.24/8.00 => ~ ! [N4: nat] :
% 12.24/8.00 ( ( X
% 12.24/8.00 = ( suc @ N4 ) )
% 12.24/8.00 => ( Y3
% 12.24/8.00 != ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] :
% 12.24/8.00 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 12.24/8.00 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 12.24/8.00 @ ( nat_prod_decode @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ( nat_sum_decode @ N4 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6936_group_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 12.24/8.00 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/8.00 => ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6937_group__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 12.24/8.00 ( ! [A8: A] :
% 12.24/8.00 ( ( F @ Z3 @ A8 )
% 12.24/8.00 = A8 )
% 12.24/8.00 => ( ! [A8: A] :
% 12.24/8.00 ( ( F @ ( Inverse @ A8 ) @ A8 )
% 12.24/8.00 = Z3 )
% 12.24/8.00 => ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6938_group__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( group_axioms @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
% 12.24/8.00 ( ! [A5: A] :
% 12.24/8.00 ( ( F2 @ Z7 @ A5 )
% 12.24/8.00 = A5 )
% 12.24/8.00 & ! [A5: A] :
% 12.24/8.00 ( ( F2 @ ( Inverse2 @ A5 ) @ A5 )
% 12.24/8.00 = Z7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6939_nth__item_Osimps_I1_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 12.24/8.00 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6940_nth__item_Osimps_I2_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ! [N: nat] :
% 12.24/8.00 ( ( nth_item @ A @ ( suc @ N ) )
% 12.24/8.00 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] :
% 12.24/8.00 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 12.24/8.00 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 12.24/8.00 @ ( nat_prod_decode @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ( nat_sum_decode @ N ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6941_nth__item_Opelims,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
% 12.24/8.00 ( ( ( nth_item @ A @ X )
% 12.24/8.00 = Y3 )
% 12.24/8.00 => ( ( accp @ nat @ nth_item_rel @ X )
% 12.24/8.00 => ( ( ( X
% 12.24/8.00 = ( zero_zero @ nat ) )
% 12.24/8.00 => ( ( Y3
% 12.24/8.00 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
% 12.24/8.00 => ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
% 12.24/8.00 => ~ ! [N4: nat] :
% 12.24/8.00 ( ( X
% 12.24/8.00 = ( suc @ N4 ) )
% 12.24/8.00 => ( ( Y3
% 12.24/8.00 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] :
% 12.24/8.00 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 12.24/8.00 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 12.24/8.00 @ ( nat_prod_decode @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ( nat_sum_decode @ N4 ) ) )
% 12.24/8.00 => ~ ( accp @ nat @ nth_item_rel @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6942_nth__item_Opsimps_I2_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ! [N: nat] :
% 12.24/8.00 ( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
% 12.24/8.00 => ( ( nth_item @ A @ ( suc @ N ) )
% 12.24/8.00 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 12.24/8.00 @ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
% 12.24/8.00 @ ^ [J3: nat] :
% 12.24/8.00 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 12.24/8.00 @ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
% 12.24/8.00 @ ( nat_prod_decode @ J3 ) )
% 12.24/8.00 @ ( nat_sum_decode @ I5 ) )
% 12.24/8.00 @ ( nat_sum_decode @ N ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6943_nth__item_Opsimps_I1_J,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( countable @ A )
% 12.24/8.00 => ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 12.24/8.00 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 12.24/8.00 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6944_group_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 12.24/8.00 ( ( semigroup @ A @ F )
% 12.24/8.00 => ( ( group_axioms @ A @ F @ Z3 @ Inverse )
% 12.24/8.00 => ( group @ A @ F @ Z3 @ Inverse ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6945_group__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( group @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
% 12.24/8.00 ( ( semigroup @ A @ F2 )
% 12.24/8.00 & ( group_axioms @ A @ F2 @ Z7 @ Inverse2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6946_sup_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_sup @ A )
% 12.24/8.00 => ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6947_semigroup_Oassoc,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( semigroup @ A @ F )
% 12.24/8.00 => ( ( F @ ( F @ A3 @ B2 ) @ C2 )
% 12.24/8.00 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6948_semigroup_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A] :
% 12.24/8.00 ( ! [A8: A,B5: A,C3: A] :
% 12.24/8.00 ( ( F @ ( F @ A8 @ B5 ) @ C3 )
% 12.24/8.00 = ( F @ A8 @ ( F @ B5 @ C3 ) ) )
% 12.24/8.00 => ( semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6949_semigroup__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semigroup @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A] :
% 12.24/8.00 ! [A5: A,B3: A,C4: A] :
% 12.24/8.00 ( ( F2 @ ( F2 @ A5 @ B3 ) @ C4 )
% 12.24/8.00 = ( F2 @ A5 @ ( F2 @ B3 @ C4 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6950_group_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
% 12.24/8.00 ( ( group @ A @ F @ Z3 @ Inverse )
% 12.24/8.00 => ( semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6951_monoid_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( monoid @ A @ F @ Z3 )
% 12.24/8.00 => ( semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6952_min_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6953_max_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6954_add_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semigroup_add @ A )
% 12.24/8.00 => ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6955_mult_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semigroup_mult @ A )
% 12.24/8.00 => ( semigroup @ A @ ( times_times @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6956_inf_Osemigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_inf @ A )
% 12.24/8.00 => ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6957_monoid_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( semigroup @ A @ F )
% 12.24/8.00 => ( ( monoid_axioms @ A @ F @ Z3 )
% 12.24/8.00 => ( monoid @ A @ F @ Z3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6958_monoid__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( monoid @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A] :
% 12.24/8.00 ( ( semigroup @ A @ F2 )
% 12.24/8.00 & ( monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6959_monoid__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( monoid_axioms @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A] :
% 12.24/8.00 ( ! [A5: A] :
% 12.24/8.00 ( ( F2 @ Z7 @ A5 )
% 12.24/8.00 = A5 )
% 12.24/8.00 & ! [A5: A] :
% 12.24/8.00 ( ( F2 @ A5 @ Z7 )
% 12.24/8.00 = A5 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6960_monoid__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ! [A8: A] :
% 12.24/8.00 ( ( F @ Z3 @ A8 )
% 12.24/8.00 = A8 )
% 12.24/8.00 => ( ! [A8: A] :
% 12.24/8.00 ( ( F @ A8 @ Z3 )
% 12.24/8.00 = A8 )
% 12.24/8.00 => ( monoid_axioms @ A @ F @ Z3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6961_monoid_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( monoid @ A @ F @ Z3 )
% 12.24/8.00 => ( monoid_axioms @ A @ F @ Z3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6962_nths__nths,axiom,
% 12.24/8.00 ! [A: $tType,Xs: list @ A,A6: set @ nat,B7: set @ nat] :
% 12.24/8.00 ( ( nths @ A @ ( nths @ A @ Xs @ A6 ) @ B7 )
% 12.24/8.00 = ( nths @ A @ Xs
% 12.24/8.00 @ ( collect @ nat
% 12.24/8.00 @ ^ [I5: nat] :
% 12.24/8.00 ( ( member @ nat @ I5 @ A6 )
% 12.24/8.00 & ( member @ nat
% 12.24/8.00 @ ( finite_card @ nat
% 12.24/8.00 @ ( collect @ nat
% 12.24/8.00 @ ^ [I9: nat] :
% 12.24/8.00 ( ( member @ nat @ I9 @ A6 )
% 12.24/8.00 & ( ord_less @ nat @ I9 @ I5 ) ) ) )
% 12.24/8.00 @ B7 ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6963_ordering_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ordering_axioms @ A @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6964_ordering__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 12.24/8.00 ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less @ A8 @ B5 )
% 12.24/8.00 = ( ( Less_eq @ A8 @ B5 )
% 12.24/8.00 & ( A8 != B5 ) ) )
% 12.24/8.00 => ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less_eq @ A8 @ B5 )
% 12.24/8.00 => ( ( Less_eq @ B5 @ A8 )
% 12.24/8.00 => ( A8 = B5 ) ) )
% 12.24/8.00 => ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6965_ordering__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ordering_axioms @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ! [A5: A,B3: A] :
% 12.24/8.00 ( ( Less2 @ A5 @ B3 )
% 12.24/8.00 = ( ( Less_eq2 @ A5 @ B3 )
% 12.24/8.00 & ( A5 != B3 ) ) )
% 12.24/8.00 & ! [A5: A,B3: A] :
% 12.24/8.00 ( ( Less_eq2 @ A5 @ B3 )
% 12.24/8.00 => ( ( Less_eq2 @ B3 @ A5 )
% 12.24/8.00 => ( A5 = B3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6966_Bex__def__raw,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( bex @ A )
% 12.24/8.00 = ( ^ [A7: set @ A,P4: A > $o] :
% 12.24/8.00 ? [X2: A] :
% 12.24/8.00 ( ( member @ A @ X2 @ A7 )
% 12.24/8.00 & ( P4 @ X2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6967_ordering__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ordering @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ( partial_preordering @ A @ Less_eq2 )
% 12.24/8.00 & ( ordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6968_ordering_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( partial_preordering @ A @ Less_eq )
% 12.24/8.00 => ( ( ordering_axioms @ A @ Less_eq @ Less )
% 12.24/8.00 => ( ordering @ A @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6969_dual__order_Opartial__preordering__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( partial_preordering @ A
% 12.24/8.00 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6970_order_Opartial__preordering__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( preorder @ A )
% 12.24/8.00 => ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6971_partial__preordering__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( partial_preordering @ A )
% 12.24/8.00 = ( ^ [Less_eq2: A > A > $o] :
% 12.24/8.00 ( ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
% 12.24/8.00 & ! [A5: A,B3: A,C4: A] :
% 12.24/8.00 ( ( Less_eq2 @ A5 @ B3 )
% 12.24/8.00 => ( ( Less_eq2 @ B3 @ C4 )
% 12.24/8.00 => ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6972_partial__preordering_Otrans,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.00 ( ( partial_preordering @ A @ Less_eq )
% 12.24/8.00 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.00 => ( ( Less_eq @ B2 @ C2 )
% 12.24/8.00 => ( Less_eq @ A3 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6973_partial__preordering_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o] :
% 12.24/8.00 ( ! [A8: A] : ( Less_eq @ A8 @ A8 )
% 12.24/8.00 => ( ! [A8: A,B5: A,C3: A] :
% 12.24/8.00 ( ( Less_eq @ A8 @ B5 )
% 12.24/8.00 => ( ( Less_eq @ B5 @ C3 )
% 12.24/8.00 => ( Less_eq @ A8 @ C3 ) ) )
% 12.24/8.00 => ( partial_preordering @ A @ Less_eq ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6974_partial__preordering_Orefl,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,A3: A] :
% 12.24/8.00 ( ( partial_preordering @ A @ Less_eq )
% 12.24/8.00 => ( Less_eq @ A3 @ A3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6975_ordering_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( ordering @ A @ Less_eq @ Less )
% 12.24/8.00 => ( partial_preordering @ A @ Less_eq ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6976_Abs__int__cases,axiom,
% 12.24/8.00 ! [X: int] :
% 12.24/8.00 ~ ! [Y5: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ( ( X
% 12.24/8.00 = ( abs_int @ Y5 ) )
% 12.24/8.00 => ~ ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6977_Abs__int__induct,axiom,
% 12.24/8.00 ! [P: int > $o,X: int] :
% 12.24/8.00 ( ! [Y5: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ( P @ ( abs_int @ Y5 ) ) )
% 12.24/8.00 => ( P @ X ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6978_Abs__int__inject,axiom,
% 12.24/8.00 ! [X: set @ ( product_prod @ nat @ nat ),Y3: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ X
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ( ( ( abs_int @ X )
% 12.24/8.00 = ( abs_int @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6979_Abs__Integ__def,axiom,
% 12.24/8.00 ( abs_Integ
% 12.24/8.00 = ( quot_abs @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6980_Abs__int__inverse,axiom,
% 12.24/8.00 ! [Y3: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ( ( rep_int @ ( abs_int @ Y3 ) )
% 12.24/8.00 = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6981_Rep__int__inject,axiom,
% 12.24/8.00 ! [X: int,Y3: int] :
% 12.24/8.00 ( ( ( rep_int @ X )
% 12.24/8.00 = ( rep_int @ Y3 ) )
% 12.24/8.00 = ( X = Y3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6982_Rep__int__inverse,axiom,
% 12.24/8.00 ! [X: int] :
% 12.24/8.00 ( ( abs_int @ ( rep_int @ X ) )
% 12.24/8.00 = X ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6983_Rep__int,axiom,
% 12.24/8.00 ! [X: int] :
% 12.24/8.00 ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ ( rep_int @ X )
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6984_Rep__int__cases,axiom,
% 12.24/8.00 ! [Y3: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ~ ! [X3: int] :
% 12.24/8.00 ( Y3
% 12.24/8.00 != ( rep_int @ X3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6985_Rep__int__induct,axiom,
% 12.24/8.00 ! [Y3: set @ ( product_prod @ nat @ nat ),P: ( set @ ( product_prod @ nat @ nat ) ) > $o] :
% 12.24/8.00 ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
% 12.24/8.00 => ( ! [X3: int] : ( P @ ( rep_int @ X3 ) )
% 12.24/8.00 => ( P @ Y3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6986_Rep__Integ__def,axiom,
% 12.24/8.00 ( rep_Integ
% 12.24/8.00 = ( quot_rep @ int @ ( product_prod @ nat @ nat ) @ rep_int ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6987_type__definition__int,axiom,
% 12.24/8.00 ( type_definition @ int @ ( set @ ( product_prod @ nat @ nat ) ) @ rep_int @ abs_int
% 12.24/8.00 @ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
% 12.24/8.00 @ ^ [C4: set @ ( product_prod @ nat @ nat )] :
% 12.24/8.00 ? [X2: product_prod @ nat @ nat] :
% 12.24/8.00 ( ( intrel @ X2 @ X2 )
% 12.24/8.00 & ( C4
% 12.24/8.00 = ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6988_Domainp__cr__rat,axiom,
% 12.24/8.00 ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
% 12.24/8.00 = ( ^ [X2: product_prod @ int @ int] :
% 12.24/8.00 ( ( product_snd @ int @ int @ X2 )
% 12.24/8.00 != ( zero_zero @ int ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6989_semilattice__neutr_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( ( comm_monoid @ A @ F @ Z3 )
% 12.24/8.00 => ( semilattice_neutr @ A @ F @ Z3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6990_inf_Osemilattice__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_inf @ A )
% 12.24/8.00 => ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6991_semilattice__order_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( semilattice @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6992_max_Osemilattice__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6993_min_Osemilattice__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6994_semilattice_Oright__idem,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( ( F @ ( F @ A3 @ B2 ) @ B2 )
% 12.24/8.00 = ( F @ A3 @ B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6995_semilattice_Oleft__idem,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( ( F @ A3 @ ( F @ A3 @ B2 ) )
% 12.24/8.00 = ( F @ A3 @ B2 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6996_semilattice_Oidem,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,A3: A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( ( F @ A3 @ A3 )
% 12.24/8.00 = A3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6997_semilattice__set_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( lattic149705377957585745ce_set @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6998_semilattice__set_Oaxioms,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A] :
% 12.24/8.00 ( ( lattic149705377957585745ce_set @ A @ F )
% 12.24/8.00 => ( semilattice @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_6999_semilattice__set__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( lattic149705377957585745ce_set @ A )
% 12.24/8.00 = ( semilattice @ A ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7000_semilattice__neutr_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( semilattice_neutr @ A @ F @ Z3 )
% 12.24/8.00 => ( semilattice @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7001_sup_Osemilattice__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_sup @ A )
% 12.24/8.00 => ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7002_semilattice__neutr__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_neutr @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A] :
% 12.24/8.00 ( ( semilattice @ A @ F2 )
% 12.24/8.00 & ( comm_monoid @ A @ F2 @ Z7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7003_semilattice__order_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7004_semilattice__order__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_order @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ( semilattice @ A @ F2 )
% 12.24/8.00 & ( semila6385135966242565138axioms @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7005_semilattice__order__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,Less_eq: A > A > $o,F: A > A > A,Less: A > A > $o] :
% 12.24/8.00 ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less_eq @ A8 @ B5 )
% 12.24/8.00 = ( A8
% 12.24/8.00 = ( F @ A8 @ B5 ) ) )
% 12.24/8.00 => ( ! [A8: A,B5: A] :
% 12.24/8.00 ( ( Less @ A8 @ B5 )
% 12.24/8.00 = ( ( A8
% 12.24/8.00 = ( F @ A8 @ B5 ) )
% 12.24/8.00 & ( A8 != B5 ) ) )
% 12.24/8.00 => ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7006_semilattice__order__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semila6385135966242565138axioms @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.00 ( ! [A5: A,B3: A] :
% 12.24/8.00 ( ( Less_eq2 @ A5 @ B3 )
% 12.24/8.00 = ( A5
% 12.24/8.00 = ( F2 @ A5 @ B3 ) ) )
% 12.24/8.00 & ! [A5: A,B3: A] :
% 12.24/8.00 ( ( Less2 @ A5 @ B3 )
% 12.24/8.00 = ( ( A5
% 12.24/8.00 = ( F2 @ A5 @ B3 ) )
% 12.24/8.00 & ( A5 != B3 ) ) ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7007_semilattice__order_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.00 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 12.24/8.00 => ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7008_prod__list_Omonoid__list__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( monoid_mult @ A )
% 12.24/8.00 => ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7009_comm__monoid_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( comm_monoid @ A @ F @ Z3 )
% 12.24/8.00 => ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7010_comm__monoid__axioms__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid_axioms @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A] :
% 12.24/8.00 ! [A5: A] :
% 12.24/8.00 ( ( F2 @ A5 @ Z7 )
% 12.24/8.00 = A5 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7011_comm__monoid__axioms_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ! [A8: A] :
% 12.24/8.00 ( ( F @ A8 @ Z3 )
% 12.24/8.00 = A8 )
% 12.24/8.00 => ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7012_sum__list_Omonoid__list__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( monoid_add @ A )
% 12.24/8.00 => ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7013_comm__monoid__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( comm_monoid @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A,Z7: A] :
% 12.24/8.00 ( ( abel_semigroup @ A @ F2 )
% 12.24/8.00 & ( comm_monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7014_comm__monoid_Ointro,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( abel_semigroup @ A @ F )
% 12.24/8.00 => ( ( comm_monoid_axioms @ A @ F @ Z3 )
% 12.24/8.00 => ( comm_monoid @ A @ F @ Z3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7015_semilattice_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A] :
% 12.24/8.00 ( ( semilattice @ A @ F )
% 12.24/8.00 => ( abel_semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7016_sup_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_sup @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7017_comm__monoid_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,Z3: A] :
% 12.24/8.00 ( ( comm_monoid @ A @ F @ Z3 )
% 12.24/8.00 => ( abel_semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7018_abel__semigroup_Ocommute,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 12.24/8.00 ( ( abel_semigroup @ A @ F )
% 12.24/8.00 => ( ( F @ A3 @ B2 )
% 12.24/8.00 = ( F @ B2 @ A3 ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7019_abel__semigroup_Oleft__commute,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
% 12.24/8.00 ( ( abel_semigroup @ A @ F )
% 12.24/8.00 => ( ( F @ B2 @ ( F @ A3 @ C2 ) )
% 12.24/8.00 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7020_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/8.00 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/8.00 => ( abel_semigroup @ A @ Conj ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7021_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
% 12.24/8.00 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/8.00 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/8.00 => ( abel_semigroup @ A @ Disj ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7022_min_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7023_add_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ab_semigroup_add @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7024_max_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( linorder @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7025_mult_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( ab_semigroup_mult @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7026_inf_Oabel__semigroup__axioms,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( semilattice_inf @ A )
% 12.24/8.00 => ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7027_abel__semigroup_Oaxioms_I1_J,axiom,
% 12.24/8.00 ! [A: $tType,F: A > A > A] :
% 12.24/8.00 ( ( abel_semigroup @ A @ F )
% 12.24/8.00 => ( semigroup @ A @ F ) ) ).
% 12.24/8.00
% 12.24/8.00 thf(fact_7028_abel__semigroup__def,axiom,
% 12.24/8.00 ! [A: $tType] :
% 12.24/8.00 ( ( abel_semigroup @ A )
% 12.24/8.00 = ( ^ [F2: A > A > A] :
% 12.24/8.00 ( ( semigroup @ A @ F2 )
% 12.24/8.00 & ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7029_abel__semigroup_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ( semigroup @ A @ F )
% 12.24/8.01 => ( ( abel_s757365448890700780axioms @ A @ F )
% 12.24/8.01 => ( abel_semigroup @ A @ F ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7030_abel__semigroup__axioms__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( abel_s757365448890700780axioms @ A )
% 12.24/8.01 = ( ^ [F2: A > A > A] :
% 12.24/8.01 ! [A5: A,B3: A] :
% 12.24/8.01 ( ( F2 @ A5 @ B3 )
% 12.24/8.01 = ( F2 @ B3 @ A5 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7031_abel__semigroup__axioms_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ! [A8: A,B5: A] :
% 12.24/8.01 ( ( F @ A8 @ B5 )
% 12.24/8.01 = ( F @ B5 @ A8 ) )
% 12.24/8.01 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7032_abel__semigroup_Oaxioms_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ( abel_semigroup @ A @ F )
% 12.24/8.01 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7033_semilattice_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ( abel_semigroup @ A @ F )
% 12.24/8.01 => ( ( semilattice_axioms @ A @ F )
% 12.24/8.01 => ( semilattice @ A @ F ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7034_semilattice__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( semilattice @ A )
% 12.24/8.01 = ( ^ [F2: A > A > A] :
% 12.24/8.01 ( ( abel_semigroup @ A @ F2 )
% 12.24/8.01 & ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7035_semilattice__axioms_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ! [A8: A] :
% 12.24/8.01 ( ( F @ A8 @ A8 )
% 12.24/8.01 = A8 )
% 12.24/8.01 => ( semilattice_axioms @ A @ F ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7036_semilattice__axioms__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( semilattice_axioms @ A )
% 12.24/8.01 = ( ^ [F2: A > A > A] :
% 12.24/8.01 ! [A5: A] :
% 12.24/8.01 ( ( F2 @ A5 @ A5 )
% 12.24/8.01 = A5 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7037_semilattice_Oaxioms_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A] :
% 12.24/8.01 ( ( semilattice @ A @ F )
% 12.24/8.01 => ( semilattice_axioms @ A @ F ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7038_abstract__boolean__algebra__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( boolea2506097494486148201lgebra @ A )
% 12.24/8.01 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
% 12.24/8.01 ( ( abel_semigroup @ A @ Conj2 )
% 12.24/8.01 & ( abel_semigroup @ A @ Disj2 )
% 12.24/8.01 & ( boolea6902313364301356556axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7039_abstract__boolean__algebra_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/8.01 ( ( abel_semigroup @ A @ Conj )
% 12.24/8.01 => ( ( abel_semigroup @ A @ Disj )
% 12.24/8.01 => ( ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/8.01 => ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7040_abstract__boolean__algebra__axioms__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( boolea6902313364301356556axioms @ A )
% 12.24/8.01 = ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
% 12.24/8.01 ( ! [X2: A,Y6: A,Z7: A] :
% 12.24/8.01 ( ( Conj2 @ X2 @ ( Disj2 @ Y6 @ Z7 ) )
% 12.24/8.01 = ( Disj2 @ ( Conj2 @ X2 @ Y6 ) @ ( Conj2 @ X2 @ Z7 ) ) )
% 12.24/8.01 & ! [X2: A,Y6: A,Z7: A] :
% 12.24/8.01 ( ( Disj2 @ X2 @ ( Conj2 @ Y6 @ Z7 ) )
% 12.24/8.01 = ( Conj2 @ ( Disj2 @ X2 @ Y6 ) @ ( Disj2 @ X2 @ Z7 ) ) )
% 12.24/8.01 & ! [X2: A] :
% 12.24/8.01 ( ( Conj2 @ X2 @ One )
% 12.24/8.01 = X2 )
% 12.24/8.01 & ! [X2: A] :
% 12.24/8.01 ( ( Disj2 @ X2 @ Zero2 )
% 12.24/8.01 = X2 )
% 12.24/8.01 & ! [X2: A] :
% 12.24/8.01 ( ( Conj2 @ X2 @ ( Compl2 @ X2 ) )
% 12.24/8.01 = Zero2 )
% 12.24/8.01 & ! [X2: A] :
% 12.24/8.01 ( ( Disj2 @ X2 @ ( Compl2 @ X2 ) )
% 12.24/8.01 = One ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7041_abstract__boolean__algebra__axioms_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,One2: A,Zero: A,Compl: A > A] :
% 12.24/8.01 ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/8.01 ( ( Conj @ X3 @ ( Disj @ Y5 @ Z4 ) )
% 12.24/8.01 = ( Disj @ ( Conj @ X3 @ Y5 ) @ ( Conj @ X3 @ Z4 ) ) )
% 12.24/8.01 => ( ! [X3: A,Y5: A,Z4: A] :
% 12.24/8.01 ( ( Disj @ X3 @ ( Conj @ Y5 @ Z4 ) )
% 12.24/8.01 = ( Conj @ ( Disj @ X3 @ Y5 ) @ ( Disj @ X3 @ Z4 ) ) )
% 12.24/8.01 => ( ! [X3: A] :
% 12.24/8.01 ( ( Conj @ X3 @ One2 )
% 12.24/8.01 = X3 )
% 12.24/8.01 => ( ! [X3: A] :
% 12.24/8.01 ( ( Disj @ X3 @ Zero )
% 12.24/8.01 = X3 )
% 12.24/8.01 => ( ! [X3: A] :
% 12.24/8.01 ( ( Conj @ X3 @ ( Compl @ X3 ) )
% 12.24/8.01 = Zero )
% 12.24/8.01 => ( ! [X3: A] :
% 12.24/8.01 ( ( Disj @ X3 @ ( Compl @ X3 ) )
% 12.24/8.01 = One2 )
% 12.24/8.01 => ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7042_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
% 12.24/8.01 ! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
% 12.24/8.01 ( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
% 12.24/8.01 => ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7043_typerep_Osize__neq,axiom,
% 12.24/8.01 ! [X: typerep] :
% 12.24/8.01 ( ( size_size @ typerep @ X )
% 12.24/8.01 != ( zero_zero @ nat ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7044_tuple__isomorphism_Osize__neq,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism @ A @ B @ C] :
% 12.24/8.01 ( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X )
% 12.24/8.01 != ( zero_zero @ nat ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7045_tuple__isomorphism_Osize_I2_J,axiom,
% 12.24/8.01 ! [B: $tType,C: $tType,A: $tType,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
% 12.24/8.01 ( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
% 12.24/8.01 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7046_typerep_Osize_I2_J,axiom,
% 12.24/8.01 ! [X13: literal,X22: list @ typerep] :
% 12.24/8.01 ( ( size_size @ typerep @ ( typerep3 @ X13 @ X22 ) )
% 12.24/8.01 = ( plus_plus @ nat @ ( size_list @ typerep @ ( size_size @ typerep ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7047_typerep_Osize__gen,axiom,
% 12.24/8.01 ! [X13: literal,X22: list @ typerep] :
% 12.24/8.01 ( ( size_typerep @ ( typerep3 @ X13 @ X22 ) )
% 12.24/8.01 = ( plus_plus @ nat @ ( size_list @ typerep @ size_typerep @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7048_tuple__isomorphism_Osize__gen,axiom,
% 12.24/8.01 ! [B: $tType,C: $tType,A: $tType,Xb3: A > nat,Xa3: B > nat,X: C > nat,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
% 12.24/8.01 ( ( tuple_9181185373184732606rphism @ A @ B @ C @ Xb3 @ Xa3 @ X @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
% 12.24/8.01 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7049_typerep__fun__box__def,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType] :
% 12.24/8.01 ( ( ( typerep2 @ A )
% 12.24/8.01 & ( typerep2 @ B ) )
% 12.24/8.01 => ( ( typerep_typerep @ ( fun_box @ A @ B ) )
% 12.24/8.01 = ( ^ [T7: itself @ ( fun_box @ A @ B )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( cons @ typerep @ ( typerep_typerep @ B @ ( type2 @ B ) ) @ ( nil @ typerep ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7050_typerep__word__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( typerep2 @ A )
% 12.24/8.01 => ( ( typerep_typerep @ ( word @ A ) )
% 12.24/8.01 = ( ^ [T7: itself @ ( word @ A )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( nil @ typerep ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7051_dual__order_Opreordering__axioms,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preorder @ A )
% 12.24/8.01 => ( preordering @ A
% 12.24/8.01 @ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
% 12.24/8.01 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7052_irreflp__greater,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preorder @ A )
% 12.24/8.01 => ( irreflp @ A
% 12.24/8.01 @ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7053_preordering_Oaxioms_I1_J,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( partial_preordering @ A @ Less_eq ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7054_irreflp__less,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preorder @ A )
% 12.24/8.01 => ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7055_preordering__dualI,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.01 ( ( preordering @ A
% 12.24/8.01 @ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
% 12.24/8.01 @ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
% 12.24/8.01 => ( preordering @ A @ Less_eq @ Less ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7056_preordering_Oasym,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less @ A3 @ B2 )
% 12.24/8.01 => ~ ( Less @ B2 @ A3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7057_preordering_Oirrefl,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ~ ( Less @ A3 @ A3 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7058_preordering_Ostrict__trans,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less @ A3 @ B2 )
% 12.24/8.01 => ( ( Less @ B2 @ C2 )
% 12.24/8.01 => ( Less @ A3 @ C2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7059_preordering_Ostrict__trans1,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less_eq @ A3 @ B2 )
% 12.24/8.01 => ( ( Less @ B2 @ C2 )
% 12.24/8.01 => ( Less @ A3 @ C2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7060_preordering_Ostrict__trans2,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less @ A3 @ B2 )
% 12.24/8.01 => ( ( Less_eq @ B2 @ C2 )
% 12.24/8.01 => ( Less @ A3 @ C2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7061_preordering_Ostrict__iff__not,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less @ A3 @ B2 )
% 12.24/8.01 = ( ( Less_eq @ A3 @ B2 )
% 12.24/8.01 & ~ ( Less_eq @ B2 @ A3 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7062_preordering_Ostrict__implies__order,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( ( Less @ A3 @ B2 )
% 12.24/8.01 => ( Less_eq @ A3 @ B2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7063_preordering__strictI,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.01 ( ! [A8: A,B5: A] :
% 12.24/8.01 ( ( Less_eq @ A8 @ B5 )
% 12.24/8.01 = ( ( Less @ A8 @ B5 )
% 12.24/8.01 | ( A8 = B5 ) ) )
% 12.24/8.01 => ( ! [A8: A,B5: A] :
% 12.24/8.01 ( ( Less @ A8 @ B5 )
% 12.24/8.01 => ~ ( Less @ B5 @ A8 ) )
% 12.24/8.01 => ( ! [A8: A] :
% 12.24/8.01 ~ ( Less @ A8 @ A8 )
% 12.24/8.01 => ( ! [A8: A,B5: A,C3: A] :
% 12.24/8.01 ( ( Less @ A8 @ B5 )
% 12.24/8.01 => ( ( Less @ B5 @ C3 )
% 12.24/8.01 => ( Less @ A8 @ C3 ) ) )
% 12.24/8.01 => ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7064_order_Opreordering__axioms,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preorder @ A )
% 12.24/8.01 => ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7065_preordering__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preordering @ A )
% 12.24/8.01 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.01 ( ( partial_preordering @ A @ Less_eq2 )
% 12.24/8.01 & ( preordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7066_preordering_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.01 ( ( partial_preordering @ A @ Less_eq )
% 12.24/8.01 => ( ( preordering_axioms @ A @ Less_eq @ Less )
% 12.24/8.01 => ( preordering @ A @ Less_eq @ Less ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7067_preordering__axioms__def,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( preordering_axioms @ A )
% 12.24/8.01 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 12.24/8.01 ! [A5: A,B3: A] :
% 12.24/8.01 ( ( Less2 @ A5 @ B3 )
% 12.24/8.01 = ( ( Less_eq2 @ A5 @ B3 )
% 12.24/8.01 & ~ ( Less_eq2 @ B3 @ A5 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7068_preordering__axioms_Ointro,axiom,
% 12.24/8.01 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 12.24/8.01 ( ! [A8: A,B5: A] :
% 12.24/8.01 ( ( Less @ A8 @ B5 )
% 12.24/8.01 = ( ( Less_eq @ A8 @ B5 )
% 12.24/8.01 & ~ ( Less_eq @ B5 @ A8 ) ) )
% 12.24/8.01 => ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7069_preordering_Oaxioms_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
% 12.24/8.01 ( ( preordering @ A @ Less_eq @ Less )
% 12.24/8.01 => ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7070_comm__monoid__set_Ozero__middle,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,P5: nat,K: nat,G: nat > A,H2: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ K @ P5 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ Z3 @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ P5 ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7071_Heap_Oset__map,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 12.24/8.01 = ( image @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7072_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7073_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ( comp @ nat @ A @ nat @ G
% 12.24/8.01 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7074_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ( comp @ nat @ A @ nat @ G
% 12.24/8.01 @ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 12.24/8.01 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7075_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7076_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7077_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7078_comm__monoid__add__class_Osum__def,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType] :
% 12.24/8.01 ( ( comm_monoid_add @ A )
% 12.24/8.01 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7079_comm__monoid__set_OatMost__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) )
% 12.24/8.01 = ( F @ ( G @ ( zero_zero @ nat ) )
% 12.24/8.01 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7080_comm__monoid__set_OlessThan__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ ( zero_zero @ nat ) )
% 12.24/8.01 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7081_comm__monoid__set_OatMost__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ ( zero_zero @ nat ) )
% 12.24/8.01 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7082_sum_Ocomm__monoid__set__axioms,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( comm_monoid_add @ A )
% 12.24/8.01 => ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7083_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7084_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7085_comm__monoid__set_Onested__swap,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [J3: nat] :
% 12.24/8.01 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/8.01 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7086_comm__monoid__set_OatLeast1__atMost__eq,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7087_comm__monoid__set_Oin__pairs__0,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7088_comm__monoid__set_Ocl__ivl__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/8.01 = Z3 ) )
% 12.24/8.01 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7089_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less @ nat @ M @ N )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
% 12.24/8.01 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7090_comm__monoid__set_Oop__ivl__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ( ord_less @ nat @ N @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/8.01 = Z3 ) )
% 12.24/8.01 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7091_comm__monoid__set_Ohead__if,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ( ord_less @ nat @ N @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = Z3 ) )
% 12.24/8.01 & ( ~ ( ord_less @ nat @ N @ M )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7092_comm__monoid__set_Oivl__cong,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType] :
% 12.24/8.01 ( ( ord @ B )
% 12.24/8.01 => ! [F: A > A > A,Z3: A,A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( A3 = C2 )
% 12.24/8.01 => ( ( B2 = D2 )
% 12.24/8.01 => ( ! [X3: B] :
% 12.24/8.01 ( ( ord_less_eq @ B @ C2 @ X3 )
% 12.24/8.01 => ( ( ord_less @ B @ X3 @ D2 )
% 12.24/8.01 => ( ( G @ X3 )
% 12.24/8.01 = ( H2 @ X3 ) ) ) )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7093_comm__monoid__set_Onested__swap_H,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [J3: nat] :
% 12.24/8.01 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( A3 @ I5 @ J3 )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7094_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7095_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7096_comm__monoid__set_OlessThan__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7097_comm__monoid__set_OatMost__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7098_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7099_comm__monoid__set_OatLeastLessThan__rev,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
% 12.24/8.01 @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7100_prod_Ocomm__monoid__set__axioms,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( comm_monoid_mult @ A )
% 12.24/8.01 => ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7101_comm__monoid__set_Onat__diff__reindex,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7102_comm__monoid__set_Onat__group,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,K: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [M2: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7103_comm__monoid__set_Oub__add__nat,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A,P5: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7104_comm__monoid__set_Ohead,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7105_comm__monoid__set_OatLeast__Suc__atMost,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7106_comm__monoid__set_Onat__ivl__Suc_H,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7107_comm__monoid__set_OatLeastLessThan__Suc,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,A3: nat,B2: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ A3 @ B2 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7108_comm__monoid__set_OatLeastLessThan__concat,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,P5: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ N @ P5 )
% 12.24/8.01 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7109_comm__monoid__set_Olast__plus,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
% 12.24/8.01 = ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7110_comm__monoid__set_OSuc__reindex__ivl,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( ord_less_eq @ nat @ M @ N )
% 12.24/8.01 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 12.24/8.01 = ( F @ ( G @ M )
% 12.24/8.01 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7111_Heap_Oinj__map__strong,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa3: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
% 12.24/8.01 ( ! [Z4: A,Za: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 12.24/8.01 => ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa3 ) )
% 12.24/8.01 => ( ( ( F @ Z4 )
% 12.24/8.01 = ( Fa @ Za ) )
% 12.24/8.01 => ( Z4 = Za ) ) ) )
% 12.24/8.01 => ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 12.24/8.01 = ( heap_Time_map_Heap @ A @ B @ Fa @ Xa3 ) )
% 12.24/8.01 => ( X = Xa3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7112_Heap_Omap__cong0,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 12.24/8.01 ( ! [Z4: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 12.24/8.01 => ( ( F @ Z4 )
% 12.24/8.01 = ( G @ Z4 ) ) )
% 12.24/8.01 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 12.24/8.01 = ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7113_Heap_Omap__cong,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 12.24/8.01 ( ( X = Ya )
% 12.24/8.01 => ( ! [Z4: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 12.24/8.01 => ( ( F @ Z4 )
% 12.24/8.01 = ( G @ Z4 ) ) )
% 12.24/8.01 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 12.24/8.01 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7114_comm__monoid__set_OIf__cases,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,F: A > A > A,Z3: A,A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( finite_finite2 @ B @ A6 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z3
% 12.24/8.01 @ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
% 12.24/8.01 @ A6 )
% 12.24/8.01 = ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7115_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7116_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7117_comm__monoid__mult__class_Oprod__def,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType] :
% 12.24/8.01 ( ( comm_monoid_mult @ A )
% 12.24/8.01 => ( ( groups7121269368397514597t_prod @ B @ A )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7118_comm__monoid__set_Oin__pairs,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
% 12.24/8.01 @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7119_comm__monoid__set_Otriangle__reindex,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/8.01 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.01 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.01 @ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [K3: nat] :
% 12.24/8.01 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/8.01 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7120_comm__monoid__set_Otriangle__reindex__eq,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
% 12.24/8.01 @ ( collect @ ( product_prod @ nat @ nat )
% 12.24/8.01 @ ( product_case_prod @ nat @ nat @ $o
% 12.24/8.01 @ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [K3: nat] :
% 12.24/8.01 ( groups_comm_monoid_F @ A @ nat @ F @ Z3
% 12.24/8.01 @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ K3 ) )
% 12.24/8.01 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7121_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7122_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
% 12.24/8.01 ! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
% 12.24/8.01 ( ( groups778175481326437816id_set @ A @ F @ Z3 )
% 12.24/8.01 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 12.24/8.01 = ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7123_Heap_Oin__rel,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType] :
% 12.24/8.01 ( ( heap_Time_rel_Heap @ A @ B )
% 12.24/8.01 = ( ^ [R6: A > B > $o,A5: heap_Time_Heap @ A,B3: heap_Time_Heap @ B] :
% 12.24/8.01 ? [Z7: heap_Time_Heap @ ( product_prod @ A @ B )] :
% 12.24/8.01 ( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z7
% 12.24/8.01 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 12.24/8.01 @ ^ [X2: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 12.24/8.01 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z7 )
% 12.24/8.01 = A5 )
% 12.24/8.01 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z7 )
% 12.24/8.01 = B3 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7124_Heap_Opred__set,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap_Time_pred_Heap @ A )
% 12.24/8.01 = ( ^ [P4: A > $o,X2: heap_Time_Heap @ A] :
% 12.24/8.01 ! [Y6: A] :
% 12.24/8.01 ( ( member @ A @ Y6 @ ( heap_Time_set_Heap @ A @ X2 ) )
% 12.24/8.01 => ( P4 @ Y6 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7125_Heap_ODomainp__rel,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,R2: A > B > $o] :
% 12.24/8.01 ( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) )
% 12.24/8.01 = ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7126_Heap_Opred__True,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap_Time_pred_Heap @ A
% 12.24/8.01 @ ^ [Uu: A] : $true )
% 12.24/8.01 = ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7127_Heap_Orel__eq,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap_Time_rel_Heap @ A @ A
% 12.24/8.01 @ ^ [Y: A,Z2: A] : ( Y = Z2 ) )
% 12.24/8.01 = ( ^ [Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] : ( Y = Z2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7128_Heap_Orel__refl,axiom,
% 12.24/8.01 ! [B: $tType,Ra: B > B > $o,X: heap_Time_Heap @ B] :
% 12.24/8.01 ( ! [X3: B] : ( Ra @ X3 @ X3 )
% 12.24/8.01 => ( heap_Time_rel_Heap @ B @ B @ Ra @ X @ X ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7129_Heap_Orel__transp,axiom,
% 12.24/8.01 ! [A: $tType,R2: A > A > $o] :
% 12.24/8.01 ( ( transp @ A @ R2 )
% 12.24/8.01 => ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7130_Heap_Opred__mono,axiom,
% 12.24/8.01 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 12.24/8.01 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 12.24/8.01 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7131_Heap_Orel__mono,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,R2: A > B > $o,Ra: A > B > $o] :
% 12.24/8.01 ( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra )
% 12.24/8.01 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7132_Heap_Obi__total__rel,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,R2: A > B > $o] :
% 12.24/8.01 ( ( bi_total @ A @ B @ R2 )
% 12.24/8.01 => ( bi_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7133_Heap_Orel__transfer,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 12.24/8.01 ( 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 )
% 12.24/8.01 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 12.24/8.01 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 12.24/8.01 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) ) )
% 12.24/8.01 @ ( 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 )
% 12.24/8.01 @ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
% 12.24/8.01 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) ) )
% 12.24/8.01 @ ( heap_Time_rel_Heap @ A @ B )
% 12.24/8.01 @ ( heap_Time_rel_Heap @ C @ D ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7134_Heap_Opred__transfer,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,R2: A > B > $o] :
% 12.24/8.01 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
% 12.24/8.01 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
% 12.24/8.01 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/8.01 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R2 )
% 12.24/8.01 @ ^ [Y: $o,Z2: $o] : ( Y = Z2 ) )
% 12.24/8.01 @ ( heap_Time_pred_Heap @ A )
% 12.24/8.01 @ ( heap_Time_pred_Heap @ B ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7135_Heap_Omap__transfer,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,F7: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F7 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F7 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E4 ) > ( heap_Time_Heap @ F7 ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F7 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E4 ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F7 ) @ ( heap_Time_rel_Heap @ A @ E4 @ Rb ) @ ( heap_Time_rel_Heap @ B @ F7 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E4 @ F7 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7136_Heap_Omap__cong__pred,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 12.24/8.01 ( ( X = Ya )
% 12.24/8.01 => ( ( heap_Time_pred_Heap @ A
% 12.24/8.01 @ ^ [Z7: A] :
% 12.24/8.01 ( ( F @ Z7 )
% 12.24/8.01 = ( G @ Z7 ) )
% 12.24/8.01 @ Ya )
% 12.24/8.01 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 12.24/8.01 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7137_Heap_Orel__map_I1_J,axiom,
% 12.24/8.01 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B] :
% 12.24/8.01 ( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y3 )
% 12.24/8.01 = ( heap_Time_rel_Heap @ A @ B
% 12.24/8.01 @ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
% 12.24/8.01 @ X
% 12.24/8.01 @ Y3 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7138_Heap_Orel__map_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y3: heap_Time_Heap @ B] :
% 12.24/8.01 ( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y3 ) )
% 12.24/8.01 = ( heap_Time_rel_Heap @ A @ B
% 12.24/8.01 @ ^ [X2: A,Y6: B] : ( Sa @ X2 @ ( G @ Y6 ) )
% 12.24/8.01 @ X
% 12.24/8.01 @ Y3 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7139_Heap_OQuotient,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,R2: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 12.24/8.01 ( ( quotient @ A @ B @ R2 @ Abs @ Rep @ T6 )
% 12.24/8.01 => ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7140_Heap_Opred__map,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
% 12.24/8.01 = ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7141_Heap_Opred__mono__strong,axiom,
% 12.24/8.01 ! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
% 12.24/8.01 ( ( heap_Time_pred_Heap @ A @ P @ X )
% 12.24/8.01 => ( ! [Z4: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 12.24/8.01 => ( ( P @ Z4 )
% 12.24/8.01 => ( Pa @ Z4 ) ) )
% 12.24/8.01 => ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7142_Heap_Orel__refl__strong,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Ra: A > A > $o] :
% 12.24/8.01 ( ! [Z4: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 12.24/8.01 => ( Ra @ Z4 @ Z4 ) )
% 12.24/8.01 => ( heap_Time_rel_Heap @ A @ A @ Ra @ X @ X ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7143_Heap_Orel__mono__strong,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,R2: A > B > $o,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Ra: A > B > $o] :
% 12.24/8.01 ( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
% 12.24/8.01 => ( ! [Z4: A,Yb: B] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
% 12.24/8.01 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y3 ) )
% 12.24/8.01 => ( ( R2 @ Z4 @ Yb )
% 12.24/8.01 => ( Ra @ Z4 @ Yb ) ) ) )
% 12.24/8.01 => ( heap_Time_rel_Heap @ A @ B @ Ra @ X @ Y3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7144_Heap_Opred__cong,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
% 12.24/8.01 ( ( X = Ya )
% 12.24/8.01 => ( ! [Z4: A] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 12.24/8.01 => ( ( P @ Z4 )
% 12.24/8.01 = ( Pa @ Z4 ) ) )
% 12.24/8.01 => ( ( heap_Time_pred_Heap @ A @ P @ X )
% 12.24/8.01 = ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7145_Heap_Orel__cong,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Xa3: heap_Time_Heap @ B,R2: A > B > $o,Ra: A > B > $o] :
% 12.24/8.01 ( ( X = Ya )
% 12.24/8.01 => ( ( Y3 = Xa3 )
% 12.24/8.01 => ( ! [Z4: A,Yb: B] :
% 12.24/8.01 ( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 12.24/8.01 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa3 ) )
% 12.24/8.01 => ( ( R2 @ Z4 @ Yb )
% 12.24/8.01 = ( Ra @ Z4 @ Yb ) ) ) )
% 12.24/8.01 => ( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
% 12.24/8.01 = ( heap_Time_rel_Heap @ A @ B @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7146_Fract__def,axiom,
% 12.24/8.01 ( fract
% 12.24/8.01 = ( 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 )
% 12.24/8.01 @ ^ [A5: int,B3: int] :
% 12.24/8.01 ( if @ ( product_prod @ int @ int )
% 12.24/8.01 @ ( B3
% 12.24/8.01 = ( zero_zero @ int ) )
% 12.24/8.01 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 12.24/8.01 @ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7147_subset_Osuc__Union__closed__suc,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 )
% 12.24/8.01 = Y7 )
% 12.24/8.01 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7148_predicate2I,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 12.24/8.01 ( ! [X3: A,Y5: B] :
% 12.24/8.01 ( ( P @ X3 @ Y5 )
% 12.24/8.01 => ( Q @ X3 @ Y5 ) )
% 12.24/8.01 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7149_predicate1I,axiom,
% 12.24/8.01 ! [A: $tType,P: A > $o,Q: A > $o] :
% 12.24/8.01 ( ! [X3: A] :
% 12.24/8.01 ( ( P @ X3 )
% 12.24/8.01 => ( Q @ X3 ) )
% 12.24/8.01 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7150_of__int__eq__id,axiom,
% 12.24/8.01 ( ( ring_1_of_int @ int )
% 12.24/8.01 = ( id @ int ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7151_predicate1D,axiom,
% 12.24/8.01 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 12.24/8.01 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 12.24/8.01 => ( ( P @ X )
% 12.24/8.01 => ( Q @ X ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7152_predicate2D,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
% 12.24/8.01 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 12.24/8.01 => ( ( P @ X @ Y3 )
% 12.24/8.01 => ( Q @ X @ Y3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7153_rev__predicate1D,axiom,
% 12.24/8.01 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 12.24/8.01 ( ( P @ X )
% 12.24/8.01 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 12.24/8.01 => ( Q @ X ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7154_rev__predicate2D,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
% 12.24/8.01 ( ( P @ X @ Y3 )
% 12.24/8.01 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 12.24/8.01 => ( Q @ X @ Y3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7155_subset_Osuc__Union__closed_Ocases,axiom,
% 12.24/8.01 ! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ! [X10: set @ ( set @ A )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 12.24/8.01 => ~ ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 12.24/8.01 => ~ ! [X10: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
% 12.24/8.01 => ~ ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7156_subset_Osuc__Union__closed_Osimps,axiom,
% 12.24/8.01 ! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 = ( ? [X7: set @ ( set @ A )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 12.24/8.01 & ( member @ ( set @ ( set @ A ) ) @ X7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 12.24/8.01 | ? [X7: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
% 12.24/8.01 & ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X7 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7157_subset_Osuc__Union__closed_Osuc,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( member @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7158_subset_Oeq__suc__Union,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.01 = X8 )
% 12.24/8.01 = ( X8
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7159_subset_Osuc__Union__closed__Union,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7160_subset_Osuc__Union__closed__in__carrier,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7161_subset_Osuc__Union__closed__total,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 12.24/8.01 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7162_subset_Osuc__Union__closed__induct,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Q: ( set @ ( set @ A ) ) > $o] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ! [X10: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( Q @ X10 )
% 12.24/8.01 => ( Q @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) ) ) )
% 12.24/8.01 => ( ! [X10: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ! [X4: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X4 @ X10 )
% 12.24/8.01 => ( Q @ X4 ) )
% 12.24/8.01 => ( Q @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) ) ) )
% 12.24/8.01 => ( Q @ X8 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7163_subset_Osuc__Union__closed__cases,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ! [Y10: set @ ( set @ A )] :
% 12.24/8.01 ( ( X8
% 12.24/8.01 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y10 ) )
% 12.24/8.01 => ~ ( member @ ( set @ ( set @ A ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
% 12.24/8.01 => ~ ! [Y10: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( X8
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Y10 ) )
% 12.24/8.01 => ~ ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7164_subset_Osuc__Union__closed_OUnion,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7165_subset_Osuc__Union__closed_Ointros_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,Xa5: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Xa5 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7166_subset_Osuc__Union__closed__chain,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7167_subset_Osuc__Union__closed__empty,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7168_natural_Odomain,axiom,
% 12.24/8.01 ! [P: nat > $o] :
% 12.24/8.01 ( ( ( domainp @ nat @ nat
% 12.24/8.01 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/8.01 = P )
% 12.24/8.01 => ( ( domainp @ nat @ code_natural @ code_pcr_natural )
% 12.24/8.01 = P ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7169_integer_Odomain,axiom,
% 12.24/8.01 ! [P: int > $o] :
% 12.24/8.01 ( ( ( domainp @ int @ int
% 12.24/8.01 @ ^ [Y: int,Z2: int] : ( Y = Z2 ) )
% 12.24/8.01 = P )
% 12.24/8.01 => ( ( domainp @ int @ code_integer @ code_pcr_integer )
% 12.24/8.01 = P ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7170_subset_Osuc__Union__closed__subsetD,axiom,
% 12.24/8.01 ! [A: $tType,Y7: set @ ( set @ A ),X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 )
% 12.24/8.01 => ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( X8 = Y7 )
% 12.24/8.01 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7171_subset_Osuc__Union__closed__total_H,axiom,
% 12.24/8.01 ! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ! [Z9: set @ ( set @ A )] :
% 12.24/8.01 ( ( member @ ( set @ ( set @ A ) ) @ Z9 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
% 12.24/8.01 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Z9 @ Y7 )
% 12.24/8.01 => ( ( Z9 = Y7 )
% 12.24/8.01 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Z9 ) @ Y7 ) ) ) )
% 12.24/8.01 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
% 12.24/8.01 | ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7172_length__remdups__leq,axiom,
% 12.24/8.01 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7173_subset_Osuc__Union__closed__def,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
% 12.24/8.01 = ( collect @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7174_sorted__remdups,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( linorder @ A )
% 12.24/8.01 => ! [Xs: list @ A] :
% 12.24/8.01 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 12.24/8.01 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7175_subset_Osuc__Union__closedp__suc__Union__closed__eq,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A )] :
% 12.24/8.01 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
% 12.24/8.01 = ( ^ [X2: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ X2 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7176_subset_Osuc__Union__closedp_Osuc,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
% 12.24/8.01 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
% 12.24/8.01 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7177_subset_Osuc__Union__closedp_Osimps,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
% 12.24/8.01 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
% 12.24/8.01 = ( ? [X7: set @ ( set @ A )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 12.24/8.01 & ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
% 12.24/8.01 | ? [X7: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
% 12.24/8.01 & ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X7 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7178_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A ),Xa5: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ Xa5 )
% 12.24/8.01 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7179_subset_Osuc__Union__closedp_OUnion,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X8 )
% 12.24/8.01 => ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7180_subset_Osuc__Union__closedp_Ocases,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
% 12.24/8.01 ( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
% 12.24/8.01 => ( ! [X10: set @ ( set @ A )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 12.24/8.01 => ~ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
% 12.24/8.01 => ~ ! [X10: set @ ( set @ ( set @ A ) )] :
% 12.24/8.01 ( ( A3
% 12.24/8.01 = ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
% 12.24/8.01 => ~ ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X10 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7181_int_Odomain,axiom,
% 12.24/8.01 ! [DR1: nat > $o,DR2: nat > $o] :
% 12.24/8.01 ( ( ( domainp @ nat @ nat
% 12.24/8.01 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/8.01 = DR1 )
% 12.24/8.01 => ( ( ( domainp @ nat @ nat
% 12.24/8.01 @ ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
% 12.24/8.01 = DR2 )
% 12.24/8.01 => ( ( domainp @ ( product_prod @ nat @ nat ) @ int @ pcr_int )
% 12.24/8.01 = ( basic_pred_prod @ nat @ nat @ DR1 @ DR2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7182_Heap_Ocase,axiom,
% 12.24/8.01 ! [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 ) ) )] :
% 12.24/8.01 ( ( heap_Time_case_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 12.24/8.01 = ( F @ X ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7183_Heap_Ocase__distrib,axiom,
% 12.24/8.01 ! [C: $tType,B: $tType,A: $tType,H2: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( H2 @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
% 12.24/8.01 = ( heap_Time_case_Heap @ A @ C
% 12.24/8.01 @ ^ [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H2 @ ( F @ X2 ) )
% 12.24/8.01 @ Heap ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7184_Heap_Orec,axiom,
% 12.24/8.01 ! [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 ) ) )] :
% 12.24/8.01 ( ( heap_Time_rec_Heap @ A @ C @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 12.24/8.01 = ( F @ X ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7185_enumerate__Suc__eq,axiom,
% 12.24/8.01 ! [A: $tType,N: nat,Xs: list @ A] :
% 12.24/8.01 ( ( enumerate @ A @ ( suc @ N ) @ Xs )
% 12.24/8.01 = ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7186_Heap__ordE,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
% 12.24/8.01 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 12.24/8.01 => ( ( ( heap_Time_execute @ A @ X @ H2 )
% 12.24/8.01 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/8.01 => ( ( heap_Time_execute @ A @ X @ H2 )
% 12.24/8.01 = ( heap_Time_execute @ A @ Y3 @ H2 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7187_Heap__ordI,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
% 12.24/8.01 ( ! [H6: heap_ext @ product_unit] :
% 12.24/8.01 ( ( ( heap_Time_execute @ A @ X @ H6 )
% 12.24/8.01 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 12.24/8.01 | ( ( heap_Time_execute @ A @ X @ H6 )
% 12.24/8.01 = ( heap_Time_execute @ A @ Y3 @ H6 ) ) )
% 12.24/8.01 => ( heap_Time_Heap_ord @ A @ X @ Y3 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7188_heap_Oleq__refl,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7189_heap_Oleq__trans,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,Z3: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 12.24/8.01 => ( ( heap_Time_Heap_ord @ A @ Y3 @ Z3 )
% 12.24/8.01 => ( heap_Time_Heap_ord @ A @ X @ Z3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7190_heap_Oleq__antisym,axiom,
% 12.24/8.01 ! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( heap_Time_Heap_ord @ A @ X @ Y3 )
% 12.24/8.01 => ( ( heap_Time_Heap_ord @ A @ Y3 @ X )
% 12.24/8.01 => ( X = Y3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7191_heap_Olub__least,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),Z3: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
% 12.24/8.01 => ( ! [X3: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( member @ ( heap_Time_Heap @ A ) @ X3 @ A6 )
% 12.24/8.01 => ( heap_Time_Heap_ord @ A @ X3 @ Z3 ) )
% 12.24/8.01 => ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A6 ) @ Z3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7192_heap_Olub__upper,axiom,
% 12.24/8.01 ! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
% 12.24/8.01 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
% 12.24/8.01 => ( ( member @ ( heap_Time_Heap @ A ) @ X @ A6 )
% 12.24/8.01 => ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A6 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7193_heap_Oconst__mono,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
% 12.24/8.01 ( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
% 12.24/8.01 @ ^ [F2: B] : C2 ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7194_heap__interpretation,axiom,
% 12.24/8.01 ! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7195_Heap__Time__Monad_Obind__mono,axiom,
% 12.24/8.01 ! [C: $tType,D: $tType,B: $tType,A: $tType,B7: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C6: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
% 12.24/8.01 ( ( 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 ) @ B7 )
% 12.24/8.01 => ( ! [Y5: 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 ) @ ( C6 @ Y5 ) )
% 12.24/8.01 => ( 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 )
% 12.24/8.01 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 12.24/8.01 ( heap_Time_bind @ C @ D @ ( B7 @ F2 )
% 12.24/8.01 @ ^ [Y6: C] : ( C6 @ Y6 @ F2 ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7196_admissible__heap,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
% 12.24/8.01 ( 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 ) )
% 12.24/8.01 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 12.24/8.01 ! [X2: A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: B,N3: nat] :
% 12.24/8.01 ( ( heap_Time_effect @ B @ ( F2 @ X2 ) @ H4 @ H5 @ R5 @ N3 )
% 12.24/8.01 => ( P @ X2 @ H4 @ H5 @ R5 @ N3 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7197_heap_Ofixp__induct__uc,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
% 12.24/8.01 ( ! [X3: B] :
% 12.24/8.01 ( 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 )
% 12.24/8.01 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
% 12.24/8.01 => ( ( F
% 12.24/8.01 = ( C6
% 12.24/8.01 @ ( 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 ) )
% 12.24/8.01 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
% 12.24/8.01 => ( ! [F3: B > ( heap_Time_Heap @ A )] :
% 12.24/8.01 ( ( U4 @ ( C6 @ F3 ) )
% 12.24/8.01 = F3 )
% 12.24/8.01 => ( ( 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 )
% 12.24/8.01 => ( ( P
% 12.24/8.01 @ ^ [Uu: B] :
% 12.24/8.01 ( heap_Time_Heap2 @ A
% 12.24/8.01 @ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
% 12.24/8.01 => ( ! [F3: C] :
% 12.24/8.01 ( ( P @ ( U4 @ F3 ) )
% 12.24/8.01 => ( P @ ( U4 @ ( F4 @ F3 ) ) ) )
% 12.24/8.01 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7198_heap_Omono__body__fixp,axiom,
% 12.24/8.01 ! [A: $tType,B: $tType,F4: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
% 12.24/8.01 ( ! [X3: B] :
% 12.24/8.01 ( 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 )
% 12.24/8.01 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F4 @ F2 @ X3 ) )
% 12.24/8.01 => ( ( 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 ) ) @ F4 )
% 12.24/8.01 = ( F4 @ ( 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 ) ) @ F4 ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(fact_7199_heap_Ofixp__rule__uc,axiom,
% 12.24/8.01 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
% 12.24/8.01 ( ! [X3: B] :
% 12.24/8.01 ( 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 )
% 12.24/8.01 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
% 12.24/8.01 => ( ( F
% 12.24/8.01 = ( C6
% 12.24/8.01 @ ( 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 ) )
% 12.24/8.01 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
% 12.24/8.01 => ( ! [F3: C] :
% 12.24/8.01 ( ( C6 @ ( U4 @ F3 ) )
% 12.24/8.01 = F3 )
% 12.24/8.01 => ( F
% 12.24/8.01 = ( F4 @ F ) ) ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(subcl_Heap_Oheap___HOL_Otype,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap @ A )
% 12.24/8.01 => ( type @ A ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap @ A )
% 12.24/8.01 => ( typerep2 @ A ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
% 12.24/8.01 ! [A: $tType] :
% 12.24/8.01 ( ( heap @ A )
% 12.24/8.01 => ( countable @ A ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
% 12.24/8.01 bounded_lattice @ product_unit ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
% 12.24/8.01 finite_finite @ product_unit ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ( finite_finite @ A17 )
% 12.24/8.01 & ( finite_finite @ A18 ) )
% 12.24/8.01 => ( finite_finite @ ( product_prod @ A17 @ A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
% 12.24/8.01 ! [A17: $tType] :
% 12.24/8.01 ( ( finite_finite @ A17 )
% 12.24/8.01 => ( finite_finite @ ( option @ A17 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ( finite_finite @ A17 )
% 12.24/8.01 & ( finite_finite @ A18 ) )
% 12.24/8.01 => ( finite_finite @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
% 12.24/8.01 finite_finite @ char ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_5,axiom,
% 12.24/8.01 bounded_lattice @ $o ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 12.24/8.01 finite_finite @ $o ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Set_Oset___Lattices_Obounded__lattice_7,axiom,
% 12.24/8.01 ! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 12.24/8.01 ! [A17: $tType] :
% 12.24/8.01 ( ( finite_finite @ A17 )
% 12.24/8.01 => ( finite_finite @ ( set @ A17 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Obounded__lattice_9,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bounded_lattice @ A18 )
% 12.24/8.01 => ( bounded_lattice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Finite__Set_Ofinite_10,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ( finite_finite @ A17 )
% 12.24/8.01 & ( finite_finite @ A18 ) )
% 12.24/8.01 => ( finite_finite @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( comple6319245703460814977attice @ A18 )
% 12.24/8.01 => ( condit1219197933456340205attice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( comple592849572758109894attice @ A18 )
% 12.24/8.01 => ( comple592849572758109894attice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( comple489889107523837845lgebra @ A18 )
% 12.24/8.01 => ( comple489889107523837845lgebra @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bounded_lattice @ A18 )
% 12.24/8.01 => ( bounde4967611905675639751up_bot @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bounded_lattice @ A18 )
% 12.24/8.01 => ( bounde4346867609351753570nf_top @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( comple6319245703460814977attice @ A18 )
% 12.24/8.01 => ( comple6319245703460814977attice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( boolea8198339166811842893lgebra @ A18 )
% 12.24/8.01 => ( boolea8198339166811842893lgebra @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bounded_lattice @ A18 )
% 12.24/8.01 => ( bounded_lattice_top @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bounded_lattice @ A18 )
% 12.24/8.01 => ( bounded_lattice_bot @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( comple6319245703460814977attice @ A18 )
% 12.24/8.01 => ( comple9053668089753744459l_ccpo @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( semilattice_sup @ A18 )
% 12.24/8.01 => ( semilattice_sup @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( semilattice_inf @ A18 )
% 12.24/8.01 => ( semilattice_inf @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( distrib_lattice @ A18 )
% 12.24/8.01 => ( distrib_lattice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( order_top @ A18 )
% 12.24/8.01 => ( order_top @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( order_bot @ A18 )
% 12.24/8.01 => ( order_bot @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Countable_Ocountable,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ( finite_finite @ A17 )
% 12.24/8.01 & ( countable @ A18 ) )
% 12.24/8.01 => ( countable @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Opreorder,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( preorder @ A18 )
% 12.24/8.01 => ( preorder @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Lattices_Olattice,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( lattice @ A18 )
% 12.24/8.01 => ( lattice @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Typerep_Otyperep,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ( typerep2 @ A17 )
% 12.24/8.01 & ( typerep2 @ A18 ) )
% 12.24/8.01 => ( typerep2 @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Oorder,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( order @ A18 )
% 12.24/8.01 => ( order @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Otop,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( top @ A18 )
% 12.24/8.01 => ( top @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Oord,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( ord @ A18 )
% 12.24/8.01 => ( ord @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Orderings_Obot,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( bot @ A18 )
% 12.24/8.01 => ( bot @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Groups_Ouminus,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( uminus @ A18 )
% 12.24/8.01 => ( uminus @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_fun___Groups_Ominus,axiom,
% 12.24/8.01 ! [A17: $tType,A18: $tType] :
% 12.24/8.01 ( ( minus @ A18 )
% 12.24/8.01 => ( minus @ ( A17 > A18 ) ) ) ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 12.24/8.01 condit6923001295902523014norder @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
% 12.24/8.01 condit1219197933456340205attice @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
% 12.24/8.01 bit_un5681908812861735899ations @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 12.24/8.01 semiri1453513574482234551roduct @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 12.24/8.01 euclid5411537665997757685th_nat @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
% 12.24/8.01 euclid8789492081693882211th_nat @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 12.24/8.01 ordere1937475149494474687imp_le @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
% 12.24/8.01 euclid3128863361964157862miring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
% 12.24/8.01 euclid4440199948858584721cancel @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
% 12.24/8.01 normal6328177297339901930cative @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 12.24/8.01 unique1627219031080169319umeral @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
% 12.24/8.01 euclid8851590272496341667cancel @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 12.24/8.01 semiri6575147826004484403cancel @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 12.24/8.01 strict9044650504122735259up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 12.24/8.01 ordere580206878836729694up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 12.24/8.01 ordere2412721322843649153imp_le @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
% 12.24/8.01 bit_se359711467146920520ations @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
% 12.24/8.01 linord2810124833399127020strict @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 12.24/8.01 strict7427464778891057005id_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 12.24/8.01 ordere8940638589300402666id_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
% 12.24/8.01 euclid3725896446679973847miring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
% 12.24/8.01 linord715952674999750819strict @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
% 12.24/8.01 linord4140545234300271783up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
% 12.24/8.01 bit_ri3973907225187159222ations @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 12.24/8.01 semiri2026040879449505780visors @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 12.24/8.01 linord181362715937106298miring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
% 12.24/8.01 euclid5891614535332579305n_ring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
% 12.24/8.01 semido2269285787275462019factor @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
% 12.24/8.01 linord8928482502909563296strict @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 12.24/8.01 semiri3467727345109120633visors @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 12.24/8.01 ordere6658533253407199908up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 12.24/8.01 ordere166539214618696060dd_abs @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
% 12.24/8.01 semiri6843258321239162965malize @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
% 12.24/8.01 ordere6911136660526730532id_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 12.24/8.01 linord5086331880401160121up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 12.24/8.01 cancel2418104881723323429up_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
% 12.24/8.01 ring_15535105094025558882visors @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 12.24/8.01 cancel1802427076303600483id_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
% 12.24/8.01 linord4710134922213307826strict @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
% 12.24/8.01 comm_s4317794764714335236cancel @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
% 12.24/8.01 bit_semiring_bits @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
% 12.24/8.01 ordere2520102378445227354miring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
% 12.24/8.01 normal8620421768224518004emidom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
% 12.24/8.01 linord6961819062388156250ring_1 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 12.24/8.01 ordered_ab_group_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 12.24/8.01 cancel_semigroup_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
% 12.24/8.01 linordered_semiring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
% 12.24/8.01 ordered_semiring_0 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 12.24/8.01 linordered_semidom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_12,axiom,
% 12.24/8.01 semilattice_sup @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_13,axiom,
% 12.24/8.01 semilattice_inf @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Lattices_Odistrib__lattice_14,axiom,
% 12.24/8.01 distrib_lattice @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
% 12.24/8.01 ab_semigroup_mult @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
% 12.24/8.01 semiring_1_cancel @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
% 12.24/8.01 algebraic_semidom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
% 12.24/8.01 comm_monoid_mult @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 12.24/8.01 ab_semigroup_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
% 12.24/8.01 ordered_semiring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
% 12.24/8.01 ordered_ring_abs @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
% 12.24/8.01 semiring_parity @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 12.24/8.01 comm_monoid_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
% 12.24/8.01 semiring_modulo @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
% 12.24/8.01 linordered_ring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 12.24/8.01 linordered_idom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 12.24/8.01 comm_semiring_1 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 12.24/8.01 comm_semiring_0 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
% 12.24/8.01 semigroup_mult @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
% 12.24/8.01 semidom_modulo @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 12.24/8.01 semidom_divide @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
% 12.24/8.01 semiring_numeral @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 12.24/8.01 semigroup_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
% 12.24/8.01 zero_less_one @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
% 12.24/8.01 comm_semiring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 12.24/8.01 semiring_char_0 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 12.24/8.01 ab_group_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Countable_Ocountable_15,axiom,
% 12.24/8.01 countable @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 12.24/8.01 zero_neq_one @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
% 12.24/8.01 ordered_ring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
% 12.24/8.01 idom_abs_sgn @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
% 12.24/8.01 ring_parity @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Opreorder_16,axiom,
% 12.24/8.01 preorder @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 12.24/8.01 linorder @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 12.24/8.01 monoid_mult @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
% 12.24/8.01 idom_modulo @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
% 12.24/8.01 idom_divide @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 12.24/8.01 comm_ring_1 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 12.24/8.01 monoid_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 12.24/8.01 semiring_1 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
% 12.24/8.01 semiring_0 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 12.24/8.01 no_top @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 12.24/8.01 no_bot @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Lattices_Olattice_17,axiom,
% 12.24/8.01 lattice @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 12.24/8.01 group_add @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
% 12.24/8.01 semiring_gcd @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
% 12.24/8.01 semiring_Gcd @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Typerep_Otyperep_18,axiom,
% 12.24/8.01 typerep2 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
% 12.24/8.01 mult_zero @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
% 12.24/8.01 comm_ring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Oorder_19,axiom,
% 12.24/8.01 order @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 12.24/8.01 neg_numeral @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 12.24/8.01 ring_char_0 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemiring,axiom,
% 12.24/8.01 semiring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Osemidom,axiom,
% 12.24/8.01 semidom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Orderings_Oord_20,axiom,
% 12.24/8.01 ord @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ouminus_21,axiom,
% 12.24/8.01 uminus @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 12.24/8.01 ring_1 @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
% 12.24/8.01 abs_if @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ominus_22,axiom,
% 12.24/8.01 minus @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
% 12.24/8.01 ring_gcd @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Power_Opower,axiom,
% 12.24/8.01 power @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 12.24/8.01 numeral @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 12.24/8.01 zero @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oplus,axiom,
% 12.24/8.01 plus @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oring,axiom,
% 12.24/8.01 ring @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Oidom,axiom,
% 12.24/8.01 idom @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 12.24/8.01 one @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Rings_Odvd,axiom,
% 12.24/8.01 dvd @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Int_Oint___Heap_Oheap,axiom,
% 12.24/8.01 heap @ int ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_23,axiom,
% 12.24/8.01 condit6923001295902523014norder @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
% 12.24/8.01 condit1219197933456340205attice @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_25,axiom,
% 12.24/8.01 bit_un5681908812861735899ations @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_26,axiom,
% 12.24/8.01 semiri1453513574482234551roduct @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_27,axiom,
% 12.24/8.01 euclid5411537665997757685th_nat @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28,axiom,
% 12.24/8.01 ordere1937475149494474687imp_le @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_29,axiom,
% 12.24/8.01 euclid3128863361964157862miring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_30,axiom,
% 12.24/8.01 euclid4440199948858584721cancel @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_31,axiom,
% 12.24/8.01 normal6328177297339901930cative @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_32,axiom,
% 12.24/8.01 unique1627219031080169319umeral @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_33,axiom,
% 12.24/8.01 semiri6575147826004484403cancel @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_34,axiom,
% 12.24/8.01 strict9044650504122735259up_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 12.24/8.01 ordere1170586879665033532d_diff @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_35,axiom,
% 12.24/8.01 ordere580206878836729694up_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_36,axiom,
% 12.24/8.01 ordere2412721322843649153imp_le @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_37,axiom,
% 12.24/8.01 bit_se359711467146920520ations @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_38,axiom,
% 12.24/8.01 linord2810124833399127020strict @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_39,axiom,
% 12.24/8.01 strict7427464778891057005id_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_40,axiom,
% 12.24/8.01 ordere8940638589300402666id_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 12.24/8.01 canoni5634975068530333245id_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_41,axiom,
% 12.24/8.01 euclid3725896446679973847miring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_42,axiom,
% 12.24/8.01 linord4140545234300271783up_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43,axiom,
% 12.24/8.01 semiri2026040879449505780visors @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44,axiom,
% 12.24/8.01 linord181362715937106298miring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_45,axiom,
% 12.24/8.01 semido2269285787275462019factor @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46,axiom,
% 12.24/8.01 linord8928482502909563296strict @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47,axiom,
% 12.24/8.01 semiri3467727345109120633visors @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48,axiom,
% 12.24/8.01 ordere6658533253407199908up_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49,axiom,
% 12.24/8.01 semiri6843258321239162965malize @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50,axiom,
% 12.24/8.01 ordere6911136660526730532id_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51,axiom,
% 12.24/8.01 cancel2418104881723323429up_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
% 12.24/8.01 cancel1802427076303600483id_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
% 12.24/8.01 comm_s4317794764714335236cancel @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
% 12.24/8.01 bit_semiring_bits @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55,axiom,
% 12.24/8.01 ordere2520102378445227354miring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Onormalization__semidom_56,axiom,
% 12.24/8.01 normal8620421768224518004emidom @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_57,axiom,
% 12.24/8.01 cancel_semigroup_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Olinordered__semiring_58,axiom,
% 12.24/8.01 linordered_semiring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_59,axiom,
% 12.24/8.01 ordered_semiring_0 @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_60,axiom,
% 12.24/8.01 linordered_semidom @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_61,axiom,
% 12.24/8.01 semilattice_sup @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_62,axiom,
% 12.24/8.01 semilattice_inf @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_63,axiom,
% 12.24/8.01 distrib_lattice @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
% 12.24/8.01 ab_semigroup_mult @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
% 12.24/8.01 semiring_1_cancel @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
% 12.24/8.01 algebraic_semidom @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,axiom,
% 12.24/8.01 comm_monoid_mult @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 12.24/8.01 comm_monoid_diff @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_68,axiom,
% 12.24/8.01 ab_semigroup_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
% 12.24/8.01 ordered_semiring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
% 12.24/8.01 semiring_parity @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
% 12.24/8.01 comm_monoid_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
% 12.24/8.01 semiring_modulo @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
% 12.24/8.01 comm_semiring_1 @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
% 12.24/8.01 comm_semiring_0 @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
% 12.24/8.01 semigroup_mult @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
% 12.24/8.01 semidom_modulo @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
% 12.24/8.01 semidom_divide @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
% 12.24/8.01 semiring_numeral @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
% 12.24/8.01 semigroup_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
% 12.24/8.01 zero_less_one @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
% 12.24/8.01 comm_semiring @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 12.24/8.01 wellorder @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
% 12.24/8.01 order_bot @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
% 12.24/8.01 semiring_char_0 @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
% 12.24/8.01 countable @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
% 12.24/8.01 zero_neq_one @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
% 12.24/8.01 preorder @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
% 12.24/8.01 linorder @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
% 12.24/8.01 monoid_mult @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
% 12.24/8.01 monoid_add @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
% 12.24/8.01 semiring_1 @ nat ).
% 12.24/8.01
% 12.24/8.01 thf(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
% 12.24/8.01 semiring_0 @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
% 12.24/8.02 no_top @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
% 12.24/8.02 lattice @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
% 12.24/8.02 semiring_gcd @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
% 12.24/8.02 semiring_Gcd @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Typerep_Otyperep_96,axiom,
% 12.24/8.02 typerep2 @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
% 12.24/8.02 mult_zero @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
% 12.24/8.02 order @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
% 12.24/8.02 semiring @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
% 12.24/8.02 semidom @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Orderings_Oord_101,axiom,
% 12.24/8.02 ord @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Orderings_Obot_102,axiom,
% 12.24/8.02 bot @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Groups_Ominus_103,axiom,
% 12.24/8.02 minus @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Power_Opower_104,axiom,
% 12.24/8.02 power @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Num_Onumeral_105,axiom,
% 12.24/8.02 numeral @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Groups_Ozero_106,axiom,
% 12.24/8.02 zero @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Groups_Oplus_107,axiom,
% 12.24/8.02 plus @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Groups_Oone_108,axiom,
% 12.24/8.02 one @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Rings_Odvd_109,axiom,
% 12.24/8.02 dvd @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Heap_Oheap_110,axiom,
% 12.24/8.02 heap @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 12.24/8.02 size @ nat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Orderings_Opreorder_111,axiom,
% 12.24/8.02 preorder @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Orderings_Olinorder_112,axiom,
% 12.24/8.02 linorder @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Typerep_Otyperep_113,axiom,
% 12.24/8.02 typerep2 @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Orderings_Oorder_114,axiom,
% 12.24/8.02 order @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Orderings_Oord_115,axiom,
% 12.24/8.02 ord @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Groups_Oplus_116,axiom,
% 12.24/8.02 plus @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Num_Onum___Nat_Osize_117,axiom,
% 12.24/8.02 size @ num ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_118,axiom,
% 12.24/8.02 semiri1453513574482234551roduct @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_119,axiom,
% 12.24/8.02 ordere1937475149494474687imp_le @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_120,axiom,
% 12.24/8.02 semiri6575147826004484403cancel @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_121,axiom,
% 12.24/8.02 strict9044650504122735259up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_122,axiom,
% 12.24/8.02 ordere580206878836729694up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_123,axiom,
% 12.24/8.02 ordere2412721322843649153imp_le @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_124,axiom,
% 12.24/8.02 linord2810124833399127020strict @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_125,axiom,
% 12.24/8.02 strict7427464778891057005id_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_126,axiom,
% 12.24/8.02 ordere8940638589300402666id_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
% 12.24/8.02 archim462609752435547400_field @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_127,axiom,
% 12.24/8.02 linord715952674999750819strict @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_128,axiom,
% 12.24/8.02 linord4140545234300271783up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_129,axiom,
% 12.24/8.02 semiri2026040879449505780visors @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_130,axiom,
% 12.24/8.02 linord181362715937106298miring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_131,axiom,
% 12.24/8.02 linord8928482502909563296strict @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_132,axiom,
% 12.24/8.02 semiri3467727345109120633visors @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_133,axiom,
% 12.24/8.02 ordere6658533253407199908up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_134,axiom,
% 12.24/8.02 ordere166539214618696060dd_abs @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
% 12.24/8.02 archim2362893244070406136eiling @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_135,axiom,
% 12.24/8.02 ordere6911136660526730532id_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_136,axiom,
% 12.24/8.02 linord5086331880401160121up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_137,axiom,
% 12.24/8.02 cancel2418104881723323429up_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_138,axiom,
% 12.24/8.02 ring_15535105094025558882visors @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_139,axiom,
% 12.24/8.02 cancel1802427076303600483id_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_140,axiom,
% 12.24/8.02 linord4710134922213307826strict @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_141,axiom,
% 12.24/8.02 comm_s4317794764714335236cancel @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_142,axiom,
% 12.24/8.02 ordere2520102378445227354miring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_143,axiom,
% 12.24/8.02 linord6961819062388156250ring_1 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_144,axiom,
% 12.24/8.02 ordered_ab_group_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_145,axiom,
% 12.24/8.02 cancel_semigroup_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__semiring_146,axiom,
% 12.24/8.02 linordered_semiring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_147,axiom,
% 12.24/8.02 ordered_semiring_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_148,axiom,
% 12.24/8.02 linordered_semidom @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
% 12.24/8.02 dense_linorder @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_149,axiom,
% 12.24/8.02 semilattice_sup @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_150,axiom,
% 12.24/8.02 semilattice_inf @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_151,axiom,
% 12.24/8.02 distrib_lattice @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_152,axiom,
% 12.24/8.02 ab_semigroup_mult @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_153,axiom,
% 12.24/8.02 semiring_1_cancel @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_154,axiom,
% 12.24/8.02 comm_monoid_mult @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_155,axiom,
% 12.24/8.02 ab_semigroup_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
% 12.24/8.02 linordered_field @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oordered__semiring_156,axiom,
% 12.24/8.02 ordered_semiring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_157,axiom,
% 12.24/8.02 ordered_ring_abs @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_158,axiom,
% 12.24/8.02 comm_monoid_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__ring_159,axiom,
% 12.24/8.02 linordered_ring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Olinordered__idom_160,axiom,
% 12.24/8.02 linordered_idom @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_161,axiom,
% 12.24/8.02 comm_semiring_1 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_162,axiom,
% 12.24/8.02 comm_semiring_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
% 12.24/8.02 dense_order @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_163,axiom,
% 12.24/8.02 semigroup_mult @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemidom__divide_164,axiom,
% 12.24/8.02 semidom_divide @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Num_Osemiring__numeral_165,axiom,
% 12.24/8.02 semiring_numeral @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Osemigroup__add_166,axiom,
% 12.24/8.02 semigroup_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
% 12.24/8.02 field_abs_sgn @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
% 12.24/8.02 division_ring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ozero__less__one_167,axiom,
% 12.24/8.02 zero_less_one @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_168,axiom,
% 12.24/8.02 comm_semiring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_169,axiom,
% 12.24/8.02 semiring_char_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oab__group__add_170,axiom,
% 12.24/8.02 ab_group_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
% 12.24/8.02 field_char_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Countable_Ocountable_171,axiom,
% 12.24/8.02 countable @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_172,axiom,
% 12.24/8.02 zero_neq_one @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oordered__ring_173,axiom,
% 12.24/8.02 ordered_ring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_174,axiom,
% 12.24/8.02 idom_abs_sgn @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Opreorder_175,axiom,
% 12.24/8.02 preorder @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Olinorder_176,axiom,
% 12.24/8.02 linorder @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Omonoid__mult_177,axiom,
% 12.24/8.02 monoid_mult @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oidom__divide_178,axiom,
% 12.24/8.02 idom_divide @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_179,axiom,
% 12.24/8.02 comm_ring_1 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Omonoid__add_180,axiom,
% 12.24/8.02 monoid_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__1_181,axiom,
% 12.24/8.02 semiring_1 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring__0_182,axiom,
% 12.24/8.02 semiring_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Ono__top_183,axiom,
% 12.24/8.02 no_top @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Ono__bot_184,axiom,
% 12.24/8.02 no_bot @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Lattices_Olattice_185,axiom,
% 12.24/8.02 lattice @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ogroup__add_186,axiom,
% 12.24/8.02 group_add @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Typerep_Otyperep_187,axiom,
% 12.24/8.02 typerep2 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Omult__zero_188,axiom,
% 12.24/8.02 mult_zero @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Ocomm__ring_189,axiom,
% 12.24/8.02 comm_ring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Oorder_190,axiom,
% 12.24/8.02 order @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Num_Oneg__numeral_191,axiom,
% 12.24/8.02 neg_numeral @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Nat_Oring__char__0_192,axiom,
% 12.24/8.02 ring_char_0 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemiring_193,axiom,
% 12.24/8.02 semiring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
% 12.24/8.02 inverse @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Osemidom_194,axiom,
% 12.24/8.02 semidom @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Orderings_Oord_195,axiom,
% 12.24/8.02 ord @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ouminus_196,axiom,
% 12.24/8.02 uminus @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oring__1_197,axiom,
% 12.24/8.02 ring_1 @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oabs__if_198,axiom,
% 12.24/8.02 abs_if @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ominus_199,axiom,
% 12.24/8.02 minus @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Fields_Ofield,axiom,
% 12.24/8.02 field @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Power_Opower_200,axiom,
% 12.24/8.02 power @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Num_Onumeral_201,axiom,
% 12.24/8.02 numeral @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Ozero_202,axiom,
% 12.24/8.02 zero @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oplus_203,axiom,
% 12.24/8.02 plus @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oring_204,axiom,
% 12.24/8.02 ring @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Oidom_205,axiom,
% 12.24/8.02 idom @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Groups_Oone_206,axiom,
% 12.24/8.02 one @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Rat_Orat___Rings_Odvd_207,axiom,
% 12.24/8.02 dvd @ rat ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_208,axiom,
% 12.24/8.02 ! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_209,axiom,
% 12.24/8.02 ! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_210,axiom,
% 12.24/8.02 ! [A17: $tType] : ( comple489889107523837845lgebra @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_211,axiom,
% 12.24/8.02 ! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_212,axiom,
% 12.24/8.02 ! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_213,axiom,
% 12.24/8.02 ! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_214,axiom,
% 12.24/8.02 ! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_215,axiom,
% 12.24/8.02 ! [A17: $tType] : ( bounded_lattice_top @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_216,axiom,
% 12.24/8.02 ! [A17: $tType] : ( bounded_lattice_bot @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_217,axiom,
% 12.24/8.02 ! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_218,axiom,
% 12.24/8.02 ! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_219,axiom,
% 12.24/8.02 ! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_220,axiom,
% 12.24/8.02 ! [A17: $tType] : ( distrib_lattice @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Oorder__top_221,axiom,
% 12.24/8.02 ! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Oorder__bot_222,axiom,
% 12.24/8.02 ! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Countable_Ocountable_223,axiom,
% 12.24/8.02 ! [A17: $tType] :
% 12.24/8.02 ( ( finite_finite @ A17 )
% 12.24/8.02 => ( countable @ ( set @ A17 ) ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Opreorder_224,axiom,
% 12.24/8.02 ! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Lattices_Olattice_225,axiom,
% 12.24/8.02 ! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Typerep_Otyperep_226,axiom,
% 12.24/8.02 ! [A17: $tType] :
% 12.24/8.02 ( ( typerep2 @ A17 )
% 12.24/8.02 => ( typerep2 @ ( set @ A17 ) ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Oorder_227,axiom,
% 12.24/8.02 ! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Otop_228,axiom,
% 12.24/8.02 ! [A17: $tType] : ( top @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Oord_229,axiom,
% 12.24/8.02 ! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Orderings_Obot_230,axiom,
% 12.24/8.02 ! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Groups_Ouminus_231,axiom,
% 12.24/8.02 ! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_Set_Oset___Groups_Ominus_232,axiom,
% 12.24/8.02 ! [A17: $tType] : ( minus @ ( set @ A17 ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233,axiom,
% 12.24/8.02 condit1219197933456340205attice @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_234,axiom,
% 12.24/8.02 comple592849572758109894attice @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_235,axiom,
% 12.24/8.02 comple489889107523837845lgebra @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_236,axiom,
% 12.24/8.02 bounde4967611905675639751up_bot @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_237,axiom,
% 12.24/8.02 bounde4346867609351753570nf_top @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_238,axiom,
% 12.24/8.02 comple6319245703460814977attice @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_239,axiom,
% 12.24/8.02 boolea8198339166811842893lgebra @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_240,axiom,
% 12.24/8.02 bounded_lattice_top @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_241,axiom,
% 12.24/8.02 bounded_lattice_bot @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_242,axiom,
% 12.24/8.02 comple9053668089753744459l_ccpo @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_243,axiom,
% 12.24/8.02 semilattice_sup @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_244,axiom,
% 12.24/8.02 semilattice_inf @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_245,axiom,
% 12.24/8.02 distrib_lattice @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
% 12.24/8.02 order_top @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
% 12.24/8.02 order_bot @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
% 12.24/8.02 countable @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
% 12.24/8.02 preorder @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
% 12.24/8.02 linorder @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Lattices_Olattice_251,axiom,
% 12.24/8.02 lattice @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Typerep_Otyperep_252,axiom,
% 12.24/8.02 typerep2 @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
% 12.24/8.02 order @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Otop_254,axiom,
% 12.24/8.02 top @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Oord_255,axiom,
% 12.24/8.02 ord @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Orderings_Obot_256,axiom,
% 12.24/8.02 bot @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
% 12.24/8.02 uminus @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Groups_Ominus_258,axiom,
% 12.24/8.02 minus @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_HOL_Obool___Heap_Oheap_259,axiom,
% 12.24/8.02 heap @ $o ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_List_Olist___Countable_Ocountable_260,axiom,
% 12.24/8.02 ! [A17: $tType] :
% 12.24/8.02 ( ( countable @ A17 )
% 12.24/8.02 => ( countable @ ( list @ A17 ) ) ) ).
% 12.24/8.02
% 12.24/8.02 thf(tcon_List_Olist___Typerep_Otyperep_261,axiom,
% 12.24/8.02 ! [A17: $tType] :
% 12.24/8.03 ( ( typerep2 @ A17 )
% 12.24/8.03 => ( typerep2 @ ( list @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_List_Olist___Heap_Oheap_262,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( heap @ A17 )
% 12.24/8.03 => ( heap @ ( list @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_List_Olist___Nat_Osize_263,axiom,
% 12.24/8.03 ! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap_Oarray___Countable_Ocountable_264,axiom,
% 12.24/8.03 ! [A17: $tType] : ( countable @ ( array @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap_Oarray___Typerep_Otyperep_265,axiom,
% 12.24/8.03 ! [A17: $tType] : ( typerep2 @ ( array @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap_Oarray___Heap_Oheap_266,axiom,
% 12.24/8.03 ! [A17: $tType] : ( heap @ ( array @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap_Oarray___Nat_Osize_267,axiom,
% 12.24/8.03 ! [A17: $tType] : ( size @ ( array @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Ochar___Countable_Ocountable_268,axiom,
% 12.24/8.03 countable @ char ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Ochar___Typerep_Otyperep_269,axiom,
% 12.24/8.03 typerep2 @ char ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Ochar___Heap_Oheap_270,axiom,
% 12.24/8.03 heap @ char ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Ochar___Nat_Osize_271,axiom,
% 12.24/8.03 size @ char ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Nitpick_Oword___Typerep_Otyperep_272,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( typerep2 @ A17 )
% 12.24/8.03 => ( typerep2 @ ( word @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Sum__Type_Osum___Countable_Ocountable_273,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( countable @ A17 )
% 12.24/8.03 & ( countable @ A18 ) )
% 12.24/8.03 => ( countable @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_274,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( typerep2 @ A17 )
% 12.24/8.03 & ( typerep2 @ A18 ) )
% 12.24/8.03 => ( typerep2 @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Sum__Type_Osum___Heap_Oheap_275,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( heap @ A17 )
% 12.24/8.03 & ( heap @ A18 ) )
% 12.24/8.03 => ( heap @ ( sum_sum @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Sum__Type_Osum___Nat_Osize_276,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] : ( size @ ( sum_sum @ A17 @ A18 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Option_Ooption___Countable_Ocountable_277,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( countable @ A17 )
% 12.24/8.03 => ( countable @ ( option @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Option_Ooption___Typerep_Otyperep_278,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( typerep2 @ A17 )
% 12.24/8.03 => ( typerep2 @ ( option @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Option_Ooption___Heap_Oheap_279,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( heap @ A17 )
% 12.24/8.03 => ( heap @ ( option @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Option_Ooption___Nat_Osize_280,axiom,
% 12.24/8.03 ! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Groups_Osemigroup__add_281,axiom,
% 12.24/8.03 semigroup_add @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Countable_Ocountable_282,axiom,
% 12.24/8.03 countable @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Orderings_Opreorder_283,axiom,
% 12.24/8.03 preorder @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Orderings_Olinorder_284,axiom,
% 12.24/8.03 linorder @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Groups_Omonoid__add_285,axiom,
% 12.24/8.03 monoid_add @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Typerep_Otyperep_286,axiom,
% 12.24/8.03 typerep2 @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Orderings_Oorder_287,axiom,
% 12.24/8.03 order @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Orderings_Oord_288,axiom,
% 12.24/8.03 ord @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Groups_Ozero_289,axiom,
% 12.24/8.03 zero @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Groups_Oplus_290,axiom,
% 12.24/8.03 plus @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Heap_Oheap_291,axiom,
% 12.24/8.03 heap @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_String_Oliteral___Nat_Osize_292,axiom,
% 12.24/8.03 size @ literal ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Nitpick_Ofun__box___Typerep_Otyperep_293,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( typerep2 @ A17 )
% 12.24/8.03 & ( typerep2 @ A18 ) )
% 12.24/8.03 => ( typerep2 @ ( fun_box @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Typerep_Otyperep___Countable_Ocountable_294,axiom,
% 12.24/8.03 countable @ typerep ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_295,axiom,
% 12.24/8.03 typerep2 @ typerep ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Typerep_Otyperep___Heap_Oheap_296,axiom,
% 12.24/8.03 heap @ typerep ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Typerep_Otyperep___Nat_Osize_297,axiom,
% 12.24/8.03 size @ typerep ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_298,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( typerep2 @ A17 )
% 12.24/8.03 & ( typerep2 @ A18 ) )
% 12.24/8.03 => ( typerep2 @ ( old_node @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Oprod___Countable_Ocountable_299,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( countable @ A17 )
% 12.24/8.03 & ( countable @ A18 ) )
% 12.24/8.03 => ( countable @ ( product_prod @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_300,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( typerep2 @ A17 )
% 12.24/8.03 & ( typerep2 @ A18 ) )
% 12.24/8.03 => ( typerep2 @ ( product_prod @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Oprod___Heap_Oheap_301,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] :
% 12.24/8.03 ( ( ( heap @ A17 )
% 12.24/8.03 & ( heap @ A18 ) )
% 12.24/8.03 => ( heap @ ( product_prod @ A17 @ A18 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Oprod___Nat_Osize_302,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType] : ( size @ ( product_prod @ A17 @ A18 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_303,axiom,
% 12.24/8.03 condit6923001295902523014norder @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_304,axiom,
% 12.24/8.03 condit1219197933456340205attice @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_305,axiom,
% 12.24/8.03 comple592849572758109894attice @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_306,axiom,
% 12.24/8.03 comple489889107523837845lgebra @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_307,axiom,
% 12.24/8.03 bounde4967611905675639751up_bot @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_308,axiom,
% 12.24/8.03 bounde4346867609351753570nf_top @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
% 12.24/8.03 comple5582772986160207858norder @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_309,axiom,
% 12.24/8.03 comple6319245703460814977attice @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_310,axiom,
% 12.24/8.03 boolea8198339166811842893lgebra @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_311,axiom,
% 12.24/8.03 bounded_lattice_top @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_312,axiom,
% 12.24/8.03 bounded_lattice_bot @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_313,axiom,
% 12.24/8.03 comple9053668089753744459l_ccpo @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_314,axiom,
% 12.24/8.03 semilattice_sup @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_315,axiom,
% 12.24/8.03 semilattice_inf @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_316,axiom,
% 12.24/8.03 distrib_lattice @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_317,axiom,
% 12.24/8.03 wellorder @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_318,axiom,
% 12.24/8.03 order_top @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_319,axiom,
% 12.24/8.03 order_bot @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Countable_Ocountable_320,axiom,
% 12.24/8.03 countable @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_321,axiom,
% 12.24/8.03 preorder @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_322,axiom,
% 12.24/8.03 linorder @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Lattices_Olattice_323,axiom,
% 12.24/8.03 lattice @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_324,axiom,
% 12.24/8.03 typerep2 @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Oorder_325,axiom,
% 12.24/8.03 order @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Otop_326,axiom,
% 12.24/8.03 top @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Oord_327,axiom,
% 12.24/8.03 ord @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Orderings_Obot_328,axiom,
% 12.24/8.03 bot @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Groups_Ouminus_329,axiom,
% 12.24/8.03 uminus @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Groups_Ominus_330,axiom,
% 12.24/8.03 minus @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Product__Type_Ounit___Heap_Oheap_331,axiom,
% 12.24/8.03 heap @ product_unit ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_332,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( typerep2 @ A17 )
% 12.24/8.03 => ( typerep2 @ ( heap_ext @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_333,axiom,
% 12.24/8.03 bit_un5681908812861735899ations @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_334,axiom,
% 12.24/8.03 semiri1453513574482234551roduct @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_335,axiom,
% 12.24/8.03 euclid5411537665997757685th_nat @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_336,axiom,
% 12.24/8.03 euclid8789492081693882211th_nat @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_337,axiom,
% 12.24/8.03 ordere1937475149494474687imp_le @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_338,axiom,
% 12.24/8.03 euclid3128863361964157862miring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_339,axiom,
% 12.24/8.03 euclid4440199948858584721cancel @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_340,axiom,
% 12.24/8.03 unique1627219031080169319umeral @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_341,axiom,
% 12.24/8.03 euclid8851590272496341667cancel @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_342,axiom,
% 12.24/8.03 semiri6575147826004484403cancel @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_343,axiom,
% 12.24/8.03 strict9044650504122735259up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_344,axiom,
% 12.24/8.03 ordere580206878836729694up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_345,axiom,
% 12.24/8.03 ordere2412721322843649153imp_le @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_346,axiom,
% 12.24/8.03 bit_se359711467146920520ations @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_347,axiom,
% 12.24/8.03 linord2810124833399127020strict @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_348,axiom,
% 12.24/8.03 strict7427464778891057005id_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_349,axiom,
% 12.24/8.03 ordere8940638589300402666id_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_350,axiom,
% 12.24/8.03 euclid3725896446679973847miring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_351,axiom,
% 12.24/8.03 linord715952674999750819strict @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_352,axiom,
% 12.24/8.03 linord4140545234300271783up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_353,axiom,
% 12.24/8.03 bit_ri3973907225187159222ations @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_354,axiom,
% 12.24/8.03 semiri2026040879449505780visors @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_355,axiom,
% 12.24/8.03 linord181362715937106298miring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_356,axiom,
% 12.24/8.03 euclid5891614535332579305n_ring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_357,axiom,
% 12.24/8.03 linord8928482502909563296strict @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_358,axiom,
% 12.24/8.03 semiri3467727345109120633visors @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_359,axiom,
% 12.24/8.03 ordere6658533253407199908up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_360,axiom,
% 12.24/8.03 ordere166539214618696060dd_abs @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_361,axiom,
% 12.24/8.03 ordere6911136660526730532id_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_362,axiom,
% 12.24/8.03 linord5086331880401160121up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_363,axiom,
% 12.24/8.03 cancel2418104881723323429up_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_364,axiom,
% 12.24/8.03 ring_15535105094025558882visors @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_365,axiom,
% 12.24/8.03 cancel1802427076303600483id_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_366,axiom,
% 12.24/8.03 linord4710134922213307826strict @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_367,axiom,
% 12.24/8.03 comm_s4317794764714335236cancel @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_368,axiom,
% 12.24/8.03 bit_semiring_bits @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_369,axiom,
% 12.24/8.03 ordere2520102378445227354miring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_370,axiom,
% 12.24/8.03 linord6961819062388156250ring_1 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_371,axiom,
% 12.24/8.03 ordered_ab_group_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_372,axiom,
% 12.24/8.03 cancel_semigroup_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_373,axiom,
% 12.24/8.03 linordered_semiring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_374,axiom,
% 12.24/8.03 ordered_semiring_0 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_375,axiom,
% 12.24/8.03 linordered_semidom @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_376,axiom,
% 12.24/8.03 ab_semigroup_mult @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_377,axiom,
% 12.24/8.03 semiring_1_cancel @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_378,axiom,
% 12.24/8.03 algebraic_semidom @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_379,axiom,
% 12.24/8.03 comm_monoid_mult @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_380,axiom,
% 12.24/8.03 ab_semigroup_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_381,axiom,
% 12.24/8.03 ordered_semiring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_382,axiom,
% 12.24/8.03 ordered_ring_abs @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_383,axiom,
% 12.24/8.03 semiring_parity @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_384,axiom,
% 12.24/8.03 comm_monoid_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_385,axiom,
% 12.24/8.03 semiring_modulo @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_386,axiom,
% 12.24/8.03 linordered_ring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_387,axiom,
% 12.24/8.03 linordered_idom @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_388,axiom,
% 12.24/8.03 comm_semiring_1 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_389,axiom,
% 12.24/8.03 comm_semiring_0 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_390,axiom,
% 12.24/8.03 semigroup_mult @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_391,axiom,
% 12.24/8.03 semidom_modulo @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_392,axiom,
% 12.24/8.03 semidom_divide @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_393,axiom,
% 12.24/8.03 semiring_numeral @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_394,axiom,
% 12.24/8.03 semigroup_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_395,axiom,
% 12.24/8.03 zero_less_one @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_396,axiom,
% 12.24/8.03 comm_semiring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_397,axiom,
% 12.24/8.03 semiring_char_0 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_398,axiom,
% 12.24/8.03 ab_group_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_399,axiom,
% 12.24/8.03 zero_neq_one @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_400,axiom,
% 12.24/8.03 ordered_ring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_401,axiom,
% 12.24/8.03 idom_abs_sgn @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_402,axiom,
% 12.24/8.03 ring_parity @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_403,axiom,
% 12.24/8.03 preorder @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_404,axiom,
% 12.24/8.03 linorder @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_405,axiom,
% 12.24/8.03 monoid_mult @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_406,axiom,
% 12.24/8.03 idom_modulo @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_407,axiom,
% 12.24/8.03 idom_divide @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_408,axiom,
% 12.24/8.03 comm_ring_1 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_409,axiom,
% 12.24/8.03 monoid_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_410,axiom,
% 12.24/8.03 semiring_1 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_411,axiom,
% 12.24/8.03 semiring_0 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_412,axiom,
% 12.24/8.03 group_add @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_413,axiom,
% 12.24/8.03 typerep2 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
% 12.24/8.03 mult_zero @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
% 12.24/8.03 comm_ring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
% 12.24/8.03 order @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
% 12.24/8.03 neg_numeral @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
% 12.24/8.03 ring_char_0 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
% 12.24/8.03 semiring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_420,axiom,
% 12.24/8.03 semidom @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_421,axiom,
% 12.24/8.03 ord @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_422,axiom,
% 12.24/8.03 uminus @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_423,axiom,
% 12.24/8.03 ring_1 @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_424,axiom,
% 12.24/8.03 abs_if @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_425,axiom,
% 12.24/8.03 minus @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Power_Opower_426,axiom,
% 12.24/8.03 power @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_427,axiom,
% 12.24/8.03 numeral @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_428,axiom,
% 12.24/8.03 zero @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_429,axiom,
% 12.24/8.03 plus @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_430,axiom,
% 12.24/8.03 ring @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_431,axiom,
% 12.24/8.03 idom @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_432,axiom,
% 12.24/8.03 one @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_433,axiom,
% 12.24/8.03 dvd @ code_integer ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_434,axiom,
% 12.24/8.03 bit_un5681908812861735899ations @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_435,axiom,
% 12.24/8.03 euclid5411537665997757685th_nat @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_436,axiom,
% 12.24/8.03 ordere1937475149494474687imp_le @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_437,axiom,
% 12.24/8.03 euclid3128863361964157862miring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_438,axiom,
% 12.24/8.03 euclid4440199948858584721cancel @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_439,axiom,
% 12.24/8.03 semiri6575147826004484403cancel @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_440,axiom,
% 12.24/8.03 strict9044650504122735259up_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_441,axiom,
% 12.24/8.03 ordere580206878836729694up_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_442,axiom,
% 12.24/8.03 ordere2412721322843649153imp_le @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_443,axiom,
% 12.24/8.03 bit_se359711467146920520ations @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_444,axiom,
% 12.24/8.03 linord2810124833399127020strict @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_445,axiom,
% 12.24/8.03 strict7427464778891057005id_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_446,axiom,
% 12.24/8.03 ordere8940638589300402666id_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_447,axiom,
% 12.24/8.03 euclid3725896446679973847miring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_448,axiom,
% 12.24/8.03 linord4140545234300271783up_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_449,axiom,
% 12.24/8.03 semiri2026040879449505780visors @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_450,axiom,
% 12.24/8.03 linord181362715937106298miring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_451,axiom,
% 12.24/8.03 linord8928482502909563296strict @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_452,axiom,
% 12.24/8.03 semiri3467727345109120633visors @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_453,axiom,
% 12.24/8.03 ordere6658533253407199908up_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_454,axiom,
% 12.24/8.03 ordere6911136660526730532id_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_455,axiom,
% 12.24/8.03 cancel2418104881723323429up_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_456,axiom,
% 12.24/8.03 cancel1802427076303600483id_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_457,axiom,
% 12.24/8.03 comm_s4317794764714335236cancel @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_458,axiom,
% 12.24/8.03 bit_semiring_bits @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_459,axiom,
% 12.24/8.03 ordere2520102378445227354miring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_460,axiom,
% 12.24/8.03 cancel_semigroup_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_461,axiom,
% 12.24/8.03 linordered_semiring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_462,axiom,
% 12.24/8.03 ordered_semiring_0 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_463,axiom,
% 12.24/8.03 linordered_semidom @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_464,axiom,
% 12.24/8.03 ab_semigroup_mult @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_465,axiom,
% 12.24/8.03 semiring_1_cancel @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_466,axiom,
% 12.24/8.03 algebraic_semidom @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_467,axiom,
% 12.24/8.03 comm_monoid_mult @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_468,axiom,
% 12.24/8.03 comm_monoid_diff @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_469,axiom,
% 12.24/8.03 ab_semigroup_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_470,axiom,
% 12.24/8.03 ordered_semiring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_471,axiom,
% 12.24/8.03 semiring_parity @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_472,axiom,
% 12.24/8.03 comm_monoid_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_473,axiom,
% 12.24/8.03 semiring_modulo @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_474,axiom,
% 12.24/8.03 comm_semiring_1 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_475,axiom,
% 12.24/8.03 comm_semiring_0 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_476,axiom,
% 12.24/8.03 semigroup_mult @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_477,axiom,
% 12.24/8.03 semidom_modulo @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_478,axiom,
% 12.24/8.03 semidom_divide @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_479,axiom,
% 12.24/8.03 semiring_numeral @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_480,axiom,
% 12.24/8.03 semigroup_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_481,axiom,
% 12.24/8.03 zero_less_one @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_482,axiom,
% 12.24/8.03 comm_semiring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_483,axiom,
% 12.24/8.03 semiring_char_0 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_484,axiom,
% 12.24/8.03 zero_neq_one @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_485,axiom,
% 12.24/8.03 preorder @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_486,axiom,
% 12.24/8.03 linorder @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_487,axiom,
% 12.24/8.03 monoid_mult @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_488,axiom,
% 12.24/8.03 monoid_add @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_489,axiom,
% 12.24/8.03 semiring_1 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_490,axiom,
% 12.24/8.03 semiring_0 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_491,axiom,
% 12.24/8.03 typerep2 @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_492,axiom,
% 12.24/8.03 mult_zero @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_493,axiom,
% 12.24/8.03 order @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_494,axiom,
% 12.24/8.03 semiring @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_495,axiom,
% 12.24/8.03 semidom @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_496,axiom,
% 12.24/8.03 ord @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ominus_497,axiom,
% 12.24/8.03 minus @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Power_Opower_498,axiom,
% 12.24/8.03 power @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_499,axiom,
% 12.24/8.03 numeral @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_500,axiom,
% 12.24/8.03 zero @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_501,axiom,
% 12.24/8.03 plus @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Groups_Oone_502,axiom,
% 12.24/8.03 one @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_503,axiom,
% 12.24/8.03 dvd @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Code__Numeral_Onatural___Nat_Osize_504,axiom,
% 12.24/8.03 size @ code_natural ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_505,axiom,
% 12.24/8.03 ! [A17: $tType] :
% 12.24/8.03 ( ( typerep2 @ A17 )
% 12.24/8.03 => ( typerep2 @ ( heap_Time_Heap @ A17 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_506,axiom,
% 12.24/8.03 ! [A17: $tType] : ( size @ ( heap_Time_Heap @ A17 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_507,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType,A19: $tType] :
% 12.24/8.03 ( ( ( typerep2 @ A17 )
% 12.24/8.03 & ( typerep2 @ A18 )
% 12.24/8.03 & ( typerep2 @ A19 ) )
% 12.24/8.03 => ( typerep2 @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tcon_Record_Otuple__isomorphism___Nat_Osize_508,axiom,
% 12.24/8.03 ! [A17: $tType,A18: $tType,A19: $tType] : ( size @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(help_If_3_1_T,axiom,
% 12.24/8.03 ! [P: $o] :
% 12.24/8.03 ( ( P = $true )
% 12.24/8.03 | ( P = $false ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(help_If_2_1_T,axiom,
% 12.24/8.03 ! [A: $tType,X: A,Y3: A] :
% 12.24/8.03 ( ( if @ A @ $false @ X @ Y3 )
% 12.24/8.03 = Y3 ) ).
% 12.24/8.03
% 12.24/8.03 thf(help_If_1_1_T,axiom,
% 12.24/8.03 ! [A: $tType,X: A,Y3: A] :
% 12.24/8.03 ( ( if @ A @ $true @ X @ Y3 )
% 12.24/8.03 = X ) ).
% 12.24/8.03
% 12.24/8.03 thf(help_fChoice_1_1_T,axiom,
% 12.24/8.03 ! [A: $tType,P: A > $o] :
% 12.24/8.03 ( ( P @ ( fChoice @ A @ P ) )
% 12.24/8.03 = ( ? [X7: A] : ( P @ X7 ) ) ) ).
% 12.24/8.03
% 12.24/8.03 thf(tfree_0,hypothesis,
% 12.24/8.03 heap @ a ).
% 12.24/8.03
% 12.24/8.03 thf(conj_0,hypothesis,
% 12.24/8.03 $true ).
% 12.24/8.03
% 12.24/8.03 thf(conj_1,conjecture,
% 12.24/8.03 thesis ).
% 12.24/8.03 ------------------------
% 12.24/8.61 % (31439)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_31297 for (2994ds/275Mi)
% 12.24/8.61 % (31437)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31297 for (2994ds/2Mi)
% 12.24/8.61 % (31435)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_31297 for (2994ds/4Mi)
% 12.24/8.61 % (31440)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31297 for (2994ds/18Mi)
% 12.24/8.61 % (31436)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_31297 for (2994ds/27Mi)
% 12.24/8.61 % (31438)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_31297 for (2994ds/2Mi)
% 12.24/8.61 % (31434)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31297 for (2994ds/183Mi)
% 12.24/8.61 % (31437)Instruction limit reached!
% 12.24/8.61 % (31437)------------------------------
% 12.24/8.61 % (31437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.61 % (31437)Termination reason: Unknown
% 12.24/8.61 % (31437)Termination phase: shuffling
% 12.24/8.61
% 12.24/8.61 % (31437)Memory used [KB]: 10362
% 12.24/8.61 % (31437)Time elapsed: 0.004 s
% 12.24/8.61 % (31437)Instructions burned: 2 (million)
% 12.24/8.61 % (31437)------------------------------
% 12.24/8.61 % (31437)------------------------------
% 12.24/8.61 % (31438)Instruction limit reached!
% 12.24/8.61 % (31438)------------------------------
% 12.24/8.61 % (31438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.61 % (31438)Termination reason: Unknown
% 12.24/8.61 % (31438)Termination phase: shuffling
% 12.24/8.61
% 12.24/8.61 % (31438)Memory used [KB]: 10362
% 12.24/8.61 % (31438)Time elapsed: 0.004 s
% 12.24/8.61 % (31438)Instructions burned: 2 (million)
% 12.24/8.61 % (31438)------------------------------
% 12.24/8.61 % (31438)------------------------------
% 12.24/8.62 % (31435)Instruction limit reached!
% 12.24/8.62 % (31435)------------------------------
% 12.24/8.62 % (31435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.62 % (31435)Termination reason: Unknown
% 12.24/8.62 % (31435)Termination phase: shuffling
% 12.24/8.62
% 12.24/8.62 % (31435)Memory used [KB]: 10490
% 12.24/8.62 % (31435)Time elapsed: 0.007 s
% 12.24/8.62 % (31435)Instructions burned: 4 (million)
% 12.24/8.62 % (31435)------------------------------
% 12.24/8.62 % (31435)------------------------------
% 12.24/8.62 % (31440)Instruction limit reached!
% 12.24/8.62 % (31440)------------------------------
% 12.24/8.62 % (31440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.63 % (31440)Termination reason: Unknown
% 12.24/8.63 % (31440)Termination phase: shuffling
% 12.24/8.63
% 12.24/8.63 % (31440)Memory used [KB]: 10874
% 12.24/8.63 % (31440)Time elapsed: 0.017 s
% 12.24/8.63 % (31440)Instructions burned: 18 (million)
% 12.24/8.63 % (31440)------------------------------
% 12.24/8.63 % (31440)------------------------------
% 12.24/8.63 % (31436)Instruction limit reached!
% 12.24/8.63 % (31436)------------------------------
% 12.24/8.63 % (31436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.63 % (31436)Termination reason: Unknown
% 12.24/8.63 % (31436)Termination phase: shuffling
% 12.24/8.63
% 12.24/8.63 % (31436)Memory used [KB]: 11129
% 12.24/8.63 % (31436)Time elapsed: 0.022 s
% 12.24/8.63 % (31436)Instructions burned: 27 (million)
% 12.24/8.63 % (31436)------------------------------
% 12.24/8.63 % (31436)------------------------------
% 12.24/8.63 % (31441)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31297 for (2994ds/3Mi)
% 12.24/8.63 % (31443)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_31297 for (2994ds/15Mi)
% 12.24/8.63 % (31441)Instruction limit reached!
% 12.24/8.63 % (31441)------------------------------
% 12.24/8.63 % (31441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.63 % (31441)Termination reason: Unknown
% 12.24/8.63 % (31441)Termination phase: shuffling
% 12.24/8.63
% 12.24/8.63 % (31441)Memory used [KB]: 10490
% 12.24/8.63 % (31441)Time elapsed: 0.006 s
% 12.24/8.63 % (31441)Instructions burned: 3 (million)
% 12.24/8.63 % (31441)------------------------------
% 12.24/8.63 % (31441)------------------------------
% 12.24/8.64 % (31444)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_31297 for (2993ds/3Mi)
% 12.24/8.65 % (31443)Instruction limit reached!
% 12.24/8.65 % (31443)------------------------------
% 12.24/8.65 % (31443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.65 % (31443)Termination reason: Unknown
% 12.24/8.65 % (31443)Termination phase: shuffling
% 12.24/8.65
% 12.24/8.65 % (31443)Memory used [KB]: 10874
% 12.24/8.65 % (31443)Time elapsed: 0.015 s
% 12.24/8.65 % (31443)Instructions burned: 15 (million)
% 12.24/8.65 % (31443)------------------------------
% 12.24/8.65 % (31443)------------------------------
% 12.24/8.65 % (31444)Instruction limit reached!
% 12.24/8.65 % (31444)------------------------------
% 12.24/8.65 % (31444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.65 % (31444)Termination reason: Unknown
% 12.24/8.65 % (31444)Termination phase: shuffling
% 12.24/8.65
% 12.24/8.65 % (31444)Memory used [KB]: 10490
% 12.24/8.65 % (31444)Time elapsed: 0.007 s
% 12.24/8.65 % (31444)Instructions burned: 3 (million)
% 12.24/8.65 % (31444)------------------------------
% 12.24/8.65 % (31444)------------------------------
% 12.24/8.65 % (31445)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_31297 for (2993ds/1041Mi)
% 12.24/8.65 % (31446)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/7Mi)
% 12.24/8.66 % (31442)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_31297 for (2994ds/37Mi)
% 12.24/8.66 % (31446)Instruction limit reached!
% 12.24/8.66 % (31446)------------------------------
% 12.24/8.66 % (31446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.66 % (31446)Termination reason: Unknown
% 12.24/8.66 % (31446)Termination phase: shuffling
% 12.24/8.66
% 12.24/8.66 % (31446)Memory used [KB]: 10746
% 12.24/8.66 % (31446)Time elapsed: 0.011 s
% 12.24/8.66 % (31446)Instructions burned: 8 (million)
% 12.24/8.66 % (31446)------------------------------
% 12.24/8.66 % (31446)------------------------------
% 12.24/8.66 % (31447)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/16Mi)
% 12.24/8.66 % (31448)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_31297 for (2993ds/3Mi)
% 12.24/8.67 % (31448)Instruction limit reached!
% 12.24/8.67 % (31448)------------------------------
% 12.24/8.67 % (31448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.67 % (31448)Termination reason: Unknown
% 12.24/8.67 % (31448)Termination phase: shuffling
% 12.24/8.67
% 12.24/8.67 % (31448)Memory used [KB]: 10490
% 12.24/8.67 % (31448)Time elapsed: 0.006 s
% 12.24/8.67 % (31448)Instructions burned: 3 (million)
% 12.24/8.67 % (31448)------------------------------
% 12.24/8.67 % (31448)------------------------------
% 12.24/8.68 % (31447)Instruction limit reached!
% 12.24/8.68 % (31447)------------------------------
% 12.24/8.68 % (31447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.68 % (31447)Termination reason: Unknown
% 12.24/8.68 % (31447)Termination phase: shuffling
% 12.24/8.68
% 12.24/8.68 % (31447)Memory used [KB]: 10874
% 12.24/8.68 % (31447)Time elapsed: 0.016 s
% 12.24/8.68 % (31447)Instructions burned: 16 (million)
% 12.24/8.68 % (31447)------------------------------
% 12.24/8.68 % (31447)------------------------------
% 12.24/8.68 % (31449)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_31297 for (2993ds/3Mi)
% 12.24/8.69 % (31450)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/7Mi)
% 12.24/8.69 % (31449)Instruction limit reached!
% 12.24/8.69 % (31449)------------------------------
% 12.24/8.69 % (31449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.69 % (31449)Termination reason: Unknown
% 12.24/8.69 % (31449)Termination phase: shuffling
% 12.24/8.69
% 12.24/8.69 % (31449)Memory used [KB]: 10490
% 12.24/8.69 % (31449)Time elapsed: 0.006 s
% 12.24/8.69 % (31449)Instructions burned: 3 (million)
% 12.24/8.69 % (31449)------------------------------
% 12.24/8.69 % (31449)------------------------------
% 12.24/8.69 % (31450)Instruction limit reached!
% 12.24/8.69 % (31450)------------------------------
% 12.24/8.69 % (31450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.69 % (31450)Termination reason: Unknown
% 12.24/8.69 % (31450)Termination phase: shuffling
% 12.24/8.69
% 12.24/8.69 % (31450)Memory used [KB]: 10618
% 12.24/8.69 % (31450)Time elapsed: 0.010 s
% 12.24/8.69 % (31450)Instructions burned: 7 (million)
% 12.24/8.69 % (31450)------------------------------
% 12.24/8.69 % (31450)------------------------------
% 12.24/8.70 % (31451)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/3Mi)
% 12.24/8.70 % (31451)Instruction limit reached!
% 12.24/8.70 % (31451)------------------------------
% 12.24/8.70 % (31451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.70 % (31451)Termination reason: Unknown
% 12.24/8.70 % (31451)Termination phase: shuffling
% 12.24/8.70
% 12.24/8.70 % (31451)Memory used [KB]: 10490
% 12.24/8.70 % (31451)Time elapsed: 0.006 s
% 12.24/8.70 % (31451)Instructions burned: 3 (million)
% 12.24/8.70 % (31451)------------------------------
% 12.24/8.70 % (31451)------------------------------
% 12.24/8.70 % (31452)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/4Mi)
% 12.24/8.71 % (31442)Instruction limit reached!
% 12.24/8.71 % (31442)------------------------------
% 12.24/8.71 % (31442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.71 % (31442)Termination reason: Unknown
% 12.24/8.71 % (31442)Termination phase: shuffling
% 12.24/8.71
% 12.24/8.71 % (31442)Memory used [KB]: 12281
% 12.24/8.71 % (31442)Time elapsed: 0.028 s
% 12.24/8.71 % (31442)Instructions burned: 38 (million)
% 12.24/8.71 % (31442)------------------------------
% 12.24/8.71 % (31442)------------------------------
% 12.24/8.71 % (31452)Instruction limit reached!
% 12.24/8.71 % (31452)------------------------------
% 12.24/8.71 % (31452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.71 % (31452)Termination reason: Unknown
% 12.24/8.71 % (31452)Termination phase: shuffling
% 12.24/8.71
% 12.24/8.71 % (31452)Memory used [KB]: 10490
% 12.24/8.71 % (31452)Time elapsed: 0.008 s
% 12.24/8.71 % (31452)Instructions burned: 4 (million)
% 12.24/8.71 % (31452)------------------------------
% 12.24/8.71 % (31452)------------------------------
% 12.24/8.71 % (31453)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/18Mi)
% 12.24/8.72 % (31454)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/710Mi)
% 12.24/8.73 % (31453)Instruction limit reached!
% 12.24/8.73 % (31453)------------------------------
% 12.24/8.73 % (31453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.73 % (31453)Termination reason: Unknown
% 12.24/8.73 % (31453)Termination phase: shuffling
% 12.24/8.73
% 12.24/8.73 % (31453)Memory used [KB]: 10874
% 12.24/8.73 % (31453)Time elapsed: 0.017 s
% 12.24/8.73 % (31453)Instructions burned: 19 (million)
% 12.24/8.73 % (31453)------------------------------
% 12.24/8.73 % (31453)------------------------------
% 12.24/8.73 % (31456)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_31297 for (2993ds/902Mi)
% 12.24/8.74 % (31457)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_31297 for (2993ds/21Mi)
% 12.24/8.75 % (31455)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31297 for (2993ds/6Mi)
% 12.24/8.76 % (31439)Instruction limit reached!
% 12.24/8.76 % (31439)------------------------------
% 12.24/8.76 % (31439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.76 % (31439)Termination reason: Unknown
% 12.24/8.76 % (31439)Termination phase: shuffling
% 12.24/8.76
% 12.24/8.76 % (31439)Memory used [KB]: 16119
% 12.24/8.76 % (31439)Time elapsed: 0.151 s
% 12.24/8.76 % (31439)Instructions burned: 275 (million)
% 12.24/8.76 % (31439)------------------------------
% 12.24/8.76 % (31439)------------------------------
% 12.24/8.76 % (31457)Instruction limit reached!
% 12.24/8.76 % (31457)------------------------------
% 12.24/8.76 % (31457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.76 % (31457)Termination reason: Unknown
% 12.24/8.76 % (31457)Termination phase: shuffling
% 12.24/8.76
% 12.24/8.76 % (31457)Memory used [KB]: 11001
% 12.24/8.76 % (31457)Time elapsed: 0.019 s
% 12.24/8.76 % (31457)Instructions burned: 22 (million)
% 12.24/8.76 % (31457)------------------------------
% 12.24/8.76 % (31457)------------------------------
% 12.24/8.76 % (31455)Instruction limit reached!
% 12.24/8.76 % (31455)------------------------------
% 12.24/8.76 % (31455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.76 % (31455)Termination reason: Unknown
% 12.24/8.76 % (31455)Termination phase: shuffling
% 12.24/8.76
% 12.24/8.76 % (31455)Memory used [KB]: 10618
% 12.24/8.76 % (31455)Time elapsed: 0.010 s
% 12.24/8.76 % (31455)Instructions burned: 6 (million)
% 12.24/8.76 % (31455)------------------------------
% 12.24/8.76 % (31455)------------------------------
% 12.24/8.78 % (31460)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31297 for (2992ds/377Mi)
% 12.24/8.78 % (31458)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31297 for (2992ds/5Mi)
% 12.24/8.78 % (31459)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_31297 for (2992ds/6Mi)
% 12.24/8.78 % (31434)Instruction limit reached!
% 12.24/8.78 % (31434)------------------------------
% 12.24/8.78 % (31434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.78 % (31434)Termination reason: Unknown
% 12.24/8.78 % (31434)Termination phase: shuffling
% 12.24/8.78
% 12.24/8.78 % (31434)Memory used [KB]: 14583
% 12.24/8.78 % (31434)Time elapsed: 0.111 s
% 12.24/8.78 % (31434)Instructions burned: 183 (million)
% 12.24/8.78 % (31434)------------------------------
% 12.24/8.78 % (31434)------------------------------
% 12.24/8.78 % (31458)Instruction limit reached!
% 12.24/8.78 % (31458)------------------------------
% 12.24/8.78 % (31458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.78 % (31458)Termination reason: Unknown
% 12.24/8.79 % (31458)Termination phase: shuffling
% 12.24/8.79
% 12.24/8.79 % (31458)Memory used [KB]: 10618
% 12.24/8.79 % (31458)Time elapsed: 0.009 s
% 12.24/8.79 % (31458)Instructions burned: 5 (million)
% 12.24/8.79 % (31458)------------------------------
% 12.24/8.79 % (31458)------------------------------
% 12.24/8.79 % (31459)Instruction limit reached!
% 12.24/8.79 % (31459)------------------------------
% 12.24/8.79 % (31459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.79 % (31459)Termination reason: Unknown
% 12.24/8.79 % (31459)Termination phase: shuffling
% 12.24/8.79
% 12.24/8.79 % (31459)Memory used [KB]: 10618
% 12.24/8.79 % (31459)Time elapsed: 0.010 s
% 12.24/8.79 % (31459)Instructions burned: 6 (million)
% 12.24/8.79 % (31459)------------------------------
% 12.24/8.79 % (31459)------------------------------
% 12.24/8.85 % (31461)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_31297 for (2992ds/779Mi)
% 12.24/8.87 % (31462)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_31297 for (2992ds/19Mi)
% 12.24/8.87 % (31463)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31297 for (2992ds/879Mi)
% 12.24/8.93 % (31462)Instruction limit reached!
% 12.24/8.93 % (31462)------------------------------
% 12.24/8.93 % (31462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.93 % (31462)Termination reason: Unknown
% 12.24/8.93 % (31462)Termination phase: shuffling
% 12.24/8.93
% 12.24/8.93 % (31462)Memory used [KB]: 10874
% 12.24/8.93 % (31462)Time elapsed: 0.019 s
% 12.24/8.93 % (31462)Instructions burned: 19 (million)
% 12.24/8.93 % (31462)------------------------------
% 12.24/8.93 % (31462)------------------------------
% 12.24/8.94 % (31464)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31297 for (2990ds/17Mi)
% 12.24/8.96 % (31464)Instruction limit reached!
% 12.24/8.96 % (31464)------------------------------
% 12.24/8.96 % (31464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.96 % (31464)Termination reason: Unknown
% 12.24/8.96 % (31464)Termination phase: shuffling
% 12.24/8.96
% 12.24/8.96 % (31464)Memory used [KB]: 10874
% 12.24/8.96 % (31464)Time elapsed: 0.016 s
% 12.24/8.96 % (31464)Instructions burned: 18 (million)
% 12.24/8.96 % (31464)------------------------------
% 12.24/8.96 % (31464)------------------------------
% 12.24/8.98 % (31460)Instruction limit reached!
% 12.24/8.98 % (31460)------------------------------
% 12.24/8.98 % (31460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/8.98 % (31460)Termination reason: Unknown
% 12.24/8.98 % (31460)Termination phase: shuffling
% 12.24/8.98
% 12.24/8.98 % (31460)Memory used [KB]: 17654
% 12.24/8.98 % (31460)Time elapsed: 0.205 s
% 12.24/8.98 % (31460)Instructions burned: 377 (million)
% 12.24/8.98 % (31460)------------------------------
% 12.24/8.98 % (31460)------------------------------
% 12.24/9.00 % (31466)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_31297 for (2990ds/30Mi)
% 12.24/9.02 % (31466)Instruction limit reached!
% 12.24/9.02 % (31466)------------------------------
% 12.24/9.02 % (31466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.24/9.02 % (31466)Termination reason: Unknown
% 12.24/9.02 % (31466)Termination phase: shuffling
% 12.24/9.02
% 12.24/9.02 % (31466)Memory used [KB]: 11129
% 12.24/9.02 % (31466)Time elapsed: 0.023 s
% 12.24/9.02 % (31466)Instructions burned: 30 (million)
% 12.24/9.02 % (31466)------------------------------
% 12.24/9.02 % (31466)------------------------------
% 14.92/9.04 % (31467)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_31297 for (2989ds/127Mi)
% 14.92/9.05 % (31465)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31297 for (2990ds/3Mi)
% 14.92/9.06 % (31465)Instruction limit reached!
% 14.92/9.06 % (31465)------------------------------
% 14.92/9.06 % (31465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.92/9.06 % (31465)Termination reason: Unknown
% 14.92/9.06 % (31465)Termination phase: shuffling
% 14.92/9.06
% 14.92/9.06 % (31465)Memory used [KB]: 10490
% 14.92/9.06 % (31465)Time elapsed: 0.010 s
% 14.92/9.06 % (31465)Instructions burned: 3 (million)
% 14.92/9.06 % (31465)------------------------------
% 14.92/9.06 % (31465)------------------------------
% 14.92/9.10 % (31454)Instruction limit reached!
% 14.92/9.10 % (31454)------------------------------
% 14.92/9.10 % (31454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.92/9.10 % (31454)Termination reason: Unknown
% 14.92/9.10 % (31454)Termination phase: Property scanning
% 14.92/9.10
% 14.92/9.10 % (31454)Memory used [KB]: 23027
% 14.92/9.10 % (31454)Time elapsed: 0.381 s
% 14.92/9.10 % (31454)Instructions burned: 711 (million)
% 14.92/9.10 % (31454)------------------------------
% 14.92/9.10 % (31454)------------------------------
% 14.92/9.11 % (31468)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_31297 for (2989ds/100Mi)
% 14.92/9.11 % (31467)Instruction limit reached!
% 14.92/9.11 % (31467)------------------------------
% 14.92/9.11 % (31467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.92/9.11 % (31467)Termination reason: Unknown
% 14.92/9.11 % (31467)Termination phase: shuffling
% 14.92/9.11
% 14.92/9.11 % (31467)Memory used [KB]: 13688
% 14.92/9.11 % (31467)Time elapsed: 0.074 s
% 14.92/9.11 % (31467)Instructions burned: 128 (million)
% 14.92/9.11 % (31467)------------------------------
% 14.92/9.11 % (31467)------------------------------
% 14.92/9.12 % (31469)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_31297 for (2989ds/3Mi)
% 14.92/9.12 % (31469)Instruction limit reached!
% 14.92/9.12 % (31469)------------------------------
% 14.92/9.12 % (31469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.92/9.12 % (31469)Termination reason: Unknown
% 14.92/9.12 % (31469)Termination phase: shuffling
% 14.92/9.12
% 14.92/9.12 % (31469)Memory used [KB]: 10490
% 14.92/9.12 % (31469)Time elapsed: 0.006 s
% 14.92/9.12 % (31469)Instructions burned: 3 (million)
% 14.92/9.12 % (31469)------------------------------
% 14.92/9.12 % (31469)------------------------------
% 15.40/9.14 % (31470)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_31297 for (2989ds/20Mi)
% 15.40/9.15 % (31470)Instruction limit reached!
% 15.40/9.15 % (31470)------------------------------
% 15.40/9.15 % (31470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.40/9.15 % (31470)Termination reason: Unknown
% 15.40/9.15 % (31470)Termination phase: shuffling
% 15.40/9.15
% 15.40/9.15 % (31470)Memory used [KB]: 11001
% 15.40/9.16 % (31470)Time elapsed: 0.018 s
% 15.40/9.16 % (31470)Instructions burned: 21 (million)
% 15.40/9.16 % (31470)------------------------------
% 15.40/9.16 % (31470)------------------------------
% 15.40/9.16 % (31445)Instruction limit reached!
% 15.40/9.16 % (31445)------------------------------
% 15.40/9.16 % (31445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.40/9.16 % (31445)Termination reason: Unknown
% 15.40/9.16 % (31445)Termination phase: shuffling
% 15.40/9.16
% 15.40/9.16 % (31445)Memory used [KB]: 32494
% 15.40/9.16 % (31445)Time elapsed: 0.518 s
% 15.40/9.16 % (31445)Instructions burned: 1042 (million)
% 15.40/9.16 % (31445)------------------------------
% 15.40/9.16 % (31445)------------------------------
% 15.40/9.19 % (31472)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_31297 for (2988ds/107Mi)
% 15.62/9.21 % (31471)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_31297 for (2988ds/86Mi)
% 15.62/9.24 % (31473)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_31297 for (2988ds/39Mi)
% 15.62/9.28 % (31468)Instruction limit reached!
% 15.62/9.28 % (31468)------------------------------
% 15.62/9.28 % (31468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.62/9.28 % (31468)Termination reason: Unknown
% 15.62/9.28 % (31468)Termination phase: shuffling
% 15.62/9.28
% 15.62/9.28 % (31468)Memory used [KB]: 13176
% 15.62/9.28 % (31468)Time elapsed: 0.073 s
% 15.62/9.28 % (31468)Instructions burned: 101 (million)
% 15.62/9.28 % (31468)------------------------------
% 15.62/9.28 % (31468)------------------------------
% 15.87/9.33 % (31471)Instruction limit reached!
% 15.87/9.33 % (31471)------------------------------
% 15.87/9.33 % (31471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.87/9.33 % (31471)Termination reason: Unknown
% 15.87/9.33 % (31471)Termination phase: shuffling
% 15.87/9.33
% 15.87/9.33 % (31471)Memory used [KB]: 13048
% 15.87/9.33 % (31471)Time elapsed: 0.058 s
% 15.87/9.33 % (31471)Instructions burned: 86 (million)
% 15.87/9.33 % (31471)------------------------------
% 15.87/9.33 % (31471)------------------------------
% 15.87/9.33 % (31473)Instruction limit reached!
% 15.87/9.33 % (31473)------------------------------
% 15.87/9.33 % (31473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.87/9.33 % (31473)Termination reason: Unknown
% 15.87/9.33 % (31473)Termination phase: shuffling
% 15.87/9.33
% 15.87/9.33 % (31473)Memory used [KB]: 12153
% 15.87/9.33 % (31473)Time elapsed: 0.032 s
% 15.87/9.33 % (31473)Instructions burned: 39 (million)
% 15.87/9.33 % (31473)------------------------------
% 15.87/9.33 % (31473)------------------------------
% 15.87/9.36 % (31474)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_31297 for (2987ds/448Mi)
% 15.87/9.38 % (31461)Instruction limit reached!
% 15.87/9.38 % (31461)------------------------------
% 15.87/9.38 % (31461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.87/9.38 % (31461)Termination reason: Unknown
% 15.87/9.38 % (31461)Termination phase: shuffling
% 15.87/9.38
% 15.87/9.38 % (31461)Memory used [KB]: 24690
% 15.87/9.38 % (31461)Time elapsed: 0.397 s
% 15.87/9.38 % (31461)Instructions burned: 780 (million)
% 15.87/9.38 % (31461)------------------------------
% 15.87/9.38 % (31461)------------------------------
% 15.87/9.39 % (31475)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_31297 for (2986ds/46Mi)
% 15.87/9.40 % (31476)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_31297 for (2986ds/98Mi)
% 16.16/9.41 % (31472)Instruction limit reached!
% 16.16/9.41 % (31472)------------------------------
% 16.16/9.41 % (31472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.16/9.41 % (31472)Termination reason: Unknown
% 16.16/9.41 % (31472)Termination phase: shuffling
% 16.16/9.41
% 16.16/9.41 % (31472)Memory used [KB]: 13432
% 16.16/9.41 % (31472)Time elapsed: 0.076 s
% 16.16/9.41 % (31472)Instructions burned: 108 (million)
% 16.16/9.41 % (31472)------------------------------
% 16.16/9.41 % (31472)------------------------------
% 16.16/9.41 % (31477)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_31297 for (2986ds/507Mi)
% 16.16/9.48 % (31478)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_31297 for (2986ds/149Mi)
% 16.16/9.49 % (31475)Instruction limit reached!
% 16.16/9.49 % (31475)------------------------------
% 16.16/9.49 % (31475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.16/9.49 % (31475)Termination reason: Unknown
% 16.16/9.49 % (31475)Termination phase: shuffling
% 16.16/9.49
% 16.16/9.49 % (31475)Memory used [KB]: 11385
% 16.16/9.49 % (31475)Time elapsed: 0.036 s
% 16.16/9.49 % (31475)Instructions burned: 47 (million)
% 16.16/9.49 % (31475)------------------------------
% 16.16/9.49 % (31475)------------------------------
% 16.36/9.54 % (31476)Instruction limit reached!
% 16.36/9.54 % (31476)------------------------------
% 16.36/9.54 % (31476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.36/9.54 % (31476)Termination reason: Unknown
% 16.36/9.54 % (31476)Termination phase: shuffling
% 16.36/9.54
% 16.36/9.54 % (31476)Memory used [KB]: 13176
% 16.36/9.54 % (31476)Time elapsed: 0.060 s
% 16.36/9.54 % (31476)Instructions burned: 99 (million)
% 16.36/9.54 % (31476)------------------------------
% 16.36/9.54 % (31476)------------------------------
% 16.36/9.58 % (31480)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_31297 for (2984ds/6Mi)
% 16.36/9.59 % (31479)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_31297 for (2985ds/39Mi)
% 16.36/9.59 % (31480)Instruction limit reached!
% 16.36/9.59 % (31480)------------------------------
% 16.36/9.59 % (31480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.36/9.59 % (31480)Termination reason: Unknown
% 16.36/9.59 % (31480)Termination phase: shuffling
% 16.36/9.59
% 16.36/9.59 % (31480)Memory used [KB]: 10618
% 16.36/9.59 % (31480)Time elapsed: 0.010 s
% 16.36/9.59 % (31480)Instructions burned: 6 (million)
% 16.36/9.59 % (31480)------------------------------
% 16.36/9.59 % (31480)------------------------------
% 16.53/9.64 % (31481)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_31297 for (2984ds/1087Mi)
% 16.53/9.68 % (31477)Instruction limit reached!
% 16.53/9.68 % (31477)------------------------------
% 16.53/9.68 % (31477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.53/9.68 % (31477)Termination reason: Unknown
% 16.53/9.68 % (31477)Termination phase: shuffling
% 16.53/9.68
% 16.53/9.68 % (31477)Memory used [KB]: 19445
% 16.53/9.68 % (31477)Time elapsed: 0.272 s
% 16.53/9.68 % (31477)Instructions burned: 507 (million)
% 16.53/9.68 % (31477)------------------------------
% 16.53/9.68 % (31477)------------------------------
% 16.93/9.70 % (31479)Instruction limit reached!
% 16.93/9.70 % (31479)------------------------------
% 16.93/9.70 % (31479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.93/9.70 % (31479)Termination reason: Unknown
% 16.93/9.70 % (31479)Termination phase: shuffling
% 16.93/9.70
% 16.93/9.70 % (31479)Memory used [KB]: 11257
% 16.93/9.70 % (31479)Time elapsed: 0.032 s
% 16.93/9.70 % (31479)Instructions burned: 39 (million)
% 16.93/9.70 % (31479)------------------------------
% 16.93/9.70 % (31479)------------------------------
% 16.93/9.72 % (31482)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_31297 for (2983ds/47Mi)
% 16.93/9.72 % (31483)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_31297 for (2983ds/44Mi)
% 16.93/9.74 % (31483)Instruction limit reached!
% 16.93/9.74 % (31483)------------------------------
% 16.93/9.74 % (31483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.93/9.74 % (31483)Termination reason: Unknown
% 16.93/9.74 % (31483)Termination phase: shuffling
% 16.93/9.74
% 16.93/9.74 % (31483)Memory used [KB]: 12281
% 16.93/9.74 % (31483)Time elapsed: 0.029 s
% 16.93/9.74 % (31483)Instructions burned: 44 (million)
% 16.93/9.74 % (31483)------------------------------
% 16.93/9.74 % (31483)------------------------------
% 16.93/9.75 % (31482)Instruction limit reached!
% 16.93/9.75 % (31482)------------------------------
% 16.93/9.75 % (31482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.93/9.75 % (31482)Termination reason: Unknown
% 16.93/9.75 % (31482)Termination phase: shuffling
% 16.93/9.75
% 16.93/9.75 % (31482)Memory used [KB]: 11385
% 16.93/9.75 % (31482)Time elapsed: 0.032 s
% 16.93/9.75 % (31482)Instructions burned: 48 (million)
% 16.93/9.75 % (31482)------------------------------
% 16.93/9.75 % (31482)------------------------------
% 16.93/9.76 % (31484)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_31297 for (2982ds/432Mi)
% 16.93/9.79 % (31478)Instruction limit reached!
% 16.93/9.79 % (31478)------------------------------
% 16.93/9.79 % (31478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.93/9.79 % (31478)Termination reason: Unknown
% 16.93/9.79 % (31478)Termination phase: shuffling
% 16.93/9.79
% 16.93/9.79 % (31478)Memory used [KB]: 14072
% 16.93/9.79 % (31478)Time elapsed: 0.102 s
% 16.93/9.79 % (31478)Instructions burned: 150 (million)
% 16.93/9.79 % (31478)------------------------------
% 16.93/9.79 % (31478)------------------------------
% 17.24/9.80 % (31485)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_31297 for (2982ds/62Mi)
% 17.24/9.85 % (31486)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_31297 for (2982ds/31Mi)
% 17.31/9.93 % (31486)Instruction limit reached!
% 17.31/9.93 % (31486)------------------------------
% 17.31/9.93 % (31486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.31/9.93 % (31486)Termination reason: Unknown
% 17.31/9.93 % (31486)Termination phase: shuffling
% 17.31/9.93
% 17.31/9.93 % (31486)Memory used [KB]: 11129
% 17.31/9.93 % (31486)Time elapsed: 0.027 s
% 17.31/9.93 % (31486)Instructions burned: 31 (million)
% 17.31/9.93 % (31486)------------------------------
% 17.31/9.93 % (31486)------------------------------
% 17.31/9.93 % (31485)Instruction limit reached!
% 17.31/9.93 % (31485)------------------------------
% 17.31/9.93 % (31485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.31/9.93 % (31485)Termination reason: Unknown
% 17.31/9.93 % (31485)Termination phase: shuffling
% 17.31/9.93
% 17.31/9.93 % (31485)Memory used [KB]: 12537
% 17.31/9.93 % (31485)Time elapsed: 0.046 s
% 17.31/9.93 % (31485)Instructions burned: 62 (million)
% 17.31/9.93 % (31485)------------------------------
% 17.31/9.93 % (31485)------------------------------
% 17.31/9.98 % (31484)Instruction limit reached!
% 17.31/9.98 % (31484)------------------------------
% 17.31/9.98 % (31484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.31/9.98 % (31484)Termination reason: Unknown
% 17.31/9.98 % (31484)Termination phase: shuffling
% 17.31/9.98
% 17.31/9.98 % (31484)Memory used [KB]: 19829
% 17.31/9.98 % (31484)Time elapsed: 0.222 s
% 17.31/9.98 % (31484)Instructions burned: 434 (million)
% 17.31/9.98 % (31484)------------------------------
% 17.31/9.98 % (31484)------------------------------
% 17.31/10.01 % (31487)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_31297 for (2980ds/96Mi)
% 17.31/10.02 % (31488)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_31297 for (2980ds/453Mi)
% 18.15/10.06 % (31489)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_31297 for (2980ds/71Mi)
% 18.15/10.06 % (31463)Instruction limit reached!
% 18.15/10.06 % (31463)------------------------------
% 18.15/10.06 % (31463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.15/10.06 % (31463)Termination reason: Unknown
% 18.15/10.06 % (31463)Termination phase: Property scanning
% 18.15/10.06
% 18.15/10.06 % (31463)Memory used [KB]: 22899
% 18.15/10.06 % (31463)Time elapsed: 0.525 s
% 18.15/10.06 % (31463)Instructions burned: 880 (million)
% 18.15/10.06 % (31463)------------------------------
% 18.15/10.06 % (31463)------------------------------
% 18.15/10.08 % (31490)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_31297 for (2979ds/83Mi)
% 18.15/10.08 % (31456)Instruction limit reached!
% 18.15/10.08 % (31456)------------------------------
% 18.15/10.08 % (31456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.15/10.08 % (31456)Termination reason: Unknown
% 18.15/10.08 % (31456)Termination phase: Property scanning
% 18.15/10.08
% 18.15/10.08 % (31456)Memory used [KB]: 23027
% 18.15/10.08 % (31456)Time elapsed: 0.541 s
% 18.15/10.08 % (31456)Instructions burned: 903 (million)
% 18.15/10.08 % (31456)------------------------------
% 18.15/10.08 % (31456)------------------------------
% 18.15/10.12 % (31474)Instruction limit reached!
% 18.15/10.12 % (31474)------------------------------
% 18.15/10.12 % (31474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.15/10.12 % (31474)Termination reason: Unknown
% 18.15/10.12 % (31474)Termination phase: shuffling
% 18.15/10.12
% 18.15/10.12 % (31474)Memory used [KB]: 20084
% 18.15/10.12 % (31474)Time elapsed: 0.265 s
% 18.15/10.12 % (31474)Instructions burned: 449 (million)
% 18.15/10.12 % (31474)------------------------------
% 18.15/10.12 % (31474)------------------------------
% 18.37/10.13 % (31490)Instruction limit reached!
% 18.37/10.13 % (31490)------------------------------
% 18.37/10.13 % (31490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.37/10.13 % (31490)Termination reason: Unknown
% 18.37/10.13 % (31490)Termination phase: shuffling
% 18.37/10.13
% 18.37/10.13 % (31490)Memory used [KB]: 13048
% 18.37/10.13 % (31490)Time elapsed: 0.051 s
% 18.37/10.13 % (31490)Instructions burned: 84 (million)
% 18.37/10.13 % (31490)------------------------------
% 18.37/10.13 % (31490)------------------------------
% 18.37/10.16 % (31491)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_31297 for (2979ds/11Mi)
% 18.37/10.17 % (31493)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31297 for (2978ds/6Mi)
% 18.37/10.17 % (31492)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_31297 for (2978ds/41Mi)
% 18.37/10.17 % (31489)Instruction limit reached!
% 18.37/10.17 % (31489)------------------------------
% 18.37/10.17 % (31489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.37/10.17 % (31489)Termination reason: Unknown
% 18.37/10.17 % (31489)Termination phase: shuffling
% 18.37/10.17
% 18.37/10.17 % (31489)Memory used [KB]: 11769
% 18.37/10.17 % (31489)Time elapsed: 0.052 s
% 18.37/10.17 % (31489)Instructions burned: 71 (million)
% 18.37/10.17 % (31489)------------------------------
% 18.37/10.17 % (31489)------------------------------
% 18.37/10.18 % (31493)Instruction limit reached!
% 18.37/10.18 % (31493)------------------------------
% 18.37/10.18 % (31493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.37/10.18 % (31493)Termination reason: Unknown
% 18.37/10.18 % (31493)Termination phase: shuffling
% 18.37/10.18
% 18.37/10.18 % (31493)Memory used [KB]: 10618
% 18.37/10.18 % (31493)Time elapsed: 0.017 s
% 18.37/10.18 % (31493)Instructions burned: 6 (million)
% 18.37/10.18 % (31493)------------------------------
% 18.37/10.18 % (31493)------------------------------
% 18.37/10.18 % (31491)Instruction limit reached!
% 18.37/10.18 % (31491)------------------------------
% 18.37/10.18 % (31491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.37/10.18 % (31491)Termination reason: Unknown
% 18.37/10.18 % (31491)Termination phase: shuffling
% 18.37/10.18
% 18.37/10.18 % (31491)Memory used [KB]: 10746
% 18.37/10.18 % (31491)Time elapsed: 0.014 s
% 18.37/10.18 % (31491)Instructions burned: 12 (million)
% 18.37/10.18 % (31491)------------------------------
% 18.37/10.18 % (31491)------------------------------
% 18.61/10.20 % (31487)Instruction limit reached!
% 18.61/10.20 % (31487)------------------------------
% 18.61/10.20 % (31487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.61/10.20 % (31487)Termination reason: Unknown
% 18.61/10.20 % (31487)Termination phase: shuffling
% 18.61/10.20
% 18.61/10.20 % (31487)Memory used [KB]: 13304
% 18.61/10.20 % (31487)Time elapsed: 0.067 s
% 18.61/10.20 % (31487)Instructions burned: 97 (million)
% 18.61/10.20 % (31487)------------------------------
% 18.61/10.20 % (31487)------------------------------
% 18.61/10.21 % (31494)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_31297 for (2978ds/150Mi)
% 18.61/10.22 % (31492)Instruction limit reached!
% 18.61/10.22 % (31492)------------------------------
% 18.61/10.22 % (31492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.61/10.22 % (31492)Termination reason: Unknown
% 18.61/10.22 % (31492)Termination phase: shuffling
% 18.61/10.22
% 18.61/10.22 % (31492)Memory used [KB]: 12281
% 18.61/10.22 % (31492)Time elapsed: 0.038 s
% 18.61/10.22 % (31492)Instructions burned: 41 (million)
% 18.61/10.22 % (31492)------------------------------
% 18.61/10.22 % (31492)------------------------------
% 18.61/10.23 % (31495)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_31297 for (2978ds/343Mi)
% 18.61/10.24 % (31496)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_31297 for (2978ds/109Mi)
% 18.61/10.25 % (31497)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_31297 for (2978ds/253Mi)
% 18.61/10.28 % (31498)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_31297 for (2977ds/140Mi)
% 18.94/10.42 % (31496)Instruction limit reached!
% 18.94/10.42 % (31496)------------------------------
% 18.94/10.42 % (31496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/10.42 % (31496)Termination reason: Unknown
% 18.94/10.42 % (31496)Termination phase: shuffling
% 18.94/10.42
% 18.94/10.42 % (31496)Memory used [KB]: 13432
% 18.94/10.42 % (31496)Time elapsed: 0.070 s
% 18.94/10.42 % (31496)Instructions burned: 110 (million)
% 18.94/10.42 % (31496)------------------------------
% 18.94/10.42 % (31496)------------------------------
% 18.94/10.48 % (31499)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_31297 for (2975ds/614Mi)
% 18.94/10.49 % (31494)Instruction limit reached!
% 18.94/10.49 % (31494)------------------------------
% 18.94/10.49 % (31494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.94/10.49 % (31494)Termination reason: Unknown
% 18.94/10.49 % (31494)Termination phase: shuffling
% 18.94/10.49
% 18.94/10.49 % (31494)Memory used [KB]: 13944
% 18.94/10.49 % (31494)Time elapsed: 0.105 s
% 18.94/10.49 % (31494)Instructions burned: 150 (million)
% 18.94/10.49 % (31494)------------------------------
% 18.94/10.49 % (31494)------------------------------
% 19.26/10.52 % (31498)Instruction limit reached!
% 19.26/10.52 % (31498)------------------------------
% 19.26/10.52 % (31498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.26/10.52 % (31498)Termination reason: Unknown
% 19.26/10.52 % (31498)Termination phase: shuffling
% 19.26/10.52
% 19.26/10.52 % (31498)Memory used [KB]: 13816
% 19.26/10.52 % (31498)Time elapsed: 0.088 s
% 19.26/10.52 % (31498)Instructions burned: 141 (million)
% 19.26/10.52 % (31498)------------------------------
% 19.26/10.52 % (31498)------------------------------
% 19.26/10.54 % (31481)Instruction limit reached!
% 19.26/10.54 % (31481)------------------------------
% 19.26/10.54 % (31481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.26/10.54 % (31481)Termination reason: Unknown
% 19.26/10.54 % (31481)Termination phase: Property scanning
% 19.26/10.54
% 19.26/10.54 % (31481)Memory used [KB]: 23027
% 19.26/10.54 % (31481)Time elapsed: 0.597 s
% 19.26/10.54 % (31481)Instructions burned: 1088 (million)
% 19.26/10.54 % (31481)------------------------------
% 19.26/10.54 % (31481)------------------------------
% 19.26/10.57 % (31500)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_31297 for (2975ds/202Mi)
% 19.26/10.57 % (31501)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_31297 for (2975ds/339Mi)
% 19.43/10.64 % (31502)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_31297 for (2974ds/283Mi)
% 19.46/10.76 % (31495)Instruction limit reached!
% 19.46/10.76 % (31495)------------------------------
% 19.46/10.76 % (31495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.46/10.76 % (31495)Termination reason: Unknown
% 19.46/10.76 % (31495)Termination phase: shuffling
% 19.46/10.76
% 19.46/10.76 % (31495)Memory used [KB]: 17142
% 19.46/10.76 % (31495)Time elapsed: 0.203 s
% 19.46/10.76 % (31495)Instructions burned: 344 (million)
% 19.46/10.76 % (31495)------------------------------
% 19.46/10.76 % (31495)------------------------------
% 19.46/10.76 % (31497)Instruction limit reached!
% 19.46/10.76 % (31497)------------------------------
% 19.46/10.76 % (31497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.46/10.76 % (31497)Termination reason: Unknown
% 19.46/10.76 % (31497)Termination phase: shuffling
% 19.46/10.76
% 19.46/10.76 % (31497)Memory used [KB]: 15735
% 19.46/10.76 % (31497)Time elapsed: 0.171 s
% 19.46/10.76 % (31497)Instructions burned: 253 (million)
% 19.46/10.76 % (31497)------------------------------
% 19.46/10.76 % (31497)------------------------------
% 19.46/10.77 % (31488)Instruction limit reached!
% 19.46/10.77 % (31488)------------------------------
% 19.46/10.77 % (31488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.46/10.77 % (31488)Termination reason: Unknown
% 19.46/10.77 % (31488)Termination phase: shuffling
% 19.46/10.77
% 19.46/10.77 % (31488)Memory used [KB]: 20212
% 19.46/10.77 % (31488)Time elapsed: 0.274 s
% 19.46/10.77 % (31488)Instructions burned: 454 (million)
% 19.46/10.77 % (31488)------------------------------
% 19.46/10.77 % (31488)------------------------------
% 19.86/10.83 % (31503)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_31297 for (2972ds/134Mi)
% 19.86/10.86 % (31504)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_31297 for (2972ds/298Mi)
% 19.86/10.87 % (31505)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_31297 for (2972ds/192Mi)
% 19.87/11.02 % (31500)Instruction limit reached!
% 19.87/11.02 % (31500)------------------------------
% 19.87/11.02 % (31500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.87/11.02 % (31500)Termination reason: Unknown
% 19.87/11.02 % (31500)Termination phase: shuffling
% 19.87/11.02
% 19.87/11.02 % (31500)Memory used [KB]: 14839
% 19.87/11.02 % (31500)Time elapsed: 0.141 s
% 19.87/11.02 % (31500)Instructions burned: 202 (million)
% 19.87/11.02 % (31500)------------------------------
% 19.87/11.02 % (31500)------------------------------
% 20.64/11.06 % (31503)Instruction limit reached!
% 20.64/11.06 % (31503)------------------------------
% 20.64/11.06 % (31503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.64/11.06 % (31503)Termination reason: Unknown
% 20.64/11.06 % (31503)Termination phase: shuffling
% 20.64/11.06
% 20.64/11.06 % (31503)Memory used [KB]: 13816
% 20.64/11.06 % (31503)Time elapsed: 0.085 s
% 20.64/11.06 % (31503)Instructions burned: 135 (million)
% 20.64/11.06 % (31503)------------------------------
% 20.64/11.06 % (31503)------------------------------
% 20.64/11.10 % (31506)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_31297 for (2969ds/258Mi)
% 20.64/11.12 % (31501)Instruction limit reached!
% 20.64/11.12 % (31501)------------------------------
% 20.64/11.12 % (31501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.64/11.12 % (31501)Termination reason: Unknown
% 20.64/11.12 % (31501)Termination phase: shuffling
% 20.64/11.12
% 20.64/11.12 % (31501)Memory used [KB]: 17014
% 20.64/11.12 % (31501)Time elapsed: 0.201 s
% 20.64/11.12 % (31501)Instructions burned: 339 (million)
% 20.64/11.12 % (31501)------------------------------
% 20.64/11.12 % (31501)------------------------------
% 20.63/11.16 % (31508)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_31297 for (2968ds/231Mi)
% 20.63/11.16 % (31507)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_31297 for (2969ds/90Mi)
% 20.63/11.19 % (31502)Instruction limit reached!
% 20.63/11.19 % (31502)------------------------------
% 20.63/11.19 % (31502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.63/11.19 % (31502)Termination reason: Unknown
% 20.63/11.19 % (31502)Termination phase: shuffling
% 20.63/11.19
% 20.63/11.19 % (31502)Memory used [KB]: 15735
% 20.63/11.19 % (31502)Time elapsed: 0.173 s
% 20.63/11.19 % (31502)Instructions burned: 283 (million)
% 20.63/11.19 % (31502)------------------------------
% 20.63/11.19 % (31502)------------------------------
% 20.92/11.24 % (31505)Instruction limit reached!
% 20.92/11.24 % (31505)------------------------------
% 20.92/11.24 % (31505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.92/11.24 % (31505)Termination reason: Unknown
% 20.92/11.24 % (31505)Termination phase: shuffling
% 20.92/11.24
% 20.92/11.24 % (31505)Memory used [KB]: 14711
% 20.92/11.24 % (31505)Time elapsed: 0.131 s
% 20.92/11.24 % (31505)Instructions burned: 192 (million)
% 20.92/11.24 % (31505)------------------------------
% 20.92/11.24 % (31505)------------------------------
% 20.92/11.26 % (31509)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_31297 for (2968ds/458Mi)
% 20.92/11.27 % (31508)Instruction limit reached!
% 20.92/11.27 % (31508)------------------------------
% 20.92/11.27 % (31508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.92/11.27 % (31508)Termination reason: Unknown
% 20.92/11.28 % (31508)Termination phase: shuffling
% 20.92/11.28
% 20.92/11.28 % (31508)Memory used [KB]: 14967
% 20.92/11.28 % (31508)Time elapsed: 0.119 s
% 20.92/11.28 % (31508)Instructions burned: 231 (million)
% 20.92/11.28 % (31508)------------------------------
% 20.92/11.28 % (31508)------------------------------
% 21.15/11.31 % (31510)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_31297 for (2967ds/31Mi)
% 21.15/11.31 % (31507)Instruction limit reached!
% 21.15/11.31 % (31507)------------------------------
% 21.15/11.31 % (31507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.15/11.31 % (31507)Termination reason: Unknown
% 21.15/11.31 % (31507)Termination phase: shuffling
% 21.15/11.31
% 21.15/11.31 % (31507)Memory used [KB]: 13176
% 21.15/11.31 % (31507)Time elapsed: 0.058 s
% 21.15/11.31 % (31507)Instructions burned: 90 (million)
% 21.15/11.31 % (31507)------------------------------
% 21.15/11.31 % (31507)------------------------------
% 21.15/11.35 % (31511)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_31297 for (2967ds/297Mi)
% 21.15/11.36 % (31504)Instruction limit reached!
% 21.15/11.36 % (31504)------------------------------
% 21.15/11.36 % (31504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.15/11.36 % (31504)Termination reason: Unknown
% 21.15/11.36 % (31504)Termination phase: shuffling
% 21.15/11.36
% 21.15/11.36 % (31504)Memory used [KB]: 15991
% 21.15/11.36 % (31504)Time elapsed: 0.179 s
% 21.15/11.36 % (31504)Instructions burned: 299 (million)
% 21.15/11.36 % (31504)------------------------------
% 21.15/11.36 % (31504)------------------------------
% 21.29/11.40 % (31512)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_31297 for (2967ds/987Mi)
% 21.29/11.40 % (31510)Instruction limit reached!
% 21.29/11.40 % (31510)------------------------------
% 21.29/11.40 % (31510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.29/11.40 % (31510)Termination reason: Unknown
% 21.29/11.40 % (31510)Termination phase: shuffling
% 21.29/11.40
% 21.29/11.40 % (31510)Memory used [KB]: 11129
% 21.29/11.40 % (31510)Time elapsed: 0.029 s
% 21.29/11.40 % (31510)Instructions burned: 32 (million)
% 21.29/11.40 % (31510)------------------------------
% 21.29/11.40 % (31510)------------------------------
% 21.29/11.42 % (31513)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_31297 for (2966ds/182Mi)
% 21.29/11.46 % (31499)Instruction limit reached!
% 21.29/11.46 % (31499)------------------------------
% 21.29/11.46 % (31499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.29/11.46 % (31499)Termination reason: Unknown
% 21.29/11.46 % (31499)Termination phase: shuffling
% 21.29/11.46
% 21.29/11.46 % (31499)Memory used [KB]: 21364
% 21.29/11.46 % (31499)Time elapsed: 0.360 s
% 21.29/11.46 % (31499)Instructions burned: 615 (million)
% 21.29/11.46 % (31499)------------------------------
% 21.29/11.46 % (31499)------------------------------
% 21.29/11.47 % (31514)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_31297 for (2966ds/182Mi)
% 21.37/11.51 % (31506)Instruction limit reached!
% 21.37/11.51 % (31506)------------------------------
% 21.37/11.51 % (31506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.37/11.51 % (31506)Termination reason: Unknown
% 21.37/11.51 % (31506)Termination phase: shuffling
% 21.37/11.51
% 21.37/11.51 % (31506)Memory used [KB]: 15735
% 21.37/11.51 % (31506)Time elapsed: 0.155 s
% 21.37/11.51 % (31506)Instructions burned: 259 (million)
% 21.37/11.51 % (31506)------------------------------
% 21.37/11.51 % (31506)------------------------------
% 21.37/11.52 % (31515)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_31297 for (2965ds/154Mi)
% 21.37/11.55 % (31516)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_31297 for (2965ds/79Mi)
% 21.37/11.56 % (31513)Instruction limit reached!
% 21.37/11.56 % (31513)------------------------------
% 21.37/11.56 % (31513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.37/11.56 % (31513)Termination reason: Unknown
% 21.37/11.56 % (31513)Termination phase: shuffling
% 21.37/11.56
% 21.37/11.56 % (31513)Memory used [KB]: 14583
% 21.37/11.56 % (31513)Time elapsed: 0.112 s
% 21.37/11.56 % (31513)Instructions burned: 183 (million)
% 21.37/11.56 % (31513)------------------------------
% 21.37/11.56 % (31513)------------------------------
% 21.65/11.62 % (31517)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_31297 for (2964ds/216Mi)
% 21.65/11.63 % (31516)Instruction limit reached!
% 21.65/11.63 % (31516)------------------------------
% 21.65/11.63 % (31516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.65/11.63 % (31516)Termination reason: Unknown
% 21.65/11.63 % (31516)Termination phase: shuffling
% 21.65/11.63
% 21.65/11.63 % (31516)Memory used [KB]: 12920
% 21.65/11.63 % (31516)Time elapsed: 0.050 s
% 21.65/11.63 % (31516)Instructions burned: 80 (million)
% 21.65/11.63 % (31516)------------------------------
% 21.65/11.63 % (31516)------------------------------
% 21.65/11.67 % (31511)Instruction limit reached!
% 21.65/11.67 % (31511)------------------------------
% 21.65/11.67 % (31511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.65/11.67 % (31511)Termination reason: Unknown
% 21.65/11.67 % (31511)Termination phase: shuffling
% 21.65/11.67
% 21.65/11.67 % (31511)Memory used [KB]: 15991
% 21.65/11.67 % (31511)Time elapsed: 0.160 s
% 21.65/11.67 % (31511)Instructions burned: 299 (million)
% 21.65/11.67 % (31511)------------------------------
% 21.65/11.67 % (31511)------------------------------
% 21.65/11.68 % (31518)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_31297 for (2963ds/114Mi)
% 21.65/11.68 % (31515)Instruction limit reached!
% 21.65/11.68 % (31515)------------------------------
% 21.65/11.68 % (31515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.65/11.68 % (31515)Termination reason: Unknown
% 21.65/11.68 % (31515)Termination phase: shuffling
% 21.65/11.68
% 21.65/11.68 % (31515)Memory used [KB]: 13944
% 21.65/11.68 % (31515)Time elapsed: 0.098 s
% 21.65/11.68 % (31515)Instructions burned: 154 (million)
% 21.65/11.68 % (31515)------------------------------
% 21.65/11.68 % (31515)------------------------------
% 21.99/11.73 % (31520)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_31297 for (2963ds/182Mi)
% 21.99/11.73 % (31519)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_31297 for (2963ds/922Mi)
% 21.99/11.79 % (31518)Instruction limit reached!
% 21.99/11.79 % (31518)------------------------------
% 21.99/11.79 % (31518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/11.79 % (31518)Termination reason: Unknown
% 21.99/11.79 % (31518)Termination phase: shuffling
% 21.99/11.79
% 21.99/11.79 % (31518)Memory used [KB]: 13560
% 21.99/11.79 % (31518)Time elapsed: 0.069 s
% 21.99/11.79 % (31518)Instructions burned: 115 (million)
% 21.99/11.79 % (31518)------------------------------
% 21.99/11.79 % (31518)------------------------------
% 22.07/11.80 % (31514)Instruction limit reached!
% 22.07/11.80 % (31514)------------------------------
% 22.07/11.80 % (31514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.07/11.80 % (31514)Termination reason: Unknown
% 22.07/11.80 % (31514)Termination phase: shuffling
% 22.07/11.80
% 22.07/11.80 % (31514)Memory used [KB]: 14583
% 22.07/11.80 % (31514)Time elapsed: 0.124 s
% 22.07/11.80 % (31514)Instructions burned: 182 (million)
% 22.07/11.80 % (31514)------------------------------
% 22.07/11.80 % (31514)------------------------------
% 22.07/11.85 % (31521)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_31297 for (2962ds/164Mi)
% 22.07/11.85 % (31517)Instruction limit reached!
% 22.07/11.85 % (31517)------------------------------
% 22.07/11.85 % (31517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.07/11.85 % (31517)Termination reason: Unknown
% 22.07/11.85 % (31517)Termination phase: shuffling
% 22.07/11.85
% 22.07/11.85 % (31517)Memory used [KB]: 14711
% 22.07/11.85 % (31517)Time elapsed: 0.136 s
% 22.07/11.85 % (31517)Instructions burned: 216 (million)
% 22.07/11.85 % (31517)------------------------------
% 22.07/11.85 % (31517)------------------------------
% 22.07/11.86 % (31522)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_31297 for (2962ds/50Mi)
% 22.34/11.91 % (31523)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_31297 for (2961ds/368Mi)
% 22.34/11.93 % (31520)Instruction limit reached!
% 22.34/11.93 % (31520)------------------------------
% 22.34/11.93 % (31520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.34/11.93 % (31520)Termination reason: Unknown
% 22.34/11.93 % (31520)Termination phase: shuffling
% 22.34/11.93
% 22.34/11.93 % (31520)Memory used [KB]: 14328
% 22.34/11.93 % (31520)Time elapsed: 0.100 s
% 22.34/11.93 % (31520)Instructions burned: 183 (million)
% 22.34/11.93 % (31520)------------------------------
% 22.34/11.93 % (31520)------------------------------
% 22.34/11.94 % (31522)Instruction limit reached!
% 22.34/11.94 % (31522)------------------------------
% 22.34/11.94 % (31522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.34/11.94 % (31522)Termination reason: Unknown
% 22.34/11.94 % (31522)Termination phase: shuffling
% 22.34/11.94
% 22.34/11.94 % (31522)Memory used [KB]: 12409
% 22.34/11.94 % (31522)Time elapsed: 0.033 s
% 22.34/11.94 % (31522)Instructions burned: 50 (million)
% 22.34/11.94 % (31522)------------------------------
% 22.34/11.94 % (31522)------------------------------
% 22.34/11.95 % (31509)Instruction limit reached!
% 22.34/11.95 % (31509)------------------------------
% 22.34/11.95 % (31509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.34/11.95 % (31509)Termination reason: Unknown
% 22.34/11.95 % (31509)Termination phase: shuffling
% 22.34/11.95
% 22.34/11.95 % (31509)Memory used [KB]: 18933
% 22.34/11.95 % (31509)Time elapsed: 0.291 s
% 22.34/11.95 % (31509)Instructions burned: 458 (million)
% 22.34/11.95 % (31509)------------------------------
% 22.34/11.95 % (31509)------------------------------
% 22.34/11.98 % (31525)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_31297 for (2960ds/267Mi)
% 22.34/12.00 % (31524)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_31297 for (2960ds/1021Mi)
% 22.34/12.03 % (31526)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_31297 for (2960ds/204Mi)
% 23.46/12.07 % (31521)Instruction limit reached!
% 23.46/12.07 % (31521)------------------------------
% 23.46/12.07 % (31521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.46/12.07 % (31521)Termination reason: Unknown
% 23.46/12.07 % (31521)Termination phase: shuffling
% 23.46/12.07
% 23.46/12.07 % (31521)Memory used [KB]: 14200
% 23.46/12.07 % (31521)Time elapsed: 0.093 s
% 23.46/12.07 % (31521)Instructions burned: 165 (million)
% 23.46/12.07 % (31521)------------------------------
% 23.46/12.07 % (31521)------------------------------
% 23.46/12.13 % (31527)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_31297 for (2959ds/26Mi)
% 23.83/12.20 % (31527)Instruction limit reached!
% 23.83/12.20 % (31527)------------------------------
% 23.83/12.20 % (31527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.83/12.20 % (31527)Termination reason: Unknown
% 23.83/12.20 % (31527)Termination phase: shuffling
% 23.83/12.20
% 23.83/12.20 % (31527)Memory used [KB]: 11001
% 23.83/12.20 % (31527)Time elapsed: 0.023 s
% 23.83/12.20 % (31527)Instructions burned: 26 (million)
% 23.83/12.20 % (31527)------------------------------
% 23.83/12.20 % (31527)------------------------------
% 23.83/12.26 % (31528)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_31297 for (2958ds/53Mi)
% 23.83/12.28 % (31526)Instruction limit reached!
% 23.83/12.28 % (31526)------------------------------
% 23.83/12.28 % (31526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.83/12.28 % (31526)Termination reason: Unknown
% 23.83/12.28 % (31526)Termination phase: shuffling
% 23.83/12.28
% 23.83/12.28 % (31526)Memory used [KB]: 14839
% 23.83/12.28 % (31526)Time elapsed: 0.120 s
% 23.83/12.28 % (31526)Instructions burned: 204 (million)
% 23.83/12.28 % (31526)------------------------------
% 23.83/12.28 % (31526)------------------------------
% 24.15/12.30 % (31525)Instruction limit reached!
% 24.15/12.30 % (31525)------------------------------
% 24.15/12.30 % (31525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.15/12.30 % (31525)Termination reason: Unknown
% 24.15/12.30 % (31525)Termination phase: shuffling
% 24.15/12.30
% 24.15/12.30 % (31525)Memory used [KB]: 15479
% 24.15/12.30 % (31525)Time elapsed: 0.144 s
% 24.15/12.30 % (31525)Instructions burned: 267 (million)
% 24.15/12.30 % (31525)------------------------------
% 24.15/12.30 % (31525)------------------------------
% 24.15/12.33 % (31528)Instruction limit reached!
% 24.15/12.33 % (31528)------------------------------
% 24.15/12.33 % (31528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.15/12.33 % (31528)Termination reason: Unknown
% 24.15/12.33 % (31528)Termination phase: shuffling
% 24.15/12.33
% 24.15/12.33 % (31528)Memory used [KB]: 12537
% 24.15/12.33 % (31528)Time elapsed: 0.035 s
% 24.15/12.33 % (31528)Instructions burned: 54 (million)
% 24.15/12.33 % (31528)------------------------------
% 24.15/12.33 % (31528)------------------------------
% 24.15/12.34 % (31530)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_31297 for (2957ds/1539Mi)
% 24.15/12.35 % (31529)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_31297 for (2957ds/4436Mi)
% 24.15/12.36 % (31523)Instruction limit reached!
% 24.15/12.36 % (31523)------------------------------
% 24.15/12.36 % (31523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.15/12.36 % (31523)Termination reason: Unknown
% 24.15/12.36 % (31523)Termination phase: shuffling
% 24.15/12.36
% 24.15/12.36 % (31523)Memory used [KB]: 17526
% 24.15/12.36 % (31523)Time elapsed: 0.224 s
% 24.15/12.36 % (31523)Instructions burned: 368 (million)
% 24.15/12.36 % (31523)------------------------------
% 24.15/12.36 % (31523)------------------------------
% 24.15/12.37 % (31531)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_31297 for (2956ds/408Mi)
% 24.39/12.42 % (31532)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_31297 for (2956ds/5205Mi)
% 24.54/12.59 % (31512)Instruction limit reached!
% 24.54/12.59 % (31512)------------------------------
% 24.54/12.59 % (31512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.54/12.59 % (31512)Termination reason: Unknown
% 24.54/12.59 % (31512)Termination phase: shuffling
% 24.54/12.59
% 24.54/12.59 % (31512)Memory used [KB]: 31598
% 24.54/12.59 % (31512)Time elapsed: 0.556 s
% 24.54/12.59 % (31512)Instructions burned: 987 (million)
% 24.54/12.59 % (31512)------------------------------
% 24.54/12.59 % (31512)------------------------------
% 24.17/12.64 % (31533)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_31297 for (2954ds/1913Mi)
% 24.79/12.75 % (31519)Instruction limit reached!
% 24.79/12.75 % (31519)------------------------------
% 24.79/12.75 % (31519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.79/12.75 % (31519)Termination reason: Unknown
% 24.79/12.75 % (31519)Termination phase: Property scanning
% 24.79/12.75
% 24.79/12.75 % (31519)Memory used [KB]: 23027
% 24.79/12.75 % (31519)Time elapsed: 0.532 s
% 24.79/12.75 % (31519)Instructions burned: 922 (million)
% 24.79/12.75 % (31519)------------------------------
% 24.79/12.75 % (31519)------------------------------
% 24.90/12.81 % (31534)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_31297 for (2952ds/4350Mi)
% 24.95/13.02 % (31531)Instruction limit reached!
% 24.95/13.02 % (31531)------------------------------
% 24.95/13.02 % (31531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.95/13.02 % (31531)Termination reason: Unknown
% 24.95/13.02 % (31531)Termination phase: shuffling
% 24.95/13.02
% 24.95/13.02 % (31531)Memory used [KB]: 18293
% 24.95/13.02 % (31531)Time elapsed: 0.243 s
% 24.95/13.02 % (31531)Instructions burned: 408 (million)
% 24.95/13.02 % (31531)------------------------------
% 24.95/13.02 % (31531)------------------------------
% 26.44/13.09 % (31535)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_31297 for (2950ds/68Mi)
% 26.44/13.15 % (31524)Instruction limit reached!
% 26.44/13.15 % (31524)------------------------------
% 26.44/13.15 % (31524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.44/13.15 % (31524)Termination reason: Unknown
% 26.44/13.15 % (31524)Termination phase: Property scanning
% 26.44/13.15
% 26.44/13.15 % (31524)Memory used [KB]: 23027
% 26.44/13.15 % (31524)Time elapsed: 0.591 s
% 26.44/13.15 % (31524)Instructions burned: 1022 (million)
% 26.44/13.15 % (31524)------------------------------
% 26.44/13.15 % (31524)------------------------------
% 26.78/13.20 % (31535)Instruction limit reached!
% 26.78/13.20 % (31535)------------------------------
% 26.78/13.20 % (31535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.78/13.20 % (31535)Termination reason: Unknown
% 26.78/13.20 % (31535)Termination phase: shuffling
% 26.78/13.20
% 26.78/13.20 % (31535)Memory used [KB]: 11769
% 26.78/13.20 % (31535)Time elapsed: 0.047 s
% 26.78/13.20 % (31535)Instructions burned: 68 (million)
% 26.78/13.20 % (31535)------------------------------
% 26.78/13.20 % (31535)------------------------------
% 26.78/13.21 % (31536)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_31297 for (2948ds/358Mi)
% 26.78/13.24 % (31537)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_31297 for (2948ds/31Mi)
% 26.78/13.28 % (31537)Instruction limit reached!
% 26.78/13.28 % (31537)------------------------------
% 26.78/13.28 % (31537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.78/13.28 % (31537)Termination reason: Unknown
% 26.78/13.28 % (31537)Termination phase: shuffling
% 26.78/13.28
% 26.78/13.28 % (31537)Memory used [KB]: 11129
% 26.78/13.28 % (31537)Time elapsed: 0.024 s
% 26.78/13.28 % (31537)Instructions burned: 31 (million)
% 26.78/13.28 % (31537)------------------------------
% 26.78/13.28 % (31537)------------------------------
% 27.11/13.33 % (31538)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_31297 for (2947ds/279Mi)
% 27.59/13.51 % (31536)Instruction limit reached!
% 27.59/13.51 % (31536)------------------------------
% 27.59/13.51 % (31536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.59/13.51 % (31536)Termination reason: Unknown
% 27.59/13.51 % (31536)Termination phase: shuffling
% 27.59/13.51
% 27.59/13.51 % (31536)Memory used [KB]: 17270
% 27.59/13.51 % (31536)Time elapsed: 0.199 s
% 27.59/13.51 % (31536)Instructions burned: 359 (million)
% 27.59/13.51 % (31536)------------------------------
% 27.59/13.51 % (31536)------------------------------
% 27.59/13.53 % (31539)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_31297 for (2945ds/258Mi)
% 28.06/13.61 % (31538)Instruction limit reached!
% 28.06/13.61 % (31538)------------------------------
% 28.06/13.61 % (31538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.06/13.61 % (31538)Termination reason: Unknown
% 28.06/13.61 % (31538)Termination phase: shuffling
% 28.06/13.61
% 28.06/13.61 % (31538)Memory used [KB]: 15607
% 28.06/13.61 % (31538)Time elapsed: 0.149 s
% 28.06/13.61 % (31538)Instructions burned: 279 (million)
% 28.06/13.61 % (31538)------------------------------
% 28.06/13.61 % (31538)------------------------------
% 28.06/13.67 % (31539)Instruction limit reached!
% 28.06/13.67 % (31539)------------------------------
% 28.06/13.67 % (31539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.06/13.67 % (31539)Termination reason: Unknown
% 28.06/13.67 % (31539)Termination phase: shuffling
% 28.06/13.67
% 28.06/13.67 % (31539)Memory used [KB]: 15735
% 28.06/13.67 % (31539)Time elapsed: 0.144 s
% 28.06/13.67 % (31539)Instructions burned: 258 (million)
% 28.06/13.67 % (31539)------------------------------
% 28.06/13.67 % (31539)------------------------------
% 28.06/13.67 % (31540)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_31297 for (2944ds/26Mi)
% 28.64/13.71 % (31541)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_31297 for (2943ds/995Mi)
% 28.64/13.71 % (31540)Instruction limit reached!
% 28.64/13.71 % (31540)------------------------------
% 28.64/13.71 % (31540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.64/13.71 % (31540)Termination reason: Unknown
% 28.64/13.71 % (31540)Termination phase: shuffling
% 28.64/13.71
% 28.64/13.71 % (31540)Memory used [KB]: 11001
% 28.64/13.71 % (31540)Time elapsed: 0.021 s
% 28.64/13.71 % (31540)Instructions burned: 27 (million)
% 28.64/13.71 % (31540)------------------------------
% 28.64/13.71 % (31540)------------------------------
% 28.64/13.76 % (31542)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_31297 for (2943ds/373Mi)
% 30.17/14.15 % (31542)Instruction limit reached!
% 30.17/14.15 % (31542)------------------------------
% 30.17/14.15 % (31542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.17/14.15 % (31542)Termination reason: Unknown
% 30.17/14.15 % (31542)Termination phase: shuffling
% 30.17/14.15
% 30.17/14.15 % (31542)Memory used [KB]: 17526
% 30.17/14.15 % (31542)Time elapsed: 0.217 s
% 30.17/14.15 % (31542)Instructions burned: 373 (million)
% 30.17/14.15 % (31542)------------------------------
% 30.17/14.15 % (31542)------------------------------
% 30.56/14.20 % (31543)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_31297 for (2938ds/253Mi)
% 30.62/14.24 % (31530)Instruction limit reached!
% 30.62/14.24 % (31530)------------------------------
% 30.62/14.24 % (31530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.62/14.24 % (31530)Termination reason: Unknown
% 30.62/14.24 % (31530)Termination phase: Preprocessing 3
% 30.62/14.24
% 30.62/14.24 % (31530)Memory used [KB]: 30447
% 30.62/14.24 % (31530)Time elapsed: 0.882 s
% 30.62/14.24 % (31530)Instructions burned: 1539 (million)
% 30.62/14.24 % (31530)------------------------------
% 30.62/14.24 % (31530)------------------------------
% 30.62/14.28 % (31544)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_31297 for (2937ds/832Mi)
% 31.26/14.45 % (31543)Instruction limit reached!
% 31.26/14.45 % (31543)------------------------------
% 31.26/14.45 % (31543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.26/14.45 % (31543)Termination reason: Unknown
% 31.26/14.45 % (31543)Termination phase: shuffling
% 31.26/14.45
% 31.26/14.45 % (31543)Memory used [KB]: 15351
% 31.26/14.45 % (31543)Time elapsed: 0.140 s
% 31.26/14.45 % (31543)Instructions burned: 255 (million)
% 31.26/14.45 % (31543)------------------------------
% 31.26/14.45 % (31543)------------------------------
% 31.26/14.49 % (31545)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_31297 for (2935ds/668Mi)
% 31.71/14.55 % (31533)Instruction limit reached!
% 31.71/14.55 % (31533)------------------------------
% 31.71/14.55 % (31533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.71/14.55 % (31533)Termination reason: Unknown
% 31.71/14.55 % (31533)Termination phase: Saturation
% 31.71/14.55
% 31.71/14.55 % (31533)Memory used [KB]: 25330
% 31.71/14.55 % (31533)Time elapsed: 1.950 s
% 31.71/14.55 % (31533)Instructions burned: 1913 (million)
% 31.71/14.55 % (31533)------------------------------
% 31.71/14.55 % (31533)------------------------------
% 32.02/14.62 % (31541)Instruction limit reached!
% 32.02/14.62 % (31541)------------------------------
% 32.02/14.62 % (31541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.02/14.62 % (31541)Termination reason: Unknown
% 32.02/14.62 % (31541)Termination phase: shuffling
% 32.02/14.62
% 32.02/14.62 % (31541)Memory used [KB]: 31726
% 32.02/14.62 % (31541)Time elapsed: 0.514 s
% 32.02/14.62 % (31541)Instructions burned: 995 (million)
% 32.02/14.62 % (31541)------------------------------
% 32.02/14.62 % (31541)------------------------------
% 32.02/14.64 % (31547)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_31297 for (2934ds/294Mi)
% 32.02/14.65 % (31546)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_31297 for (2934ds/40Mi)
% 32.22/14.73 % (31546)Instruction limit reached!
% 32.22/14.73 % (31546)------------------------------
% 32.22/14.73 % (31546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.22/14.73 % (31546)Termination reason: Unknown
% 32.22/14.73 % (31546)Termination phase: shuffling
% 32.22/14.73
% 32.22/14.73 % (31546)Memory used [KB]: 12281
% 32.22/14.73 % (31546)Time elapsed: 0.032 s
% 32.22/14.73 % (31546)Instructions burned: 41 (million)
% 32.22/14.73 % (31546)------------------------------
% 32.22/14.73 % (31546)------------------------------
% 32.22/14.78 % (31548)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_31297 for (2932ds/187Mi)
% 32.47/14.84 % (31547)Instruction limit reached!
% 32.47/14.84 % (31547)------------------------------
% 32.47/14.84 % (31547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.47/14.84 % (31547)Termination reason: Unknown
% 32.47/14.84 % (31547)Termination phase: shuffling
% 32.47/14.84
% 32.47/14.84 % (31547)Memory used [KB]: 15863
% 32.47/14.84 % (31547)Time elapsed: 0.152 s
% 32.47/14.84 % (31547)Instructions burned: 294 (million)
% 32.47/14.84 % (31547)------------------------------
% 32.47/14.84 % (31547)------------------------------
% 32.82/14.90 % (31549)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_31297 for (2931ds/371Mi)
% 32.82/14.91 % (31548)Instruction limit reached!
% 32.82/14.91 % (31548)------------------------------
% 32.82/14.91 % (31548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.82/14.91 % (31548)Termination reason: Unknown
% 32.82/14.91 % (31548)Termination phase: shuffling
% 32.82/14.91
% 32.82/14.91 % (31548)Memory used [KB]: 14583
% 32.82/14.91 % (31548)Time elapsed: 0.109 s
% 32.82/14.91 % (31548)Instructions burned: 188 (million)
% 32.82/14.91 % (31548)------------------------------
% 32.82/14.91 % (31548)------------------------------
% 32.82/14.93 % (31550)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_31297 for (2931ds/1827Mi)
% 34.07/15.11 % (31549)Instruction limit reached!
% 34.07/15.11 % (31549)------------------------------
% 34.07/15.11 % (31549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.07/15.11 % (31549)Termination reason: Unknown
% 34.07/15.11 % (31549)Termination phase: shuffling
% 34.07/15.11
% 34.07/15.11 % (31549)Memory used [KB]: 17526
% 34.07/15.11 % (31549)Time elapsed: 0.201 s
% 34.07/15.11 % (31549)Instructions burned: 371 (million)
% 34.07/15.11 % (31549)------------------------------
% 34.07/15.11 % (31549)------------------------------
% 34.57/15.15 % (31545)Instruction limit reached!
% 34.57/15.15 % (31545)------------------------------
% 34.57/15.15 % (31545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.57/15.15 % (31545)Termination reason: Unknown
% 34.57/15.15 % (31545)Termination phase: shuffling
% 34.57/15.15
% 34.57/15.15 % (31545)Memory used [KB]: 23155
% 34.57/15.15 % (31545)Time elapsed: 0.358 s
% 34.57/15.15 % (31545)Instructions burned: 668 (million)
% 34.57/15.15 % (31545)------------------------------
% 34.57/15.15 % (31545)------------------------------
% 34.57/15.16 % (31544)Instruction limit reached!
% 34.57/15.16 % (31544)------------------------------
% 34.57/15.16 % (31544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.57/15.16 % (31544)Termination reason: Unknown
% 34.57/15.16 % (31544)Termination phase: Property scanning
% 34.57/15.16
% 34.57/15.16 % (31544)Memory used [KB]: 23027
% 34.57/15.16 % (31544)Time elapsed: 0.478 s
% 34.57/15.16 % (31544)Instructions burned: 832 (million)
% 34.57/15.16 % (31544)------------------------------
% 34.57/15.16 % (31544)------------------------------
% 34.57/15.18 % (31553)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_31297 for (2928ds/932Mi)
% 34.57/15.18 % (31551)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_31297 for (2929ds/70Mi)
% 34.57/15.20 % (31552)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_31297 for (2928ds/316Mi)
% 35.00/15.31 % (31551)Instruction limit reached!
% 35.00/15.31 % (31551)------------------------------
% 35.00/15.31 % (31551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.00/15.31 % (31551)Termination reason: Unknown
% 35.00/15.31 % (31551)Termination phase: shuffling
% 35.00/15.31
% 35.00/15.31 % (31551)Memory used [KB]: 12665
% 35.00/15.31 % (31551)Time elapsed: 0.046 s
% 35.00/15.31 % (31551)Instructions burned: 71 (million)
% 35.00/15.31 % (31551)------------------------------
% 35.00/15.31 % (31551)------------------------------
% 35.00/15.38 % (31554)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_31297 for (2927ds/4714Mi)
% 35.50/15.51 % (31552)Instruction limit reached!
% 35.50/15.51 % (31552)------------------------------
% 35.50/15.51 % (31552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.50/15.51 % (31552)Termination reason: Unknown
% 35.50/15.51 % (31552)Termination phase: shuffling
% 35.50/15.51
% 35.50/15.51 % (31552)Memory used [KB]: 18293
% 35.50/15.51 % (31552)Time elapsed: 0.173 s
% 35.50/15.51 % (31552)Instructions burned: 316 (million)
% 35.50/15.51 % (31552)------------------------------
% 35.50/15.51 % (31552)------------------------------
% 35.50/15.55 % (31555)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_31297 for (2925ds/8250Mi)
% 35.91/15.63 % (31553)Instruction limit reached!
% 35.91/15.63 % (31553)------------------------------
% 35.91/15.63 % (31553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.91/15.63 % (31553)Termination reason: Unknown
% 35.91/15.63 % (31553)Termination phase: shuffling
% 35.91/15.63
% 35.91/15.63 % (31553)Memory used [KB]: 26865
% 35.91/15.63 % (31553)Time elapsed: 0.454 s
% 35.91/15.63 % (31553)Instructions burned: 932 (million)
% 35.91/15.63 % (31553)------------------------------
% 35.91/15.63 % (31553)------------------------------
% 35.91/15.65 % (31556)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_31297 for (2923ds/770Mi)
% 36.81/15.86 % (31550)Instruction limit reached!
% 36.81/15.86 % (31550)------------------------------
% 36.81/15.86 % (31550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.81/15.86 % (31550)Termination reason: Unknown
% 36.81/15.86 % (31550)Termination phase: Property scanning
% 36.81/15.86
% 36.81/15.86 % (31550)Memory used [KB]: 31854
% 36.81/15.86 % (31550)Time elapsed: 0.934 s
% 36.81/15.86 % (31550)Instructions burned: 1829 (million)
% 36.81/15.86 % (31550)------------------------------
% 36.81/15.86 % (31550)------------------------------
% 36.81/15.88 % (31557)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_31297 for (2921ds/794Mi)
% 38.17/16.05 % (31556)Instruction limit reached!
% 38.17/16.05 % (31556)------------------------------
% 38.17/16.05 % (31556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.17/16.05 % (31556)Termination reason: Unknown
% 38.17/16.05 % (31556)Termination phase: shuffling
% 38.17/16.05
% 38.17/16.05 % (31556)Memory used [KB]: 23411
% 38.17/16.05 % (31556)Time elapsed: 0.404 s
% 38.17/16.05 % (31556)Instructions burned: 770 (million)
% 38.17/16.05 % (31556)------------------------------
% 38.17/16.05 % (31556)------------------------------
% 38.17/16.07 % (31558)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_31297 for (2919ds/464Mi)
% 38.17/16.12 % (31532)Instruction limit reached!
% 38.17/16.12 % (31532)------------------------------
% 38.17/16.12 % (31532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.17/16.12 % (31532)Termination reason: Unknown
% 38.17/16.12 % (31532)Termination phase: Saturation
% 38.17/16.12
% 38.17/16.12 % (31532)Memory used [KB]: 46054
% 38.17/16.12 % (31532)Time elapsed: 2.541 s
% 38.17/16.12 % (31532)Instructions burned: 5206 (million)
% 38.17/16.12 % (31532)------------------------------
% 38.17/16.12 % (31532)------------------------------
% 38.51/16.17 % (31559)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_31297 for (2919ds/1376Mi)
% 38.94/16.30 % (31557)Instruction limit reached!
% 38.94/16.30 % (31557)------------------------------
% 38.94/16.30 % (31557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.94/16.30 % (31557)Termination reason: Unknown
% 38.94/16.30 % (31557)Termination phase: Property scanning
% 38.94/16.30
% 38.94/16.30 % (31557)Memory used [KB]: 22899
% 38.94/16.30 % (31557)Time elapsed: 0.422 s
% 38.94/16.30 % (31557)Instructions burned: 795 (million)
% 38.94/16.30 % (31557)------------------------------
% 38.94/16.30 % (31557)------------------------------
% 38.94/16.32 % (31558)Instruction limit reached!
% 38.94/16.32 % (31558)------------------------------
% 38.94/16.32 % (31558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.94/16.32 % (31558)Termination reason: Unknown
% 38.94/16.32 % (31558)Termination phase: shuffling
% 38.94/16.32
% 38.94/16.32 % (31558)Memory used [KB]: 18933
% 38.94/16.32 % (31558)Time elapsed: 0.248 s
% 38.94/16.32 % (31558)Instructions burned: 464 (million)
% 38.94/16.32 % (31558)------------------------------
% 38.94/16.32 % (31558)------------------------------
% 38.94/16.32 % (31560)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_31297 for (2917ds/906Mi)
% 38.94/16.34 % (31561)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_31297 for (2917ds/434Mi)
% 39.52/16.57 % (31561)Instruction limit reached!
% 39.52/16.57 % (31561)------------------------------
% 39.52/16.57 % (31561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.52/16.57 % (31561)Termination reason: Unknown
% 39.52/16.57 % (31561)Termination phase: shuffling
% 39.52/16.57
% 39.52/16.57 % (31561)Memory used [KB]: 18677
% 39.52/16.57 % (31561)Time elapsed: 0.240 s
% 39.52/16.57 % (31561)Instructions burned: 435 (million)
% 39.52/16.57 % (31561)------------------------------
% 39.52/16.57 % (31561)------------------------------
% 39.52/16.59 % (31562)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_31297 for (2914ds/865Mi)
% 40.07/16.79 % (31560)Instruction limit reached!
% 40.07/16.79 % (31560)------------------------------
% 40.07/16.79 % (31560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.07/16.79 % (31560)Termination reason: Unknown
% 40.07/16.79 % (31560)Termination phase: Property scanning
% 40.07/16.79
% 40.07/16.79 % (31560)Memory used [KB]: 22899
% 40.07/16.79 % (31560)Time elapsed: 0.471 s
% 40.07/16.79 % (31560)Instructions burned: 907 (million)
% 40.07/16.79 % (31560)------------------------------
% 40.07/16.79 % (31560)------------------------------
% 40.70/16.81 % (31563)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_31297 for (2912ds/593Mi)
% 40.88/17.02 % (31562)Instruction limit reached!
% 40.88/17.02 % (31562)------------------------------
% 40.88/17.02 % (31562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.88/17.02 % (31562)Termination reason: Unknown
% 40.88/17.02 % (31562)Termination phase: shuffling
% 40.88/17.02
% 40.88/17.02 % (31562)Memory used [KB]: 25969
% 40.88/17.02 % (31562)Time elapsed: 0.430 s
% 40.88/17.02 % (31562)Instructions burned: 866 (million)
% 40.88/17.02 % (31562)------------------------------
% 40.88/17.02 % (31562)------------------------------
% 42.19/17.04 % (31564)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_31297 for (2910ds/443Mi)
% 42.19/17.10 % (31563)Instruction limit reached!
% 42.19/17.10 % (31563)------------------------------
% 42.19/17.10 % (31563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.19/17.10 % (31563)Termination reason: Unknown
% 42.19/17.10 % (31563)Termination phase: shuffling
% 42.19/17.10
% 42.19/17.10 % (31563)Memory used [KB]: 22131
% 42.19/17.10 % (31563)Time elapsed: 0.292 s
% 42.19/17.10 % (31563)Instructions burned: 594 (million)
% 42.19/17.10 % (31563)------------------------------
% 42.19/17.10 % (31563)------------------------------
% 42.19/17.12 % (31565)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_31297 for (2909ds/594Mi)
% 42.70/17.28 % (31564)Instruction limit reached!
% 42.70/17.28 % (31564)------------------------------
% 42.70/17.28 % (31564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.70/17.28 % (31564)Termination reason: Unknown
% 42.70/17.28 % (31564)Termination phase: shuffling
% 42.70/17.28
% 42.70/17.28 % (31564)Memory used [KB]: 18677
% 42.70/17.28 % (31564)Time elapsed: 0.245 s
% 42.70/17.28 % (31564)Instructions burned: 443 (million)
% 42.70/17.28 % (31564)------------------------------
% 42.70/17.28 % (31564)------------------------------
% 43.26/17.30 % (31566)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_31297 for (2907ds/327Mi)
% 43.26/17.34 % (31529)Instruction limit reached!
% 43.26/17.34 % (31529)------------------------------
% 43.26/17.34 % (31529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.26/17.34 % (31529)Termination reason: Unknown
% 43.26/17.34 % (31529)Termination phase: Saturation
% 43.26/17.34
% 43.26/17.34 % (31529)Memory used [KB]: 28400
% 43.26/17.34 % (31529)Time elapsed: 4.806 s
% 43.26/17.34 % (31529)Instructions burned: 4438 (million)
% 43.26/17.34 % (31529)------------------------------
% 43.26/17.34 % (31529)------------------------------
% 43.26/17.37 % (31567)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_31297 for (2906ds/5411Mi)
% 43.58/17.44 % (31565)Instruction limit reached!
% 43.58/17.44 % (31565)------------------------------
% 43.58/17.44 % (31565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.58/17.44 % (31565)Termination reason: Unknown
% 43.58/17.44 % (31565)Termination phase: shuffling
% 43.58/17.44
% 43.58/17.44 % (31565)Memory used [KB]: 20980
% 43.58/17.44 % (31565)Time elapsed: 0.322 s
% 43.58/17.44 % (31565)Instructions burned: 594 (million)
% 43.58/17.44 % (31565)------------------------------
% 43.58/17.44 % (31565)------------------------------
% 43.58/17.48 % (31568)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_31297 for (2905ds/1762Mi)
% 44.19/17.50 % (31559)Instruction limit reached!
% 44.19/17.50 % (31559)------------------------------
% 44.19/17.50 % (31559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.19/17.50 % (31559)Termination reason: Unknown
% 44.19/17.50 % (31559)Termination phase: Property scanning
% 44.19/17.50
% 44.19/17.50 % (31559)Memory used [KB]: 33133
% 44.19/17.50 % (31559)Time elapsed: 0.723 s
% 44.19/17.50 % (31559)Instructions burned: 1377 (million)
% 44.19/17.50 % (31559)------------------------------
% 44.19/17.50 % (31559)------------------------------
% 44.43/17.60 % (31569)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_31297 for (2904ds/96Mi)
% 44.43/17.63 % (31566)Instruction limit reached!
% 44.43/17.63 % (31566)------------------------------
% 44.43/17.63 % (31566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.43/17.63 % (31566)Termination reason: Unknown
% 44.43/17.63 % (31566)Termination phase: shuffling
% 44.43/17.63
% 44.43/17.63 % (31566)Memory used [KB]: 16886
% 44.43/17.63 % (31566)Time elapsed: 0.205 s
% 44.43/17.63 % (31566)Instructions burned: 327 (million)
% 44.43/17.63 % (31566)------------------------------
% 44.43/17.63 % (31566)------------------------------
% 44.43/17.67 % (31570)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_31297 for (2903ds/503Mi)
% 44.86/17.72 % (31569)Instruction limit reached!
% 44.86/17.72 % (31569)------------------------------
% 44.86/17.72 % (31569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.86/17.72 % (31569)Termination reason: Unknown
% 44.86/17.72 % (31569)Termination phase: shuffling
% 44.86/17.72
% 44.86/17.72 % (31569)Memory used [KB]: 13176
% 44.86/17.72 % (31569)Time elapsed: 0.059 s
% 44.86/17.72 % (31569)Instructions burned: 96 (million)
% 44.86/17.72 % (31569)------------------------------
% 44.86/17.72 % (31569)------------------------------
% 44.86/17.78 % (31571)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_31297 for (2902ds/90Mi)
% 45.16/17.88 % (31571)Instruction limit reached!
% 45.16/17.88 % (31571)------------------------------
% 45.16/17.88 % (31571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.16/17.88 % (31571)Termination reason: Unknown
% 45.16/17.88 % (31571)Termination phase: shuffling
% 45.16/17.88
% 45.16/17.88 % (31571)Memory used [KB]: 13176
% 45.16/17.88 % (31571)Time elapsed: 0.059 s
% 45.16/17.88 % (31571)Instructions burned: 91 (million)
% 45.16/17.88 % (31571)------------------------------
% 45.16/17.88 % (31571)------------------------------
% 45.44/17.92 % (31572)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_31297 for (2901ds/593Mi)
% 46.94/18.18 % (31534)Instruction limit reached!
% 46.94/18.18 % (31534)------------------------------
% 46.94/18.18 % (31534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.94/18.18 % (31534)Termination reason: Unknown
% 46.94/18.18 % (31534)Termination phase: Saturation
% 46.94/18.18
% 46.94/18.18 % (31534)Memory used [KB]: 40681
% 46.94/18.18 % (31534)Time elapsed: 5.403 s
% 46.94/18.18 % (31534)Instructions burned: 4351 (million)
% 46.94/18.18 % (31534)------------------------------
% 46.94/18.18 % (31534)------------------------------
% 47.22/18.23 % (31573)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_31297 for (2898ds/1512Mi)
% 47.22/18.24 % (31570)Instruction limit reached!
% 47.22/18.24 % (31570)------------------------------
% 47.22/18.24 % (31570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.22/18.24 % (31570)Termination reason: Unknown
% 47.22/18.24 % (31570)Termination phase: shuffling
% 47.22/18.24
% 47.22/18.24 % (31570)Memory used [KB]: 19701
% 47.22/18.24 % (31570)Time elapsed: 0.305 s
% 47.22/18.24 % (31570)Instructions burned: 503 (million)
% 47.22/18.24 % (31570)------------------------------
% 47.22/18.24 % (31570)------------------------------
% 47.22/18.28 % (31574)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_31297 for (2897ds/569Mi)
% 48.18/18.57 % (31572)Instruction limit reached!
% 48.18/18.57 % (31572)------------------------------
% 48.18/18.57 % (31572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.18/18.57 % (31572)Termination reason: Unknown
% 48.18/18.57 % (31572)Termination phase: shuffling
% 48.18/18.57
% 48.18/18.57 % (31572)Memory used [KB]: 20980
% 48.18/18.57 % (31572)Time elapsed: 0.348 s
% 48.18/18.57 % (31572)Instructions burned: 593 (million)
% 48.18/18.57 % (31572)------------------------------
% 48.18/18.57 % (31572)------------------------------
% 48.49/18.61 % (31575)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_31297 for (2894ds/501Mi)
% 48.49/18.67 % (31567)Refutation not found, incomplete strategy
% 48.49/18.67 % (31567)------------------------------
% 48.49/18.67 % (31567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.49/18.67 % (31567)Termination reason: Refutation not found, incomplete strategy
% 48.49/18.67
% 48.49/18.67
% 48.49/18.67 % (31567)Memory used [KB]: 27760
% 48.49/18.67 % (31567)Time elapsed: 1.330 s
% 48.49/18.67 % (31567)Instructions burned: 1217 (million)
% 48.49/18.67 % (31567)------------------------------
% 48.49/18.67 % (31567)------------------------------
% 48.73/18.73 % (31576)lrs+1010_1:3_cnfonf=conj_eager:fsr=off:prag=on:sd=4:sgt=10:ss=axioms:i=562:si=on:rtra=on_0 on DTF2THF_31297 for (2893ds/562Mi)
% 48.73/18.77 % (31574)Instruction limit reached!
% 48.73/18.77 % (31574)------------------------------
% 48.73/18.77 % (31574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.73/18.77 % (31574)Termination reason: Unknown
% 48.73/18.77 % (31574)Termination phase: shuffling
% 48.73/18.77
% 48.73/18.77 % (31574)Memory used [KB]: 21748
% 48.73/18.77 % (31574)Time elapsed: 0.306 s
% 48.73/18.77 % (31574)Instructions burned: 569 (million)
% 48.73/18.77 % (31574)------------------------------
% 48.73/18.77 % (31574)------------------------------
% 48.73/18.79 % (31577)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_31297 for (2892ds/406Mi)
% 49.40/19.01 % (31577)Instruction limit reached!
% 49.40/19.01 % (31577)------------------------------
% 49.40/19.01 % (31577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.40/19.01 % (31577)Termination reason: Unknown
% 49.40/19.01 % (31577)Termination phase: shuffling
% 49.40/19.01
% 49.40/19.01 % (31577)Memory used [KB]: 17910
% 49.40/19.01 % (31577)Time elapsed: 0.223 s
% 49.40/19.01 % (31577)Instructions burned: 407 (million)
% 49.40/19.01 % (31577)------------------------------
% 49.40/19.01 % (31577)------------------------------
% 50.57/19.03 % (31578)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_31297 for (2890ds/582Mi)
% 50.85/19.16 % (31575)Instruction limit reached!
% 50.85/19.16 % (31575)------------------------------
% 50.85/19.16 % (31575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.85/19.16 % (31575)Termination reason: Unknown
% 50.85/19.16 % (31575)Termination phase: shuffling
% 50.85/19.16
% 50.85/19.16 % (31575)Memory used [KB]: 19573
% 50.85/19.16 % (31575)Time elapsed: 0.293 s
% 50.85/19.16 % (31575)Instructions burned: 502 (million)
% 50.85/19.16 % (31575)------------------------------
% 50.85/19.16 % (31575)------------------------------
% 50.85/19.18 % (31579)lrs+2_1:1_au=on:bd=off:bsr=unit_only:sd=2:sgt=32:sos=all:ss=axioms:i=456:si=on:rtra=on_0 on DTF2THF_31297 for (2888ds/456Mi)
% 51.38/19.35 % (31578)Instruction limit reached!
% 51.38/19.35 % (31578)------------------------------
% 51.38/19.35 % (31578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.38/19.35 % (31578)Termination reason: Unknown
% 51.38/19.35 % (31578)Termination phase: shuffling
% 51.38/19.35
% 51.38/19.35 % (31578)Memory used [KB]: 21108
% 51.38/19.35 % (31578)Time elapsed: 0.323 s
% 51.38/19.35 % (31578)Instructions burned: 582 (million)
% 51.38/19.35 % (31578)------------------------------
% 51.38/19.35 % (31578)------------------------------
% 51.38/19.40 % (31580)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_31297 for (2886ds/393Mi)
% 52.00/19.43 % (31579)Instruction limit reached!
% 52.00/19.43 % (31579)------------------------------
% 52.00/19.43 % (31579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.00/19.43 % (31579)Termination reason: Unknown
% 52.00/19.43 % (31579)Termination phase: shuffling
% 52.00/19.43
% 52.00/19.43 % (31579)Memory used [KB]: 18933
% 52.00/19.43 % (31579)Time elapsed: 0.251 s
% 52.00/19.43 % (31579)Instructions burned: 456 (million)
% 52.00/19.43 % (31579)------------------------------
% 52.00/19.43 % (31579)------------------------------
% 52.00/19.44 % (31581)lrs+1010_1:6_au=on:cnfonf=conj_eager:nwc=10.0:pe=on:prag=on:s2a=on:ss=axioms:st=1.5:i=1335:si=on:rtra=on_0 on DTF2THF_31297 for (2886ds/1335Mi)
% 52.51/19.55 % (31568)Instruction limit reached!
% 52.51/19.55 % (31568)------------------------------
% 52.51/19.55 % (31568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.51/19.55 % (31568)Termination reason: Unknown
% 52.51/19.55 % (31568)Termination phase: Property scanning
% 52.51/19.55
% 52.51/19.55 % (31568)Memory used [KB]: 33133
% 52.51/19.55 % (31568)Time elapsed: 0.946 s
% 52.51/19.55 % (31568)Instructions burned: 1763 (million)
% 52.51/19.55 % (31568)------------------------------
% 52.51/19.55 % (31568)------------------------------
% 52.51/19.59 % (31576)Instruction limit reached!
% 52.51/19.59 % (31576)------------------------------
% 52.51/19.59 % (31576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.51/19.59 % (31576)Termination reason: Unknown
% 52.51/19.59 % (31576)Termination phase: shuffling
% 52.51/19.59
% 52.51/19.59 % (31576)Memory used [KB]: 21748
% 52.51/19.59 % (31576)Time elapsed: 0.322 s
% 52.51/19.59 % (31576)Instructions burned: 563 (million)
% 52.51/19.59 % (31576)------------------------------
% 52.51/19.59 % (31576)------------------------------
% 52.51/19.59 % (31582)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=3373:si=on:rtra=on_0 on DTF2THF_31297 for (2884ds/3373Mi)
% 52.79/19.63 % (31554)Instruction limit reached!
% 52.79/19.63 % (31554)------------------------------
% 52.79/19.63 % (31554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.79/19.64 % (31583)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_31297 for (2884ds/474Mi)
% 52.79/19.64 % (31554)Termination reason: Unknown
% 52.79/19.64 % (31554)Termination phase: Saturation
% 52.79/19.64
% 52.79/19.64 % (31554)Memory used [KB]: 45287
% 52.79/19.64 % (31554)Time elapsed: 2.398 s
% 52.79/19.64 % (31554)Instructions burned: 4714 (million)
% 52.79/19.64 % (31554)------------------------------
% 52.79/19.64 % (31554)------------------------------
% 52.79/19.69 % (31584)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_31297 for (2883ds/157Mi)
% 52.98/19.79 % (31580)Instruction limit reached!
% 52.98/19.79 % (31580)------------------------------
% 52.98/19.79 % (31580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.98/19.79 % (31580)Termination reason: Unknown
% 52.98/19.79 % (31580)Termination phase: shuffling
% 52.98/19.79
% 52.98/19.79 % (31580)Memory used [KB]: 19317
% 52.98/19.79 % (31580)Time elapsed: 0.211 s
% 52.98/19.79 % (31580)Instructions burned: 393 (million)
% 52.98/19.79 % (31580)------------------------------
% 52.98/19.79 % (31580)------------------------------
% 53.35/19.83 % (31585)dis+1010_1:1_amm=off:cs=on:kws=precedence:nwc=10.0:piset=all:s2a=on:sac=on:sp=unary_frequency:spb=goal:ss=axioms:st=3.0:i=590:si=on:rtra=on_0 on DTF2THF_31297 for (2882ds/590Mi)
% 53.35/19.83 % (31584)Instruction limit reached!
% 53.35/19.83 % (31584)------------------------------
% 53.35/19.83 % (31584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.35/19.85 % (31584)Termination reason: Unknown
% 53.35/19.85 % (31584)Termination phase: shuffling
% 53.35/19.85
% 53.35/19.85 % (31584)Memory used [KB]: 14072
% 53.35/19.85 % (31584)Time elapsed: 0.088 s
% 53.35/19.85 % (31584)Instructions burned: 157 (million)
% 53.35/19.85 % (31584)------------------------------
% 53.35/19.85 % (31584)------------------------------
% 53.35/19.89 % (31586)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_31297 for (2881ds/104Mi)
% 53.59/20.00 % (31586)Instruction limit reached!
% 53.59/20.00 % (31586)------------------------------
% 53.59/20.00 % (31586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.59/20.02 % (31586)Termination reason: Unknown
% 53.59/20.02 % (31586)Termination phase: shuffling
% 53.59/20.02
% 53.59/20.02 % (31586)Memory used [KB]: 13432
% 53.59/20.02 % (31586)Time elapsed: 0.068 s
% 53.59/20.02 % (31586)Instructions burned: 105 (million)
% 53.59/20.02 % (31586)------------------------------
% 53.59/20.02 % (31586)------------------------------
% 54.87/20.08 % (31587)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_31297 for (2879ds/414Mi)
% 54.87/20.16 % (31583)Instruction limit reached!
% 54.87/20.16 % (31583)------------------------------
% 54.87/20.16 % (31583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.87/20.16 % (31583)Termination reason: Unknown
% 54.87/20.16 % (31583)Termination phase: shuffling
% 54.87/20.16
% 54.87/20.16 % (31583)Memory used [KB]: 20468
% 54.87/20.16 % (31583)Time elapsed: 0.261 s
% 54.87/20.16 % (31583)Instructions burned: 475 (million)
% 54.87/20.16 % (31583)------------------------------
% 54.87/20.16 % (31583)------------------------------
% 55.35/20.22 % (31588)dis+10_1:1_av=off:bsd=on:bsr=unit_only:fd=off:hud=8:nm=16:prag=on:i=715:si=on:rtra=on_0 on DTF2THF_31297 for (2878ds/715Mi)
% 55.35/20.22 % (31573)Instruction limit reached!
% 55.35/20.22 % (31573)------------------------------
% 55.35/20.22 % (31573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.35/20.22 % (31573)Termination reason: Unknown
% 55.35/20.22 % (31573)Termination phase: Preprocessing 3
% 55.35/20.22
% 55.35/20.22 % (31573)Memory used [KB]: 29040
% 55.35/20.22 % (31573)Time elapsed: 0.912 s
% 55.35/20.22 % (31573)Instructions burned: 1512 (million)
% 55.35/20.22 % (31573)------------------------------
% 55.35/20.22 % (31573)------------------------------
% 55.35/20.27 % (31589)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=961:si=on:rtra=on_0 on DTF2THF_31297 for (2877ds/961Mi)
% 55.35/20.29 % (31585)Instruction limit reached!
% 55.35/20.29 % (31585)------------------------------
% 55.35/20.29 % (31585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.63/20.30 % (31585)Termination reason: Unknown
% 55.63/20.30 % (31585)Termination phase: shuffling
% 55.63/20.30
% 55.63/20.30 % (31585)Memory used [KB]: 20980
% 55.63/20.30 % (31585)Time elapsed: 0.330 s
% 55.63/20.30 % (31585)Instructions burned: 590 (million)
% 55.63/20.30 % (31585)------------------------------
% 55.63/20.30 % (31585)------------------------------
% 55.63/20.35 % (31590)lrs-1002_1:128_abs=on:bd=preordered:cbe=off:e2e=on:fde=unused:fe=off:tgt=full:i=1213:si=on:rtra=on_0 on DTF2THF_31297 for (2877ds/1213Mi)
% 56.03/20.53 % (31587)Instruction limit reached!
% 56.03/20.53 % (31587)------------------------------
% 56.03/20.53 % (31587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.03/20.53 % (31587)Termination reason: Unknown
% 56.03/20.53 % (31587)Termination phase: shuffling
% 56.03/20.53
% 56.03/20.53 % (31587)Memory used [KB]: 19701
% 56.03/20.53 % (31587)Time elapsed: 0.228 s
% 56.03/20.53 % (31587)Instructions burned: 415 (million)
% 56.03/20.53 % (31587)------------------------------
% 56.03/20.53 % (31587)------------------------------
% 56.03/20.55 % (31591)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_31297 for (2874ds/1979Mi)
% 56.46/20.65 % (31581)Instruction limit reached!
% 56.46/20.65 % (31581)------------------------------
% 56.46/20.65 % (31581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.46/20.65 % (31581)Termination reason: Unknown
% 56.46/20.65 % (31581)Termination phase: Property scanning
% 56.46/20.66
% 56.46/20.66 % (31581)Memory used [KB]: 33133
% 56.46/20.66 % (31581)Time elapsed: 0.691 s
% 56.46/20.66 % (31581)Instructions burned: 1336 (million)
% 56.46/20.66 % (31581)------------------------------
% 56.46/20.66 % (31581)------------------------------
% 56.66/20.71 % (31592)dis+1002_1:1_amm=off:au=on:bsd=on:cbe=off:cnfonf=lazy_gen:sp=unary_frequency:i=481:si=on:rtra=on_0 on DTF2THF_31297 for (2873ds/481Mi)
% 56.66/20.77 % (31555)Instruction limit reached!
% 56.66/20.77 % (31555)------------------------------
% 56.66/20.77 % (31555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.66/20.77 % (31555)Termination reason: Unknown
% 56.66/20.77 % (31555)Termination phase: Saturation
% 56.66/20.77
% 56.66/20.77 % (31555)Memory used [KB]: 58847
% 56.66/20.77 % (31555)Time elapsed: 5.258 s
% 56.66/20.77 % (31555)Instructions burned: 8250 (million)
% 56.66/20.77 % (31555)------------------------------
% 56.66/20.77 % (31555)------------------------------
% 56.80/20.83 % (31593)dis+10_8:1_apa=on:cnfonf=off:e2e=on:plsq=on:plsqc=1:plsqr=32,1:ss=axioms:i=783:si=on:rtra=on_0 on DTF2THF_31297 for (2872ds/783Mi)
% 56.85/20.96 % (31588)Instruction limit reached!
% 56.85/20.96 % (31588)------------------------------
% 56.85/20.96 % (31588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.85/20.96 % (31588)Termination reason: Unknown
% 56.85/20.96 % (31588)Termination phase: shuffling
% 56.85/20.96
% 56.85/20.96 % (31588)Memory used [KB]: 22899
% 56.85/20.96 % (31588)Time elapsed: 0.413 s
% 56.85/20.96 % (31588)Instructions burned: 716 (million)
% 56.85/20.96 % (31588)------------------------------
% 56.85/20.96 % (31588)------------------------------
% 56.85/21.01 % (31589)Instruction limit reached!
% 56.85/21.01 % (31589)------------------------------
% 56.85/21.01 % (31589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.85/21.01 % (31589)Termination reason: Unknown
% 56.85/21.01 % (31589)Termination phase: Property scanning
% 56.85/21.01
% 56.85/21.01 % (31589)Memory used [KB]: 22899
% 56.85/21.01 % (31589)Time elapsed: 0.520 s
% 56.85/21.01 % (31589)Instructions burned: 961 (million)
% 56.85/21.01 % (31589)------------------------------
% 56.85/21.01 % (31589)------------------------------
% 56.85/21.01 % (31594)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=432:si=on:rtra=on_0 on DTF2THF_31297 for (2870ds/432Mi)
% 56.85/21.03 % (31595)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=1670:si=on:rtra=on_0 on DTF2THF_31297 for (2870ds/1670Mi)
% 59.64/21.24 % (31592)Instruction limit reached!
% 59.64/21.24 % (31592)------------------------------
% 59.64/21.24 % (31592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.64/21.24 % (31592)Termination reason: Unknown
% 59.64/21.24 % (31592)Termination phase: shuffling
% 59.64/21.24
% 59.64/21.24 % (31592)Memory used [KB]: 20596
% 59.64/21.24 % (31592)Time elapsed: 0.269 s
% 59.64/21.24 % (31592)Instructions burned: 481 (million)
% 59.64/21.24 % (31592)------------------------------
% 59.64/21.24 % (31592)------------------------------
% 59.64/21.28 % (31596)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_31297 for (2867ds/90Mi)
% 60.03/21.34 % (31593)Instruction limit reached!
% 60.03/21.34 % (31593)------------------------------
% 60.03/21.34 % (31593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.03/21.34 % (31593)Termination reason: Unknown
% 60.03/21.34 % (31593)Termination phase: shuffling
% 60.03/21.34
% 60.03/21.34 % (31593)Memory used [KB]: 23922
% 60.03/21.34 % (31593)Time elapsed: 0.429 s
% 60.03/21.34 % (31593)Instructions burned: 783 (million)
% 60.03/21.34 % (31593)------------------------------
% 60.03/21.34 % (31593)------------------------------
% 60.03/21.38 % (31596)Instruction limit reached!
% 60.03/21.38 % (31596)------------------------------
% 60.03/21.38 % (31596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.03/21.38 % (31596)Termination reason: Unknown
% 60.03/21.38 % (31596)Termination phase: shuffling
% 60.03/21.38
% 60.03/21.38 % (31596)Memory used [KB]: 13176
% 60.03/21.38 % (31596)Time elapsed: 0.058 s
% 60.03/21.38 % (31596)Instructions burned: 90 (million)
% 60.03/21.38 % (31596)------------------------------
% 60.03/21.38 % (31596)------------------------------
% 60.03/21.38 % (31597)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_31297 for (2866ds/346Mi)
% 60.38/21.42 % (31598)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=401:si=on:rtra=on_0 on DTF2THF_31297 for (2866ds/401Mi)
% 60.38/21.49 % (31594)Instruction limit reached!
% 60.38/21.49 % (31594)------------------------------
% 60.38/21.49 % (31594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.38/21.49 % (31594)Termination reason: Unknown
% 60.38/21.49 % (31594)Termination phase: shuffling
% 60.38/21.49
% 60.38/21.49 % (31594)Memory used [KB]: 18549
% 60.38/21.49 % (31594)Time elapsed: 0.251 s
% 60.38/21.49 % (31594)Instructions burned: 432 (million)
% 60.38/21.49 % (31594)------------------------------
% 60.38/21.49 % (31594)------------------------------
% 60.59/21.54 % (31599)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_31297 for (2865ds/6454Mi)
% 60.59/21.55 % (31590)Instruction limit reached!
% 60.59/21.55 % (31590)------------------------------
% 60.59/21.55 % (31590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.59/21.55 % (31590)Termination reason: Unknown
% 60.59/21.55 % (31590)Termination phase: shuffling
% 60.59/21.55
% 60.59/21.55 % (31590)Memory used [KB]: 26993
% 60.59/21.55 % (31590)Time elapsed: 0.682 s
% 60.59/21.55 % (31590)Instructions burned: 1213 (million)
% 60.59/21.55 % (31590)------------------------------
% 60.59/21.55 % (31590)------------------------------
% 60.59/21.58 % (31600)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:kws=arity:prag=on:i=283:si=on:rtra=on_0 on DTF2THF_31297 for (2864ds/283Mi)
% 60.84/21.71 % (31591)Instruction limit reached!
% 60.84/21.71 % (31591)------------------------------
% 60.84/21.71 % (31591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.84/21.71 % (31591)Termination reason: Unknown
% 60.84/21.71 % (31591)Termination phase: Function definition elimination
% 60.84/21.71
% 60.84/21.71 % (31591)Memory used [KB]: 32110
% 60.84/21.71 % (31591)Time elapsed: 1.001 s
% 60.84/21.71 % (31591)Instructions burned: 1979 (million)
% 60.84/21.71 % (31591)------------------------------
% 60.84/21.71 % (31591)------------------------------
% 60.84/21.72 % (31597)Instruction limit reached!
% 60.84/21.72 % (31597)------------------------------
% 60.84/21.72 % (31597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.84/21.72 % (31597)Termination reason: Unknown
% 60.84/21.72 % (31597)Termination phase: shuffling
% 60.84/21.72
% 60.84/21.72 % (31597)Memory used [KB]: 18677
% 60.84/21.72 % (31597)Time elapsed: 0.186 s
% 60.84/21.72 % (31597)Instructions burned: 348 (million)
% 60.84/21.72 % (31597)------------------------------
% 60.84/21.72 % (31597)------------------------------
% 60.84/21.75 % (31601)ott+10_1:1_aac=none:apa=on:au=on:cnfonf=off:ntd=on:nwc=12.0:sos=on:i=865:si=on:rtra=on_0 on DTF2THF_31297 for (2862ds/865Mi)
% 60.84/21.75 % (31600)Instruction limit reached!
% 60.84/21.75 % (31600)------------------------------
% 60.84/21.75 % (31600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.84/21.75 % (31600)Termination reason: Unknown
% 60.84/21.75 % (31600)Termination phase: shuffling
% 60.84/21.75
% 60.84/21.75 % (31600)Memory used [KB]: 16247
% 60.84/21.75 % (31600)Time elapsed: 0.157 s
% 60.84/21.75 % (31600)Instructions burned: 283 (million)
% 60.84/21.75 % (31600)------------------------------
% 60.84/21.75 % (31600)------------------------------
% 60.84/21.78 % (31602)lrs+1002_3:4_au=on:av=off:cnfonf=off:fd=off:fsr=off:ins=2:sd=1:sos=on:ss=axioms:i=4528:si=on:rtra=on_0 on DTF2THF_31297 for (2862ds/4528Mi)
% 61.28/21.81 % (31603)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=1001:si=on:rtra=on_0 on DTF2THF_31297 for (2862ds/1001Mi)
% 61.28/21.82 % (31595)Instruction limit reached!
% 61.28/21.82 % (31595)------------------------------
% 61.28/21.82 % (31595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.28/21.82 % (31595)Termination reason: Unknown
% 61.28/21.82 % (31595)Termination phase: Property scanning
% 61.28/21.82
% 61.28/21.82 % (31595)Memory used [KB]: 33133
% 61.28/21.82 % (31595)Time elapsed: 0.801 s
% 61.28/21.82 % (31595)Instructions burned: 1671 (million)
% 61.28/21.82 % (31595)------------------------------
% 61.28/21.82 % (31595)------------------------------
% 61.28/21.83 % (31598)Instruction limit reached!
% 61.28/21.83 % (31598)------------------------------
% 61.28/21.83 % (31598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.28/21.83 % (31598)Termination reason: Unknown
% 61.28/21.83 % (31598)Termination phase: shuffling
% 61.28/21.83
% 61.28/21.83 % (31598)Memory used [KB]: 19445
% 61.28/21.83 % (31598)Time elapsed: 0.218 s
% 61.28/21.83 % (31598)Instructions burned: 402 (million)
% 61.28/21.83 % (31598)------------------------------
% 61.28/21.83 % (31598)------------------------------
% 61.28/21.87 % (31604)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_31297 for (2861ds/102Mi)
% 61.28/21.88 % (31605)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_31297 for (2861ds/952Mi)
% 62.38/21.98 % (31604)Instruction limit reached!
% 62.38/21.98 % (31604)------------------------------
% 62.38/21.98 % (31604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.38/21.98 % (31604)Termination reason: Unknown
% 62.38/21.98 % (31604)Termination phase: shuffling
% 62.38/21.98
% 62.38/21.98 % (31604)Memory used [KB]: 13176
% 62.38/21.98 % (31604)Time elapsed: 0.062 s
% 62.38/21.98 % (31604)Instructions burned: 103 (million)
% 62.38/21.98 % (31604)------------------------------
% 62.38/21.98 % (31604)------------------------------
% 62.38/22.07 % (31606)lrs+1010_1:1024_add=large:c=on:cbe=off:cnfonf=lazy_gen:fde=none:nm=30:piset=and:sac=on:uhcvi=on:i=578:si=on:rtra=on_0 on DTF2THF_31297 for (2860ds/578Mi)
% 65.48/22.67 % (31602)Refutation not found, incomplete strategy
% 65.48/22.67 % (31602)------------------------------
% 65.48/22.67 % (31602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 65.48/22.67 % (31602)Termination reason: Refutation not found, incomplete strategy
% 65.48/22.67
% 65.48/22.67
% 65.48/22.67 % (31602)Memory used [KB]: 24050
% 65.48/22.67 % (31602)Time elapsed: 0.925 s
% 65.48/22.67 % (31602)Instructions burned: 1326 (million)
% 65.48/22.67 % (31602)------------------------------
% 65.48/22.67 % (31602)------------------------------
% 65.87/22.76 % (31607)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_31297 for (2853ds/643Mi)
% 66.13/22.80 % (31601)Instruction limit reached!
% 66.13/22.80 % (31601)------------------------------
% 66.13/22.80 % (31601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.13/22.80 % (31601)Termination reason: Unknown
% 66.13/22.80 % (31601)Termination phase: Property scanning
% 66.13/22.80
% 66.13/22.80 % (31601)Memory used [KB]: 23794
% 66.13/22.80 % (31601)Time elapsed: 0.487 s
% 66.13/22.80 % (31601)Instructions burned: 867 (million)
% 66.13/22.80 % (31601)------------------------------
% 66.13/22.80 % (31601)------------------------------
% 66.13/22.87 % (31608)lrs+1010_1:1_cnfonf=off:ntd=on:nwc=3.0:s2a=on:sp=const_frequency:i=865:si=on:rtra=on_0 on DTF2THF_31297 for (2852ds/865Mi)
% 66.38/22.94 % (31606)Instruction limit reached!
% 66.38/22.94 % (31606)------------------------------
% 66.38/22.94 % (31606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.38/22.94 % (31606)Termination reason: Unknown
% 66.38/22.94 % (31606)Termination phase: shuffling
% 66.38/22.94
% 66.38/22.94 % (31606)Memory used [KB]: 21875
% 66.38/22.94 % (31606)Time elapsed: 0.327 s
% 66.38/22.94 % (31606)Instructions burned: 578 (million)
% 66.38/22.94 % (31606)------------------------------
% 66.38/22.94 % (31606)------------------------------
% 66.38/23.01 % (31609)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_31297 for (2850ds/1174Mi)
% 66.38/23.04 % (31603)Instruction limit reached!
% 66.38/23.04 % (31603)------------------------------
% 66.38/23.04 % (31603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.38/23.04 % (31603)Termination reason: Unknown
% 66.38/23.04 % (31603)Termination phase: Property scanning
% 66.38/23.04
% 66.38/23.04 % (31603)Memory used [KB]: 22899
% 66.38/23.04 % (31603)Time elapsed: 0.550 s
% 66.38/23.04 % (31603)Instructions burned: 1002 (million)
% 66.38/23.04 % (31603)------------------------------
% 66.38/23.04 % (31603)------------------------------
% 66.92/23.11 % (31610)ott+1010_1:32_apa=on:atotf=0.3:au=on:c=on:cnfonf=off:fe=off:fsr=off:nm=6:piset=or:sac=on:sos=on:i=644:si=on:rtra=on_0 on DTF2THF_31297 for (2849ds/644Mi)
% 67.55/23.32 % (31605)Instruction limit reached!
% 67.55/23.32 % (31605)------------------------------
% 67.55/23.32 % (31605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.55/23.32 % (31605)Termination reason: Unknown
% 67.55/23.32 % (31605)Termination phase: Property scanning
% 67.55/23.32
% 67.55/23.32 % (31605)Memory used [KB]: 22899
% 67.55/23.32 % (31605)Time elapsed: 0.561 s
% 67.55/23.32 % (31605)Instructions burned: 952 (million)
% 67.55/23.32 % (31605)------------------------------
% 67.55/23.32 % (31605)------------------------------
% 67.84/23.42 % (31611)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_31297 for (2846ds/1201Mi)
% 67.84/23.42 % (31582)Instruction limit reached!
% 67.84/23.42 % (31582)------------------------------
% 67.84/23.42 % (31582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.84/23.42 % (31582)Termination reason: Unknown
% 67.84/23.42 % (31582)Termination phase: Saturation
% 67.84/23.42
% 67.84/23.42 % (31582)Memory used [KB]: 39018
% 67.84/23.42 % (31582)Time elapsed: 1.831 s
% 67.84/23.42 % (31582)Instructions burned: 3374 (million)
% 67.84/23.42 % (31582)------------------------------
% 67.84/23.42 % (31582)------------------------------
% 67.84/23.48 % (31599)Refutation not found, non-redundant clauses discarded% (31599)------------------------------
% 67.84/23.48 % (31599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.84/23.48 % (31599)Termination reason: Refutation not found, non-redundant clauses discarded
% 67.84/23.48
% 67.84/23.48 % (31599)Memory used [KB]: 33005
% 67.84/23.48 % (31599)Time elapsed: 1.975 s
% 67.84/23.48 % (31599)Instructions burned: 3545 (million)
% 67.84/23.48 % (31599)------------------------------
% 67.84/23.48 % (31599)------------------------------
% 67.84/23.48 % (31612)lrs+1666_1:64_cnfonf=lazy_simp:hud=5:plsq=on:plsql=on:plsqr=152565,524288:tnu=2:i=1046:si=on:rtra=on_0 on DTF2THF_31297 for (2845ds/1046Mi)
% 68.09/23.55 % (31613)lrs+1010_1:1_c=on:cnfonf=lazy_simp:fsr=off:hud=5:kws=arity:nm=32:plsq=on:plsqc=1:plsqr=642,29:prag=on:i=222:si=on:rtra=on_0 on DTF2THF_31297 for (2845ds/222Mi)
% 68.26/23.74 % (31607)Instruction limit reached!
% 68.26/23.74 % (31607)------------------------------
% 68.26/23.74 % (31607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.26/23.74 % (31607)Termination reason: Unknown
% 68.26/23.74 % (31607)Termination phase: shuffling
% 68.26/23.74
% 68.26/23.74 % (31607)Memory used [KB]: 22771
% 68.26/23.74 % (31607)Time elapsed: 0.347 s
% 68.26/23.74 % (31607)Instructions burned: 643 (million)
% 68.26/23.74 % (31607)------------------------------
% 68.26/23.74 % (31607)------------------------------
% 68.26/23.77 % (31610)Instruction limit reached!
% 68.26/23.77 % (31610)------------------------------
% 68.26/23.77 % (31610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.26/23.77 % (31610)Termination reason: Unknown
% 68.26/23.77 % (31610)Termination phase: shuffling
% 68.26/23.77
% 68.26/23.77 % (31610)Memory used [KB]: 21492
% 68.26/23.77 % (31610)Time elapsed: 0.361 s
% 68.26/23.77 % (31610)Instructions burned: 645 (million)
% 68.26/23.77 % (31610)------------------------------
% 68.26/23.77 % (31610)------------------------------
% 68.73/23.81 % (31613)Instruction limit reached!
% 68.73/23.81 % (31613)------------------------------
% 68.73/23.81 % (31613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.73/23.81 % (31613)Termination reason: Unknown
% 68.73/23.81 % (31613)Termination phase: shuffling
% 68.73/23.81
% 68.73/23.81 % (31613)Memory used [KB]: 14839
% 68.73/23.81 % (31613)Time elapsed: 0.123 s
% 68.73/23.81 % (31613)Instructions burned: 223 (million)
% 68.73/23.81 % (31613)------------------------------
% 68.73/23.81 % (31613)------------------------------
% 68.73/23.82 % (31614)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_31297 for (2842ds/68Mi)
% 68.73/23.84 % (31615)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=454:si=on:rtra=on_0 on DTF2THF_31297 for (2842ds/454Mi)
% 68.73/23.86 % (31614)Instruction limit reached!
% 68.73/23.86 % (31614)------------------------------
% 68.73/23.86 % (31614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.73/23.86 % (31614)Termination reason: Unknown
% 68.73/23.86 % (31614)Termination phase: shuffling
% 68.73/23.86
% 68.73/23.86 % (31614)Memory used [KB]: 11769
% 68.73/23.86 % (31614)Time elapsed: 0.043 s
% 68.73/23.86 % (31614)Instructions burned: 69 (million)
% 68.73/23.86 % (31614)------------------------------
% 68.73/23.86 % (31614)------------------------------
% 68.73/23.87 % (31616)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=360:si=on:rtra=on_0 on DTF2THF_31297 for (2842ds/360Mi)
% 68.98/23.92 % (31617)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_31297 for (2841ds/157Mi)
% 68.98/24.00 % (31617)Instruction limit reached!
% 68.98/24.00 % (31617)------------------------------
% 68.98/24.00 % (31617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.98/24.00 % (31617)Termination reason: Unknown
% 68.98/24.00 % (31617)Termination phase: shuffling
% 68.98/24.00
% 68.98/24.00 % (31617)Memory used [KB]: 14072
% 68.98/24.00 % (31617)Time elapsed: 0.082 s
% 68.98/24.00 % (31617)Instructions burned: 158 (million)
% 68.98/24.00 % (31617)------------------------------
% 68.98/24.00 % (31617)------------------------------
% 68.98/24.02 % (31618)ott+1004_1:24_anc=none:cbe=off:cnfonf=lazy_not_gen:fe=off:piset=not:prag=on:rp=on:i=5061:si=on:rtra=on_0 on DTF2THF_31297 for (2840ds/5061Mi)
% 70.19/24.36 % (31608)Instruction limit reached!
% 70.19/24.36 % (31608)------------------------------
% 70.19/24.36 % (31608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.19/24.36 % (31608)Termination reason: Unknown
% 70.19/24.36 % (31608)Termination phase: Property scanning
% 70.19/24.36
% 70.19/24.36 % (31608)Memory used [KB]: 23794
% 70.19/24.36 % (31608)Time elapsed: 0.512 s
% 70.19/24.36 % (31608)Instructions burned: 866 (million)
% 70.19/24.36 % (31608)------------------------------
% 70.19/24.36 % (31608)------------------------------
% 70.40/24.42 % (31619)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_31297 for (2836ds/2401Mi)
% 70.40/24.47 % (31616)Instruction limit reached!
% 70.40/24.47 % (31616)------------------------------
% 70.40/24.47 % (31616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.40/24.47 % (31616)Termination reason: Unknown
% 70.40/24.47 % (31616)Termination phase: shuffling
% 70.40/24.47
% 70.40/24.47 % (31616)Memory used [KB]: 18805
% 70.40/24.47 % (31616)Time elapsed: 0.216 s
% 70.40/24.47 % (31616)Instructions burned: 361 (million)
% 70.40/24.47 % (31616)------------------------------
% 70.40/24.47 % (31616)------------------------------
% 70.57/24.52 % (31620)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_31297 for (2835ds/5798Mi)
% 70.57/24.56 % (31615)Instruction limit reached!
% 70.57/24.56 % (31615)------------------------------
% 70.57/24.56 % (31615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.57/24.56 % (31615)Termination reason: Unknown
% 70.57/24.56 % (31615)Termination phase: shuffling
% 70.57/24.56
% 70.57/24.56 % (31615)Memory used [KB]: 20212
% 70.57/24.56 % (31615)Time elapsed: 0.256 s
% 70.57/24.56 % (31615)Instructions burned: 455 (million)
% 70.57/24.56 % (31615)------------------------------
% 70.57/24.56 % (31615)------------------------------
% 70.70/24.64 % (31621)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_31297 for (2834ds/346Mi)
% 70.80/24.70 % (31609)Instruction limit reached!
% 70.80/24.70 % (31609)------------------------------
% 70.80/24.70 % (31609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.80/24.70 % (31609)Termination reason: Unknown
% 70.80/24.70 % (31609)Termination phase: SInE selection
% 70.80/24.70
% 70.80/24.70 % (31609)Memory used [KB]: 22899
% 70.80/24.70 % (31609)Time elapsed: 0.679 s
% 70.80/24.70 % (31609)Instructions burned: 1174 (million)
% 70.80/24.70 % (31609)------------------------------
% 70.80/24.70 % (31609)------------------------------
% 70.80/24.77 % (31622)lrs+1002_1:1_av=off:bd=off:bet=on:e2e=on:fde=unused:fe=off:plsq=on:prag=on:sos=on:i=1321:si=on:rtra=on_0 on DTF2THF_31297 for (2832ds/1321Mi)
% 70.96/25.03 % (31612)Instruction limit reached!
% 70.96/25.03 % (31612)------------------------------
% 70.96/25.03 % (31612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.96/25.03 % (31612)Termination reason: Unknown
% 70.96/25.03 % (31612)Termination phase: shuffling
% 70.96/25.03
% 70.96/25.03 % (31612)Memory used [KB]: 32494
% 70.96/25.03 % (31612)Time elapsed: 0.589 s
% 70.96/25.03 % (31612)Instructions burned: 1046 (million)
% 70.96/25.03 % (31612)------------------------------
% 70.96/25.03 % (31612)------------------------------
% 72.79/25.09 % (31623)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_31297 for (2829ds/1312Mi)
% 72.79/25.17 % (31621)Instruction limit reached!
% 72.79/25.17 % (31621)------------------------------
% 72.79/25.17 % (31621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.79/25.17 % (31621)Termination reason: Unknown
% 72.79/25.17 % (31621)Termination phase: shuffling
% 72.79/25.17
% 72.79/25.17 % (31621)Memory used [KB]: 18677
% 72.79/25.17 % (31621)Time elapsed: 0.221 s
% 72.79/25.17 % (31621)Instructions burned: 347 (million)
% 72.79/25.17 % (31621)------------------------------
% 72.79/25.17 % (31621)------------------------------
% 73.09/25.24 % (31624)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_31297 for (2828ds/672Mi)
% 73.34/25.31 % (31611)Instruction limit reached!
% 73.34/25.31 % (31611)------------------------------
% 73.34/25.31 % (31611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.34/25.31 % (31611)Termination reason: Unknown
% 73.34/25.31 % (31611)Termination phase: Preprocessing 1
% 73.34/25.31
% 73.34/25.31 % (31611)Memory used [KB]: 23539
% 73.34/25.31 % (31611)Time elapsed: 0.685 s
% 73.34/25.31 % (31611)Instructions burned: 1201 (million)
% 73.34/25.31 % (31611)------------------------------
% 73.34/25.31 % (31611)------------------------------
% 73.34/25.39 % (31625)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_31297 for (2826ds/1897Mi)
% 76.15/26.12 % (31623)Instruction limit reached!
% 76.15/26.12 % (31623)------------------------------
% 76.15/26.12 % (31623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.15/26.12 % (31623)Termination reason: Unknown
% 76.15/26.12 % (31623)Termination phase: Property scanning
% 76.15/26.12
% 76.15/26.12 % (31623)Memory used [KB]: 33133
% 76.15/26.12 % (31623)Time elapsed: 0.664 s
% 76.15/26.12 % (31623)Instructions burned: 1312 (million)
% 76.15/26.12 % (31623)------------------------------
% 76.15/26.12 % (31623)------------------------------
% 76.15/26.18 % (31626)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_31297 for (2818ds/1202Mi)
% 76.62/26.26 % (31624)Instruction limit reached!
% 76.62/26.26 % (31624)------------------------------
% 76.62/26.26 % (31624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.62/26.26 % (31624)Termination reason: Unknown
% 76.62/26.26 % (31624)Termination phase: shuffling
% 76.62/26.26
% 76.62/26.26 % (31624)Memory used [KB]: 22131
% 76.62/26.26 % (31624)Time elapsed: 0.395 s
% 76.62/26.26 % (31624)Instructions burned: 672 (million)
% 76.62/26.26 % (31624)------------------------------
% 76.62/26.26 % (31624)------------------------------
% 76.99/26.32 % (31627)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_31297 for (2817ds/1493Mi)
% 77.30/26.44 % (31618)Instruction limit reached!
% 77.30/26.44 % (31618)------------------------------
% 77.30/26.44 % (31618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 77.30/26.44 % (31618)Termination reason: Unknown
% 77.30/26.44 % (31618)Termination phase: Saturation
% 77.30/26.44
% 77.30/26.44 % (31618)Memory used [KB]: 44519
% 77.30/26.44 % (31618)Time elapsed: 2.422 s
% 77.30/26.44 % (31618)Instructions burned: 5062 (million)
% 77.30/26.44 % (31618)------------------------------
% 77.30/26.44 % (31618)------------------------------
% 77.57/26.51 % (31628)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_31297 for (2815ds/7679Mi)
% 77.79/26.68 % (31622)Instruction limit reached!
% 77.79/26.68 % (31622)------------------------------
% 77.79/26.68 % (31622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 77.79/26.68 % (31622)Termination reason: Unknown
% 77.79/26.68 % (31622)Termination phase: Preprocessing 3
% 77.79/26.68
% 77.79/26.68 % (31622)Memory used [KB]: 28528
% 77.79/26.68 % (31622)Time elapsed: 0.789 s
% 77.79/26.68 % (31622)Instructions burned: 1321 (million)
% 77.79/26.68 % (31622)------------------------------
% 77.79/26.68 % (31622)------------------------------
% 77.28/26.76 % (31629)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_31297 for (2813ds/340Mi)
% 78.21/26.87 % (31625)Instruction limit reached!
% 78.21/26.87 % (31625)------------------------------
% 78.21/26.87 % (31625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 78.21/26.87 % (31625)Termination reason: Unknown
% 78.21/26.87 % (31625)Termination phase: Saturation
% 78.21/26.87
% 78.21/26.87 % (31625)Memory used [KB]: 30958
% 78.21/26.87 % (31625)Time elapsed: 1.008 s
% 78.21/26.87 % (31625)Instructions burned: 1897 (million)
% 78.21/26.87 % (31625)------------------------------
% 78.21/26.87 % (31625)------------------------------
% 78.38/26.91 % (31630)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_31297 for (2811ds/3672Mi)
% 80.52/27.32 % (31629)Instruction limit reached!
% 80.52/27.32 % (31629)------------------------------
% 80.52/27.32 % (31629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 80.52/27.32 % (31629)Termination reason: Unknown
% 80.52/27.32 % (31629)Termination phase: shuffling
% 80.52/27.32
% 80.52/27.32 % (31629)Memory used [KB]: 17014
% 80.52/27.32 % (31629)Time elapsed: 0.209 s
% 80.52/27.32 % (31629)Instructions burned: 341 (million)
% 80.52/27.32 % (31629)------------------------------
% 80.52/27.32 % (31629)------------------------------
% 80.52/27.39 % (31631)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_31297 for (2806ds/1535Mi)
% 81.07/27.50 % (31619)Instruction limit reached!
% 81.07/27.50 % (31619)------------------------------
% 81.07/27.50 % (31619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.07/27.50 % (31619)Termination reason: Unknown
% 81.07/27.50 % (31619)Termination phase: Property scanning
% 81.07/27.50
% 81.07/27.50 % (31619)Memory used [KB]: 33517
% 81.07/27.50 % (31619)Time elapsed: 1.416 s
% 81.07/27.50 % (31619)Instructions burned: 2402 (million)
% 81.07/27.50 % (31619)------------------------------
% 81.07/27.50 % (31619)------------------------------
% 81.07/27.54 % (31632)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_31297 for (2805ds/4637Mi)
% 81.52/27.78 % (31627)Instruction limit reached!
% 81.52/27.78 % (31627)------------------------------
% 81.52/27.78 % (31627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.52/27.78 % (31627)Termination reason: Unknown
% 81.52/27.78 % (31627)Termination phase: Property scanning
% 81.52/27.78
% 81.52/27.78 % (31627)Memory used [KB]: 33133
% 81.52/27.78 % (31627)Time elapsed: 0.833 s
% 81.52/27.78 % (31627)Instructions burned: 1495 (million)
% 81.52/27.78 % (31627)------------------------------
% 81.52/27.78 % (31627)------------------------------
% 81.75/27.84 % (31633)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_31297 for (2802ds/3174Mi)
% 81.92/28.06 % (31626)Instruction limit reached!
% 81.92/28.06 % (31626)------------------------------
% 81.92/28.06 % (31626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.92/28.06 % (31626)Termination reason: Unknown
% 81.92/28.06 % (31626)Termination phase: Preprocessing 1
% 81.92/28.06
% 81.92/28.06 % (31626)Memory used [KB]: 23539
% 81.92/28.06 % (31626)Time elapsed: 0.699 s
% 81.92/28.06 % (31626)Instructions burned: 1203 (million)
% 81.92/28.06 % (31626)------------------------------
% 81.92/28.06 % (31626)------------------------------
% 82.81/28.12 % (31634)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_31297 for (2799ds/307Mi)
% 84.20/28.60 % (31634)Instruction limit reached!
% 84.20/28.60 % (31634)------------------------------
% 84.20/28.60 % (31634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.20/28.60 % (31634)Termination reason: Unknown
% 84.20/28.60 % (31634)Termination phase: shuffling
% 84.20/28.60
% 84.20/28.60 % (31634)Memory used [KB]: 16119
% 84.20/28.60 % (31634)Time elapsed: 0.173 s
% 84.20/28.60 % (31634)Instructions burned: 308 (million)
% 84.20/28.60 % (31634)------------------------------
% 84.20/28.60 % (31634)------------------------------
% 84.20/28.67 % (31635)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_31297 for (2794ds/3619Mi)
% 85.36/29.00 % (31631)Instruction limit reached!
% 85.36/29.00 % (31631)------------------------------
% 85.36/29.00 % (31631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 85.36/29.00 % (31631)Termination reason: Unknown
% 85.36/29.00 % (31631)Termination phase: Property scanning
% 85.36/29.00
% 85.36/29.00 % (31631)Memory used [KB]: 33133
% 85.36/29.00 % (31631)Time elapsed: 0.800 s
% 85.36/29.00 % (31631)Instructions burned: 1537 (million)
% 85.36/29.00 % (31631)------------------------------
% 85.36/29.00 % (31631)------------------------------
% 86.07/29.07 % (31636)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_31297 for (2790ds/2325Mi)
% 87.48/29.61 % (31633)First to succeed.
% 87.48/29.63 % (31633)Refutation found. Thanks to Tanya!
% 87.48/29.63 % SZS status Theorem for DTF2THF_31297
% 87.48/29.63 % SZS output start Proof for DTF2THF_31297
% 87.48/29.63 thf(type_def_5, type, int: $tType).
% 87.48/29.63 thf(type_def_7, type, list: ($tType) > $tType).
% 87.48/29.63 thf(type_def_8, type, product_unit: $tType).
% 87.48/29.63 thf(type_def_9, type, heap_ext: ($tType) > $tType).
% 87.48/29.63 thf(type_def_10, type, array: ($tType) > $tType).
% 87.48/29.63 thf(type_def_11, type, product_prod: ($tType * $tType) > $tType).
% 87.48/29.63 thf(type_def_12, type, heap_Time_Heap: ($tType) > $tType).
% 87.48/29.63 thf(type_def_13, type, nat: $tType).
% 87.48/29.63 thf(type_def_14, type, set: ($tType) > $tType).
% 87.48/29.63 thf(type_def_15, type, sum_sum: ($tType * $tType) > $tType).
% 87.48/29.63 thf(type_def_16, type, num: $tType).
% 87.48/29.63 thf(type_def_17, type, option: ($tType) > $tType).
% 87.48/29.63 thf(type_def_18, type, itself: ($tType) > $tType).
% 87.48/29.63 thf(type_def_19, type, code_integer: $tType).
% 87.48/29.63 thf(type_def_20, type, code_natural: $tType).
% 87.48/29.63 thf(type_def_21, type, old_node: ($tType * $tType) > $tType).
% 87.48/29.63 thf(type_def_22, type, typerep: $tType).
% 87.48/29.63 thf(type_def_23, type, char: $tType).
% 87.48/29.63 thf(type_def_24, type, rat: $tType).
% 87.48/29.63 thf(type_def_25, type, tuple_isomorphism: ($tType * $tType * $tType) > $tType).
% 87.48/29.63 thf(type_def_26, type, literal: $tType).
% 87.48/29.63 thf(type_def_27, type, a: $tType).
% 87.48/29.63 thf(type_def_28, type, fun_box: ($tType * $tType) > $tType).
% 87.48/29.63 thf(type_def_29, type, word: ($tType) > $tType).
% 87.48/29.63 thf(func_def_0, type, tuple_isomorphism: ($tType * $tType * $tType) > $tType).
% 87.48/29.63 thf(func_def_1, type, heap_Time_Heap: ($tType) > $tType).
% 87.48/29.63 thf(func_def_2, type, code_natural: $tType).
% 87.48/29.63 thf(func_def_3, type, code_integer: $tType).
% 87.48/29.63 thf(func_def_4, type, heap_ext: ($tType) > $tType).
% 87.48/29.63 thf(func_def_5, type, product_unit: $tType).
% 87.48/29.63 thf(func_def_6, type, product_prod: ($tType * $tType) > $tType).
% 87.48/29.63 thf(func_def_7, type, old_node: ($tType * $tType) > $tType).
% 87.48/29.63 thf(func_def_8, type, fun_box: ($tType * $tType) > $tType).
% 87.48/29.63 thf(func_def_9, type, typerep: $tType).
% 87.48/29.63 thf(func_def_10, type, literal: $tType).
% 87.48/29.63 thf(func_def_11, type, sum_sum: ($tType * $tType) > $tType).
% 87.48/29.63 thf(func_def_12, type, option: ($tType) > $tType).
% 87.48/29.63 thf(func_def_13, type, word: ($tType) > $tType).
% 87.48/29.63 thf(func_def_14, type, char: $tType).
% 87.48/29.63 thf(func_def_15, type, array: ($tType) > $tType).
% 87.48/29.63 thf(func_def_16, type, list: ($tType) > $tType).
% 87.48/29.63 thf(func_def_17, type, set: ($tType) > $tType).
% 87.48/29.63 thf(func_def_18, type, rat: $tType).
% 87.48/29.63 thf(func_def_19, type, num: $tType).
% 87.48/29.63 thf(func_def_20, type, nat: $tType).
% 87.48/29.63 thf(func_def_21, type, int: $tType).
% 87.48/29.63 thf(func_def_22, type, itself: ($tType) > $tType).
% 87.48/29.63 thf(func_def_23, type, a: $tType).
% 87.48/29.63 thf(func_def_24, type, finite_finite: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_25, type, bounded_lattice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_26, type, type: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_27, type, size: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_28, type, heap: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_29, type, dvd: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_30, type, one: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_31, type, idom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_32, type, ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_33, type, plus: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_34, type, zero: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_35, type, numeral: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_36, type, power: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_37, type, field: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_38, type, ring_gcd: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_39, type, minus: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_40, type, abs_if: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_41, type, ring_1: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_42, type, uminus: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_43, type, bot: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_44, type, ord: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_45, type, top: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_46, type, semidom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_47, type, inverse: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_48, type, semiring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_49, type, ring_char_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_50, type, neg_numeral: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_51, type, order: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_52, type, comm_ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_53, type, mult_zero: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_54, type, typerep2: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_55, type, semiring_Gcd: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_56, type, semiring_gcd: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_57, type, group_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_58, type, lattice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_59, type, no_bot: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_60, type, no_top: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_61, type, semiring_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_62, type, semiring_1: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_63, type, monoid_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_64, type, comm_ring_1: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_65, type, idom_divide: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_66, type, idom_modulo: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_67, type, monoid_mult: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_68, type, linorder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_69, type, preorder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_70, type, ring_parity: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_71, type, idom_abs_sgn: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_72, type, ordered_ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_73, type, zero_neq_one: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_74, type, countable: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_75, type, field_char_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_76, type, ab_group_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_77, type, semiring_char_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_78, type, order_bot: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_79, type, order_top: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_80, type, wellorder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_81, type, comm_semiring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_82, type, zero_less_one: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_83, type, division_ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_84, type, field_abs_sgn: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_85, type, semigroup_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_86, type, semiring_numeral: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_87, type, semidom_divide: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_88, type, semidom_modulo: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_89, type, semigroup_mult: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_90, type, dense_order: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_91, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_92, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_93, type, linordered_idom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_94, type, linordered_ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_95, type, semiring_modulo: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_96, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_97, type, semiring_parity: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_98, type, ordered_ring_abs: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_99, type, ordered_semiring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_100, type, linordered_field: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_101, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_102, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_103, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_104, type, algebraic_semidom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_105, type, semiring_1_cancel: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_106, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_107, type, distrib_lattice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_108, type, semilattice_inf: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_109, type, semilattice_sup: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_110, type, dense_linorder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_111, type, linordered_semidom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_112, type, ordered_semiring_0: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_113, type, linordered_semiring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_114, type, comple9053668089753744459l_ccpo: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_115, type, finite8700451911770168679attice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_116, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_117, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_118, type, linord6961819062388156250ring_1: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_119, type, normal8620421768224518004emidom: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_120, type, ordere2520102378445227354miring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_121, type, bit_semiring_bits: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_122, type, bounded_lattice_bot: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_123, type, bounded_lattice_top: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_124, type, comm_s4317794764714335236cancel: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_125, type, linord4710134922213307826strict: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_126, type, cancel1802427076303600483id_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_127, type, ring_15535105094025558882visors: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_128, type, cancel2418104881723323429up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_129, type, linord5086331880401160121up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_130, type, ordere6911136660526730532id_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_131, type, archim2362893244070406136eiling: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_132, type, semiri6843258321239162965malize: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_133, type, ordere166539214618696060dd_abs: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_134, type, ordere6658533253407199908up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_135, type, semiri3467727345109120633visors: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_136, type, boolea8198339166811842893lgebra: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_137, type, linord8928482502909563296strict: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_138, type, semido2269285787275462019factor: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_139, type, euclid5891614535332579305n_ring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_140, type, linord181362715937106298miring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_141, type, semiri2026040879449505780visors: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_142, type, bit_ri3973907225187159222ations: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_143, type, comple6319245703460814977attice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_144, type, linord4140545234300271783up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_145, type, linord715952674999750819strict: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_146, type, archim462609752435547400_field: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_147, type, comple5582772986160207858norder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_148, type, bounde4346867609351753570nf_top: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_149, type, bounde4967611905675639751up_bot: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_150, type, euclid3725896446679973847miring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_151, type, canoni5634975068530333245id_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_152, type, ordere8940638589300402666id_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_153, type, strict7427464778891057005id_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_154, type, linord2810124833399127020strict: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_155, type, bit_se359711467146920520ations: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_156, type, ordere2412721322843649153imp_le: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_157, type, ordere580206878836729694up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_158, type, ordere1170586879665033532d_diff: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_159, type, strict9044650504122735259up_add: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_160, type, semiri6575147826004484403cancel: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_161, type, euclid8851590272496341667cancel: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_162, type, unique1627219031080169319umeral: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_163, type, comple489889107523837845lgebra: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_164, type, comple592849572758109894attice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_165, type, normal6328177297339901930cative: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_166, type, euclid4440199948858584721cancel: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_167, type, euclid3128863361964157862miring: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_168, type, ordere1937475149494474687imp_le: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_169, type, condit5016429287641298734tinuum: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_170, type, euclid8789492081693882211th_nat: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_171, type, euclid5411537665997757685th_nat: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_172, type, semiri1453513574482234551roduct: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_173, type, bit_un5681908812861735899ations: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_174, type, condit1219197933456340205attice: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_175, type, condit6923001295902523014norder: !>[X0: $tType]:($o)).
% 87.48/29.63 thf(func_def_176, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 87.48/29.63 thf(func_def_177, type, archim6421214686448440834_floor: !>[X0: $tType]:(X0 > int)).
% 87.48/29.63 thf(func_def_178, type, archimedean_frac: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_179, type, archimedean_round: !>[X0: $tType]:(X0 > int)).
% 87.48/29.63 thf(func_def_180, type, array_alloc: !>[X0: $tType]:(list @ X0 > heap_ext @ product_unit > product_prod @ array @ X0 @ heap_ext @ product_unit)).
% 87.48/29.63 thf(func_def_181, type, array_freeze: !>[X0: $tType]:(array @ X0 > heap_Time_Heap @ list @ X0)).
% 87.48/29.63 thf(func_def_182, type, array_get: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_183, type, array_len: !>[X0: $tType]:(array @ X0 > heap_Time_Heap @ nat)).
% 87.48/29.63 thf(func_def_184, type, array_length: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > nat)).
% 87.48/29.63 thf(func_def_185, type, array_make: !>[X0: $tType]:(nat > (nat > X0) > heap_Time_Heap @ array @ X0)).
% 87.48/29.63 thf(func_def_186, type, array_map_entry: !>[X0: $tType]:(nat > (X0 > X0) > array @ X0 > heap_Time_Heap @ array @ X0)).
% 87.48/29.63 thf(func_def_187, type, array_new: !>[X0: $tType]:(nat > X0 > heap_Time_Heap @ array @ X0)).
% 87.48/29.63 thf(func_def_188, type, array_noteq: !>[X0: $tType, X1: $tType]:(array @ X0 > array @ X1 > $o)).
% 87.48/29.63 thf(func_def_189, type, array_nth: !>[X0: $tType]:(array @ X0 > nat > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_190, type, array_of_list: !>[X0: $tType]:(list @ X0 > heap_Time_Heap @ array @ X0)).
% 87.48/29.63 thf(func_def_191, type, array_present: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > $o)).
% 87.48/29.63 thf(func_def_192, type, array_set: !>[X0: $tType]:(array @ X0 > list @ X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 87.48/29.63 thf(func_def_193, type, array_swap: !>[X0: $tType]:(nat > X0 > array @ X0 > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_194, type, array_upd: !>[X0: $tType]:(nat > X0 > array @ X0 > heap_Time_Heap @ array @ X0)).
% 87.48/29.63 thf(func_def_195, type, array_update: !>[X0: $tType]:(array @ X0 > nat > X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 87.48/29.63 thf(func_def_196, type, bNF_Ca8665028551170535155natLeq: set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_197, type, bNF_Ca8459412986667044542atLess: set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_198, type, bNF_Ca3754400796208372196lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_199, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 87.48/29.63 thf(func_def_200, type, bNF_Wellorder_bsqr: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ product_prod @ X0 @ X0 @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_201, type, bNF_We413866401316099525erIncl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 87.48/29.63 thf(func_def_202, type, bNF_Wellorder_ordIso: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X1 @ X1)).
% 87.48/29.63 thf(func_def_203, type, bNF_We4044943003108391690rdLess: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X1 @ X1)).
% 87.48/29.63 thf(func_def_204, type, bNF_Wellorder_embed: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_205, type, bNF_Wellorder_embedS: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_206, type, bNF_Wellorder_iso: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_207, type, bNF_Wellorder_wo_rel: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 87.48/29.63 thf(func_def_208, type, basic_BNF_size_prod: !>[X0: $tType, X1: $tType]:((X0 > nat) > (X1 > nat) > product_prod @ X0 @ X1 > nat)).
% 87.48/29.63 thf(func_def_209, type, basic_BNF_size_sum: !>[X0: $tType, X1: $tType]:((X0 > nat) > (X1 > nat) > sum_sum @ X0 @ X1 > nat)).
% 87.48/29.63 thf(func_def_210, type, basic_pred_prod: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > $o) > product_prod @ X0 @ X1 > $o)).
% 87.48/29.63 thf(func_def_211, type, binomial: nat > nat > nat).
% 87.48/29.63 thf(func_def_212, type, gbinomial: !>[X0: $tType]:(X0 > nat > X0)).
% 87.48/29.63 thf(func_def_213, type, bit_and_int_rel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 87.48/29.63 thf(func_def_214, type, bit_and_not_num: num > num > option @ num).
% 87.48/29.63 thf(func_def_215, type, bit_concat_bit: nat > int > int > int).
% 87.48/29.63 thf(func_def_216, type, bit_or_not_num_neg: num > num > num).
% 87.48/29.63 thf(func_def_217, type, bit_ri4277139882892585799ns_not: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_218, type, bit_ri4674362597316999326ke_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_219, type, bit_se5824344872417868541ns_and: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_220, type, bit_se4197421643247451524op_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_221, type, bit_se8732182000553998342ip_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_222, type, bit_se2239418461657761734s_mask: !>[X0: $tType]:(nat > X0)).
% 87.48/29.63 thf(func_def_223, type, bit_se1065995026697491101ons_or: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_224, type, bit_se4730199178511100633sh_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_225, type, bit_se5668285175392031749et_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_226, type, bit_se2584673776208193580ke_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_227, type, bit_se2638667681897837118et_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_228, type, bit_se5824344971392196577ns_xor: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_229, type, bit_se5641148757651400278ts_bit: !>[X0: $tType]:(X0 > nat > $o)).
% 87.48/29.63 thf(func_def_230, type, bit_se6407376104438227557le_bit: !>[X0: $tType]:(itself @ X0 > nat > $o)).
% 87.48/29.63 thf(func_def_231, type, bit_take_bit_num: nat > num > option @ num).
% 87.48/29.63 thf(func_def_232, type, bit_un7362597486090784418nd_num: num > num > option @ num).
% 87.48/29.63 thf(func_def_233, type, bit_un2480387367778600638or_num: num > num > option @ num).
% 87.48/29.63 thf(func_def_234, type, boolea2506097494486148201lgebra: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_235, type, boolea6902313364301356556axioms: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_236, type, boolea3799213064322606851m_diff: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > X0 > X0 > (X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_237, type, boolea5476839437570043046axioms: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > (X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_238, type, code_Nat: code_integer > code_natural).
% 87.48/29.63 thf(func_def_239, type, code_Neg: num > code_integer).
% 87.48/29.63 thf(func_def_240, type, code_Pos: num > code_integer).
% 87.48/29.63 thf(func_def_241, type, code_Suc: code_natural > code_natural).
% 87.48/29.63 thf(func_def_242, type, code_bit_cut_integer: code_integer > product_prod @ code_integer @ $o).
% 87.48/29.63 thf(func_def_243, type, code_cr_integer: int > code_integer > $o).
% 87.48/29.63 thf(func_def_244, type, code_cr_natural: nat > code_natural > $o).
% 87.48/29.63 thf(func_def_245, type, code_divmod_abs: code_integer > code_integer > product_prod @ code_integer @ code_integer).
% 87.48/29.63 thf(func_def_246, type, code_divmod_integer: code_integer > code_integer > product_prod @ code_integer @ code_integer).
% 87.48/29.63 thf(func_def_247, type, code_dup: code_integer > code_integer).
% 87.48/29.63 thf(func_def_248, type, code_int_of_integer: code_integer > int).
% 87.48/29.63 thf(func_def_249, type, code_integer_of_int: int > code_integer).
% 87.48/29.63 thf(func_def_250, type, code_integer_of_nat: nat > code_integer).
% 87.48/29.63 thf(func_def_251, type, code_i5400310926305786745atural: code_natural > code_integer).
% 87.48/29.63 thf(func_def_252, type, code_integer_of_num: num > code_integer).
% 87.48/29.63 thf(func_def_253, type, code_nat_of_integer: code_integer > nat).
% 87.48/29.63 thf(func_def_254, type, code_case_natural: !>[X0: $tType]:(X0 > (code_natural > X0) > code_natural > X0)).
% 87.48/29.63 thf(func_def_255, type, code_nat_of_natural: code_natural > nat).
% 87.48/29.63 thf(func_def_256, type, code_natural_of_nat: nat > code_natural).
% 87.48/29.63 thf(func_def_257, type, code_rec_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0)).
% 87.48/29.63 thf(func_def_258, type, code_rec_set_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0 > $o)).
% 87.48/29.63 thf(func_def_259, type, code_n4118661773612635043nteger: code_integer > code_natural).
% 87.48/29.63 thf(func_def_260, type, code_negative: num > code_integer).
% 87.48/29.63 thf(func_def_261, type, code_num_of_integer: code_integer > num).
% 87.48/29.63 thf(func_def_262, type, code_pcr_integer: int > code_integer > $o).
% 87.48/29.63 thf(func_def_263, type, code_pcr_natural: nat > code_natural > $o).
% 87.48/29.63 thf(func_def_264, type, code_positive: num > code_integer).
% 87.48/29.63 thf(func_def_265, type, code_size_nat: code_natural > nat).
% 87.48/29.63 thf(func_def_266, type, code_sub: num > num > code_integer).
% 87.48/29.63 thf(func_def_267, type, complete_Inf_Inf: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_268, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_269, type, comple1908693960933563346ssible: !>[X0: $tType]:((set @ X0 > X0) > (X0 > X0 > $o) > (X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_270, type, comple187402453842119260l_fixp: !>[X0: $tType]:((set @ X0 > X0) > (X0 > X0 > $o) > (X0 > X0) > X0)).
% 87.48/29.63 thf(func_def_271, type, comple115746919287870866o_fixp: !>[X0: $tType]:((X0 > X0) > X0)).
% 87.48/29.63 thf(func_def_272, type, comple6359979572994053840erates: !>[X0: $tType]:((X0 > X0) > set @ X0)).
% 87.48/29.63 thf(func_def_273, type, comple7512665784863727008ratesp: !>[X0: $tType]:((X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_274, type, comple1602240252501008431_chain: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_275, type, comple7038119648293358887notone: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X1 > $o) > (X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_276, type, condit941137186595557371_above: !>[X0: $tType]:(set @ X0 > $o)).
% 87.48/29.63 thf(func_def_277, type, condit1013018076250108175_below: !>[X0: $tType]:(set @ X0 > $o)).
% 87.48/29.63 thf(func_def_278, type, condit622319405099724424ng_bdd: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_279, type, condit16957441358409770ng_bdd: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_280, type, from_nat: !>[X0: $tType]:(nat > X0)).
% 87.48/29.63 thf(func_def_281, type, nth_item: !>[X0: $tType]:(nat > set @ old_node @ X0 @ product_unit)).
% 87.48/29.63 thf(func_def_282, type, nth_item_rel: nat > nat > $o).
% 87.48/29.63 thf(func_def_283, type, adjust_div: product_prod @ int @ int > int).
% 87.48/29.63 thf(func_def_284, type, adjust_mod: int > int > int).
% 87.48/29.63 thf(func_def_285, type, divmod_nat: nat > nat > product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_286, type, eucl_rel_int: int > int > product_prod @ int @ int > $o).
% 87.48/29.63 thf(func_def_287, type, unique5940410009612947441es_aux: !>[X0: $tType]:(product_prod @ X0 @ X0 > $o)).
% 87.48/29.63 thf(func_def_288, type, unique8689654367752047608divmod: !>[X0: $tType]:(num > num > product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_289, type, unique1321980374590559556d_step: !>[X0: $tType]:(num > product_prod @ X0 @ X0 > product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_290, type, euclid6346220572633701492n_size: !>[X0: $tType]:(X0 > nat)).
% 87.48/29.63 thf(func_def_291, type, euclid7384307370059645450egment: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_292, type, comm_s3205402744901411588hammer: !>[X0: $tType]:(X0 > nat > X0)).
% 87.48/29.63 thf(func_def_293, type, semiring_char_0_fact: !>[X0: $tType]:(nat > X0)).
% 87.48/29.63 thf(func_def_294, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_295, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 87.48/29.63 thf(func_def_296, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 87.48/29.63 thf(func_def_297, type, finite_fold: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_298, type, finite_folding_on: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1 > X1) > $o)).
% 87.48/29.63 thf(func_def_299, type, finite_folding_F: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_300, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 87.48/29.63 thf(func_def_301, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 87.48/29.63 thf(func_def_302, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 87.48/29.63 thf(func_def_303, type, id: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_304, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_305, type, map_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X2) > X0 > X3)).
% 87.48/29.63 thf(func_def_306, type, fun_is_measure: !>[X0: $tType]:((X0 > nat) > $o)).
% 87.48/29.63 thf(func_def_307, type, fun_max_strict: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_308, type, fun_max_weak: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_309, type, fun_min_strict: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_310, type, fun_min_weak: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_311, type, fun_pair_leq: set @ product_prod @ product_prod @ nat @ nat @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_312, type, fun_pair_less: set @ product_prod @ product_prod @ nat @ nat @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_313, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 87.48/29.63 thf(func_def_314, 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)).
% 87.48/29.63 thf(func_def_315, type, gcd_Gcd: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_316, type, gcd_Lcm: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_317, type, bezw: nat > nat > product_prod @ int @ int).
% 87.48/29.63 thf(func_def_318, type, bezw_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 87.48/29.63 thf(func_def_319, type, bounde8507323023520639062attice: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > (X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_320, type, bounde6485984586167503788ce_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > (X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_321, type, bounde2362111253966948842tice_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_322, type, gcd_gcd: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_323, type, gcd_lcm: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_324, type, gcd_nat_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 87.48/29.63 thf(func_def_325, type, semiri4206861660011772517g_char: !>[X0: $tType]:(itself @ X0 > nat)).
% 87.48/29.63 thf(func_def_326, type, semiring_gcd_Gcd_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_327, type, semiring_gcd_Lcm_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_328, type, abel_semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_329, type, abel_s757365448890700780axioms: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_330, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_331, type, comm_monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_332, type, comm_monoid_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_333, type, group: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_334, type, group_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_335, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_336, type, monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_337, type, monoid_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_338, type, one_one: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(func_def_339, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_340, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_341, type, sgn_sgn: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_342, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_343, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_344, type, zero_zero: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(func_def_345, type, groups7311177749621191930dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_346, type, groups1027152243600224163dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_347, type, groups7121269368397514597t_prod: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_348, type, groups1962203154675924110t_prod: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 87.48/29.63 thf(func_def_349, type, groups778175481326437816id_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_350, type, groups_comm_monoid_F: !>[X0: $tType, X1: $tType]:((X0 > X0 > X0) > X0 > (X1 > X0) > set @ X1 > X0)).
% 87.48/29.63 thf(func_def_351, type, groups_comm_monoid_G: !>[X0: $tType, X1: $tType]:((X0 > X0 > X0) > X0 > (X1 > X0) > set @ X1 > X0)).
% 87.48/29.63 thf(func_def_352, type, groups1828464146339083142d_list: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_353, type, groups4802862169904069756st_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_354, type, groups4207007520872428315er_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1 > list @ X0 > X1)).
% 87.48/29.63 thf(func_def_355, type, groups8242544230860333062m_list: !>[X0: $tType]:(list @ X0 > X0)).
% 87.48/29.63 thf(func_def_356, type, groups_monoid_list: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_357, type, groups_monoid_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > list @ X0 > X0)).
% 87.48/29.63 thf(func_def_358, type, groups5270119922927024881d_list: !>[X0: $tType]:(list @ X0 > X0)).
% 87.48/29.63 thf(func_def_359, type, nO_MATCH: !>[X0: $tType, X1: $tType]:(X0 > X1 > $o)).
% 87.48/29.63 thf(func_def_360, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_361, type, uniq: !>[X0: $tType]:((X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_362, type, undefined: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(func_def_363, type, addr_of_array: !>[X0: $tType]:(array @ X0 > nat)).
% 87.48/29.63 thf(func_def_364, type, array2: !>[X0: $tType]:(nat > array @ X0)).
% 87.48/29.63 thf(func_def_365, type, size_array: !>[X0: $tType]:((X0 > nat) > array @ X0 > nat)).
% 87.48/29.63 thf(func_def_366, type, empty: heap_ext @ product_unit).
% 87.48/29.63 thf(func_def_367, type, arrays: !>[X0: $tType]:(heap_ext @ X0 > typerep > nat > list @ nat)).
% 87.48/29.63 thf(func_def_368, type, heap_ext2: !>[X0: $tType]:((typerep > nat > list @ nat) > (typerep > nat > nat) > nat > X0 > heap_ext @ X0)).
% 87.48/29.63 thf(func_def_369, type, lim: !>[X0: $tType]:(heap_ext @ X0 > nat)).
% 87.48/29.63 thf(func_def_370, type, lim_update: !>[X0: $tType]:((nat > nat) > heap_ext @ X0 > heap_ext @ X0)).
% 87.48/29.63 thf(func_def_371, 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)).
% 87.48/29.63 thf(func_def_372, 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)).
% 87.48/29.63 thf(func_def_373, type, heap_Time_map_Heap: !>[X0: $tType, X1: $tType]:((X0 > X1) > heap_Time_Heap @ X0 > heap_Time_Heap @ X1)).
% 87.48/29.63 thf(func_def_374, type, heap_Time_pred_Heap: !>[X0: $tType]:((X0 > $o) > heap_Time_Heap @ X0 > $o)).
% 87.48/29.63 thf(func_def_375, 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)).
% 87.48/29.63 thf(func_def_376, type, heap_Time_rel_Heap: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > heap_Time_Heap @ X0 > heap_Time_Heap @ X1 > $o)).
% 87.48/29.63 thf(func_def_377, type, heap_Time_set_Heap: !>[X0: $tType]:(heap_Time_Heap @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_378, type, heap_Time_size_Heap: !>[X0: $tType]:((X0 > nat) > heap_Time_Heap @ X0 > nat)).
% 87.48/29.63 thf(func_def_379, type, heap_Time_Heap_lub: !>[X0: $tType]:(set @ heap_Time_Heap @ X0 > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_380, type, heap_Time_Heap_ord: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_Time_Heap @ X0 > $o)).
% 87.48/29.63 thf(func_def_381, type, heap_Time_assert: !>[X0: $tType]:((X0 > $o) > X0 > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_382, type, heap_Time_bind: !>[X0: $tType, X1: $tType]:(heap_Time_Heap @ X0 > (X0 > heap_Time_Heap @ X1) > heap_Time_Heap @ X1)).
% 87.48/29.63 thf(func_def_383, type, heap_Time_effect: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > heap_ext @ product_unit > X0 > nat > $o)).
% 87.48/29.63 thf(func_def_384, 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)).
% 87.48/29.63 thf(func_def_385, 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)).
% 87.48/29.63 thf(func_def_386, type, heap_Time_heap: !>[X0: $tType]:((heap_ext @ product_unit > product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_387, type, heap_Time_lift: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > heap_Time_Heap @ X1)).
% 87.48/29.63 thf(func_def_388, type, heap_Time_raise: !>[X0: $tType]:(list @ char > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_389, type, heap_Time_return: !>[X0: $tType]:(X0 > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_390, type, heap_Time_success: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > $o)).
% 87.48/29.63 thf(func_def_391, type, heap_Time_tap: !>[X0: $tType]:((heap_ext @ product_unit > X0) > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_392, 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)).
% 87.48/29.63 thf(func_def_393, 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)).
% 87.48/29.63 thf(func_def_394, type, heap_Time_ureturn: !>[X0: $tType]:(X0 > heap_Time_Heap @ X0)).
% 87.48/29.63 thf(func_def_395, type, heap_Time_wait: nat > heap_Time_Heap @ product_unit).
% 87.48/29.63 thf(func_def_396, type, hilbert_bijection: !>[X0: $tType]:((X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_397, type, hilbert_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 87.48/29.63 thf(func_def_398, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_399, type, complete_lattice_gfp: !>[X0: $tType]:((X0 > X0) > X0)).
% 87.48/29.63 thf(func_def_400, type, complete_lattice_lfp: !>[X0: $tType]:((X0 > X0) > X0)).
% 87.48/29.63 thf(func_def_401, type, abs_Integ: product_prod @ nat @ nat > int).
% 87.48/29.63 thf(func_def_402, type, neg: num > int).
% 87.48/29.63 thf(func_def_403, type, pos: num > int).
% 87.48/29.63 thf(func_def_404, type, rep_Integ: int > product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_405, type, cr_int: product_prod @ nat @ nat > int > $o).
% 87.48/29.63 thf(func_def_406, type, dup: int > int).
% 87.48/29.63 thf(func_def_407, type, abs_int: set @ product_prod @ nat @ nat > int).
% 87.48/29.63 thf(func_def_408, type, rep_int: int > set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_409, type, int_ge_less_than: int > set @ product_prod @ int @ int).
% 87.48/29.63 thf(func_def_410, type, int_ge_less_than2: int > set @ product_prod @ int @ int).
% 87.48/29.63 thf(func_def_411, type, intrel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 87.48/29.63 thf(func_def_412, type, nat2: int > nat).
% 87.48/29.63 thf(func_def_413, type, pcr_int: product_prod @ nat @ nat > int > $o).
% 87.48/29.63 thf(func_def_414, type, power_int: !>[X0: $tType]:(X0 > int > X0)).
% 87.48/29.63 thf(func_def_415, type, ring_1_Ints: !>[X0: $tType]:(set @ X0)).
% 87.48/29.63 thf(func_def_416, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 87.48/29.63 thf(func_def_417, type, sub: num > num > int).
% 87.48/29.63 thf(func_def_418, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_419, type, semilattice: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_420, type, semilattice_axioms: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_421, type, semilattice_neutr: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_422, type, semila1105856199041335345_order: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_423, type, semilattice_order: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_424, type, semila6385135966242565138axioms: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_425, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_426, type, lattices_Max: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_427, type, lattices_Min: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_428, type, lattic643756798349783984er_Max: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_429, type, lattic643756798350308766er_Min: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_430, type, lattices_ord_arg_max: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_431, type, lattic1883929316492267755max_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_432, type, lattices_ord_arg_min: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_433, type, lattic7623131987881927897min_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_434, type, lattic501386751176901750rg_max: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_435, type, lattic501386751177426532rg_min: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_436, type, lattic7752659483105999362nf_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_437, type, lattic5652469242046573047tr_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 87.48/29.63 thf(func_def_438, type, lattic5214292709420241887eutr_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_439, type, lattic3600114342068043075tr_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_440, type, lattic4895041142388067077er_set: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_441, type, lattic149705377957585745ce_set: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_442, type, lattic1715443433743089157tice_F: !>[X0: $tType]:((X0 > X0 > X0) > set @ X0 > X0)).
% 87.48/29.63 thf(func_def_443, type, lattic5882676163264333800up_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 87.48/29.63 thf(func_def_444, type, quotient: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > (X0 > X1 > $o) > $o)).
% 87.48/29.63 thf(func_def_445, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_446, type, arg_min_list: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > X0)).
% 87.48/29.63 thf(func_def_447, type, arg_min_list_rel: !>[X0: $tType, X1: $tType]:(product_prod @ (X0 > X1) @ list @ X0 > product_prod @ (X0 > X1) @ list @ X0 > $o)).
% 87.48/29.63 thf(func_def_448, type, butlast: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_449, type, concat: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_450, type, coset: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_451, type, count_list: !>[X0: $tType]:(list @ X0 > X0 > nat)).
% 87.48/29.63 thf(func_def_452, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 87.48/29.63 thf(func_def_453, type, drop: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_454, type, dropWhile: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_455, type, enumerate: !>[X0: $tType]:(nat > list @ X0 > list @ product_prod @ nat @ X0)).
% 87.48/29.63 thf(func_def_456, type, filter: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_457, type, find: !>[X0: $tType]:((X0 > $o) > list @ X0 > option @ X0)).
% 87.48/29.63 thf(func_def_458, type, fold: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > list @ X0 > X1 > X1)).
% 87.48/29.63 thf(func_def_459, type, folding_insort_key: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > set @ X1 > (X1 > X0) > $o)).
% 87.48/29.63 thf(func_def_460, type, foldr: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > list @ X0 > X1 > X1)).
% 87.48/29.63 thf(func_def_461, type, gen_length: !>[X0: $tType]:(nat > list @ X0 > nat)).
% 87.48/29.63 thf(func_def_462, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 87.48/29.63 thf(func_def_463, type, lenlex: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 87.48/29.63 thf(func_def_464, type, lex: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 87.48/29.63 thf(func_def_465, type, lexord: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 87.48/29.63 thf(func_def_466, type, linord329482645794927042rt_key: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_467, type, linorder_insort_key: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_468, type, linorder_sort_key: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_469, type, linord4507533701916653071of_set: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_470, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_471, type, nil: !>[X0: $tType]:(list @ X0)).
% 87.48/29.63 thf(func_def_472, type, hd: !>[X0: $tType]:(list @ X0 > X0)).
% 87.48/29.63 thf(func_def_473, type, list_all: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 87.48/29.63 thf(func_def_474, type, list_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > list @ X0 > list @ X1 > $o)).
% 87.48/29.63 thf(func_def_475, type, map: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1)).
% 87.48/29.63 thf(func_def_476, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_477, type, size_list: !>[X0: $tType]:((X0 > nat) > list @ X0 > nat)).
% 87.48/29.63 thf(func_def_478, type, tl: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_479, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 87.48/29.63 thf(func_def_480, type, list_update: !>[X0: $tType]:(list @ X0 > nat > X0 > list @ X0)).
% 87.48/29.63 thf(func_def_481, type, listrel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ list @ X0 @ list @ X1)).
% 87.48/29.63 thf(func_def_482, type, listrel1: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 87.48/29.63 thf(func_def_483, type, measures: !>[X0: $tType]:(list @ (X0 > nat) > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_484, type, n_lists: !>[X0: $tType]:(nat > list @ X0 > list @ list @ X0)).
% 87.48/29.63 thf(func_def_485, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 87.48/29.63 thf(func_def_486, type, nths: !>[X0: $tType]:(list @ X0 > set @ nat > list @ X0)).
% 87.48/29.63 thf(func_def_487, type, ord_lexordp: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 87.48/29.63 thf(func_def_488, type, ord_lexordp_eq: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 87.48/29.63 thf(func_def_489, type, product: !>[X0: $tType, X1: $tType]:(list @ X0 > list @ X1 > list @ product_prod @ X0 @ X1)).
% 87.48/29.63 thf(func_def_490, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 87.48/29.63 thf(func_def_491, type, remdups: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_492, type, remdups_adj: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_493, type, remove1: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_494, type, removeAll: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_495, type, replicate: !>[X0: $tType]:(nat > X0 > list @ X0)).
% 87.48/29.63 thf(func_def_496, type, rev: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_497, type, rotate: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_498, type, rotate1: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_499, type, sorted_wrt: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 87.48/29.63 thf(func_def_500, type, take: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_501, type, takeWhile: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 87.48/29.63 thf(func_def_502, type, transpose: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 87.48/29.63 thf(func_def_503, type, upt: nat > nat > list @ nat).
% 87.48/29.63 thf(func_def_504, type, upto: int > int > list @ int).
% 87.48/29.63 thf(func_def_505, type, upto_aux: int > int > list @ int > list @ int).
% 87.48/29.63 thf(func_def_506, type, upto_rel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 87.48/29.63 thf(func_def_507, type, zip: !>[X0: $tType, X1: $tType]:(list @ X0 > list @ X1 > list @ product_prod @ X0 @ X1)).
% 87.48/29.63 thf(func_def_508, type, map_of: !>[X0: $tType, X1: $tType]:(list @ product_prod @ X0 @ X1 > X0 > option @ X1)).
% 87.48/29.63 thf(func_def_509, type, map_upds: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > list @ X0 > list @ X1 > X0 > option @ X1)).
% 87.48/29.63 thf(func_def_510, type, restrict_map: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X0 > X0 > option @ X1)).
% 87.48/29.63 thf(func_def_511, type, suc: nat > nat).
% 87.48/29.63 thf(func_def_512, type, compow: !>[X0: $tType]:(nat > X0 > X0)).
% 87.48/29.63 thf(func_def_513, type, funpow: !>[X0: $tType]:(nat > (X0 > X0) > X0 > X0)).
% 87.48/29.63 thf(func_def_514, type, case_nat: !>[X0: $tType]:(X0 > (nat > X0) > nat > X0)).
% 87.48/29.63 thf(func_def_515, type, pred: nat > nat).
% 87.48/29.63 thf(func_def_516, type, rec_nat: !>[X0: $tType]:(X0 > (nat > X0 > X0) > nat > X0)).
% 87.48/29.63 thf(func_def_517, type, rec_set_nat: !>[X0: $tType]:(X0 > (nat > X0 > X0) > nat > X0 > $o)).
% 87.48/29.63 thf(func_def_518, type, semiring_1_Nats: !>[X0: $tType]:(set @ X0)).
% 87.48/29.63 thf(func_def_519, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 87.48/29.63 thf(func_def_520, type, semiri8178284476397505188at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 87.48/29.63 thf(func_def_521, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 87.48/29.63 thf(func_def_522, type, nat_int_decode: nat > int).
% 87.48/29.63 thf(func_def_523, type, nat_int_encode: int > nat).
% 87.48/29.63 thf(func_def_524, type, nat_list_decode: nat > list @ nat).
% 87.48/29.63 thf(func_def_525, type, nat_list_decode_rel: nat > nat > $o).
% 87.48/29.63 thf(func_def_526, type, nat_list_encode: list @ nat > nat).
% 87.48/29.63 thf(func_def_527, type, nat_list_encode_rel: list @ nat > list @ nat > $o).
% 87.48/29.63 thf(func_def_528, type, nat_prod_decode: nat > product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_529, type, nat_prod_decode_aux: nat > nat > product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_530, type, nat_pr5047031295181774490ux_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 87.48/29.63 thf(func_def_531, type, nat_prod_encode: product_prod @ nat @ nat > nat).
% 87.48/29.63 thf(func_def_532, type, nat_set_decode: nat > set @ nat).
% 87.48/29.63 thf(func_def_533, type, nat_set_encode: set @ nat > nat).
% 87.48/29.63 thf(func_def_534, type, nat_sum_decode: nat > sum_sum @ nat @ nat).
% 87.48/29.63 thf(func_def_535, type, nat_sum_encode: sum_sum @ nat @ nat > nat).
% 87.48/29.63 thf(func_def_536, type, nat_triangle: nat > nat).
% 87.48/29.63 thf(func_def_537, type, bitM: num > num).
% 87.48/29.63 thf(func_def_538, type, inc: num > num).
% 87.48/29.63 thf(func_def_539, type, nat_of_num: num > nat).
% 87.48/29.63 thf(func_def_540, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_541, type, neg_numeral_dbl_dec: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_542, type, neg_numeral_dbl_inc: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_543, type, neg_numeral_is_num: !>[X0: $tType]:(X0 > $o)).
% 87.48/29.63 thf(func_def_544, type, neg_numeral_sub: !>[X0: $tType]:(num > num > X0)).
% 87.48/29.63 thf(func_def_545, type, bit0: num > num).
% 87.48/29.63 thf(func_def_546, type, bit1: num > num).
% 87.48/29.63 thf(func_def_547, type, one2: num).
% 87.48/29.63 thf(func_def_548, type, case_num: !>[X0: $tType]:(X0 > (num > X0) > (num > X0) > num > X0)).
% 87.48/29.63 thf(func_def_549, type, rec_num: !>[X0: $tType]:(X0 > (num > X0 > X0) > (num > X0 > X0) > num > X0)).
% 87.48/29.63 thf(func_def_550, type, size_num: num > nat).
% 87.48/29.63 thf(func_def_551, type, num_of_nat: nat > num).
% 87.48/29.63 thf(func_def_552, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 87.48/29.63 thf(func_def_553, type, pow: num > num > num).
% 87.48/29.63 thf(func_def_554, type, pred_numeral: num > nat).
% 87.48/29.63 thf(func_def_555, type, ring_1_iszero: !>[X0: $tType]:(X0 > $o)).
% 87.48/29.63 thf(func_def_556, type, sqr: num > num).
% 87.48/29.63 thf(func_def_557, type, old_Atom: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ nat > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_558, type, old_In0: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_559, type, old_In1: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_560, type, old_Leaf: !>[X0: $tType, X1: $tType]:(X0 > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_561, type, old_Node: !>[X0: $tType, X1: $tType]:(set @ product_prod @ (nat > sum_sum @ X0 @ nat) @ sum_sum @ X1 @ nat)).
% 87.48/29.63 thf(func_def_562, type, old_Numb: !>[X0: $tType, X1: $tType]:(nat > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_563, type, old_Push: !>[X0: $tType]:(sum_sum @ X0 @ nat > (nat > sum_sum @ X0 @ nat) > nat > sum_sum @ X0 @ nat)).
% 87.48/29.63 thf(func_def_564, type, old_Push_Node: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ nat > old_node @ X1 @ X0 > old_node @ X1 @ X0)).
% 87.48/29.63 thf(func_def_565, type, old_Scons: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_566, type, old_ndepth: !>[X0: $tType, X1: $tType]:(old_node @ X0 @ X1 > nat)).
% 87.48/29.63 thf(func_def_567, type, old_Abs_Node: !>[X0: $tType, X1: $tType]:(product_prod @ (nat > sum_sum @ X0 @ nat) @ sum_sum @ X1 @ nat > old_node @ X1 @ X0)).
% 87.48/29.63 thf(func_def_568, type, old_Rep_Node: !>[X0: $tType, X1: $tType]:(old_node @ X0 @ X1 > product_prod @ (nat > sum_sum @ X1 @ nat) @ sum_sum @ X0 @ nat)).
% 87.48/29.63 thf(func_def_569, type, old_ntrunc: !>[X0: $tType, X1: $tType]:(nat > set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 87.48/29.63 thf(func_def_570, type, none: !>[X0: $tType]:(option @ X0)).
% 87.48/29.63 thf(func_def_571, type, some: !>[X0: $tType]:(X0 > option @ X0)).
% 87.48/29.63 thf(func_def_572, type, case_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 87.48/29.63 thf(func_def_573, type, size_option: !>[X0: $tType]:((X0 > nat) > option @ X0 > nat)).
% 87.48/29.63 thf(func_def_574, type, the2: !>[X0: $tType]:(option @ X0 > X0)).
% 87.48/29.63 thf(func_def_575, type, order_ofilter: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_576, type, order_under: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_577, type, order_underS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_578, type, order_well_order_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 87.48/29.63 thf(func_def_579, type, bot_bot: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(func_def_580, type, least: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_581, type, max: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_582, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_583, type, ord_Least: !>[X0: $tType]:((X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_584, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_585, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_586, type, ord_max: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_587, type, ord_min: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_588, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_589, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_590, type, order_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 87.48/29.63 thf(func_def_591, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_592, type, ordering_axioms: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_593, type, ordering_top: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_594, type, ordering_top_axioms: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_595, type, partial_preordering: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_596, type, preordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_597, type, preordering_axioms: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_598, type, top_top: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(func_def_599, type, partial_fun_lub: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ X0 > X1) > set @ (X2 > X0) > X2 > X1)).
% 87.48/29.63 thf(func_def_600, type, partial_fun_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X0) > (X2 > X1) > $o)).
% 87.48/29.63 thf(func_def_601, type, partia7178651479351089652itions: !>[X0: $tType]:((X0 > X0 > $o) > (set @ X0 > X0) > $o)).
% 87.48/29.63 thf(func_def_602, type, power2: !>[X0: $tType]:(X0 > (X0 > X0 > X0) > X0 > nat > X0)).
% 87.48/29.63 thf(func_def_603, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 87.48/29.63 thf(func_def_604, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 87.48/29.63 thf(func_def_605, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 87.48/29.63 thf(func_def_606, type, product_Unity: product_unit).
% 87.48/29.63 thf(func_def_607, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 87.48/29.63 thf(func_def_608, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 87.48/29.63 thf(func_def_609, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 87.48/29.63 thf(func_def_610, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 87.48/29.63 thf(func_def_611, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 87.48/29.63 thf(func_def_612, type, type2: !>[X0: $tType]:(itself @ X0)).
% 87.48/29.63 thf(func_def_613, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 87.48/29.63 thf(func_def_614, type, quot_abs: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (set @ X0 > X1) > X0 > X1)).
% 87.48/29.63 thf(func_def_615, type, quot_rep: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > X0 > X1)).
% 87.48/29.63 thf(func_def_616, type, log: code_natural > code_natural > code_natural).
% 87.48/29.63 thf(func_def_617, type, log_rel: product_prod @ code_natural @ code_natural > product_prod @ code_natural @ code_natural > $o).
% 87.48/29.63 thf(func_def_618, type, minus_shift: code_natural > code_natural > code_natural > code_natural).
% 87.48/29.63 thf(func_def_619, type, pick: !>[X0: $tType]:(list @ product_prod @ code_natural @ X0 > code_natural > X0)).
% 87.48/29.63 thf(func_def_620, type, range: code_natural > product_prod @ code_natural @ code_natural > product_prod @ code_natural @ product_prod @ code_natural @ code_natural).
% 87.48/29.63 thf(func_def_621, 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)).
% 87.48/29.63 thf(func_def_622, type, abs_Rat: product_prod @ int @ int > rat).
% 87.48/29.63 thf(func_def_623, type, fract: int > int > rat).
% 87.48/29.63 thf(func_def_624, type, frct: product_prod @ int @ int > rat).
% 87.48/29.63 thf(func_def_625, type, rep_Rat: rat > product_prod @ int @ int).
% 87.48/29.63 thf(func_def_626, type, field_char_0_Rats: !>[X0: $tType]:(set @ X0)).
% 87.48/29.63 thf(func_def_627, type, field_char_0_of_rat: !>[X0: $tType]:(rat > X0)).
% 87.48/29.63 thf(func_def_628, type, normalize: product_prod @ int @ int > product_prod @ int @ int).
% 87.48/29.63 thf(func_def_629, type, of_int: int > rat).
% 87.48/29.63 thf(func_def_630, type, pcr_rat: product_prod @ int @ int > rat > $o).
% 87.48/29.63 thf(func_def_631, type, positive: rat > $o).
% 87.48/29.63 thf(func_def_632, type, quotient_of: rat > product_prod @ int @ int).
% 87.48/29.63 thf(func_def_633, type, ratrel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 87.48/29.63 thf(func_def_634, type, tuple_1188178415141063261rphism: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > product_prod @ X1 @ X2) > (product_prod @ X1 @ X2 > X0) > tuple_isomorphism @ X0 @ X1 @ X2)).
% 87.48/29.63 thf(func_def_635, type, tuple_9181185373184732606rphism: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > nat) > (X1 > nat) > (X2 > nat) > tuple_isomorphism @ X0 @ X1 @ X2 > nat)).
% 87.48/29.63 thf(func_def_636, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_637, type, field2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_638, type, id2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_639, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_640, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 87.48/29.63 thf(func_def_641, type, irreflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_642, type, relcomp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X0 @ X2)).
% 87.48/29.63 thf(func_def_643, type, transp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_644, type, algebr8660921524188924756oprime: !>[X0: $tType]:(X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_645, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_646, type, dvd_dvd: !>[X0: $tType]:(X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_647, type, modulo_modulo: !>[X0: $tType]:(X0 > X0 > X0)).
% 87.48/29.63 thf(func_def_648, type, normal6383669964737779283malize: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_649, type, unit_f5069060285200089521factor: !>[X0: $tType]:(X0 > X0)).
% 87.48/29.63 thf(func_def_650, type, zero_neq_one_of_bool: !>[X0: $tType]:($o > X0)).
% 87.48/29.63 thf(func_def_651, type, ball: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_652, type, bex: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 87.48/29.63 thf(func_def_653, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 87.48/29.63 thf(func_def_654, type, pow2: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 87.48/29.63 thf(func_def_655, type, disjnt: !>[X0: $tType]:(set @ X0 > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_656, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 87.48/29.63 thf(func_def_657, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_658, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 87.48/29.63 thf(func_def_659, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 87.48/29.63 thf(func_def_660, type, set_fo6178422350223883121st_nat: !>[X0: $tType]:((nat > X0 > X0) > nat > nat > X0 > X0)).
% 87.48/29.63 thf(func_def_661, 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)).
% 87.48/29.63 thf(func_def_662, type, set_ord_atLeast: !>[X0: $tType]:(X0 > set @ X0)).
% 87.48/29.63 thf(func_def_663, type, set_or1337092689740270186AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_664, type, set_or7035219750837199246ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_665, type, set_ord_atMost: !>[X0: $tType]:(X0 > set @ X0)).
% 87.48/29.63 thf(func_def_666, type, set_ord_greaterThan: !>[X0: $tType]:(X0 > set @ X0)).
% 87.48/29.63 thf(func_def_667, type, set_or3652927894154168847AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_668, type, set_or5935395276787703475ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 87.48/29.63 thf(func_def_669, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 87.48/29.63 thf(func_def_670, type, literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal).
% 87.48/29.63 thf(func_def_671, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 87.48/29.63 thf(func_def_672, type, size_char: char > nat).
% 87.48/29.63 thf(func_def_673, type, comm_s6883823935334413003f_char: !>[X0: $tType]:(char > X0)).
% 87.48/29.63 thf(func_def_674, type, unique5772411509450598832har_of: !>[X0: $tType]:(X0 > char)).
% 87.48/29.63 thf(func_def_675, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 87.48/29.63 thf(func_def_676, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 87.48/29.63 thf(func_def_677, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 87.48/29.63 thf(func_def_678, type, sum_case_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 87.48/29.63 thf(func_def_679, type, bi_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 87.48/29.63 thf(func_def_680, type, transitive_ntrancl: !>[X0: $tType]:(nat > set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_681, type, transitive_rtrancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_682, type, transitive_trancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_683, type, transitive_tranclp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 87.48/29.63 thf(func_def_684, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 87.48/29.63 thf(func_def_685, type, typerep3: literal > list @ typerep > typerep).
% 87.48/29.63 thf(func_def_686, type, size_typerep: typerep > nat).
% 87.48/29.63 thf(func_def_687, type, typerep_typerep: !>[X0: $tType]:(itself @ X0 > typerep)).
% 87.48/29.63 thf(func_def_688, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 87.48/29.63 thf(func_def_689, type, finite_psubset: !>[X0: $tType]:(set @ product_prod @ set @ X0 @ set @ X0)).
% 87.48/29.63 thf(func_def_690, type, less_than: set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_691, 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)).
% 87.48/29.63 thf(func_def_692, type, max_ext: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 87.48/29.63 thf(func_def_693, type, measure: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_694, type, min_ext: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 87.48/29.63 thf(func_def_695, type, mlex_prod: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 87.48/29.63 thf(func_def_696, type, pred_nat: set @ product_prod @ nat @ nat).
% 87.48/29.63 thf(func_def_697, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 87.48/29.63 thf(func_def_698, type, chain_subset: !>[X0: $tType]:(set @ set @ X0 > $o)).
% 87.48/29.63 thf(func_def_699, type, chains: !>[X0: $tType]:(set @ set @ X0 > set @ set @ set @ X0)).
% 87.48/29.63 thf(func_def_700, type, pred_chain: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_701, type, pred_maxchain: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_702, type, pred_suc: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > set @ X0)).
% 87.48/29.63 thf(func_def_703, type, pred_s596693808085603175closed: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ set @ X0)).
% 87.48/29.63 thf(func_def_704, type, pred_s7749564232668923593losedp: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_705, type, fChoice: !>[X0: $tType]:((X0 > $o) > X0)).
% 87.48/29.63 thf(func_def_706, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 87.48/29.63 thf(func_def_707, type, a2: array @ a).
% 87.48/29.63 thf(func_def_708, type, h: heap_ext @ product_unit).
% 87.48/29.63 thf(func_def_709, type, h2: heap_ext @ product_unit).
% 87.48/29.63 thf(func_def_710, type, n: nat).
% 87.48/29.63 thf(func_def_711, type, r: nat).
% 87.48/29.63 thf(func_def_732, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 87.48/29.63 thf(func_def_734, type, ph2: !>[X0: $tType]:(X0)).
% 87.48/29.63 thf(f22700,plain,(
% 87.48/29.63 $false),
% 87.48/29.63 inference(avatar_sat_refutation,[status(thm)],[f21424,f21425,f22406,f22692,f22699])).
% 87.48/29.63 thf(f22699,plain,(
% 87.48/29.63 spl1_1 | ~spl1_2 | ~spl1_4),
% 87.48/29.63 inference(avatar_contradiction_clause,[status(thm)],[f22698])).
% 87.48/29.63 thf(f22698,plain,(
% 87.48/29.63 $false | (spl1_1 | ~spl1_2 | ~spl1_4)),
% 87.48/29.63 inference(subsumption_resolution,[status(thm)],[f22694,f22691])).
% 87.48/29.63 thf(f22691,plain,(
% 87.48/29.63 (r = (array_length @ a @ h @ a2)) | ~spl1_4),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f22677])).
% 87.48/29.63 thf(f22677,plain,(
% 87.48/29.63 ($true != $true) | (r = (array_length @ a @ h @ a2)) | ~spl1_4),
% 87.48/29.63 inference(superposition,[status(thm)],[f21826,f22410])).
% 87.48/29.63 thf(f22410,plain,(
% 87.48/29.63 ( ! [X0 : array @ a,X1 : heap_ext @ product_unit] : (($true = (heap_Time_effect @ nat @ (array_len @ a @ X0) @ X1 @ X1 @ (array_length @ a @ X1 @ X0) @ n))) ) | ~spl1_4),
% 87.48/29.63 inference(backward_demodulation,[status(thm)],[f21732,f21422])).
% 87.48/29.63 thf(f21422,plain,(
% 87.48/29.63 (n = one_one @ nat) | ~spl1_4),
% 87.48/29.63 inference(avatar_component_clause,[status(thm)],[f21421])).
% 87.48/29.63 thf(f21421,definition,(
% 87.48/29.63 spl1_4 <=> (n = one_one @ nat)),
% 87.48/29.63 introduced(definition,[new_symbols(naming,[spl1_4])],[avatar_definition])).
% 87.48/29.63 thf(f21732,plain,(
% 87.48/29.63 ( ! [X0 : array @ a,X1 : heap_ext @ product_unit] : (($true = (heap_Time_effect @ nat @ (array_len @ a @ X0) @ X1 @ X1 @ (array_length @ a @ X1 @ X0) @ one_one @ nat))) )),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f21728])).
% 87.48/29.63 thf(f21728,plain,(
% 87.48/29.63 ( ! [X0 : array @ a,X1 : heap_ext @ product_unit] : (($true = (heap_Time_effect @ nat @ (array_len @ a @ X0) @ X1 @ X1 @ (array_length @ a @ X1 @ X0) @ one_one @ nat)) | ($true != $true)) )),
% 87.48/29.63 inference(superposition,[status(thm)],[f21400,f21368])).
% 87.48/29.63 thf(f21368,plain,(
% 87.48/29.63 (heap @ a = $true)),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f11173])).
% 87.48/29.63 thf(f11173,plain,(
% 87.48/29.63 (heap @ a = $true)),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f11172])).
% 87.48/29.63 thf(f11172,plain,(
% 87.48/29.63 heap @ a),
% 87.48/29.63 inference(rectify,[status(thm)],[f7865])).
% 87.48/29.63 thf(f7865,axiom,(
% 87.48/29.63 heap @ a),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',tfree_0)).
% 87.48/29.63 thf(f21400,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X4 : array @ X0] : ((heap @ X0 != $true) | ($true = (heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X2 @ (array_length @ X0 @ X2 @ X4) @ one_one @ nat))) )),
% 87.48/29.63 inference(equality_resolution,[status(thm)],[f21399])).
% 87.48/29.63 thf(f21399,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X4 : array @ X0,X5 : nat] : ((heap @ X0 != $true) | (one_one @ nat != X5) | ($true = (heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X2 @ (array_length @ X0 @ X2 @ X4) @ X5))) )),
% 87.48/29.63 inference(equality_resolution,[status(thm)],[f21398])).
% 87.48/29.63 thf(f21398,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X1 : heap_ext @ product_unit,X4 : array @ X0,X5 : nat] : ((heap @ X0 != $true) | (X1 != X2) | (one_one @ nat != X5) | ($true = (heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ (array_length @ X0 @ X2 @ X4) @ X5))) )),
% 87.48/29.63 inference(equality_resolution,[status(thm)],[f21356])).
% 87.48/29.63 thf(f21356,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X3 : nat,X1 : heap_ext @ product_unit,X4 : array @ X0,X5 : nat] : ((heap @ X0 != $true) | ((array_length @ X0 @ X2 @ X4) != X3) | (X1 != X2) | (one_one @ nat != X5) | ((heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ X3 @ X5) = $true)) )),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21337])).
% 87.48/29.63 thf(f21337,plain,(
% 87.48/29.63 ! [X0 : $tType] : ((heap @ X0 != $true) | ! [X1 : heap_ext @ product_unit,X2 : heap_ext @ product_unit,X3 : nat,X4 : array @ X0,X5 : nat] : (((array_length @ X0 @ X2 @ X4) != X3) | (X1 != X2) | (one_one @ nat != X5) | ((heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ X3 @ X5) = $true)))),
% 87.48/29.63 inference(flattening,[status(thm)],[f21336])).
% 87.48/29.63 thf(f21336,plain,(
% 87.48/29.63 ! [X0 : $tType] : (! [X1 : heap_ext @ product_unit,X2 : heap_ext @ product_unit,X3 : nat,X4 : array @ X0,X5 : nat] : (((((heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ X3 @ X5) = $true) | (one_one @ nat != X5)) | ((array_length @ X0 @ X2 @ X4) != X3)) | (X1 != X2)) | (heap @ X0 != $true))),
% 87.48/29.63 inference(ennf_transformation,[status(thm)],[f9555])).
% 87.48/29.63 thf(f9555,plain,(
% 87.48/29.63 ! [X0 : $tType] : ((heap @ X0 = $true) => ! [X1 : heap_ext @ product_unit,X2 : heap_ext @ product_unit,X3 : nat,X4 : array @ X0,X5 : nat] : ((X1 = X2) => (((array_length @ X0 @ X2 @ X4) = X3) => ((one_one @ nat = X5) => ((heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ X3 @ X5) = $true)))))),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f9554])).
% 87.48/29.63 thf(f9554,plain,(
% 87.48/29.63 ! [X0 : $tType] : (heap @ X0 => ! [X1 : heap_ext @ product_unit,X2 : heap_ext @ product_unit,X3 : nat,X4 : array @ X0,X5 : nat] : ((X1 = X2) => (((array_length @ X0 @ X2 @ X4) = X3) => ((one_one @ nat = X5) => (heap_Time_effect @ nat @ (array_len @ X0 @ X4) @ X2 @ X1 @ X3 @ X5)))))),
% 87.48/29.63 inference(rectify,[status(thm)],[f3])).
% 87.48/29.63 thf(f3,axiom,(
% 87.48/29.63 ! [X0 : $tType] : (heap @ X0 => ! [X8 : heap_ext @ product_unit,X9 : heap_ext @ product_unit,X10 : nat,X11 : array @ X0,X12 : nat] : ((X8 = X9) => (((array_length @ X0 @ X9 @ X11) = X10) => ((one_one @ nat = X12) => (heap_Time_effect @ nat @ (array_len @ X0 @ X11) @ X9 @ X8 @ X10 @ X12)))))),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_2_effect__lengthI)).
% 87.48/29.63 thf(f21826,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : heap_ext @ product_unit,X1 : nat] : (($true != (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X0 @ X1 @ X2)) | (r = X1)) )),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f21825])).
% 87.48/29.63 thf(f21825,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : heap_ext @ product_unit,X1 : nat] : (($true != (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X0 @ X1 @ X2)) | (r = X1) | ($true != $true)) )),
% 87.48/29.63 inference(superposition,[status(thm)],[f21351,f21379])).
% 87.48/29.63 thf(f21379,plain,(
% 87.48/29.63 ((heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ h2 @ r @ n) = $true)),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f16883])).
% 87.48/29.63 thf(f16883,plain,(
% 87.48/29.63 ((heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ h2 @ r @ n) = $true)),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f16882])).
% 87.48/29.63 thf(f16882,plain,(
% 87.48/29.63 (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ h2 @ r @ n)),
% 87.48/29.63 inference(rectify,[status(thm)],[f1])).
% 87.48/29.63 thf(f1,axiom,(
% 87.48/29.63 (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ h2 @ r @ n)),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_0_assms)).
% 87.48/29.63 thf(f21351,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X1 : heap_Time_Heap @ X0,X8 : nat,X6 : heap_ext @ product_unit,X7 : X0,X4 : X0,X5 : nat] : (($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true) | (X4 = X7)) )),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21348])).
% 87.48/29.63 thf(f21348,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true) | (X4 = X7) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(flattening,[status(thm)],[f21347])).
% 87.48/29.63 thf(f21347,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (((X4 = X7) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true)) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(ennf_transformation,[status(thm)],[f12955])).
% 87.48/29.63 thf(f12955,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (($true = (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) => (((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) = $true) => (X4 = X7)))),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f12954])).
% 87.48/29.63 thf(f12954,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : ((heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5) => ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) => (X4 = X7)))),
% 87.48/29.63 inference(rectify,[status(thm)],[f23])).
% 87.48/29.63 thf(f23,axiom,(
% 87.48/29.63 ! [X0 : $tType,X20 : heap_Time_Heap @ X0,X9 : heap_ext @ product_unit,X8 : heap_ext @ product_unit,X11 : X0,X12 : nat,X21 : heap_ext @ product_unit,X14 : X0,X22 : nat] : ((heap_Time_effect @ X0 @ X20 @ X9 @ X8 @ X11 @ X12) => ((heap_Time_effect @ X0 @ X20 @ X9 @ X21 @ X14 @ X22) => (X11 = X14)))),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_22_effect__deterministic_I1_J)).
% 87.48/29.63 thf(f22694,plain,(
% 87.48/29.63 (r != (array_length @ a @ h @ a2)) | (spl1_1 | ~spl1_2)),
% 87.48/29.63 inference(backward_demodulation,[status(thm)],[f21411,f21414])).
% 87.48/29.63 thf(f21414,plain,(
% 87.48/29.63 (h = h2) | ~spl1_2),
% 87.48/29.63 inference(avatar_component_clause,[status(thm)],[f21413])).
% 87.48/29.63 thf(f21413,definition,(
% 87.48/29.63 spl1_2 <=> (h = h2)),
% 87.48/29.63 introduced(definition,[new_symbols(naming,[spl1_2])],[avatar_definition])).
% 87.48/29.63 thf(f21411,plain,(
% 87.48/29.63 (r != (array_length @ a @ h2 @ a2)) | spl1_1),
% 87.48/29.63 inference(avatar_component_clause,[status(thm)],[f21409])).
% 87.48/29.63 thf(f21409,definition,(
% 87.48/29.63 spl1_1 <=> (r = (array_length @ a @ h2 @ a2))),
% 87.48/29.63 introduced(definition,[new_symbols(naming,[spl1_1])],[avatar_definition])).
% 87.48/29.63 thf(f22692,plain,(
% 87.48/29.63 spl1_2 | ~spl1_4),
% 87.48/29.63 inference(avatar_split_clause,[status(thm)],[f22689,f21421,f21413])).
% 87.48/29.63 thf(f22689,plain,(
% 87.48/29.63 (h = h2) | ~spl1_4),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f22675])).
% 87.48/29.63 thf(f22675,plain,(
% 87.48/29.63 ($true != $true) | (h = h2) | ~spl1_4),
% 87.48/29.63 inference(superposition,[status(thm)],[f21866,f22410])).
% 87.48/29.63 thf(f21866,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : heap_ext @ product_unit,X1 : nat] : (($true != (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X0 @ X1 @ X2)) | (h2 = X0)) )),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f21865])).
% 87.48/29.63 thf(f21865,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : heap_ext @ product_unit,X1 : nat] : (($true != (heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X0 @ X1 @ X2)) | (h2 = X0) | ($true != $true)) )),
% 87.48/29.63 inference(superposition,[status(thm)],[f21360,f21379])).
% 87.48/29.63 thf(f21360,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X1 : heap_Time_Heap @ X0,X8 : nat,X6 : heap_ext @ product_unit,X7 : X0,X4 : X0,X5 : nat] : (($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) | (X3 = X6) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true)) )),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21344])).
% 87.48/29.63 thf(f21344,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : ((X3 = X6) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(flattening,[status(thm)],[f21343])).
% 87.48/29.63 thf(f21343,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (((X3 = X6) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true)) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(ennf_transformation,[status(thm)],[f18887])).
% 87.48/29.63 thf(f18887,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (($true = (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) => (((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) = $true) => (X3 = X6)))),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f18886])).
% 87.48/29.63 thf(f18886,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : ((heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5) => ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) => (X3 = X6)))),
% 87.48/29.63 inference(rectify,[status(thm)],[f22])).
% 87.48/29.63 thf(f22,axiom,(
% 87.48/29.63 ! [X0 : $tType,X20 : heap_Time_Heap @ X0,X9 : heap_ext @ product_unit,X8 : heap_ext @ product_unit,X11 : X0,X12 : nat,X21 : heap_ext @ product_unit,X14 : X0,X22 : nat] : ((heap_Time_effect @ X0 @ X20 @ X9 @ X8 @ X11 @ X12) => ((heap_Time_effect @ X0 @ X20 @ X9 @ X21 @ X14 @ X22) => (X8 = X21)))),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_21_effect__deterministic_I2_J)).
% 87.48/29.63 thf(f22406,plain,(
% 87.48/29.63 spl1_4),
% 87.48/29.63 inference(avatar_split_clause,[status(thm)],[f22405,f21421])).
% 87.48/29.63 thf(f22405,plain,(
% 87.48/29.63 (n = one_one @ nat)),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f22403])).
% 87.48/29.63 thf(f22403,plain,(
% 87.48/29.63 (n = one_one @ nat) | ($true != $true)),
% 87.48/29.63 inference(superposition,[status(thm)],[f21846,f21732])).
% 87.48/29.63 thf(f21846,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : nat,X1 : heap_ext @ product_unit] : (((heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X1 @ X2 @ X0) != $true) | (n = X0)) )),
% 87.48/29.63 inference(trivial_inequality_removal,[status(thm)],[f21845])).
% 87.48/29.63 thf(f21845,plain,(
% 87.48/29.63 ( ! [X2 : nat,X0 : nat,X1 : heap_ext @ product_unit] : ((n = X0) | ($true != $true) | ((heap_Time_effect @ nat @ (array_len @ a @ a2) @ h @ X1 @ X2 @ X0) != $true)) )),
% 87.48/29.63 inference(superposition,[status(thm)],[f21363,f21379])).
% 87.48/29.63 thf(f21363,plain,(
% 87.48/29.63 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X1 : heap_Time_Heap @ X0,X8 : nat,X6 : heap_ext @ product_unit,X7 : X0,X4 : X0,X5 : nat] : (($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) | (X5 = X8) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true)) )),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21323])).
% 87.48/29.63 thf(f21323,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : ((X5 = X8) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(flattening,[status(thm)],[f21322])).
% 87.48/29.63 thf(f21322,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (((X5 = X8) | ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) != $true)) | ($true != (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)))),
% 87.48/29.63 inference(ennf_transformation,[status(thm)],[f18999])).
% 87.48/29.63 thf(f18999,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : (($true = (heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5)) => (((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) = $true) => (X5 = X8)))),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f18998])).
% 87.48/29.63 thf(f18998,plain,(
% 87.48/29.63 ! [X0 : $tType,X1 : heap_Time_Heap @ X0,X2 : heap_ext @ product_unit,X3 : heap_ext @ product_unit,X4 : X0,X5 : nat,X6 : heap_ext @ product_unit,X7 : X0,X8 : nat] : ((heap_Time_effect @ X0 @ X1 @ X2 @ X3 @ X4 @ X5) => ((heap_Time_effect @ X0 @ X1 @ X2 @ X6 @ X7 @ X8) => (X5 = X8)))),
% 87.48/29.63 inference(rectify,[status(thm)],[f21])).
% 87.48/29.63 thf(f21,axiom,(
% 87.48/29.63 ! [X0 : $tType,X20 : heap_Time_Heap @ X0,X9 : heap_ext @ product_unit,X8 : heap_ext @ product_unit,X11 : X0,X12 : nat,X21 : heap_ext @ product_unit,X14 : X0,X22 : nat] : ((heap_Time_effect @ X0 @ X20 @ X9 @ X8 @ X11 @ X12) => ((heap_Time_effect @ X0 @ X20 @ X9 @ X21 @ X14 @ X22) => (X12 = X22)))),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_20_effect__deterministic_I3_J)).
% 87.48/29.63 thf(f21425,plain,(
% 87.48/29.63 ~spl1_3),
% 87.48/29.63 inference(avatar_split_clause,[status(thm)],[f21394,f21417])).
% 87.48/29.63 thf(f21417,definition,(
% 87.48/29.63 spl1_3 <=> (thesis = $true)),
% 87.48/29.63 introduced(definition,[new_symbols(naming,[spl1_3])],[avatar_definition])).
% 87.48/29.63 thf(f21394,plain,(
% 87.48/29.63 (thesis != $true)),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21308])).
% 87.48/29.63 thf(f21308,plain,(
% 87.48/29.63 (thesis != $true)),
% 87.48/29.63 inference(flattening,[status(thm)],[f14581])).
% 87.48/29.63 thf(f14581,plain,(
% 87.48/29.63 ~(thesis = $true)),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f14580])).
% 87.48/29.63 thf(f14580,plain,(
% 87.48/29.63 ~thesis),
% 87.48/29.63 inference(rectify,[status(thm)],[f7868])).
% 87.48/29.63 thf(f7868,negated_conjecture,(
% 87.48/29.63 ~thesis),
% 87.48/29.63 inference(negated_conjecture,[status(cth)],[f7867])).
% 87.48/29.63 thf(f7867,conjecture,(
% 87.48/29.63 thesis),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',conj_1)).
% 87.48/29.63 thf(f21424,plain,(
% 87.48/29.63 ~spl1_1 | ~spl1_2 | spl1_3 | ~spl1_4),
% 87.48/29.63 inference(avatar_split_clause,[status(thm)],[f21366,f21421,f21417,f21413,f21409])).
% 87.48/29.63 thf(f21366,plain,(
% 87.48/29.63 (h != h2) | (r != (array_length @ a @ h2 @ a2)) | (thesis = $true) | (n != one_one @ nat)),
% 87.48/29.63 inference(cnf_transformation,[status(thm)],[f21334])).
% 87.48/29.63 thf(f21334,plain,(
% 87.48/29.63 (h != h2) | (n != one_one @ nat) | (thesis = $true) | (r != (array_length @ a @ h2 @ a2))),
% 87.48/29.63 inference(flattening,[status(thm)],[f21333])).
% 87.48/29.63 thf(f21333,plain,(
% 87.48/29.63 (((thesis = $true) | (n != one_one @ nat)) | (h != h2)) | (r != (array_length @ a @ h2 @ a2))),
% 87.48/29.63 inference(ennf_transformation,[status(thm)],[f17157])).
% 87.48/29.63 thf(f17157,plain,(
% 87.48/29.63 (r = (array_length @ a @ h2 @ a2)) => ((h = h2) => ((n = one_one @ nat) => (thesis = $true)))),
% 87.48/29.63 inference(fool_elimination,[status(thm)],[f17156])).
% 87.48/29.63 thf(f17156,plain,(
% 87.48/29.63 (r = (array_length @ a @ h2 @ a2)) => ((h = h2) => ((n = one_one @ nat) => thesis))),
% 87.48/29.63 inference(rectify,[status(thm)],[f2])).
% 87.48/29.63 thf(f2,axiom,(
% 87.48/29.63 (r = (array_length @ a @ h2 @ a2)) => ((h = h2) => ((n = one_one @ nat) => thesis))),
% 87.48/29.63 file('/export/starexec/sandbox2/tmp/tmp.lP78sG7DkC/DTF2THF_31297.p',fact_1_that)).
% 87.48/29.63 % SZS output end Proof for DTF2THF_31297
% 87.48/29.63 % (31633)------------------------------
% 87.48/29.63 % (31633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 87.48/29.63 % (31633)Termination reason: Refutation
% 87.48/29.63
% 87.48/29.63 % (31633)Memory used [KB]: 31470
% 87.48/29.63 % (31633)Time elapsed: 1.807 s
% 87.48/29.63 % (31633)Instructions burned: 1640 (million)
% 87.48/29.63 % (31633)------------------------------
% 87.48/29.63 % (31633)------------------------------
% 87.48/29.63 % (31433)Success in time 21.607 s
%------------------------------------------------------------------------------