%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT119^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 : Wed Feb 25 08:44:50 AM UTC 2026
% Result : Theorem 7.60s 2.45s
% Output : Refutation 7.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.08 % Problem : DAT119^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.10/0.29 % Computer : n029.cluster.edu
% 0.10/0.29 % Model : x86_64 x86_64
% 0.10/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.29 % Memory : 8042.1875MB
% 0.10/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.29 % CPULimit : 300
% 0.10/0.29 % WCLimit : 300
% 0.10/0.29 % DateTime : Tue Feb 24 06:07:44 EST 2026
% 0.10/0.29 % CPUTime :
% 0.10/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.38 ---- Original DTF file ---
% 0.15/0.38 thf(spec,logic,$$dhol).
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 % File : DAT119^1 : TPTP v9.2.1. Released v7.0.0.
% 0.15/0.38 % Domain : Data Structures
% 0.15/0.38 % Problem : Coinductive list 1279
% 0.15/0.38 % Version : [Bla16] axioms : Especial.
% 0.15/0.38 % English :
% 0.15/0.38
% 0.15/0.38 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.15/0.38 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.15/0.38 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.15/0.38 % Source : [Bla16]
% 0.15/0.38 % Names : coinductive_list__1279.p [Bla16]
% 0.15/0.38
% 0.15/0.38 % Status : Theorem
% 0.15/0.38 % Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% 0.15/0.38 % Syntax : Number of formulae : 355 ( 169 unt; 67 typ; 0 def)
% 0.15/0.38 % Number of atoms : 673 ( 393 equ; 0 cnn)
% 0.15/0.38 % Maximal formula atoms : 24 ( 2 avg)
% 0.15/0.38 % Number of connectives : 2810 ( 94 ~; 31 |; 45 &;2378 @)
% 0.15/0.38 % ( 0 <=>; 262 =>; 0 <=; 0 <~>)
% 0.15/0.38 % Maximal formula depth : 22 ( 6 avg)
% 0.15/0.38 % Number of types : 4 ( 3 usr)
% 0.15/0.38 % Number of type conns : 211 ( 211 >; 0 *; 0 +; 0 <<)
% 0.15/0.38 % Number of symbols : 67 ( 64 usr; 6 con; 0-8 aty)
% 0.15/0.38 % Number of variables : 802 ( 32 ^; 660 !; 57 ?; 802 :)
% 0.15/0.38 % ( 53 !>; 0 ?*; 0 @-; 0 @+)
% 0.15/0.38 % SPC : TH1_THM_EQU_NAR
% 0.15/0.38
% 0.15/0.38 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.15/0.38 % 2016-07-13 14:49:09.100
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 %----Could-be-implicit typings (6)
% 0.15/0.38 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 0.15/0.38 coindu1593790203_llist: $tType > $tType ).
% 0.15/0.38
% 0.15/0.38 thf(ty_t_Extended__Nat_Oenat,type,
% 0.15/0.38 extended_enat: $tType ).
% 0.15/0.38
% 0.15/0.38 thf(ty_t_Set_Oset,type,
% 0.15/0.38 set: $tType > $tType ).
% 0.15/0.38
% 0.15/0.38 thf(ty_t_Nat_Onat,type,
% 0.15/0.38 nat: $tType ).
% 0.15/0.38
% 0.15/0.38 thf(ty_t_itself,type,
% 0.15/0.38 itself: $tType > $tType ).
% 0.15/0.38
% 0.15/0.38 thf(ty_tf_a,type,
% 0.15/0.38 a: $tType ).
% 0.15/0.38
% 0.15/0.38 %----Explicit typings (61)
% 0.15/0.38 thf(sy_cl_HOL_Otype,type,
% 0.15/0.38 type:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ozero,type,
% 0.15/0.38 zero:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Orderings_Oorder,type,
% 0.15/0.38 order:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Omult__zero,type,
% 0.15/0.38 mult_zero:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ogroup__add,type,
% 0.15/0.38 group_add:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Osemiring__1,type,
% 0.15/0.38 semiring_1:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Orderings_Olinorder,type,
% 0.15/0.38 linorder:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Oordered__ring,type,
% 0.15/0.38 ordered_ring:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.15/0.38 semigroup_mult:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.15/0.38 comm_semiring_1:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.15/0.38 linordered_ring:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Oordered__semiring,type,
% 0.15/0.38 ordered_semiring:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.15/0.38 comm_monoid_diff:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.15/0.38 ab_semigroup_mult:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 0.15/0.38 ordered_semiring_0:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.15/0.38 ordered_ab_group_add:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 0.15/0.38 linord581940658strict:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.15/0.38 cancel1352612707id_add:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.15/0.38 cancel146912293up_add:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.15/0.38 semiri1193490041visors:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.15/0.38 canoni770627133id_add:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.15/0.38 semiri1923998003cancel:
% 0.15/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
% 0.15/0.38 coindu1928975208n_lset:
% 0.15/0.38 !>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates,type,
% 0.15/0.38 coindu1529956631erates:
% 0.15/0.38 !>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
% 0.15/0.38 coindu268472904append:
% 0.15/0.38 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 0.15/0.38 coindu1213758845finite:
% 0.15/0.38 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 0.15/0.38 coindu1018505716length:
% 0.15/0.38 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
% 0.15/0.38 coindu300403952lexord:
% 0.15/0.38 !>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
% 0.15/0.38 coindu1121789889_LCons:
% 0.15/0.38 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 0.15/0.38 coindu1598213697e_LNil:
% 0.15/0.38 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist,type,
% 0.15/0.38 coindu882539134_llist:
% 0.15/0.38 !>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
% 0.15/0.38 coindu1244876290_llist:
% 0.15/0.38 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist,type,
% 0.15/0.38 coindu44415597_llist:
% 0.15/0.38 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
% 0.15/0.38 coindu567634248member:
% 0.15/0.38 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olsetp,type,
% 0.15/0.38 coindu1055736764_lsetp:
% 0.15/0.38 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
% 0.15/0.38 coindu1599971794_llist:
% 0.15/0.38 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__Nat_Oenat__set,type,
% 0.15/0.38 coinductive_enat_set: set @ extended_enat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__Nat_Oenat__setp,type,
% 0.15/0.38 coindu530039314t_setp: extended_enat > $o ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 0.15/0.38 coindu1491768222unfold:
% 0.15/0.38 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_OeSuc,type,
% 0.15/0.38 extended_eSuc: extended_enat > extended_enat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Oenat,type,
% 0.15/0.38 extended_enat2: nat > extended_enat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.15/0.38 extended_case_enat:
% 0.15/0.38 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.15/0.38 extended_rec_enat:
% 0.15/0.38 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 0.15/0.38 extended_size_enat: extended_enat > nat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.15/0.38 extend1396239628finity:
% 0.15/0.38 !>[A: $tType] : A ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.15/0.38 extended_the_enat: extended_enat > nat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.15/0.38 minus_minus:
% 0.15/0.38 !>[A: $tType] : ( A > A > A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.15/0.38 times_times:
% 0.15/0.38 !>[A: $tType] : ( A > A > A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.15/0.38 zero_zero:
% 0.15/0.38 !>[A: $tType] : A ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_If,type,
% 0.15/0.38 if:
% 0.15/0.38 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Nat_OSuc,type,
% 0.15/0.38 suc: nat > nat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.15/0.38 semiri532925092at_aux:
% 0.15/0.38 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.15/0.38 size_size:
% 0.15/0.38 !>[A: $tType] : ( A > nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.15/0.38 nat_triangle: nat > nat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.15/0.38 ord_less_eq:
% 0.15/0.38 !>[A: $tType] : ( A > A > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Pure_Otype,type,
% 0.15/0.38 type2:
% 0.15/0.38 !>[A: $tType] : ( itself @ A ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_Set_OCollect,type,
% 0.15/0.38 collect:
% 0.15/0.38 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_c_member,type,
% 0.15/0.38 member:
% 0.15/0.38 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(sy_v_na____,type,
% 0.15/0.38 na: nat ).
% 0.15/0.38
% 0.15/0.38 thf(sy_v_thesis____,type,
% 0.15/0.38 thesis: $o ).
% 0.15/0.38
% 0.15/0.38 thf(sy_v_xsa____,type,
% 0.15/0.38 xsa: coindu1593790203_llist @ a ).
% 0.15/0.38
% 0.15/0.38 %----Relevant facts (252)
% 0.15/0.38 thf(fact_0_len,axiom,
% 0.15/0.38 ( ( coindu1018505716length @ a @ xsa )
% 0.15/0.38 = ( extended_enat2 @ ( suc @ na ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % len
% 0.15/0.38 thf(fact_1_llist_Oinject,axiom,
% 0.15/0.38 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
% 0.15/0.38 = ( ( X21 = Y21 )
% 0.15/0.38 & ( X22 = Y22 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.inject
% 0.15/0.38 thf(fact_2_lmember__code_I2_J,axiom,
% 0.15/0.38 ! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
% 0.15/0.38 = ( ( X = Y )
% 0.15/0.38 | ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lmember_code(2)
% 0.15/0.38 thf(fact_3_iterates_Ocode,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu1529956631erates @ A )
% 0.15/0.38 = ( ^ [F: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F @ ( F @ X2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % iterates.code
% 0.15/0.38 thf(fact_4_llist_Opred__inject_I2_J,axiom,
% 0.15/0.38 ! [A: $tType,P: A > $o,A2: A,Aa: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu44415597_llist @ A @ P @ ( coindu1121789889_LCons @ A @ A2 @ Aa ) )
% 0.15/0.38 = ( ( P @ A2 )
% 0.15/0.38 & ( coindu44415597_llist @ A @ P @ Aa ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.pred_inject(2)
% 0.15/0.38 thf(fact_5_llexord__LCons__eq,axiom,
% 0.15/0.38 ! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
% 0.15/0.38 ( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
% 0.15/0.38 => ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llexord_LCons_eq
% 0.15/0.38 thf(fact_6_llexord__LCons__less,axiom,
% 0.15/0.38 ! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( R @ X @ Y )
% 0.15/0.38 => ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llexord_LCons_less
% 0.15/0.38 thf(fact_7_unfold__llist__eq__LCons,axiom,
% 0.15/0.38 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X @ Xs ) )
% 0.15/0.38 = ( ~ ( IS_LNIL @ B2 )
% 0.15/0.38 & ( X
% 0.15/0.38 = ( LHD @ B2 ) )
% 0.15/0.38 & ( Xs
% 0.15/0.38 = ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % unfold_llist_eq_LCons
% 0.15/0.38 thf(fact_8_lsetp_Ointros_I2_J,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,X: A,X3: A] :
% 0.15/0.38 ( ( coindu1055736764_lsetp @ A @ Xs @ X )
% 0.15/0.38 => ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ X ) ) ).
% 0.15/0.38
% 0.15/0.38 % lsetp.intros(2)
% 0.15/0.38 thf(fact_9_lsetp_Ointros_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ X ) ).
% 0.15/0.38
% 0.15/0.38 % lsetp.intros(1)
% 0.15/0.38 thf(fact_10_lsetp_Ocases,axiom,
% 0.15/0.38 ! [A: $tType,A1: coindu1593790203_llist @ A,A22: A] :
% 0.15/0.38 ( ( coindu1055736764_lsetp @ A @ A1 @ A22 )
% 0.15/0.38 => ( ! [X4: A] :
% 0.15/0.38 ( ? [Xs2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( A1
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 0.15/0.38 => ( A22 != X4 ) )
% 0.15/0.38 => ~ ! [Xs2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ? [X5: A] :
% 0.15/0.38 ( A1
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
% 0.15/0.38 => ~ ( coindu1055736764_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lsetp.cases
% 0.15/0.38 thf(fact_11_lsetp_Osimps,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu1055736764_lsetp @ A )
% 0.15/0.38 = ( ^ [A12: coindu1593790203_llist @ A,A23: A] :
% 0.15/0.38 ( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( A12
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 0.15/0.38 & ( A23 = X2 ) )
% 0.15/0.38 | ? [Xs3: coindu1593790203_llist @ A,X2: A,X6: A] :
% 0.15/0.38 ( ( A12
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X6 @ Xs3 ) )
% 0.15/0.38 & ( A23 = X2 )
% 0.15/0.38 & ( coindu1055736764_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lsetp.simps
% 0.15/0.38 thf(fact_12_lsetp_Oinducts,axiom,
% 0.15/0.38 ! [A: $tType,X1: coindu1593790203_llist @ A,X23: A,P: ( coindu1593790203_llist @ A ) > A > $o] :
% 0.15/0.38 ( ( coindu1055736764_lsetp @ A @ X1 @ X23 )
% 0.15/0.38 => ( ! [X4: A,Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) @ X4 )
% 0.15/0.38 => ( ! [Xs2: coindu1593790203_llist @ A,X4: A,X5: A] :
% 0.15/0.38 ( ( coindu1055736764_lsetp @ A @ Xs2 @ X4 )
% 0.15/0.38 => ( ( P @ Xs2 @ X4 )
% 0.15/0.38 => ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) @ X4 ) ) )
% 0.15/0.38 => ( P @ X1 @ X23 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lsetp.inducts
% 0.15/0.38 thf(fact_13_llist_Osimps_I5_J,axiom,
% 0.15/0.38 ! [B: $tType,A: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
% 0.15/0.38 = ( F2 @ X21 @ X22 ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.simps(5)
% 0.15/0.38 thf(fact_14_unfold__llist_Octr_I2_J,axiom,
% 0.15/0.38 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.38 ( ~ ( P2 @ A2 )
% 0.15/0.38 => ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 0.15/0.38 = ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % unfold_llist.ctr(2)
% 0.15/0.38 thf(fact_15_Suc_Ohyps,axiom,
% 0.15/0.38 ! [Xs: coindu1593790203_llist @ a] :
% 0.15/0.38 ( ( ( coindu1018505716length @ a @ Xs )
% 0.15/0.38 = ( extended_enat2 @ na ) )
% 0.15/0.38 => ( coindu1213758845finite @ a @ Xs ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc.hyps
% 0.15/0.38 thf(fact_16_enat_Oinject,axiom,
% 0.15/0.38 ! [Nat: nat,Nat2: nat] :
% 0.15/0.38 ( ( ( extended_enat2 @ Nat )
% 0.15/0.38 = ( extended_enat2 @ Nat2 ) )
% 0.15/0.38 = ( Nat = Nat2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.inject
% 0.15/0.38 thf(fact_17_nat_Oinject,axiom,
% 0.15/0.38 ! [X23: nat,Y2: nat] :
% 0.15/0.38 ( ( ( suc @ X23 )
% 0.15/0.38 = ( suc @ Y2 ) )
% 0.15/0.38 = ( X23 = Y2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat.inject
% 0.15/0.38 thf(fact_18_old_Onat_Oinject,axiom,
% 0.15/0.38 ! [Nat: nat,Nat2: nat] :
% 0.15/0.38 ( ( ( suc @ Nat )
% 0.15/0.38 = ( suc @ Nat2 ) )
% 0.15/0.38 = ( Nat = Nat2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % old.nat.inject
% 0.15/0.38 thf(fact_19_Suc__inject,axiom,
% 0.15/0.38 ! [X: nat,Y: nat] :
% 0.15/0.38 ( ( ( suc @ X )
% 0.15/0.38 = ( suc @ Y ) )
% 0.15/0.38 => ( X = Y ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_inject
% 0.15/0.38 thf(fact_20_n__not__Suc__n,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( N
% 0.15/0.38 != ( suc @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % n_not_Suc_n
% 0.15/0.38 thf(fact_21_llength__LCons,axiom,
% 0.15/0.38 ! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
% 0.15/0.38 ( ( coindu1018505716length @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
% 0.15/0.38 = ( extended_eSuc @ ( coindu1018505716length @ B @ Xs ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llength_LCons
% 0.15/0.38 thf(fact_22_llexord_Ocoinduct,axiom,
% 0.15/0.38 ! [A: $tType,X7: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R: A > A > $o] :
% 0.15/0.38 ( ( X7 @ X @ Xa )
% 0.15/0.38 => ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( X7 @ X4 @ Xa2 )
% 0.15/0.38 => ( ? [Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Xb: A] :
% 0.15/0.38 ( ( X4
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
% 0.15/0.38 & ( Xa2
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Xb @ Ys2 ) )
% 0.15/0.38 & ( ( X7 @ Xs4 @ Ys2 )
% 0.15/0.38 | ( coindu300403952lexord @ A @ R @ Xs4 @ Ys2 ) ) )
% 0.15/0.38 | ? [Xb: A,Y3: A,Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( X4
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
% 0.15/0.38 & ( Xa2
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
% 0.15/0.38 & ( R @ Xb @ Y3 ) )
% 0.15/0.38 | ? [Ys2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( X4
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 & ( Xa2 = Ys2 ) ) ) )
% 0.15/0.38 => ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llexord.coinduct
% 0.15/0.38 thf(fact_23_llexord_Osimps,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu300403952lexord @ A )
% 0.15/0.38 = ( ^ [R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ? [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
% 0.15/0.38 ( ( A12
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 0.15/0.38 & ( A23
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
% 0.15/0.38 & ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys3 ) )
% 0.15/0.38 | ? [X2: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( A12
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 0.15/0.38 & ( A23
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Y4 @ Ys3 ) )
% 0.15/0.38 & ( R2 @ X2 @ Y4 ) )
% 0.15/0.38 | ? [Ys3: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( A12
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 & ( A23 = Ys3 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llexord.simps
% 0.15/0.38 thf(fact_24_llexord_Ocases,axiom,
% 0.15/0.38 ! [A: $tType,R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu300403952lexord @ A @ R @ A1 @ A22 )
% 0.15/0.38 => ( ! [Xs2: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X4: A] :
% 0.15/0.38 ( ( A1
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 0.15/0.38 => ( ( A22
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X4 @ Ys4 ) )
% 0.15/0.38 => ~ ( coindu300403952lexord @ A @ R @ Xs2 @ Ys4 ) ) )
% 0.15/0.38 => ( ! [X4: A] :
% 0.15/0.38 ( ? [Xs2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( A1
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 0.15/0.38 => ! [Y5: A] :
% 0.15/0.38 ( ? [Ys4: coindu1593790203_llist @ A] :
% 0.15/0.38 ( A22
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Y5 @ Ys4 ) )
% 0.15/0.38 => ~ ( R @ X4 @ Y5 ) ) )
% 0.15/0.38 => ~ ( ( A1
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 => ! [Ys4: coindu1593790203_llist @ A] : ( A22 != Ys4 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llexord.cases
% 0.15/0.38 thf(fact_25_eSuc__inject,axiom,
% 0.15/0.38 ! [M: extended_enat,N: extended_enat] :
% 0.15/0.38 ( ( ( extended_eSuc @ M )
% 0.15/0.38 = ( extended_eSuc @ N ) )
% 0.15/0.38 = ( M = N ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_inject
% 0.15/0.38 thf(fact_26_lfinite__code_I2_J,axiom,
% 0.15/0.38 ! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
% 0.15/0.38 ( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
% 0.15/0.38 = ( coindu1213758845finite @ B @ Xs ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_code(2)
% 0.15/0.38 thf(fact_27_lfinite__LCons,axiom,
% 0.15/0.38 ! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
% 0.15/0.38 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_LCons
% 0.15/0.38 thf(fact_28_lfinite__code_I1_J,axiom,
% 0.15/0.38 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_code(1)
% 0.15/0.38 thf(fact_29_lfinite__LNil,axiom,
% 0.15/0.38 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_LNil
% 0.15/0.38 thf(fact_30_lfinite_Ocases,axiom,
% 0.15/0.38 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ A2 )
% 0.15/0.38 => ( ( A2
% 0.15/0.38 != ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 => ~ ! [Xs2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ? [X4: A] :
% 0.15/0.38 ( A2
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 0.15/0.38 => ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite.cases
% 0.15/0.38 thf(fact_31_lfinite_Osimps,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu1213758845finite @ A )
% 0.15/0.38 = ( ^ [A3: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( A3
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 | ? [Xs3: coindu1593790203_llist @ A,X2: A] :
% 0.15/0.38 ( ( A3
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 0.15/0.38 & ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite.simps
% 0.15/0.38 thf(fact_32_lfinite_Oinducts,axiom,
% 0.15/0.38 ! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ X )
% 0.15/0.38 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 => ( ! [Xs2: coindu1593790203_llist @ A,X4: A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ Xs2 )
% 0.15/0.38 => ( ( P @ Xs2 )
% 0.15/0.38 => ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
% 0.15/0.38 => ( P @ X ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite.inducts
% 0.15/0.38 thf(fact_33_llist_Odistinct_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu1598213697e_LNil @ A )
% 0.15/0.38 != ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.distinct(1)
% 0.15/0.38 thf(fact_34_llist_Oexhaust,axiom,
% 0.15/0.38 ! [A: $tType,Y: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( Y
% 0.15/0.38 != ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 => ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
% 0.15/0.38 ( Y
% 0.15/0.38 != ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.exhaust
% 0.15/0.38 thf(fact_35_neq__LNil__conv,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( Xs
% 0.15/0.38 != ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = ( ? [X2: A,Xs5: coindu1593790203_llist @ A] :
% 0.15/0.38 ( Xs
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % neq_LNil_conv
% 0.15/0.38 thf(fact_36_lfinite__LConsI,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ Xs )
% 0.15/0.38 => ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_LConsI
% 0.15/0.38 thf(fact_37_llist_Osimps_I4_J,axiom,
% 0.15/0.38 ! [A: $tType,B: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B] :
% 0.15/0.38 ( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = F1 ) ).
% 0.15/0.38
% 0.15/0.38 % llist.simps(4)
% 0.15/0.38 thf(fact_38_enat__eSuc__iff,axiom,
% 0.15/0.38 ! [Y: nat,X: extended_enat] :
% 0.15/0.38 ( ( ( extended_enat2 @ Y )
% 0.15/0.38 = ( extended_eSuc @ X ) )
% 0.15/0.38 = ( ? [N2: nat] :
% 0.15/0.38 ( ( Y
% 0.15/0.38 = ( suc @ N2 ) )
% 0.15/0.38 & ( ( extended_enat2 @ N2 )
% 0.15/0.38 = X ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_eSuc_iff
% 0.15/0.38 thf(fact_39_eSuc__enat__iff,axiom,
% 0.15/0.38 ! [X: extended_enat,Y: nat] :
% 0.15/0.38 ( ( ( extended_eSuc @ X )
% 0.15/0.38 = ( extended_enat2 @ Y ) )
% 0.15/0.38 = ( ? [N2: nat] :
% 0.15/0.38 ( ( Y
% 0.15/0.38 = ( suc @ N2 ) )
% 0.15/0.38 & ( X
% 0.15/0.38 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_enat_iff
% 0.15/0.38 thf(fact_40_eSuc__enat,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 0.15/0.38 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_enat
% 0.15/0.38 thf(fact_41_unfold__llist_Octr_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 0.15/0.38 ( ( P2 @ A2 )
% 0.15/0.38 => ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 0.15/0.38 = ( coindu1598213697e_LNil @ B ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % unfold_llist.ctr(1)
% 0.15/0.38 thf(fact_42_llist_Opred__inject_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,P: A > $o] : ( coindu44415597_llist @ A @ P @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.pred_inject(1)
% 0.15/0.38 thf(fact_43_llexord__LNil,axiom,
% 0.15/0.38 ! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% 0.15/0.38
% 0.15/0.38 % llexord_LNil
% 0.15/0.38 thf(fact_44_mem__Collect__eq,axiom,
% 0.15/0.38 ! [A: $tType,A2: A,P: A > $o] :
% 0.15/0.38 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.15/0.38 = ( P @ A2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % mem_Collect_eq
% 0.15/0.38 thf(fact_45_Collect__mem__eq,axiom,
% 0.15/0.38 ! [A: $tType,A4: set @ A] :
% 0.15/0.38 ( ( collect @ A
% 0.15/0.38 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.15/0.38 = A4 ) ).
% 0.15/0.38
% 0.15/0.38 % Collect_mem_eq
% 0.15/0.38 thf(fact_46_Collect__cong,axiom,
% 0.15/0.38 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.38 ( ! [X4: A] :
% 0.15/0.38 ( ( P @ X4 )
% 0.15/0.38 = ( Q @ X4 ) )
% 0.15/0.38 => ( ( collect @ A @ P )
% 0.15/0.38 = ( collect @ A @ Q ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % Collect_cong
% 0.15/0.38 thf(fact_47_ext,axiom,
% 0.15/0.38 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.15/0.38 ( ! [X4: A] :
% 0.15/0.38 ( ( F3 @ X4 )
% 0.15/0.38 = ( G @ X4 ) )
% 0.15/0.38 => ( F3 = G ) ) ).
% 0.15/0.38
% 0.15/0.38 % ext
% 0.15/0.38 thf(fact_48_lmember__code_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,X: A] :
% 0.15/0.38 ~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 % lmember_code(1)
% 0.15/0.38 thf(fact_49_unfold__llist_Ocode,axiom,
% 0.15/0.38 ! [B: $tType,A: $tType] :
% 0.15/0.38 ( ( coindu1599971794_llist @ A @ B )
% 0.15/0.38 = ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % unfold_llist.code
% 0.15/0.38 thf(fact_50_co_Oenat_Oinject,axiom,
% 0.15/0.38 ! [X23: extended_enat,Y2: extended_enat] :
% 0.15/0.38 ( ( ( extended_eSuc @ X23 )
% 0.15/0.38 = ( extended_eSuc @ Y2 ) )
% 0.15/0.38 = ( X23 = Y2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % co.enat.inject
% 0.15/0.38 thf(fact_51_enat_Osimps_I6_J,axiom,
% 0.15/0.38 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.15/0.38 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.15/0.38 = ( F1 @ Nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.simps(6)
% 0.15/0.38 thf(fact_52_enat_Osimps_I4_J,axiom,
% 0.15/0.38 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.15/0.38 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.15/0.38 = ( F1 @ Nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.simps(4)
% 0.15/0.38 thf(fact_53_lfinite__rev__induct,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ Xs )
% 0.15/0.38 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 => ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ Xs2 )
% 0.15/0.38 => ( ( P @ Xs2 )
% 0.15/0.38 => ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X4 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
% 0.15/0.38 => ( P @ Xs ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_rev_induct
% 0.15/0.38 thf(fact_54_llength__LNil,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % llength_LNil
% 0.15/0.38 thf(fact_55_gen__lset__code_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,A4: set @ A] :
% 0.15/0.38 ( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = A4 ) ).
% 0.15/0.38
% 0.15/0.38 % gen_lset_code(1)
% 0.15/0.38 thf(fact_56_of__nat__aux_Osimps_I2_J,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [Inc: A > A,N: nat,I: A] :
% 0.15/0.38 ( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 0.15/0.38 = ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % of_nat_aux.simps(2)
% 0.15/0.38 thf(fact_57_llist_Ocorec__code,axiom,
% 0.15/0.38 ! [A: $tType,C: $tType] :
% 0.15/0.38 ( ( coindu1244876290_llist @ C @ A )
% 0.15/0.38 = ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.corec_code
% 0.15/0.38 thf(fact_58_the__enat_Osimps,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 0.15/0.38 = N ) ).
% 0.15/0.38
% 0.15/0.38 % the_enat.simps
% 0.15/0.38 thf(fact_59_lappend__code_I2_J,axiom,
% 0.15/0.38 ! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_code(2)
% 0.15/0.38 thf(fact_60_lappend__code_I1_J,axiom,
% 0.15/0.38 ! [A: $tType,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
% 0.15/0.38 = Ys ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_code(1)
% 0.15/0.38 thf(fact_61_lappend__LNil2,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = Xs ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_LNil2
% 0.15/0.38 thf(fact_62_lfinite__lappend,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.15/0.38 = ( ( coindu1213758845finite @ A @ Xs )
% 0.15/0.38 & ( coindu1213758845finite @ A @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lfinite_lappend
% 0.15/0.38 thf(fact_63_lappend__assoc,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
% 0.15/0.38 = ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_assoc
% 0.15/0.38 thf(fact_64_co_Oenat_Odistinct_I1_J,axiom,
% 0.15/0.38 ! [X23: extended_enat] :
% 0.15/0.38 ( ( zero_zero @ extended_enat )
% 0.15/0.38 != ( extended_eSuc @ X23 ) ) ).
% 0.15/0.38
% 0.15/0.38 % co.enat.distinct(1)
% 0.15/0.38 thf(fact_65_co_Oenat_OdiscI,axiom,
% 0.15/0.38 ! [Enat: extended_enat,X23: extended_enat] :
% 0.15/0.38 ( ( Enat
% 0.15/0.38 = ( extended_eSuc @ X23 ) )
% 0.15/0.38 => ( Enat
% 0.15/0.38 != ( zero_zero @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % co.enat.discI
% 0.15/0.38 thf(fact_66_enat__coexhaust,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( zero_zero @ extended_enat ) )
% 0.15/0.38 => ~ ! [N3: extended_enat] :
% 0.15/0.38 ( N
% 0.15/0.38 != ( extended_eSuc @ N3 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_coexhaust
% 0.15/0.38 thf(fact_67_co_Oenat_Oexhaust,axiom,
% 0.15/0.38 ! [Y: extended_enat] :
% 0.15/0.38 ( ( Y
% 0.15/0.38 != ( zero_zero @ extended_enat ) )
% 0.15/0.38 => ~ ! [X24: extended_enat] :
% 0.15/0.38 ( Y
% 0.15/0.38 != ( extended_eSuc @ X24 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % co.enat.exhaust
% 0.15/0.38 thf(fact_68_neq__zero__conv__eSuc,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( ? [N4: extended_enat] :
% 0.15/0.38 ( N
% 0.15/0.38 = ( extended_eSuc @ N4 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % neq_zero_conv_eSuc
% 0.15/0.38 thf(fact_69_lappend__LNil__LNil,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_LNil_LNil
% 0.15/0.38 thf(fact_70_LNil__eq__lappend__iff,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( ( coindu1598213697e_LNil @ A )
% 0.15/0.38 = ( coindu268472904append @ A @ Xs @ Ys ) )
% 0.15/0.38 = ( ( Xs
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 & ( Ys
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % LNil_eq_lappend_iff
% 0.15/0.38 thf(fact_71_lappend__eq__LNil__iff,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 = ( ( Xs
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) )
% 0.15/0.38 & ( Ys
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_eq_LNil_iff
% 0.15/0.38 thf(fact_72_lappend__inf,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ~ ( coindu1213758845finite @ A @ Xs )
% 0.15/0.38 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 0.15/0.38 = Xs ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_inf
% 0.15/0.38 thf(fact_73_zero__ne__eSuc,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( zero_zero @ extended_enat )
% 0.15/0.38 != ( extended_eSuc @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_ne_eSuc
% 0.15/0.38 thf(fact_74_llist_Ocorec_I2_J,axiom,
% 0.15/0.38 ! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
% 0.15/0.38 ( ~ ( P2 @ A2 )
% 0.15/0.38 => ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
% 0.15/0.38 = ( coindu1121789889_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.corec(2)
% 0.15/0.38 thf(fact_75_llist_Ocorec_I1_J,axiom,
% 0.15/0.38 ! [C: $tType,A: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
% 0.15/0.38 ( ( P2 @ A2 )
% 0.15/0.38 => ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
% 0.15/0.38 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % llist.corec(1)
% 0.15/0.38 thf(fact_76_lappend__snocL1__conv__LCons2,axiom,
% 0.15/0.38 ! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
% 0.15/0.38 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
% 0.15/0.38 = ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lappend_snocL1_conv_LCons2
% 0.15/0.38 thf(fact_77_enat__set_Ocoinduct,axiom,
% 0.15/0.38 ! [X7: extended_enat > $o,X: extended_enat] :
% 0.15/0.38 ( ( X7 @ X )
% 0.15/0.38 => ( ! [X4: extended_enat] :
% 0.15/0.38 ( ( X7 @ X4 )
% 0.15/0.38 => ( ( X4
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 | ? [N5: extended_enat] :
% 0.15/0.38 ( ( X4
% 0.15/0.38 = ( extended_eSuc @ N5 ) )
% 0.15/0.38 & ( ( X7 @ N5 )
% 0.15/0.38 | ( member @ extended_enat @ N5 @ coinductive_enat_set ) ) ) ) )
% 0.15/0.38 => ( member @ extended_enat @ X @ coinductive_enat_set ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_set.coinduct
% 0.15/0.38 thf(fact_78_enat__set_Osimps,axiom,
% 0.15/0.38 ! [A2: extended_enat] :
% 0.15/0.38 ( ( member @ extended_enat @ A2 @ coinductive_enat_set )
% 0.15/0.38 = ( ( A2
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 | ? [N2: extended_enat] :
% 0.15/0.38 ( ( A2
% 0.15/0.38 = ( extended_eSuc @ N2 ) )
% 0.15/0.38 & ( member @ extended_enat @ N2 @ coinductive_enat_set ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_set.simps
% 0.15/0.38 thf(fact_79_enat__set_Ocases,axiom,
% 0.15/0.38 ! [A2: extended_enat] :
% 0.15/0.38 ( ( member @ extended_enat @ A2 @ coinductive_enat_set )
% 0.15/0.38 => ( ( A2
% 0.15/0.38 != ( zero_zero @ extended_enat ) )
% 0.15/0.38 => ~ ! [N6: extended_enat] :
% 0.15/0.38 ( ( A2
% 0.15/0.38 = ( extended_eSuc @ N6 ) )
% 0.15/0.38 => ~ ( member @ extended_enat @ N6 @ coinductive_enat_set ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_set.cases
% 0.15/0.38 thf(fact_80_enat__setp_Ocoinduct,axiom,
% 0.15/0.38 ! [X7: extended_enat > $o,X: extended_enat] :
% 0.15/0.38 ( ( X7 @ X )
% 0.15/0.38 => ( ! [X4: extended_enat] :
% 0.15/0.38 ( ( X7 @ X4 )
% 0.15/0.38 => ( ( X4
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 | ? [N5: extended_enat] :
% 0.15/0.38 ( ( X4
% 0.15/0.38 = ( extended_eSuc @ N5 ) )
% 0.15/0.38 & ( ( X7 @ N5 )
% 0.15/0.38 | ( coindu530039314t_setp @ N5 ) ) ) ) )
% 0.15/0.38 => ( coindu530039314t_setp @ X ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp.coinduct
% 0.15/0.38 thf(fact_81_enat__setp_Osimps,axiom,
% 0.15/0.38 ( coindu530039314t_setp
% 0.15/0.38 = ( ^ [A3: extended_enat] :
% 0.15/0.38 ( ( A3
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 | ? [N2: extended_enat] :
% 0.15/0.38 ( ( A3
% 0.15/0.38 = ( extended_eSuc @ N2 ) )
% 0.15/0.38 & ( coindu530039314t_setp @ N2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp.simps
% 0.15/0.38 thf(fact_82_enat__setp_Ocases,axiom,
% 0.15/0.38 ! [A2: extended_enat] :
% 0.15/0.38 ( ( coindu530039314t_setp @ A2 )
% 0.15/0.38 => ( ( A2
% 0.15/0.38 != ( zero_zero @ extended_enat ) )
% 0.15/0.38 => ~ ! [N6: extended_enat] :
% 0.15/0.38 ( ( A2
% 0.15/0.38 = ( extended_eSuc @ N6 ) )
% 0.15/0.38 => ~ ( coindu530039314t_setp @ N6 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp.cases
% 0.15/0.38 thf(fact_83_case__enat__0,axiom,
% 0.15/0.38 ! [A: $tType,F3: nat > A,I: A] :
% 0.15/0.38 ( ( extended_case_enat @ A @ F3 @ I @ ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( F3 @ ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % case_enat_0
% 0.15/0.38 thf(fact_84_the__enat__0,axiom,
% 0.15/0.38 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % the_enat_0
% 0.15/0.38 thf(fact_85_enat__setp__enat__set__eq,axiom,
% 0.15/0.38 ( coindu530039314t_setp
% 0.15/0.38 = ( ^ [X2: extended_enat] : ( member @ extended_enat @ X2 @ coinductive_enat_set ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp_enat_set_eq
% 0.15/0.38 thf(fact_86_not0__implies__Suc,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( zero_zero @ nat ) )
% 0.15/0.38 => ? [M2: nat] :
% 0.15/0.38 ( N
% 0.15/0.38 = ( suc @ M2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % not0_implies_Suc
% 0.15/0.38 thf(fact_87_old_Onat_Oinducts,axiom,
% 0.15/0.38 ! [P: nat > $o,Nat: nat] :
% 0.15/0.38 ( ( P @ ( zero_zero @ nat ) )
% 0.15/0.38 => ( ! [Nat3: nat] :
% 0.15/0.38 ( ( P @ Nat3 )
% 0.15/0.38 => ( P @ ( suc @ Nat3 ) ) )
% 0.15/0.38 => ( P @ Nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % old.nat.inducts
% 0.15/0.38 thf(fact_88_old_Onat_Oexhaust,axiom,
% 0.15/0.38 ! [Y: nat] :
% 0.15/0.38 ( ( Y
% 0.15/0.38 != ( zero_zero @ nat ) )
% 0.15/0.38 => ~ ! [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 != ( suc @ Nat3 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % old.nat.exhaust
% 0.15/0.38 thf(fact_89_Zero__not__Suc,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( zero_zero @ nat )
% 0.15/0.38 != ( suc @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % Zero_not_Suc
% 0.15/0.38 thf(fact_90_Zero__neq__Suc,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( zero_zero @ nat )
% 0.15/0.38 != ( suc @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % Zero_neq_Suc
% 0.15/0.38 thf(fact_91_Suc__neq__Zero,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( suc @ M )
% 0.15/0.38 != ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_neq_Zero
% 0.15/0.38 thf(fact_92_zero__induct,axiom,
% 0.15/0.38 ! [P: nat > $o,K: nat] :
% 0.15/0.38 ( ( P @ K )
% 0.15/0.38 => ( ! [N6: nat] :
% 0.15/0.38 ( ( P @ ( suc @ N6 ) )
% 0.15/0.38 => ( P @ N6 ) )
% 0.15/0.38 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_induct
% 0.15/0.38 thf(fact_93_diff__induct,axiom,
% 0.15/0.38 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.15/0.38 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 0.15/0.38 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 0.15/0.38 => ( ! [X4: nat,Y5: nat] :
% 0.15/0.38 ( ( P @ X4 @ Y5 )
% 0.15/0.38 => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 0.15/0.38 => ( P @ M @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_induct
% 0.15/0.38 thf(fact_94_nat__induct,axiom,
% 0.15/0.38 ! [P: nat > $o,N: nat] :
% 0.15/0.38 ( ( P @ ( zero_zero @ nat ) )
% 0.15/0.38 => ( ! [N6: nat] :
% 0.15/0.38 ( ( P @ N6 )
% 0.15/0.38 => ( P @ ( suc @ N6 ) ) )
% 0.15/0.38 => ( P @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat_induct
% 0.15/0.38 thf(fact_95_nat_OdiscI,axiom,
% 0.15/0.38 ! [Nat: nat,X23: nat] :
% 0.15/0.38 ( ( Nat
% 0.15/0.38 = ( suc @ X23 ) )
% 0.15/0.38 => ( Nat
% 0.15/0.38 != ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat.discI
% 0.15/0.38 thf(fact_96_old_Onat_Odistinct_I1_J,axiom,
% 0.15/0.38 ! [Nat2: nat] :
% 0.15/0.38 ( ( zero_zero @ nat )
% 0.15/0.38 != ( suc @ Nat2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % old.nat.distinct(1)
% 0.15/0.38 thf(fact_97_old_Onat_Odistinct_I2_J,axiom,
% 0.15/0.38 ! [Nat4: nat] :
% 0.15/0.38 ( ( suc @ Nat4 )
% 0.15/0.38 != ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % old.nat.distinct(2)
% 0.15/0.38 thf(fact_98_nat_Odistinct_I1_J,axiom,
% 0.15/0.38 ! [X23: nat] :
% 0.15/0.38 ( ( zero_zero @ nat )
% 0.15/0.38 != ( suc @ X23 ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat.distinct(1)
% 0.15/0.38 thf(fact_99_of__nat__aux_Osimps_I1_J,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [Inc: A > A,I: A] :
% 0.15/0.38 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 0.15/0.38 = I ) ) ).
% 0.15/0.38
% 0.15/0.38 % of_nat_aux.simps(1)
% 0.15/0.38 thf(fact_100_enat__setp_Ointros_I1_J,axiom,
% 0.15/0.38 coindu530039314t_setp @ ( zero_zero @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp.intros(1)
% 0.15/0.38 thf(fact_101_enat__setp_Ointros_I2_J,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( coindu530039314t_setp @ N )
% 0.15/0.38 => ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_setp.intros(2)
% 0.15/0.38 thf(fact_102_enat__set_Ointros_I1_J,axiom,
% 0.15/0.38 member @ extended_enat @ ( zero_zero @ extended_enat ) @ coinductive_enat_set ).
% 0.15/0.38
% 0.15/0.38 % enat_set.intros(1)
% 0.15/0.38 thf(fact_103_enat__set_Ointros_I2_J,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( member @ extended_enat @ N @ coinductive_enat_set )
% 0.15/0.38 => ( member @ extended_enat @ ( extended_eSuc @ N ) @ coinductive_enat_set ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_set.intros(2)
% 0.15/0.38 thf(fact_104_enat__0__iff_I2_J,axiom,
% 0.15/0.38 ! [X: nat] :
% 0.15/0.38 ( ( ( zero_zero @ extended_enat )
% 0.15/0.38 = ( extended_enat2 @ X ) )
% 0.15/0.38 = ( X
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_0_iff(2)
% 0.15/0.38 thf(fact_105_enat__0__iff_I1_J,axiom,
% 0.15/0.38 ! [X: nat] :
% 0.15/0.38 ( ( ( extended_enat2 @ X )
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( X
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_0_iff(1)
% 0.15/0.38 thf(fact_106_zero__enat__def,axiom,
% 0.15/0.38 ( ( zero_zero @ extended_enat )
% 0.15/0.38 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_enat_def
% 0.15/0.38 thf(fact_107_zero__reorient,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( zero @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [X: A] :
% 0.15/0.38 ( ( ( zero_zero @ A )
% 0.15/0.38 = X )
% 0.15/0.38 = ( X
% 0.15/0.38 = ( zero_zero @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_reorient
% 0.15/0.38 thf(fact_108_enat_Osize_I1_J,axiom,
% 0.15/0.38 ! [Nat: nat] :
% 0.15/0.38 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.size(1)
% 0.15/0.38 thf(fact_109_enat_Osize_I3_J,axiom,
% 0.15/0.38 ! [Nat: nat] :
% 0.15/0.38 ( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.size(3)
% 0.15/0.38 thf(fact_110_enat__unfold_Osimps,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( coindu1491768222unfold @ A )
% 0.15/0.38 = ( ^ [Stop: A > $o,Next: A > A,A3: A] : ( if @ extended_enat @ ( Stop @ A3 ) @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A3 ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_unfold.simps
% 0.15/0.38 thf(fact_111_enat__unfold__unique,axiom,
% 0.15/0.38 ! [A: $tType,Stop2: A > $o,H: A > extended_enat,Next2: A > A,X: A] :
% 0.15/0.38 ( ! [X4: A] :
% 0.15/0.38 ( ( ( Stop2 @ X4 )
% 0.15/0.38 => ( ( H @ X4 )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) )
% 0.15/0.38 & ( ~ ( Stop2 @ X4 )
% 0.15/0.38 => ( ( H @ X4 )
% 0.15/0.38 = ( extended_eSuc @ ( H @ ( Next2 @ X4 ) ) ) ) ) )
% 0.15/0.38 => ( ( H @ X )
% 0.15/0.38 = ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ X ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_unfold_unique
% 0.15/0.38 thf(fact_112_the__enat__eSuc,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ( extended_the_enat @ ( extended_eSuc @ N ) )
% 0.15/0.38 = ( suc @ ( extended_the_enat @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % the_enat_eSuc
% 0.15/0.38 thf(fact_113_list__decode_Ocases,axiom,
% 0.15/0.38 ! [X: nat] :
% 0.15/0.38 ( ( X
% 0.15/0.38 != ( zero_zero @ nat ) )
% 0.15/0.38 => ~ ! [N6: nat] :
% 0.15/0.38 ( X
% 0.15/0.38 != ( suc @ N6 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % list_decode.cases
% 0.15/0.38 thf(fact_114_not__enat__eq,axiom,
% 0.15/0.38 ! [X: extended_enat] :
% 0.15/0.38 ( ( ! [Y4: nat] :
% 0.15/0.38 ( X
% 0.15/0.38 != ( extended_enat2 @ Y4 ) ) )
% 0.15/0.38 = ( X
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % not_enat_eq
% 0.15/0.38 thf(fact_115_not__infinity__eq,axiom,
% 0.15/0.38 ! [X: extended_enat] :
% 0.15/0.38 ( ( X
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( ? [I2: nat] :
% 0.15/0.38 ( X
% 0.15/0.38 = ( extended_enat2 @ I2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % not_infinity_eq
% 0.15/0.38 thf(fact_116_eSuc__infinity,axiom,
% 0.15/0.38 ( ( extended_eSuc @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_infinity
% 0.15/0.38 thf(fact_117_enat_Osimps_I5_J,axiom,
% 0.15/0.38 ! [T: $tType,F1: nat > T,F2: T] :
% 0.15/0.38 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = F2 ) ).
% 0.15/0.38
% 0.15/0.38 % enat.simps(5)
% 0.15/0.38 thf(fact_118_enat__unfold__eq__0,axiom,
% 0.15/0.38 ! [A: $tType,Stop2: A > $o,Next2: A > A,A2: A] :
% 0.15/0.38 ( ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( Stop2 @ A2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_unfold_eq_0
% 0.15/0.38 thf(fact_119_enat__unfold__stop,axiom,
% 0.15/0.38 ! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
% 0.15/0.38 ( ( Stop2 @ A2 )
% 0.15/0.38 => ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_unfold_stop
% 0.15/0.38 thf(fact_120_enat_Osimps_I7_J,axiom,
% 0.15/0.38 ! [T: $tType,F1: nat > T,F2: T] :
% 0.15/0.38 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = F2 ) ).
% 0.15/0.38
% 0.15/0.38 % enat.simps(7)
% 0.15/0.38 thf(fact_121_enat_Osize_I2_J,axiom,
% 0.15/0.38 ( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.size(2)
% 0.15/0.38 thf(fact_122_enat_Osize_I4_J,axiom,
% 0.15/0.38 ( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.size(4)
% 0.15/0.38 thf(fact_123_infinity__ne__i0,axiom,
% 0.15/0.38 ( ( extend1396239628finity @ extended_enat )
% 0.15/0.38 != ( zero_zero @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % infinity_ne_i0
% 0.15/0.38 thf(fact_124_enat_Odistinct_I2_J,axiom,
% 0.15/0.38 ! [Nat5: nat] :
% 0.15/0.38 ( ( extend1396239628finity @ extended_enat )
% 0.15/0.38 != ( extended_enat2 @ Nat5 ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.distinct(2)
% 0.15/0.38 thf(fact_125_enat_Odistinct_I1_J,axiom,
% 0.15/0.38 ! [Nat: nat] :
% 0.15/0.38 ( ( extended_enat2 @ Nat )
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.distinct(1)
% 0.15/0.38 thf(fact_126_enat2__cases,axiom,
% 0.15/0.38 ! [Y: extended_enat,Ya: extended_enat] :
% 0.15/0.38 ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ! [Nata: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 != ( extended_enat2 @ Nata ) ) )
% 0.15/0.38 => ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( Ya
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) )
% 0.15/0.38 => ( ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ! [Nat3: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 != ( extended_enat2 @ Nat3 ) ) )
% 0.15/0.38 => ~ ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( Ya
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat2_cases
% 0.15/0.38 thf(fact_127_enat3__cases,axiom,
% 0.15/0.38 ! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 0.15/0.38 ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( ? [Nata: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 = ( extended_enat2 @ Nata ) )
% 0.15/0.38 => ! [Natb: nat] :
% 0.15/0.38 ( Yb
% 0.15/0.38 != ( extended_enat2 @ Natb ) ) ) )
% 0.15/0.38 => ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( ? [Nata: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 = ( extended_enat2 @ Nata ) )
% 0.15/0.38 => ( Yb
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.15/0.38 => ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( ( Ya
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ! [Nata: nat] :
% 0.15/0.38 ( Yb
% 0.15/0.38 != ( extended_enat2 @ Nata ) ) ) )
% 0.15/0.38 => ( ( ? [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( ( Ya
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( Yb
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.15/0.38 => ( ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ? [Nat3: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ! [Nata: nat] :
% 0.15/0.38 ( Yb
% 0.15/0.38 != ( extended_enat2 @ Nata ) ) ) )
% 0.15/0.38 => ( ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ? [Nat3: nat] :
% 0.15/0.38 ( Ya
% 0.15/0.38 = ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( Yb
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.15/0.38 => ( ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ( Ya
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ! [Nat3: nat] :
% 0.15/0.38 ( Yb
% 0.15/0.38 != ( extended_enat2 @ Nat3 ) ) ) )
% 0.15/0.38 => ~ ( ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ( Ya
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( Yb
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat3_cases
% 0.15/0.38 thf(fact_128_enat_Oexhaust,axiom,
% 0.15/0.38 ! [Y: extended_enat] :
% 0.15/0.38 ( ! [Nat3: nat] :
% 0.15/0.38 ( Y
% 0.15/0.38 != ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( Y
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.exhaust
% 0.15/0.38 thf(fact_129_enat_Oinducts,axiom,
% 0.15/0.38 ! [P: extended_enat > $o,Enat: extended_enat] :
% 0.15/0.38 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 0.15/0.38 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( P @ Enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat.inducts
% 0.15/0.38 thf(fact_130_enat__ex__split,axiom,
% 0.15/0.38 ( ( ^ [P4: extended_enat > $o] :
% 0.15/0.38 ? [X8: extended_enat] : ( P4 @ X8 ) )
% 0.15/0.38 = ( ^ [P5: extended_enat > $o] :
% 0.15/0.38 ( ( P5 @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 | ? [X2: nat] : ( P5 @ ( extended_enat2 @ X2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_ex_split
% 0.15/0.38 thf(fact_131_eSuc__eq__infinity__iff,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( ( extended_eSuc @ N )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( N
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_eq_infinity_iff
% 0.15/0.38 thf(fact_132_infinity__eq__eSuc__iff,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( ( extend1396239628finity @ extended_enat )
% 0.15/0.38 = ( extended_eSuc @ N ) )
% 0.15/0.38 = ( N
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % infinity_eq_eSuc_iff
% 0.15/0.38 thf(fact_133_enat__unfold__next,axiom,
% 0.15/0.38 ! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
% 0.15/0.38 ( ~ ( Stop2 @ A2 )
% 0.15/0.38 => ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 0.15/0.38 = ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ ( Next2 @ A2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_unfold_next
% 0.15/0.38 thf(fact_134_enat__the__enat,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ( extended_enat2 @ ( extended_the_enat @ N ) )
% 0.15/0.38 = N ) ) ).
% 0.15/0.38
% 0.15/0.38 % enat_the_enat
% 0.15/0.38 thf(fact_135_exists__least__lemma,axiom,
% 0.15/0.38 ! [P: nat > $o] :
% 0.15/0.38 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.15/0.38 => ( ? [X12: nat] : ( P @ X12 )
% 0.15/0.38 => ? [N6: nat] :
% 0.15/0.38 ( ~ ( P @ N6 )
% 0.15/0.38 & ( P @ ( suc @ N6 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % exists_least_lemma
% 0.15/0.38 thf(fact_136_dependent__nat__choice,axiom,
% 0.15/0.38 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 0.15/0.38 ( ? [X12: A] : ( P @ ( zero_zero @ nat ) @ X12 )
% 0.15/0.38 => ( ! [X4: A,N6: nat] :
% 0.15/0.38 ( ( P @ N6 @ X4 )
% 0.15/0.38 => ? [Y3: A] :
% 0.15/0.38 ( ( P @ ( suc @ N6 ) @ Y3 )
% 0.15/0.38 & ( Q @ N6 @ X4 @ Y3 ) ) )
% 0.15/0.38 => ? [F4: nat > A] :
% 0.15/0.38 ! [N5: nat] :
% 0.15/0.38 ( ( P @ N5 @ ( F4 @ N5 ) )
% 0.15/0.38 & ( Q @ N5 @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % dependent_nat_choice
% 0.15/0.38 thf(fact_137_times__enat__simps_I3_J,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( ( N
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) )
% 0.15/0.38 & ( ( N
% 0.15/0.38 != ( zero_zero @ nat ) )
% 0.15/0.38 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % times_enat_simps(3)
% 0.15/0.38 thf(fact_138_times__enat__simps_I4_J,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( ( M
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) )
% 0.15/0.38 & ( ( M
% 0.15/0.38 != ( zero_zero @ nat ) )
% 0.15/0.38 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % times_enat_simps(4)
% 0.15/0.38 thf(fact_139_times__enat__simps_I2_J,axiom,
% 0.15/0.38 ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % times_enat_simps(2)
% 0.15/0.38 thf(fact_140_imult__is__0,axiom,
% 0.15/0.38 ! [M: extended_enat,N: extended_enat] :
% 0.15/0.38 ( ( ( times_times @ extended_enat @ M @ N )
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 = ( ( M
% 0.15/0.38 = ( zero_zero @ extended_enat ) )
% 0.15/0.38 | ( N
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % imult_is_0
% 0.15/0.38 thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.38 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.15/0.38 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % ab_semigroup_mult_class.mult_ac(1)
% 0.15/0.38 thf(fact_142_mult_Oassoc,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semigroup_mult @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.38 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.15/0.38 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult.assoc
% 0.15/0.38 thf(fact_143_mult_Ocommute,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.15/0.38 => ( ( times_times @ A )
% 0.15/0.38 = ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult.commute
% 0.15/0.38 thf(fact_144_mult_Oleft__commute,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [B2: A,A2: A,C2: A] :
% 0.15/0.38 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 0.15/0.38 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult.left_commute
% 0.15/0.38 thf(fact_145_imult__is__infinity,axiom,
% 0.15/0.38 ! [A2: extended_enat,B2: extended_enat] :
% 0.15/0.38 ( ( ( times_times @ extended_enat @ A2 @ B2 )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( ( ( A2
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 & ( B2
% 0.15/0.38 != ( zero_zero @ extended_enat ) ) )
% 0.15/0.38 | ( ( B2
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 & ( A2
% 0.15/0.38 != ( zero_zero @ extended_enat ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % imult_is_infinity
% 0.15/0.38 thf(fact_146_mult__zero__left,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( mult_zero @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_zero_left
% 0.15/0.38 thf(fact_147_mult__zero__right,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( mult_zero @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_zero_right
% 0.15/0.38 thf(fact_148_mult__eq__0__iff,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ( times_times @ A @ A2 @ B2 )
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 = ( ( A2
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 | ( B2
% 0.15/0.38 = ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_eq_0_iff
% 0.15/0.38 thf(fact_149_mult__cancel__left,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.38 ( ( ( times_times @ A @ C2 @ A2 )
% 0.15/0.38 = ( times_times @ A @ C2 @ B2 ) )
% 0.15/0.38 = ( ( C2
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 | ( A2 = B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_cancel_left
% 0.15/0.38 thf(fact_150_mult__is__0,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ M @ N )
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 = ( ( M
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 | ( N
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_is_0
% 0.15/0.38 thf(fact_151_mult__0__right,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_0_right
% 0.15/0.38 thf(fact_152_mult__cancel1,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ K @ M )
% 0.15/0.38 = ( times_times @ nat @ K @ N ) )
% 0.15/0.38 = ( ( M = N )
% 0.15/0.38 | ( K
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_cancel1
% 0.15/0.38 thf(fact_153_mult__cancel2,axiom,
% 0.15/0.38 ! [M: nat,K: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ M @ K )
% 0.15/0.38 = ( times_times @ nat @ N @ K ) )
% 0.15/0.38 = ( ( M = N )
% 0.15/0.38 | ( K
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_cancel2
% 0.15/0.38 thf(fact_154_mult__cancel__right,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,C2: A,B2: A] :
% 0.15/0.38 ( ( ( times_times @ A @ A2 @ C2 )
% 0.15/0.38 = ( times_times @ A @ B2 @ C2 ) )
% 0.15/0.38 = ( ( C2
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 | ( A2 = B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_cancel_right
% 0.15/0.38 thf(fact_155_mult__eq__1__iff,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ M @ N )
% 0.15/0.38 = ( suc @ ( zero_zero @ nat ) ) )
% 0.15/0.38 = ( ( M
% 0.15/0.38 = ( suc @ ( zero_zero @ nat ) ) )
% 0.15/0.38 & ( N
% 0.15/0.38 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_eq_1_iff
% 0.15/0.38 thf(fact_156_one__eq__mult__iff,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.15/0.38 = ( times_times @ nat @ M @ N ) )
% 0.15/0.38 = ( ( M
% 0.15/0.38 = ( suc @ ( zero_zero @ nat ) ) )
% 0.15/0.38 & ( N
% 0.15/0.38 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % one_eq_mult_iff
% 0.15/0.38 thf(fact_157_times__enat__simps_I1_J,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.15/0.38 = ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % times_enat_simps(1)
% 0.15/0.38 thf(fact_158_mult__0,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_0
% 0.15/0.38 thf(fact_159_Suc__mult__cancel1,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ ( suc @ K ) @ M )
% 0.15/0.38 = ( times_times @ nat @ ( suc @ K ) @ N ) )
% 0.15/0.38 = ( M = N ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_mult_cancel1
% 0.15/0.38 thf(fact_160_mult__right__cancel,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.38 ( ( C2
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 => ( ( ( times_times @ A @ A2 @ C2 )
% 0.15/0.38 = ( times_times @ A @ B2 @ C2 ) )
% 0.15/0.38 = ( A2 = B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_right_cancel
% 0.15/0.38 thf(fact_161_mult__left__cancel,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.38 ( ( C2
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 => ( ( ( times_times @ A @ C2 @ A2 )
% 0.15/0.38 = ( times_times @ A @ C2 @ B2 ) )
% 0.15/0.38 = ( A2 = B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_left_cancel
% 0.15/0.38 thf(fact_162_no__zero__divisors,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( A2
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 => ( ( B2
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 => ( ( times_times @ A @ A2 @ B2 )
% 0.15/0.38 != ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % no_zero_divisors
% 0.15/0.38 thf(fact_163_divisors__zero,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ( times_times @ A @ A2 @ B2 )
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 => ( ( A2
% 0.15/0.38 = ( zero_zero @ A ) )
% 0.15/0.38 | ( B2
% 0.15/0.38 = ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % divisors_zero
% 0.15/0.38 thf(fact_164_mult__not__zero,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( mult_zero @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ( times_times @ A @ A2 @ B2 )
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 => ( ( A2
% 0.15/0.38 != ( zero_zero @ A ) )
% 0.15/0.38 & ( B2
% 0.15/0.38 != ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_not_zero
% 0.15/0.38 thf(fact_165_semiring__normalization__rules_I9_J,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % semiring_normalization_rules(9)
% 0.15/0.38 thf(fact_166_semiring__normalization__rules_I10_J,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % semiring_normalization_rules(10)
% 0.15/0.38 thf(fact_167_idiff__infinity__right,axiom,
% 0.15/0.38 ! [A2: nat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_infinity_right
% 0.15/0.38 thf(fact_168_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ A2 @ A2 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % cancel_comm_monoid_add_class.diff_cancel
% 0.15/0.38 thf(fact_169_diff__zero,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 = A2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_zero
% 0.15/0.38 thf(fact_170_zero__diff,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_diff
% 0.15/0.38 thf(fact_171_diff__0__right,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 = A2 ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_0_right
% 0.15/0.38 thf(fact_172_diff__self,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ A2 @ A2 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_self
% 0.15/0.38 thf(fact_173_idiff__0__right,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 0.15/0.38 = N ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_0_right
% 0.15/0.38 thf(fact_174_idiff__0,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_0
% 0.15/0.38 thf(fact_175_eSuc__minus__eSuc,axiom,
% 0.15/0.38 ! [N: extended_enat,M: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.15/0.38 = ( minus_minus @ extended_enat @ N @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % eSuc_minus_eSuc
% 0.15/0.38 thf(fact_176_idiff__infinity,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N )
% 0.15/0.38 = ( extend1396239628finity @ extended_enat ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_infinity
% 0.15/0.38 thf(fact_177_idiff__enat__0__right,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
% 0.15/0.38 = N ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_enat_0_right
% 0.15/0.38 thf(fact_178_idiff__enat__0,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
% 0.15/0.38 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_enat_0
% 0.15/0.38 thf(fact_179_idiff__self,axiom,
% 0.15/0.38 ! [N: extended_enat] :
% 0.15/0.38 ( ( N
% 0.15/0.38 != ( extend1396239628finity @ extended_enat ) )
% 0.15/0.38 => ( ( minus_minus @ extended_enat @ N @ N )
% 0.15/0.38 = ( zero_zero @ extended_enat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_self
% 0.15/0.38 thf(fact_180_eq__iff__diff__eq__0,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ( ( ^ [Y6: A,Z: A] : ( Y6 = Z ) )
% 0.15/0.38 = ( ^ [A3: A,B3: A] :
% 0.15/0.38 ( ( minus_minus @ A @ A3 @ B3 )
% 0.15/0.38 = ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % eq_iff_diff_eq_0
% 0.15/0.38 thf(fact_181_diff__eq__diff__eq,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.15/0.38 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.15/0.38 = ( minus_minus @ A @ C2 @ D ) )
% 0.15/0.38 => ( ( A2 = B2 )
% 0.15/0.38 = ( C2 = D ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_eq_diff_eq
% 0.15/0.38 thf(fact_182_diff__right__commute,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,C2: A,B2: A] :
% 0.15/0.38 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 0.15/0.38 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_right_commute
% 0.15/0.38 thf(fact_183_nat__mult__eq__cancel__disj,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ( times_times @ nat @ K @ M )
% 0.15/0.38 = ( times_times @ nat @ K @ N ) )
% 0.15/0.38 = ( ( K
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 | ( M = N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat_mult_eq_cancel_disj
% 0.15/0.38 thf(fact_184_one__le__mult__iff,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
% 0.15/0.38 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 0.15/0.38 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % one_le_mult_iff
% 0.15/0.38 thf(fact_185_triangle__0,axiom,
% 0.15/0.38 ( ( nat_triangle @ ( zero_zero @ nat ) )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % triangle_0
% 0.15/0.38 thf(fact_186_diff__self__eq__0,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( minus_minus @ nat @ M @ M )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_self_eq_0
% 0.15/0.38 thf(fact_187_diff__0__eq__0,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.15/0.38 = ( zero_zero @ nat ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_0_eq_0
% 0.15/0.38 thf(fact_188_diff__Suc__Suc,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.15/0.38 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_Suc_Suc
% 0.15/0.38 thf(fact_189_Suc__diff__diff,axiom,
% 0.15/0.38 ! [M: nat,N: nat,K: nat] :
% 0.15/0.38 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.15/0.38 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_diff_diff
% 0.15/0.38 thf(fact_190_diff__diff__cancel,axiom,
% 0.15/0.38 ! [I: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ I @ N )
% 0.15/0.38 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.15/0.38 = I ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_diff_cancel
% 0.15/0.38 thf(fact_191_le__zero__eq,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [N: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.15/0.38 = ( N
% 0.15/0.38 = ( zero_zero @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_zero_eq
% 0.15/0.38 thf(fact_192_diff__is__0__eq_H,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( ( minus_minus @ nat @ M @ N )
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_is_0_eq'
% 0.15/0.38 thf(fact_193_diff__is__0__eq,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ( minus_minus @ nat @ M @ N )
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_is_0_eq
% 0.15/0.38 thf(fact_194_le0,axiom,
% 0.15/0.38 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.15/0.38
% 0.15/0.38 % le0
% 0.15/0.38 thf(fact_195_Suc__le__mono,axiom,
% 0.15/0.38 ! [N: nat,M: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.15/0.38 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_le_mono
% 0.15/0.38 thf(fact_196_idiff__enat__enat,axiom,
% 0.15/0.38 ! [A2: nat,B2: nat] :
% 0.15/0.38 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
% 0.15/0.38 = ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % idiff_enat_enat
% 0.15/0.38 thf(fact_197_diff__ge__0__iff__ge,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.15/0.38 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_ge_0_iff_ge
% 0.15/0.38 thf(fact_198_le__cube,axiom,
% 0.15/0.38 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_cube
% 0.15/0.38 thf(fact_199_le__square,axiom,
% 0.15/0.38 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_square
% 0.15/0.38 thf(fact_200_mult__le__mono,axiom,
% 0.15/0.38 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ I @ J )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ K @ L )
% 0.15/0.38 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_le_mono
% 0.15/0.38 thf(fact_201_mult__le__mono1,axiom,
% 0.15/0.38 ! [I: nat,J: nat,K: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ I @ J )
% 0.15/0.38 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_le_mono1
% 0.15/0.38 thf(fact_202_mult__le__mono2,axiom,
% 0.15/0.38 ! [I: nat,J: nat,K: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ I @ J )
% 0.15/0.38 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_le_mono2
% 0.15/0.38 thf(fact_203_diff__mult__distrib,axiom,
% 0.15/0.38 ! [M: nat,N: nat,K: nat] :
% 0.15/0.38 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 0.15/0.38 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_mult_distrib
% 0.15/0.38 thf(fact_204_diff__mult__distrib2,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 0.15/0.38 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_mult_distrib2
% 0.15/0.38 thf(fact_205_minus__nat_Odiff__0,axiom,
% 0.15/0.38 ! [M: nat] :
% 0.15/0.38 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.15/0.38 = M ) ).
% 0.15/0.38
% 0.15/0.38 % minus_nat.diff_0
% 0.15/0.38 thf(fact_206_diffs0__imp__equal,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ( minus_minus @ nat @ M @ N )
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 => ( ( ( minus_minus @ nat @ N @ M )
% 0.15/0.38 = ( zero_zero @ nat ) )
% 0.15/0.38 => ( M = N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diffs0_imp_equal
% 0.15/0.38 thf(fact_207_prod__decode__aux_Oinduct,axiom,
% 0.15/0.38 ! [P: nat > nat > $o,A0: nat,A1: nat] :
% 0.15/0.38 ( ! [K2: nat,M2: nat] :
% 0.15/0.38 ( ( ~ ( ord_less_eq @ nat @ M2 @ K2 )
% 0.15/0.38 => ( P @ ( suc @ K2 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K2 ) ) ) )
% 0.15/0.38 => ( P @ K2 @ M2 ) )
% 0.15/0.38 => ( P @ A0 @ A1 ) ) ).
% 0.15/0.38
% 0.15/0.38 % prod_decode_aux.induct
% 0.15/0.38 thf(fact_208_zero__induct__lemma,axiom,
% 0.15/0.38 ! [P: nat > $o,K: nat,I: nat] :
% 0.15/0.38 ( ( P @ K )
% 0.15/0.38 => ( ! [N6: nat] :
% 0.15/0.38 ( ( P @ ( suc @ N6 ) )
% 0.15/0.38 => ( P @ N6 ) )
% 0.15/0.38 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_induct_lemma
% 0.15/0.38 thf(fact_209_eq__diff__iff,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ K @ M )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ K @ N )
% 0.15/0.38 => ( ( ( minus_minus @ nat @ M @ K )
% 0.15/0.38 = ( minus_minus @ nat @ N @ K ) )
% 0.15/0.38 = ( M = N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % eq_diff_iff
% 0.15/0.38 thf(fact_210_le__diff__iff,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ K @ M )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ K @ N )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.15/0.38 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_diff_iff
% 0.15/0.38 thf(fact_211_Nat_Odiff__diff__eq,axiom,
% 0.15/0.38 ! [K: nat,M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ K @ M )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ K @ N )
% 0.15/0.38 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.15/0.38 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % Nat.diff_diff_eq
% 0.15/0.38 thf(fact_212_diff__le__mono,axiom,
% 0.15/0.38 ! [M: nat,N: nat,L: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_le_mono
% 0.15/0.38 thf(fact_213_diff__le__self,axiom,
% 0.15/0.38 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.15/0.38
% 0.15/0.38 % diff_le_self
% 0.15/0.38 thf(fact_214_le__diff__iff_H,axiom,
% 0.15/0.38 ! [A2: nat,C2: nat,B2: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ A2 @ C2 )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 0.15/0.38 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_diff_iff'
% 0.15/0.38 thf(fact_215_diff__le__mono2,axiom,
% 0.15/0.38 ! [M: nat,N: nat,L: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_le_mono2
% 0.15/0.38 thf(fact_216_Suc__diff__le,axiom,
% 0.15/0.38 ! [N: nat,M: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ N @ M )
% 0.15/0.38 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 0.15/0.38 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_diff_le
% 0.15/0.38 thf(fact_217_diff__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,D: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ D @ C2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_mono
% 0.15/0.38 thf(fact_218_diff__left__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [B2: A,A2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_left_mono
% 0.15/0.38 thf(fact_219_diff__right__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_right_mono
% 0.15/0.38 thf(fact_220_diff__eq__diff__less__eq,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.15/0.38 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.15/0.38 = ( minus_minus @ A @ C2 @ D ) )
% 0.15/0.38 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % diff_eq_diff_less_eq
% 0.15/0.38 thf(fact_221_lift__Suc__mono__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( order @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [F3: nat > A,N: nat,N7: nat] :
% 0.15/0.38 ( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ N @ N7 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lift_Suc_mono_le
% 0.15/0.38 thf(fact_222_lift__Suc__antimono__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( order @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [F3: nat > A,N: nat,N7: nat] :
% 0.15/0.38 ( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ N @ N7 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % lift_Suc_antimono_le
% 0.15/0.38 thf(fact_223_nat__le__linear,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % nat_le_linear
% 0.15/0.38 thf(fact_224_le__antisym,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ N @ M )
% 0.15/0.38 => ( M = N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_antisym
% 0.15/0.38 thf(fact_225_eq__imp__le,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( M = N )
% 0.15/0.38 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % eq_imp_le
% 0.15/0.38 thf(fact_226_le__trans,axiom,
% 0.15/0.38 ! [I: nat,J: nat,K: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ I @ J )
% 0.15/0.38 => ( ( ord_less_eq @ nat @ J @ K )
% 0.15/0.38 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_trans
% 0.15/0.38 thf(fact_227_le__refl,axiom,
% 0.15/0.38 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.15/0.38
% 0.15/0.38 % le_refl
% 0.15/0.38 thf(fact_228_wlog__linorder__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( linorder @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.15/0.38 ( ! [A5: A,B4: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.15/0.38 => ( P @ A5 @ B4 ) )
% 0.15/0.38 => ( ( ( P @ B2 @ A2 )
% 0.15/0.38 => ( P @ A2 @ B2 ) )
% 0.15/0.38 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % wlog_linorder_le
% 0.15/0.38 thf(fact_229_full__nat__induct,axiom,
% 0.15/0.38 ! [P: nat > $o,N: nat] :
% 0.15/0.38 ( ! [N6: nat] :
% 0.15/0.38 ( ! [M3: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N6 )
% 0.15/0.38 => ( P @ M3 ) )
% 0.15/0.38 => ( P @ N6 ) )
% 0.15/0.38 => ( P @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % full_nat_induct
% 0.15/0.38 thf(fact_230_not__less__eq__eq,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.15/0.38 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.15/0.38
% 0.15/0.38 % not_less_eq_eq
% 0.15/0.38 thf(fact_231_Suc__n__not__le__n,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_n_not_le_n
% 0.15/0.38 thf(fact_232_le__Suc__eq,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.15/0.38 = ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 | ( M
% 0.15/0.38 = ( suc @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_Suc_eq
% 0.15/0.38 thf(fact_233_Suc__le__D,axiom,
% 0.15/0.38 ! [N: nat,M4: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 0.15/0.38 => ? [M2: nat] :
% 0.15/0.38 ( M4
% 0.15/0.38 = ( suc @ M2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_le_D
% 0.15/0.38 thf(fact_234_le__SucI,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_SucI
% 0.15/0.38 thf(fact_235_le__SucE,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.15/0.38 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.15/0.38 => ( M
% 0.15/0.38 = ( suc @ N ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_SucE
% 0.15/0.38 thf(fact_236_Suc__leD,axiom,
% 0.15/0.38 ! [M: nat,N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.15/0.38 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.15/0.38
% 0.15/0.38 % Suc_leD
% 0.15/0.38 thf(fact_237_zero__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_le
% 0.15/0.38 thf(fact_238_less__eq__nat_Osimps_I1_J,axiom,
% 0.15/0.38 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.15/0.38
% 0.15/0.38 % less_eq_nat.simps(1)
% 0.15/0.38 thf(fact_239_le__0__eq,axiom,
% 0.15/0.38 ! [N: nat] :
% 0.15/0.38 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.15/0.38 = ( N
% 0.15/0.38 = ( zero_zero @ nat ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % le_0_eq
% 0.15/0.38 thf(fact_240_mult__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ C2 @ D )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_mono
% 0.15/0.38 thf(fact_241_mult__mono_H,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A,D: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ C2 @ D )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_mono'
% 0.15/0.38 thf(fact_242_zero__le__square,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( linordered_ring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % zero_le_square
% 0.15/0.38 thf(fact_243_split__mult__pos__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.15/0.38 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 0.15/0.38 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % split_mult_pos_le
% 0.15/0.38 thf(fact_244_mult__left__mono__neg,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [B2: A,A2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_left_mono_neg
% 0.15/0.38 thf(fact_245_mult__nonpos__nonpos,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.15/0.38 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_nonpos_nonpos
% 0.15/0.38 thf(fact_246_mult__left__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_left_mono
% 0.15/0.38 thf(fact_247_mult__right__mono__neg,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [B2: A,A2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_right_mono_neg
% 0.15/0.38 thf(fact_248_mult__right__mono,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_right_mono
% 0.15/0.38 thf(fact_249_mult__le__0__iff,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( linord581940658strict @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 0.15/0.38 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.15/0.38 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_le_0_iff
% 0.15/0.38 thf(fact_250_split__mult__neg__le,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.15/0.38 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.38 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
% 0.15/0.38 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % split_mult_neg_le
% 0.15/0.38 thf(fact_251_mult__nonneg__nonneg,axiom,
% 0.15/0.38 ! [A: $tType] :
% 0.15/0.38 ( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
% 0.15/0.38 => ! [A2: A,B2: A] :
% 0.15/0.38 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.38 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.15/0.38 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 % mult_nonneg_nonneg
% 0.15/0.38
% 0.15/0.38 %----Type constructors (31)
% 0.15/0.38 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.15/0.38 ! [A6: $tType,A7: $tType] :
% 0.15/0.38 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.15/0.38 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.15/0.38 semiri1923998003cancel @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.15/0.38 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 0.15/0.38 semiri1193490041visors @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.15/0.38 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.15/0.38 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
% 0.15/0.38 ordered_semiring_0 @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.15/0.38 ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.15/0.38 comm_monoid_diff @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 0.15/0.38 ordered_semiring @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.15/0.38 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.15/0.38 semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.15/0.38 linorder @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 0.15/0.38 semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 0.15/0.38 mult_zero @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.15/0.38 order @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.15/0.38 zero @ nat @ ( type2 @ nat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.15/0.38 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
% 0.15/0.38 linorder @ $o @ ( type2 @ $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
% 0.15/0.38 order @ $o @ ( type2 @ $o ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_5,axiom,
% 0.15/0.38 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_6,axiom,
% 0.15/0.38 semiri1193490041visors @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_7,axiom,
% 0.15/0.38 ab_semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_8,axiom,
% 0.15/0.38 ordered_semiring @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_9,axiom,
% 0.15/0.38 comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_10,axiom,
% 0.15/0.38 semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_11,axiom,
% 0.15/0.38 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_12,axiom,
% 0.15/0.38 semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_13,axiom,
% 0.15/0.38 mult_zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_14,axiom,
% 0.15/0.38 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_15,axiom,
% 0.15/0.38 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.15/0.38
% 0.15/0.38 %----Helper facts (3)
% 0.15/0.38 thf(help_If_3_1_T,axiom,
% 0.15/0.38 ! [P: $o] :
% 0.15/0.38 ( ( P = $true )
% 0.15/0.38 | ( P = $false ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(help_If_2_1_T,axiom,
% 0.15/0.38 ! [A: $tType,X: A,Y: A] :
% 0.15/0.38 ( ( if @ A @ $false @ X @ Y )
% 0.15/0.38 = Y ) ).
% 0.15/0.38
% 0.15/0.38 thf(help_If_1_1_T,axiom,
% 0.15/0.38 ! [A: $tType,X: A,Y: A] :
% 0.15/0.38 ( ( if @ A @ $true @ X @ Y )
% 0.15/0.38 = X ) ).
% 0.15/0.38
% 0.15/0.38 %----Conjectures (2)
% 0.15/0.38 thf(conj_0,hypothesis,
% 0.15/0.38 ! [X9: a,Xs6: coindu1593790203_llist @ a] :
% 0.15/0.38 ( ( xsa
% 0.15/0.38 = ( coindu1121789889_LCons @ a @ X9 @ Xs6 ) )
% 0.15/0.38 => thesis ) ).
% 0.15/0.38
% 0.15/0.38 thf(conj_1,conjecture,
% 0.15/0.38 thesis ).
% 0.15/0.38
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 ------------------------
% 2.75/1.59 ---- Embedded in THF ---
% 2.75/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.75/1.59 %%% Generated on Tue Feb 24 06:07:45 EST 2026
% 2.75/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.75/1.59 %%% Logic specification used:
% 2.75/1.59 %%% thf(spec, logic, $$dhol).
% 2.75/1.59
% 2.75/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2DTF10986.p
% 2.75/1.59 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist, type, coindu1593790203_llist: ($tType > $tType)).
% 2.75/1.59 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.75/1.59 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.75/1.59 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.75/1.59 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.75/1.59 thf(ty_tf_a, type, a: $tType).
% 2.75/1.59 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Oordered__ring, type, ordered_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Oordered__semiring, type, ordered_semiring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Oordered__semiring__0, type, ordered_semiring_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord581940658strict: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri1923998003cancel: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset, type, coindu1928975208n_lset: (!> [A:$tType]: (((set @ A) > ((coindu1593790203_llist @ A) > (set @ A)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates, type, coindu1529956631erates: (!> [A:$tType]: (((A > A) > (A > (coindu1593790203_llist @ A)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend, type, coindu268472904append: (!> [A:$tType]: (((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite, type, coindu1213758845finite: (!> [A:$tType]: (((coindu1593790203_llist @ A) > $o)))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength, type, coindu1018505716length: (!> [A:$tType]: (((coindu1593790203_llist @ A) > extended_enat)))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord, type, coindu300403952lexord: (!> [A:$tType]: (((A > (A > $o)) > ((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons, type, coindu1121789889_LCons: (!> [A:$tType]: ((A > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil, type, coindu1598213697e_LNil: (!> [A:$tType]: ((coindu1593790203_llist @ A)))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist, type, coindu882539134_llist: (!> [B:$tType,A:$tType]: ((B > ((A > ((coindu1593790203_llist @ A) > B)) > ((coindu1593790203_llist @ A) > B)))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist, type, coindu1244876290_llist: (!> [C:$tType,A:$tType]: (((C > $o) > ((C > A) > ((C > $o) > ((C > (coindu1593790203_llist @ A)) > ((C > C) > (C > (coindu1593790203_llist @ A)))))))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist, type, coindu44415597_llist: (!> [A:$tType]: (((A > $o) > ((coindu1593790203_llist @ A) > $o))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember, type, coindu567634248member: (!> [A:$tType]: ((A > ((coindu1593790203_llist @ A) > $o))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olsetp, type, coindu1055736764_lsetp: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (A > $o))))).
% 2.75/1.59 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist, type, coindu1599971794_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coindu1593790203_llist @ B)))))))).
% 2.75/1.59 thf(sy_c_Coinductive__Nat_Oenat__set, type, coinductive_enat_set: (set @ extended_enat)).
% 2.75/1.59 thf(sy_c_Coinductive__Nat_Oenat__setp, type, coindu530039314t_setp: (extended_enat > $o)).
% 2.75/1.59 thf(sy_c_Coinductive__Nat_Oenat__unfold, type, coindu1491768222unfold: (!> [A:$tType]: (((A > $o) > ((A > A) > (A > extended_enat)))))).
% 2.75/1.59 thf(sy_c_Extended__Nat_OeSuc, type, extended_eSuc: (extended_enat > extended_enat)).
% 2.75/1.59 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.75/1.59 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.75/1.59 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.75/1.59 thf(sy_c_Extended__Nat_Oenat_Osize__enat, type, extended_size_enat: (extended_enat > nat)).
% 2.75/1.59 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.75/1.59 thf(sy_c_Extended__Nat_Othe__enat, type, extended_the_enat: (extended_enat > nat)).
% 2.75/1.59 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.75/1.59 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 2.75/1.59 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.75/1.59 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.75/1.59 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.75/1.59 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri532925092at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 2.75/1.59 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.75/1.59 thf(sy_c_Nat__Bijection_Otriangle, type, nat_triangle: (nat > nat)).
% 2.75/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.75/1.59 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.75/1.59 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.75/1.59 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.75/1.59 thf(sy_v_na____, type, na: nat).
% 2.75/1.59 thf(sy_v_thesis____, type, thesis: $o).
% 2.75/1.59 thf(sy_v_xsa____, type, xsa: (coindu1593790203_llist @ a)).
% 2.75/1.59 thf(fact_0_len, axiom, (((coindu1018505716length @ a) @ xsa) = (extended_enat2 @ (suc @ na)))).
% 2.75/1.59 thf(fact_1_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coindu1593790203_llist @ A),Y21:A,Y22:(coindu1593790203_llist @ A)]: ((((((coindu1121789889_LCons @ A) @ X21) @ X22) = (((coindu1121789889_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.75/1.59 thf(fact_2_lmember__code_I2_J, axiom, (! [A:$tType,X:A,Y:A,Ys:(coindu1593790203_llist @ A)]: (((((coindu567634248member @ A) @ X) @ (((coindu1121789889_LCons @ A) @ Y) @ Ys)) = ((X = Y) | (((coindu567634248member @ A) @ X) @ Ys)))))).
% 2.75/1.59 thf(fact_3_iterates_Ocode, axiom, (! [A:$tType]: (((coindu1529956631erates @ A) = (^ [F:(A > A),X2:A]: ((((coindu1121789889_LCons @ A) @ X2) @ (((coindu1529956631erates @ A) @ F) @ (F @ X2))))))))).
% 2.75/1.59 thf(fact_4_llist_Opred__inject_I2_J, axiom, (! [A:$tType,P:(A > $o),A2:A,Aa:(coindu1593790203_llist @ A)]: (((((coindu44415597_llist @ A) @ P) @ (((coindu1121789889_LCons @ A) @ A2) @ Aa)) = ((P @ A2) & (((coindu44415597_llist @ A) @ P) @ Aa)))))).
% 2.75/1.59 thf(fact_5_llexord__LCons__eq, axiom, (! [A:$tType,R:(A > (A > $o)),Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),X:A]: ((((((coindu300403952lexord @ A) @ R) @ Xs) @ Ys) => ((((coindu300403952lexord @ A) @ R) @ (((coindu1121789889_LCons @ A) @ X) @ Xs)) @ (((coindu1121789889_LCons @ A) @ X) @ Ys)))))).
% 2.75/1.59 thf(fact_6_llexord__LCons__less, axiom, (! [A:$tType,R:(A > (A > $o)),X:A,Y:A,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((R @ X) @ Y) => ((((coindu300403952lexord @ A) @ R) @ (((coindu1121789889_LCons @ A) @ X) @ Xs)) @ (((coindu1121789889_LCons @ A) @ Y) @ Ys)))))).
% 2.75/1.59 thf(fact_7_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B2:B,X:A,Xs:(coindu1593790203_llist @ A)]: (((((((((coindu1599971794_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B2) = (((coindu1121789889_LCons @ A) @ X) @ Xs)) = ((~ ((IS_LNIL @ B2))) & ((X = (LHD @ B2)) & (Xs = ((((((coindu1599971794_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B2))))))))).
% 2.75/1.59 thf(fact_8_lsetp_Ointros_I2_J, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),X:A,X3:A]: (((((coindu1055736764_lsetp @ A) @ Xs) @ X) => (((coindu1055736764_lsetp @ A) @ (((coindu1121789889_LCons @ A) @ X3) @ Xs)) @ X))))).
% 2.75/1.59 thf(fact_9_lsetp_Ointros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coindu1593790203_llist @ A)]: ((((coindu1055736764_lsetp @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs)) @ X)))).
% 2.75/1.59 thf(fact_10_lsetp_Ocases, axiom, (! [A:$tType,A1:(coindu1593790203_llist @ A),A22:A]: (((((coindu1055736764_lsetp @ A) @ A1) @ A22) => ((! [X4:A]: (((? [Xs2:(coindu1593790203_llist @ A)]: ((A1 = (((coindu1121789889_LCons @ A) @ X4) @ Xs2)))) => (A22 != X4)))) => (~ ((! [Xs2:(coindu1593790203_llist @ A)]: (((? [X5:A]: ((A1 = (((coindu1121789889_LCons @ A) @ X5) @ Xs2)))) => (~ ((((coindu1055736764_lsetp @ A) @ Xs2) @ A22))))))))))))).
% 2.75/1.59 thf(fact_11_lsetp_Osimps, axiom, (! [A:$tType]: (((coindu1055736764_lsetp @ A) = (^ [A12:(coindu1593790203_llist @ A),A23:A]: (((? [X2:A,Xs3:(coindu1593790203_llist @ A)]: (((A12 = (((coindu1121789889_LCons @ A) @ X2) @ Xs3)) & (A23 = X2)))) | (? [Xs3:(coindu1593790203_llist @ A),X2:A,X6:A]: (((A12 = (((coindu1121789889_LCons @ A) @ X6) @ Xs3)) & ((A23 = X2) & (((coindu1055736764_lsetp @ A) @ Xs3) @ X2)))))))))))).
% 2.75/1.59 thf(fact_12_lsetp_Oinducts, axiom, (! [A:$tType,X1:(coindu1593790203_llist @ A),X23:A,P:((coindu1593790203_llist @ A) > (A > $o))]: (((((coindu1055736764_lsetp @ A) @ X1) @ X23) => ((! [X4:A,Xs2:(coindu1593790203_llist @ A)]: (((P @ (((coindu1121789889_LCons @ A) @ X4) @ Xs2)) @ X4))) => ((! [Xs2:(coindu1593790203_llist @ A),X4:A,X5:A]: (((((coindu1055736764_lsetp @ A) @ Xs2) @ X4) => (((P @ Xs2) @ X4) => ((P @ (((coindu1121789889_LCons @ A) @ X5) @ Xs2)) @ X4))))) => ((P @ X1) @ X23))))))).
% 2.75/1.59 thf(fact_13_llist_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F2:(A > ((coindu1593790203_llist @ A) > B)),X21:A,X22:(coindu1593790203_llist @ A)]: (((((((coindu882539134_llist @ B) @ A) @ F1) @ F2) @ (((coindu1121789889_LCons @ A) @ X21) @ X22)) = ((F2 @ X21) @ X22))))).
% 2.75/1.59 thf(fact_14_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((~ ((P2 @ A2))) => (((((((coindu1599971794_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2) = (((coindu1121789889_LCons @ B) @ (G21 @ A2)) @ ((((((coindu1599971794_llist @ A) @ B) @ P2) @ G21) @ G22) @ (G22 @ A2)))))))).
% 2.75/1.59 thf(fact_15_Suc_Ohyps, axiom, (! [Xs:(coindu1593790203_llist @ a)]: (((((coindu1018505716length @ a) @ Xs) = (extended_enat2 @ na)) => ((coindu1213758845finite @ a) @ Xs))))).
% 2.75/1.59 thf(fact_16_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 2.75/1.59 thf(fact_17_nat_Oinject, axiom, (! [X23:nat,Y2:nat]: ((((suc @ X23) = (suc @ Y2)) = (X23 = Y2))))).
% 2.75/1.59 thf(fact_18_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.75/1.59 thf(fact_19_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 2.75/1.59 thf(fact_20_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.75/1.59 thf(fact_21_llength__LCons, axiom, (! [B:$tType,X:B,Xs:(coindu1593790203_llist @ B)]: ((((coindu1018505716length @ B) @ (((coindu1121789889_LCons @ B) @ X) @ Xs)) = (extended_eSuc @ ((coindu1018505716length @ B) @ Xs)))))).
% 2.75/1.59 thf(fact_22_llexord_Ocoinduct, axiom, (! [A:$tType,X7:((coindu1593790203_llist @ A) > ((coindu1593790203_llist @ A) > $o)),X:(coindu1593790203_llist @ A),Xa:(coindu1593790203_llist @ A),R:(A > (A > $o))]: ((((X7 @ X) @ Xa) => ((! [X4:(coindu1593790203_llist @ A),Xa2:(coindu1593790203_llist @ A)]: ((((X7 @ X4) @ Xa2) => ((? [Xs4:(coindu1593790203_llist @ A),Ys2:(coindu1593790203_llist @ A),Xb:A]: (((X4 = (((coindu1121789889_LCons @ A) @ Xb) @ Xs4)) & ((Xa2 = (((coindu1121789889_LCons @ A) @ Xb) @ Ys2)) & (((X7 @ Xs4) @ Ys2) | ((((coindu300403952lexord @ A) @ R) @ Xs4) @ Ys2)))))) | ((? [Xb:A,Y3:A,Xs4:(coindu1593790203_llist @ A),Ys2:(coindu1593790203_llist @ A)]: (((X4 = (((coindu1121789889_LCons @ A) @ Xb) @ Xs4)) & ((Xa2 = (((coindu1121789889_LCons @ A) @ Y3) @ Ys2)) & ((R @ Xb) @ Y3))))) | (? [Ys2:(coindu1593790203_llist @ A)]: (((X4 = (coindu1598213697e_LNil @ A)) & (Xa2 = Ys2))))))))) => ((((coindu300403952lexord @ A) @ R) @ X) @ Xa)))))).
% 2.75/1.59 thf(fact_23_llexord_Osimps, axiom, (! [A:$tType]: (((coindu300403952lexord @ A) = (^ [R2:(A > (A > $o)),A12:(coindu1593790203_llist @ A),A23:(coindu1593790203_llist @ A)]: (((? [Xs3:(coindu1593790203_llist @ A),Ys3:(coindu1593790203_llist @ A),X2:A]: (((A12 = (((coindu1121789889_LCons @ A) @ X2) @ Xs3)) & ((A23 = (((coindu1121789889_LCons @ A) @ X2) @ Ys3)) & ((((coindu300403952lexord @ A) @ R2) @ Xs3) @ Ys3))))) | ((? [X2:A,Y4:A,Xs3:(coindu1593790203_llist @ A),Ys3:(coindu1593790203_llist @ A)]: (((A12 = (((coindu1121789889_LCons @ A) @ X2) @ Xs3)) & ((A23 = (((coindu1121789889_LCons @ A) @ Y4) @ Ys3)) & ((R2 @ X2) @ Y4))))) | (? [Ys3:(coindu1593790203_llist @ A)]: (((A12 = (coindu1598213697e_LNil @ A)) & (A23 = Ys3)))))))))))).
% 2.75/1.59 thf(fact_24_llexord_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A1:(coindu1593790203_llist @ A),A22:(coindu1593790203_llist @ A)]: ((((((coindu300403952lexord @ A) @ R) @ A1) @ A22) => ((! [Xs2:(coindu1593790203_llist @ A),Ys4:(coindu1593790203_llist @ A),X4:A]: (((A1 = (((coindu1121789889_LCons @ A) @ X4) @ Xs2)) => ((A22 = (((coindu1121789889_LCons @ A) @ X4) @ Ys4)) => (~ (((((coindu300403952lexord @ A) @ R) @ Xs2) @ Ys4))))))) => ((! [X4:A]: (((? [Xs2:(coindu1593790203_llist @ A)]: ((A1 = (((coindu1121789889_LCons @ A) @ X4) @ Xs2)))) => (! [Y5:A]: (((? [Ys4:(coindu1593790203_llist @ A)]: ((A22 = (((coindu1121789889_LCons @ A) @ Y5) @ Ys4)))) => (~ (((R @ X4) @ Y5))))))))) => (~ (((A1 = (coindu1598213697e_LNil @ A)) => (! [Ys4:(coindu1593790203_llist @ A)]: ((A22 != Ys4)))))))))))).
% 2.75/1.59 thf(fact_25_eSuc__inject, axiom, (! [M:extended_enat,N:extended_enat]: ((((extended_eSuc @ M) = (extended_eSuc @ N)) = (M = N))))).
% 2.75/1.59 thf(fact_26_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coindu1593790203_llist @ B)]: ((((coindu1213758845finite @ B) @ (((coindu1121789889_LCons @ B) @ X) @ Xs)) = ((coindu1213758845finite @ B) @ Xs))))).
% 2.75/1.59 thf(fact_27_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs)) = ((coindu1213758845finite @ A) @ Xs))))).
% 2.75/1.59 thf(fact_28_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) @ (coindu1598213697e_LNil @ A))))).
% 2.75/1.59 thf(fact_29_lfinite__LNil, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) @ (coindu1598213697e_LNil @ A))))).
% 2.75/1.59 thf(fact_30_lfinite_Ocases, axiom, (! [A:$tType,A2:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ A2) => ((A2 != (coindu1598213697e_LNil @ A)) => (~ ((! [Xs2:(coindu1593790203_llist @ A)]: (((? [X4:A]: ((A2 = (((coindu1121789889_LCons @ A) @ X4) @ Xs2)))) => (~ (((coindu1213758845finite @ A) @ Xs2))))))))))))).
% 2.75/1.59 thf(fact_31_lfinite_Osimps, axiom, (! [A:$tType]: (((coindu1213758845finite @ A) = (^ [A3:(coindu1593790203_llist @ A)]: (((A3 = (coindu1598213697e_LNil @ A)) | (? [Xs3:(coindu1593790203_llist @ A),X2:A]: (((A3 = (((coindu1121789889_LCons @ A) @ X2) @ Xs3)) & ((coindu1213758845finite @ A) @ Xs3))))))))))).
% 2.75/1.59 thf(fact_32_lfinite_Oinducts, axiom, (! [A:$tType,X:(coindu1593790203_llist @ A),P:((coindu1593790203_llist @ A) > $o)]: ((((coindu1213758845finite @ A) @ X) => ((P @ (coindu1598213697e_LNil @ A)) => ((! [Xs2:(coindu1593790203_llist @ A),X4:A]: ((((coindu1213758845finite @ A) @ Xs2) => ((P @ Xs2) => (P @ (((coindu1121789889_LCons @ A) @ X4) @ Xs2)))))) => (P @ X))))))).
% 2.75/1.59 thf(fact_33_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coindu1593790203_llist @ A)]: (((coindu1598213697e_LNil @ A) != (((coindu1121789889_LCons @ A) @ X21) @ X22))))).
% 2.75/1.59 thf(fact_34_llist_Oexhaust, axiom, (! [A:$tType,Y:(coindu1593790203_llist @ A)]: (((Y != (coindu1598213697e_LNil @ A)) => (~ ((! [X212:A,X222:(coindu1593790203_llist @ A)]: ((Y != (((coindu1121789889_LCons @ A) @ X212) @ X222)))))))))).
% 2.75/1.59 thf(fact_35_neq__LNil__conv, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((Xs != (coindu1598213697e_LNil @ A)) = (? [X2:A,Xs5:(coindu1593790203_llist @ A)]: ((Xs = (((coindu1121789889_LCons @ A) @ X2) @ Xs5)))))))).
% 2.75/1.59 thf(fact_36_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),X:A]: ((((coindu1213758845finite @ A) @ Xs) => ((coindu1213758845finite @ A) @ (((coindu1121789889_LCons @ A) @ X) @ Xs)))))).
% 2.75/1.59 thf(fact_37_llist_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F2:(A > ((coindu1593790203_llist @ A) > B))]: (((((((coindu882539134_llist @ B) @ A) @ F1) @ F2) @ (coindu1598213697e_LNil @ A)) = F1)))).
% 2.75/1.59 thf(fact_38_enat__eSuc__iff, axiom, (! [Y:nat,X:extended_enat]: ((((extended_enat2 @ Y) = (extended_eSuc @ X)) = (? [N2:nat]: (((Y = (suc @ N2)) & ((extended_enat2 @ N2) = X)))))))).
% 2.75/1.59 thf(fact_39_eSuc__enat__iff, axiom, (! [X:extended_enat,Y:nat]: ((((extended_eSuc @ X) = (extended_enat2 @ Y)) = (? [N2:nat]: (((Y = (suc @ N2)) & (X = (extended_enat2 @ N2))))))))).
% 2.75/1.59 thf(fact_40_eSuc__enat, axiom, (! [N:nat]: (((extended_eSuc @ (extended_enat2 @ N)) = (extended_enat2 @ (suc @ N)))))).
% 2.75/1.59 thf(fact_41_unfold__llist_Octr_I1_J, axiom, (! [A:$tType,B:$tType,P2:(A > $o),A2:A,G21:(A > B),G22:(A > A)]: (((P2 @ A2) => (((((((coindu1599971794_llist @ A) @ B) @ P2) @ G21) @ G22) @ A2) = (coindu1598213697e_LNil @ B)))))).
% 2.75/1.59 thf(fact_42_llist_Opred__inject_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((((coindu44415597_llist @ A) @ P) @ (coindu1598213697e_LNil @ A))))).
% 2.75/1.59 thf(fact_43_llexord__LNil, axiom, (! [A:$tType,R:(A > (A > $o)),Ys:(coindu1593790203_llist @ A)]: (((((coindu300403952lexord @ A) @ R) @ (coindu1598213697e_LNil @ A)) @ Ys)))).
% 2.75/1.59 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.75/1.59 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.75/1.59 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.75/1.59 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X4:A]: (((F3 @ X4) = (G @ X4)))) => (F3 = G))))).
% 2.75/1.59 thf(fact_48_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coindu567634248member @ A) @ X) @ (coindu1598213697e_LNil @ A))))))).
% 2.75/1.59 thf(fact_49_unfold__llist_Ocode, axiom, (! [B:$tType,A:$tType]: ((((coindu1599971794_llist @ A) @ B) = (^ [P3:(A > $o),G212:(A > B),G222:(A > A),A3:A]: (((((if @ (coindu1593790203_llist @ B)) @ (P3 @ A3)) @ (coindu1598213697e_LNil @ B)) @ (((coindu1121789889_LCons @ B) @ (G212 @ A3)) @ ((((((coindu1599971794_llist @ A) @ B) @ P3) @ G212) @ G222) @ (G222 @ A3)))))))))).
% 2.75/1.59 thf(fact_50_co_Oenat_Oinject, axiom, (! [X23:extended_enat,Y2:extended_enat]: ((((extended_eSuc @ X23) = (extended_eSuc @ Y2)) = (X23 = Y2))))).
% 2.75/1.59 thf(fact_51_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.75/1.59 thf(fact_52_enat_Osimps_I4_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.75/1.59 thf(fact_53_lfinite__rev__induct, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),P:((coindu1593790203_llist @ A) > $o)]: ((((coindu1213758845finite @ A) @ Xs) => ((P @ (coindu1598213697e_LNil @ A)) => ((! [X4:A,Xs2:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ Xs2) => ((P @ Xs2) => (P @ (((coindu268472904append @ A) @ Xs2) @ (((coindu1121789889_LCons @ A) @ X4) @ (coindu1598213697e_LNil @ A)))))))) => (P @ Xs))))))).
% 2.75/1.59 thf(fact_54_llength__LNil, axiom, (! [A:$tType]: ((((coindu1018505716length @ A) @ (coindu1598213697e_LNil @ A)) = (zero_zero @ extended_enat))))).
% 2.75/1.59 thf(fact_55_gen__lset__code_I1_J, axiom, (! [A:$tType,A4:(set @ A)]: (((((coindu1928975208n_lset @ A) @ A4) @ (coindu1598213697e_LNil @ A)) = A4)))).
% 2.75/1.59 thf(fact_56_of__nat__aux_Osimps_I2_J, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [Inc:(A > A),N:nat,I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (suc @ N)) @ I) = ((((semiri532925092at_aux @ A) @ Inc) @ N) @ (Inc @ I))))))))).
% 2.75/1.59 thf(fact_57_llist_Ocorec__code, axiom, (! [A:$tType,C:$tType]: ((((coindu1244876290_llist @ C) @ A) = (^ [P3:(C > $o),G212:(C > A),Q22:(C > $o),G221:(C > (coindu1593790203_llist @ A)),G2222:(C > C),A3:C]: (((((if @ (coindu1593790203_llist @ A)) @ (P3 @ A3)) @ (coindu1598213697e_LNil @ A)) @ (((coindu1121789889_LCons @ A) @ (G212 @ A3)) @ ((((if @ (coindu1593790203_llist @ A)) @ (Q22 @ A3)) @ (G221 @ A3)) @ ((((((((coindu1244876290_llist @ C) @ A) @ P3) @ G212) @ Q22) @ G221) @ G2222) @ (G2222 @ A3))))))))))).
% 2.75/1.59 thf(fact_58_the__enat_Osimps, axiom, (! [N:nat]: (((extended_the_enat @ (extended_enat2 @ N)) = N)))).
% 2.75/1.59 thf(fact_59_lappend__code_I2_J, axiom, (! [A:$tType,Xa:A,X:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (((coindu1121789889_LCons @ A) @ Xa) @ X)) @ Ys) = (((coindu1121789889_LCons @ A) @ Xa) @ (((coindu268472904append @ A) @ X) @ Ys)))))).
% 2.75/1.59 thf(fact_60_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (coindu1598213697e_LNil @ A)) @ Ys) = Ys)))).
% 2.75/1.59 thf(fact_61_lappend__LNil2, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ Xs) @ (coindu1598213697e_LNil @ A)) = Xs)))).
% 2.75/1.59 thf(fact_62_lfinite__lappend, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu1213758845finite @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) = (((coindu1213758845finite @ A) @ Xs) & ((coindu1213758845finite @ A) @ Ys)))))).
% 2.75/1.59 thf(fact_63_lappend__assoc, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),Zs:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (((coindu268472904append @ A) @ Xs) @ Ys)) @ Zs) = (((coindu268472904append @ A) @ Xs) @ (((coindu268472904append @ A) @ Ys) @ Zs)))))).
% 2.75/1.59 thf(fact_64_co_Oenat_Odistinct_I1_J, axiom, (! [X23:extended_enat]: (((zero_zero @ extended_enat) != (extended_eSuc @ X23))))).
% 2.75/1.59 thf(fact_65_co_Oenat_OdiscI, axiom, (! [Enat:extended_enat,X23:extended_enat]: (((Enat = (extended_eSuc @ X23)) => (Enat != (zero_zero @ extended_enat)))))).
% 2.75/1.59 thf(fact_66_enat__coexhaust, axiom, (! [N:extended_enat]: (((N != (zero_zero @ extended_enat)) => (~ ((! [N3:extended_enat]: ((N != (extended_eSuc @ N3)))))))))).
% 2.75/1.59 thf(fact_67_co_Oenat_Oexhaust, axiom, (! [Y:extended_enat]: (((Y != (zero_zero @ extended_enat)) => (~ ((! [X24:extended_enat]: ((Y != (extended_eSuc @ X24)))))))))).
% 2.75/1.59 thf(fact_68_neq__zero__conv__eSuc, axiom, (! [N:extended_enat]: (((N != (zero_zero @ extended_enat)) = (? [N4:extended_enat]: ((N = (extended_eSuc @ N4)))))))).
% 2.75/1.59 thf(fact_69_lappend__LNil__LNil, axiom, (! [A:$tType]: (((((coindu268472904append @ A) @ (coindu1598213697e_LNil @ A)) @ (coindu1598213697e_LNil @ A)) = (coindu1598213697e_LNil @ A))))).
% 2.75/1.59 thf(fact_70_LNil__eq__lappend__iff, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((coindu1598213697e_LNil @ A) = (((coindu268472904append @ A) @ Xs) @ Ys)) = ((Xs = (coindu1598213697e_LNil @ A)) & (Ys = (coindu1598213697e_LNil @ A))))))).
% 2.75/1.59 thf(fact_71_lappend__eq__LNil__iff, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: ((((((coindu268472904append @ A) @ Xs) @ Ys) = (coindu1598213697e_LNil @ A)) = ((Xs = (coindu1598213697e_LNil @ A)) & (Ys = (coindu1598213697e_LNil @ A))))))).
% 2.75/1.59 thf(fact_72_lappend__inf, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A)]: (((~ (((coindu1213758845finite @ A) @ Xs))) => ((((coindu268472904append @ A) @ Xs) @ Ys) = Xs))))).
% 2.75/1.59 thf(fact_73_zero__ne__eSuc, axiom, (! [N:extended_enat]: (((zero_zero @ extended_enat) != (extended_eSuc @ N))))).
% 2.75/1.59 thf(fact_74_llist_Ocorec_I2_J, axiom, (! [A:$tType,C:$tType,P2:(C > $o),A2:C,G21:(C > A),Q222:(C > $o),G2212:(C > (coindu1593790203_llist @ A)),G2223:(C > C)]: (((~ ((P2 @ A2))) => (((((((((coindu1244876290_llist @ C) @ A) @ P2) @ G21) @ Q222) @ G2212) @ G2223) @ A2) = (((coindu1121789889_LCons @ A) @ (G21 @ A2)) @ ((((if @ (coindu1593790203_llist @ A)) @ (Q222 @ A2)) @ (G2212 @ A2)) @ ((((((((coindu1244876290_llist @ C) @ A) @ P2) @ G21) @ Q222) @ G2212) @ G2223) @ (G2223 @ A2))))))))).
% 2.75/1.59 thf(fact_75_llist_Ocorec_I1_J, axiom, (! [C:$tType,A:$tType,P2:(C > $o),A2:C,G21:(C > A),Q222:(C > $o),G2212:(C > (coindu1593790203_llist @ A)),G2223:(C > C)]: (((P2 @ A2) => (((((((((coindu1244876290_llist @ C) @ A) @ P2) @ G21) @ Q222) @ G2212) @ G2223) @ A2) = (coindu1598213697e_LNil @ A)))))).
% 2.75/1.59 thf(fact_76_lappend__snocL1__conv__LCons2, axiom, (! [A:$tType,Xs:(coindu1593790203_llist @ A),Y:A,Ys:(coindu1593790203_llist @ A)]: (((((coindu268472904append @ A) @ (((coindu268472904append @ A) @ Xs) @ (((coindu1121789889_LCons @ A) @ Y) @ (coindu1598213697e_LNil @ A)))) @ Ys) = (((coindu268472904append @ A) @ Xs) @ (((coindu1121789889_LCons @ A) @ Y) @ Ys)))))).
% 2.75/1.59 thf(fact_77_enat__set_Ocoinduct, axiom, (! [X7:(extended_enat > $o),X:extended_enat]: (((X7 @ X) => ((! [X4:extended_enat]: (((X7 @ X4) => ((X4 = (zero_zero @ extended_enat)) | (? [N5:extended_enat]: (((X4 = (extended_eSuc @ N5)) & ((X7 @ N5) | (((member @ extended_enat) @ N5) @ coinductive_enat_set))))))))) => (((member @ extended_enat) @ X) @ coinductive_enat_set)))))).
% 2.75/1.59 thf(fact_78_enat__set_Osimps, axiom, (! [A2:extended_enat]: (((((member @ extended_enat) @ A2) @ coinductive_enat_set) = ((A2 = (zero_zero @ extended_enat)) | (? [N2:extended_enat]: (((A2 = (extended_eSuc @ N2)) & (((member @ extended_enat) @ N2) @ coinductive_enat_set))))))))).
% 2.75/1.59 thf(fact_79_enat__set_Ocases, axiom, (! [A2:extended_enat]: (((((member @ extended_enat) @ A2) @ coinductive_enat_set) => ((A2 != (zero_zero @ extended_enat)) => (~ ((! [N6:extended_enat]: (((A2 = (extended_eSuc @ N6)) => (~ ((((member @ extended_enat) @ N6) @ coinductive_enat_set))))))))))))).
% 2.75/1.59 thf(fact_80_enat__setp_Ocoinduct, axiom, (! [X7:(extended_enat > $o),X:extended_enat]: (((X7 @ X) => ((! [X4:extended_enat]: (((X7 @ X4) => ((X4 = (zero_zero @ extended_enat)) | (? [N5:extended_enat]: (((X4 = (extended_eSuc @ N5)) & ((X7 @ N5) | (coindu530039314t_setp @ N5))))))))) => (coindu530039314t_setp @ X)))))).
% 2.75/1.59 thf(fact_81_enat__setp_Osimps, axiom, (coindu530039314t_setp = (^ [A3:extended_enat]: (((A3 = (zero_zero @ extended_enat)) | (? [N2:extended_enat]: (((A3 = (extended_eSuc @ N2)) & (coindu530039314t_setp @ N2))))))))).
% 2.75/1.59 thf(fact_82_enat__setp_Ocases, axiom, (! [A2:extended_enat]: (((coindu530039314t_setp @ A2) => ((A2 != (zero_zero @ extended_enat)) => (~ ((! [N6:extended_enat]: (((A2 = (extended_eSuc @ N6)) => (~ ((coindu530039314t_setp @ N6))))))))))))).
% 2.75/1.59 thf(fact_83_case__enat__0, axiom, (! [A:$tType,F3:(nat > A),I:A]: ((((((extended_case_enat @ A) @ F3) @ I) @ (zero_zero @ extended_enat)) = (F3 @ (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_84_the__enat__0, axiom, ((extended_the_enat @ (zero_zero @ extended_enat)) = (zero_zero @ nat))).
% 2.75/1.59 thf(fact_85_enat__setp__enat__set__eq, axiom, (coindu530039314t_setp = (^ [X2:extended_enat]: ((((member @ extended_enat) @ X2) @ coinductive_enat_set))))).
% 2.75/1.59 thf(fact_86_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M2:nat]: ((N = (suc @ M2)))))))).
% 2.75/1.59 thf(fact_87_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat3:nat]: (((P @ Nat3) => (P @ (suc @ Nat3))))) => (P @ Nat)))))).
% 2.75/1.59 thf(fact_88_old_Onat_Oexhaust, axiom, (! [Y:nat]: (((Y != (zero_zero @ nat)) => (~ ((! [Nat3:nat]: ((Y != (suc @ Nat3)))))))))).
% 2.75/1.59 thf(fact_89_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.75/1.59 thf(fact_90_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.75/1.59 thf(fact_91_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_92_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N6:nat]: (((P @ (suc @ N6)) => (P @ N6)))) => (P @ (zero_zero @ nat))))))).
% 2.75/1.59 thf(fact_93_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X4:nat]: (((P @ X4) @ (zero_zero @ nat)))) => ((! [Y5:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y5)))) => ((! [X4:nat,Y5:nat]: ((((P @ X4) @ Y5) => ((P @ (suc @ X4)) @ (suc @ Y5))))) => ((P @ M) @ N))))))).
% 2.75/1.59 thf(fact_94_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N6:nat]: (((P @ N6) => (P @ (suc @ N6))))) => (P @ N)))))).
% 2.75/1.59 thf(fact_95_nat_OdiscI, axiom, (! [Nat:nat,X23:nat]: (((Nat = (suc @ X23)) => (Nat != (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_96_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.75/1.59 thf(fact_97_old_Onat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((suc @ Nat4) != (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_98_nat_Odistinct_I1_J, axiom, (! [X23:nat]: (((zero_zero @ nat) != (suc @ X23))))).
% 2.75/1.59 thf(fact_99_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [Inc:(A > A),I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))).
% 2.75/1.59 thf(fact_100_enat__setp_Ointros_I1_J, axiom, (coindu530039314t_setp @ (zero_zero @ extended_enat))).
% 2.75/1.59 thf(fact_101_enat__setp_Ointros_I2_J, axiom, (! [N:extended_enat]: (((coindu530039314t_setp @ N) => (coindu530039314t_setp @ (extended_eSuc @ N)))))).
% 2.75/1.59 thf(fact_102_enat__set_Ointros_I1_J, axiom, (((member @ extended_enat) @ (zero_zero @ extended_enat)) @ coinductive_enat_set)).
% 2.75/1.59 thf(fact_103_enat__set_Ointros_I2_J, axiom, (! [N:extended_enat]: (((((member @ extended_enat) @ N) @ coinductive_enat_set) => (((member @ extended_enat) @ (extended_eSuc @ N)) @ coinductive_enat_set))))).
% 2.75/1.59 thf(fact_104_enat__0__iff_I2_J, axiom, (! [X:nat]: ((((zero_zero @ extended_enat) = (extended_enat2 @ X)) = (X = (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_105_enat__0__iff_I1_J, axiom, (! [X:nat]: ((((extended_enat2 @ X) = (zero_zero @ extended_enat)) = (X = (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_106_zero__enat__def, axiom, ((zero_zero @ extended_enat) = (extended_enat2 @ (zero_zero @ nat)))).
% 2.75/1.59 thf(fact_107_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.75/1.59 thf(fact_108_enat_Osize_I1_J, axiom, (! [Nat:nat]: (((extended_size_enat @ (extended_enat2 @ Nat)) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_109_enat_Osize_I3_J, axiom, (! [Nat:nat]: ((((size_size @ extended_enat) @ (extended_enat2 @ Nat)) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_110_enat__unfold_Osimps, axiom, (! [A:$tType]: (((coindu1491768222unfold @ A) = (^ [Stop:(A > $o),Next:(A > A),A3:A]: (((((if @ extended_enat) @ (Stop @ A3)) @ (zero_zero @ extended_enat)) @ (extended_eSuc @ ((((coindu1491768222unfold @ A) @ Stop) @ Next) @ (Next @ A3)))))))))).
% 2.75/1.59 thf(fact_111_enat__unfold__unique, axiom, (! [A:$tType,Stop2:(A > $o),H:(A > extended_enat),Next2:(A > A),X:A]: (((! [X4:A]: ((((Stop2 @ X4) => ((H @ X4) = (zero_zero @ extended_enat))) & ((~ ((Stop2 @ X4))) => ((H @ X4) = (extended_eSuc @ (H @ (Next2 @ X4)))))))) => ((H @ X) = ((((coindu1491768222unfold @ A) @ Stop2) @ Next2) @ X)))))).
% 2.75/1.59 thf(fact_112_the__enat__eSuc, axiom, (! [N:extended_enat]: (((N != (extend1396239628finity @ extended_enat)) => ((extended_the_enat @ (extended_eSuc @ N)) = (suc @ (extended_the_enat @ N))))))).
% 2.75/1.59 thf(fact_113_list__decode_Ocases, axiom, (! [X:nat]: (((X != (zero_zero @ nat)) => (~ ((! [N6:nat]: ((X != (suc @ N6)))))))))).
% 2.75/1.59 thf(fact_114_not__enat__eq, axiom, (! [X:extended_enat]: (((! [Y4:nat]: ((X != (extended_enat2 @ Y4)))) = (X = (extend1396239628finity @ extended_enat)))))).
% 2.75/1.59 thf(fact_115_not__infinity__eq, axiom, (! [X:extended_enat]: (((X != (extend1396239628finity @ extended_enat)) = (? [I2:nat]: ((X = (extended_enat2 @ I2)))))))).
% 2.75/1.59 thf(fact_116_eSuc__infinity, axiom, ((extended_eSuc @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat))).
% 2.75/1.59 thf(fact_117_enat_Osimps_I5_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.75/1.59 thf(fact_118_enat__unfold__eq__0, axiom, (! [A:$tType,Stop2:(A > $o),Next2:(A > A),A2:A]: (((((((coindu1491768222unfold @ A) @ Stop2) @ Next2) @ A2) = (zero_zero @ extended_enat)) = (Stop2 @ A2))))).
% 2.75/1.59 thf(fact_119_enat__unfold__stop, axiom, (! [A:$tType,Stop2:(A > $o),A2:A,Next2:(A > A)]: (((Stop2 @ A2) => (((((coindu1491768222unfold @ A) @ Stop2) @ Next2) @ A2) = (zero_zero @ extended_enat)))))).
% 2.75/1.59 thf(fact_120_enat_Osimps_I7_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.75/1.59 thf(fact_121_enat_Osize_I2_J, axiom, ((extended_size_enat @ (extend1396239628finity @ extended_enat)) = (zero_zero @ nat))).
% 2.75/1.59 thf(fact_122_enat_Osize_I4_J, axiom, (((size_size @ extended_enat) @ (extend1396239628finity @ extended_enat)) = (zero_zero @ nat))).
% 2.75/1.59 thf(fact_123_infinity__ne__i0, axiom, ((extend1396239628finity @ extended_enat) != (zero_zero @ extended_enat))).
% 2.75/1.59 thf(fact_124_enat_Odistinct_I2_J, axiom, (! [Nat5:nat]: (((extend1396239628finity @ extended_enat) != (extended_enat2 @ Nat5))))).
% 2.75/1.59 thf(fact_125_enat_Odistinct_I1_J, axiom, (! [Nat:nat]: (((extended_enat2 @ Nat) != (extend1396239628finity @ extended_enat))))).
% 2.75/1.59 thf(fact_126_enat2__cases, axiom, (! [Y:extended_enat,Ya:extended_enat]: ((((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Ya != (extended_enat2 @ Nata))))) => (((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => (Ya != (extend1396239628finity @ extended_enat))) => (((Y = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Ya != (extended_enat2 @ Nat3))))) => (~ (((Y = (extend1396239628finity @ extended_enat)) => (Ya != (extend1396239628finity @ extended_enat))))))))))).
% 2.75/1.59 thf(fact_127_enat3__cases, axiom, (! [Y:extended_enat,Ya:extended_enat,Yb:extended_enat]: ((((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (! [Natb:nat]: ((Yb != (extended_enat2 @ Natb)))))) => (((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((? [Nat3:nat]: ((Y = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((Y = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Yb != (extended_enat2 @ Nat3)))))) => (~ (((Y = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))))))))))))))).
% 2.75/1.59 thf(fact_128_enat_Oexhaust, axiom, (! [Y:extended_enat]: (((! [Nat3:nat]: ((Y != (extended_enat2 @ Nat3)))) => (Y = (extend1396239628finity @ extended_enat)))))).
% 2.75/1.59 thf(fact_129_enat_Oinducts, axiom, (! [P:(extended_enat > $o),Enat:extended_enat]: (((! [Nat3:nat]: ((P @ (extended_enat2 @ Nat3)))) => ((P @ (extend1396239628finity @ extended_enat)) => (P @ Enat)))))).
% 2.75/1.59 thf(fact_130_enat__ex__split, axiom, ((^ [P4:(extended_enat > $o)]: ((? [X8:extended_enat]: ((P4 @ X8))))) = (^ [P5:(extended_enat > $o)]: (((P5 @ (extend1396239628finity @ extended_enat)) | (? [X2:nat]: ((P5 @ (extended_enat2 @ X2))))))))).
% 2.75/1.59 thf(fact_131_eSuc__eq__infinity__iff, axiom, (! [N:extended_enat]: ((((extended_eSuc @ N) = (extend1396239628finity @ extended_enat)) = (N = (extend1396239628finity @ extended_enat)))))).
% 2.75/1.59 thf(fact_132_infinity__eq__eSuc__iff, axiom, (! [N:extended_enat]: ((((extend1396239628finity @ extended_enat) = (extended_eSuc @ N)) = (N = (extend1396239628finity @ extended_enat)))))).
% 2.75/1.59 thf(fact_133_enat__unfold__next, axiom, (! [A:$tType,Stop2:(A > $o),A2:A,Next2:(A > A)]: (((~ ((Stop2 @ A2))) => (((((coindu1491768222unfold @ A) @ Stop2) @ Next2) @ A2) = (extended_eSuc @ ((((coindu1491768222unfold @ A) @ Stop2) @ Next2) @ (Next2 @ A2)))))))).
% 2.75/1.59 thf(fact_134_enat__the__enat, axiom, (! [N:extended_enat]: (((N != (extend1396239628finity @ extended_enat)) => ((extended_enat2 @ (extended_the_enat @ N)) = N))))).
% 2.75/1.59 thf(fact_135_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X12:nat]: ((P @ X12))) => (? [N6:nat]: (((~ ((P @ N6))) & (P @ (suc @ N6)))))))))).
% 2.75/1.59 thf(fact_136_dependent__nat__choice, axiom, (! [A:$tType,P:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X12:A]: (((P @ (zero_zero @ nat)) @ X12))) => ((! [X4:A,N6:nat]: ((((P @ N6) @ X4) => (? [Y3:A]: ((((P @ (suc @ N6)) @ Y3) & (((Q @ N6) @ X4) @ Y3))))))) => (? [F4:(nat > A)]: ((! [N5:nat]: ((((P @ N5) @ (F4 @ N5)) & (((Q @ N5) @ (F4 @ N5)) @ (F4 @ (suc @ N5))))))))))))).
% 2.75/1.59 thf(fact_137_times__enat__simps_I3_J, axiom, (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ N)) = (zero_zero @ extended_enat))) & ((N != (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ N)) = (extend1396239628finity @ extended_enat))))))).
% 2.75/1.59 thf(fact_138_times__enat__simps_I4_J, axiom, (! [M:nat]: ((((M = (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat)) = (zero_zero @ extended_enat))) & ((M != (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat))))))).
% 2.75/1.59 thf(fact_139_times__enat__simps_I2_J, axiom, ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat))).
% 2.75/1.59 thf(fact_140_imult__is__0, axiom, (! [M:extended_enat,N:extended_enat]: ((((((times_times @ extended_enat) @ M) @ N) = (zero_zero @ extended_enat)) = ((M = (zero_zero @ extended_enat)) | (N = (zero_zero @ extended_enat))))))).
% 2.75/1.59 thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.75/1.59 thf(fact_142_mult_Oassoc, axiom, (! [A:$tType]: ((((semigroup_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.75/1.59 thf(fact_143_mult_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => ((times_times @ A) = (^ [A3:A,B3:A]: ((((times_times @ A) @ B3) @ A3)))))))).
% 2.75/1.59 thf(fact_144_mult_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.75/1.59 thf(fact_145_imult__is__infinity, axiom, (! [A2:extended_enat,B2:extended_enat]: ((((((times_times @ extended_enat) @ A2) @ B2) = (extend1396239628finity @ extended_enat)) = (((A2 = (extend1396239628finity @ extended_enat)) & (B2 != (zero_zero @ extended_enat))) | ((B2 = (extend1396239628finity @ extended_enat)) & (A2 != (zero_zero @ extended_enat)))))))).
% 2.75/1.59 thf(fact_146_mult__zero__left, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_147_mult__zero__right, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_148_mult__eq__0__iff, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.75/1.59 thf(fact_149_mult__cancel__left, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type2 @ A)) => (! [C2:A,A2:A,B2:A]: ((((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.75/1.59 thf(fact_150_mult__is__0, axiom, (! [M:nat,N:nat]: ((((((times_times @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N = (zero_zero @ nat))))))).
% 2.75/1.59 thf(fact_151_mult__0__right, axiom, (! [M:nat]: (((((times_times @ nat) @ M) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_152_mult__cancel1, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 2.75/1.59 thf(fact_153_mult__cancel2, axiom, (! [M:nat,K:nat,N:nat]: ((((((times_times @ nat) @ M) @ K) = (((times_times @ nat) @ N) @ K)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 2.75/1.59 thf(fact_154_mult__cancel__right, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type2 @ A)) => (! [A2:A,C2:A,B2:A]: ((((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.75/1.59 thf(fact_155_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)))))))).
% 2.75/1.59 thf(fact_156_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)))))))).
% 2.75/1.59 thf(fact_157_times__enat__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (extended_enat2 @ (((times_times @ nat) @ M) @ N)))))).
% 2.75/1.59 thf(fact_158_mult__0, axiom, (! [N:nat]: (((((times_times @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_159_Suc__mult__cancel1, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ (suc @ K)) @ M) = (((times_times @ nat) @ (suc @ K)) @ N)) = (M = N))))).
% 2.75/1.59 thf(fact_160_mult__right__cancel, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type2 @ A)) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = (A2 = B2))))))))).
% 2.75/1.59 thf(fact_161_mult__left__cancel, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type2 @ A)) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = (A2 = B2))))))))).
% 2.75/1.59 thf(fact_162_no__zero__divisors, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)))))))))).
% 2.75/1.59 thf(fact_163_divisors__zero, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) => ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.75/1.59 thf(fact_164_mult__not__zero, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)) => ((A2 != (zero_zero @ A)) & (B2 != (zero_zero @ A)))))))))).
% 2.75/1.59 thf(fact_165_semiring__normalization__rules_I9_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_166_semiring__normalization__rules_I10_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_167_idiff__infinity__right, axiom, (! [A2:nat]: (((((minus_minus @ extended_enat) @ (extended_enat2 @ A2)) @ (extend1396239628finity @ extended_enat)) = (zero_zero @ extended_enat))))).
% 2.75/1.59 thf(fact_168_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_169_diff__zero, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.75/1.59 thf(fact_170_zero__diff, axiom, (! [A:$tType]: ((((comm_monoid_diff @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_171_diff__0__right, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.75/1.59 thf(fact_172_diff__self, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.75/1.59 thf(fact_173_idiff__0__right, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ N) @ (zero_zero @ extended_enat)) = N)))).
% 2.75/1.59 thf(fact_174_idiff__0, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ (zero_zero @ extended_enat)) @ N) = (zero_zero @ extended_enat))))).
% 2.75/1.59 thf(fact_175_eSuc__minus__eSuc, axiom, (! [N:extended_enat,M:extended_enat]: (((((minus_minus @ extended_enat) @ (extended_eSuc @ N)) @ (extended_eSuc @ M)) = (((minus_minus @ extended_enat) @ N) @ M))))).
% 2.75/1.59 thf(fact_176_idiff__infinity, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ N) = (extend1396239628finity @ extended_enat))))).
% 2.75/1.59 thf(fact_177_idiff__enat__0__right, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ N) @ (extended_enat2 @ (zero_zero @ nat))) = N)))).
% 2.75/1.59 thf(fact_178_idiff__enat__0, axiom, (! [N:extended_enat]: (((((minus_minus @ extended_enat) @ (extended_enat2 @ (zero_zero @ nat))) @ N) = (extended_enat2 @ (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_179_idiff__self, axiom, (! [N:extended_enat]: (((N != (extend1396239628finity @ extended_enat)) => ((((minus_minus @ extended_enat) @ N) @ N) = (zero_zero @ extended_enat)))))).
% 2.75/1.59 thf(fact_180_eq__iff__diff__eq__0, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => ((^ [Y6:A,Z:A]: ((Y6 = Z))) = (^ [A3:A,B3:A]: (((((minus_minus @ A) @ A3) @ B3) = (zero_zero @ A))))))))).
% 2.75/1.59 thf(fact_181_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((A2 = B2) = (C2 = D))))))))).
% 2.75/1.59 thf(fact_182_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2)))))))).
% 2.75/1.59 thf(fact_183_nat__mult__eq__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((K = (zero_zero @ nat)) | (M = N)))))).
% 2.75/1.59 thf(fact_184_one__le__mult__iff, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((times_times @ nat) @ M) @ N)) = ((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ M) & (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ N)))))).
% 2.75/1.59 thf(fact_185_triangle__0, axiom, ((nat_triangle @ (zero_zero @ nat)) = (zero_zero @ nat))).
% 2.75/1.59 thf(fact_186_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_187_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.75/1.59 thf(fact_188_diff__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M)) @ (suc @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 2.75/1.59 thf(fact_189_Suc__diff__diff, axiom, (! [M:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ K))))).
% 2.75/1.59 thf(fact_190_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.75/1.59 thf(fact_191_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.75/1.59 thf(fact_192_diff__is__0__eq_H, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_193_diff__is__0__eq, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.59 thf(fact_194_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.75/1.59 thf(fact_195_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.75/1.59 thf(fact_196_idiff__enat__enat, axiom, (! [A2:nat,B2:nat]: (((((minus_minus @ extended_enat) @ (extended_enat2 @ A2)) @ (extended_enat2 @ B2)) = (extended_enat2 @ (((minus_minus @ nat) @ A2) @ B2)))))).
% 2.75/1.59 thf(fact_197_diff__ge__0__iff__ge, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.75/1.59 thf(fact_198_le__cube, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ (((times_times @ nat) @ M) @ M)))))).
% 2.75/1.59 thf(fact_199_le__square, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ M))))).
% 2.75/1.59 thf(fact_200_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))))))).
% 2.75/1.59 thf(fact_201_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)))))).
% 2.75/1.59 thf(fact_202_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)))))).
% 2.75/1.59 thf(fact_203_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)))))).
% 2.75/1.59 thf(fact_204_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)))))).
% 2.75/1.59 thf(fact_205_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.75/1.59 thf(fact_206_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)))))).
% 2.75/1.59 thf(fact_207_prod__decode__aux_Oinduct, axiom, (! [P:(nat > (nat > $o)),A0:nat,A1:nat]: (((! [K2:nat,M2:nat]: ((((~ ((((ord_less_eq @ nat) @ M2) @ K2))) => ((P @ (suc @ K2)) @ (((minus_minus @ nat) @ M2) @ (suc @ K2)))) => ((P @ K2) @ M2)))) => ((P @ A0) @ A1))))).
% 2.75/1.59 thf(fact_208_zero__induct__lemma, axiom, (! [P:(nat > $o),K:nat,I:nat]: (((P @ K) => ((! [N6:nat]: (((P @ (suc @ N6)) => (P @ N6)))) => (P @ (((minus_minus @ nat) @ K) @ I))))))).
% 2.75/1.59 thf(fact_209_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.75/1.59 thf(fact_210_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.75/1.59 thf(fact_211_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.75/1.59 thf(fact_212_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.75/1.59 thf(fact_213_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.75/1.59 thf(fact_214_le__diff__iff_H, axiom, (! [A2:nat,C2:nat,B2:nat]: (((((ord_less_eq @ nat) @ A2) @ C2) => ((((ord_less_eq @ nat) @ B2) @ C2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C2) @ A2)) @ (((minus_minus @ nat) @ C2) @ B2)) = (((ord_less_eq @ nat) @ B2) @ A2))))))).
% 2.75/1.59 thf(fact_215_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.75/1.59 thf(fact_216_Suc__diff__le, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((minus_minus @ nat) @ (suc @ M)) @ N) = (suc @ (((minus_minus @ nat) @ M) @ N))))))).
% 2.75/1.59 thf(fact_217_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,D:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ D) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ D)))))))))).
% 2.75/1.59 thf(fact_218_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 2.75/1.59 thf(fact_219_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 2.75/1.59 thf(fact_220_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A,D:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D)) => ((((ord_less_eq @ A) @ A2) @ B2) = (((ord_less_eq @ A) @ C2) @ D))))))))).
% 2.75/1.59 thf(fact_221_lift__Suc__mono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F3:(nat > A),N:nat,N7:nat]: (((! [N6:nat]: ((((ord_less_eq @ A) @ (F3 @ N6)) @ (F3 @ (suc @ N6))))) => ((((ord_less_eq @ nat) @ N) @ N7) => (((ord_less_eq @ A) @ (F3 @ N)) @ (F3 @ N7)))))))))).
% 2.75/1.59 thf(fact_222_lift__Suc__antimono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F3:(nat > A),N:nat,N7:nat]: (((! [N6:nat]: ((((ord_less_eq @ A) @ (F3 @ (suc @ N6))) @ (F3 @ N6)))) => ((((ord_less_eq @ nat) @ N) @ N7) => (((ord_less_eq @ A) @ (F3 @ N7)) @ (F3 @ N)))))))))).
% 2.75/1.59 thf(fact_223_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.75/1.59 thf(fact_224_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.75/1.59 thf(fact_225_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.59 thf(fact_226_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.75/1.59 thf(fact_227_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.75/1.59 thf(fact_228_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((P @ A5) @ B4)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.75/1.59 thf(fact_229_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N6:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N6) => (P @ M3)))) => (P @ N6)))) => (P @ N))))).
% 2.75/1.59 thf(fact_230_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 2.75/1.59 thf(fact_231_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.75/1.59 thf(fact_232_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 2.75/1.59 thf(fact_233_Suc__le__D, axiom, (! [N:nat,M4:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M4) => (? [M2:nat]: ((M4 = (suc @ M2)))))))).
% 2.75/1.59 thf(fact_234_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 2.75/1.59 thf(fact_235_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 2.75/1.59 thf(fact_236_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.75/1.59 thf(fact_237_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.75/1.59 thf(fact_238_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.75/1.59 thf(fact_239_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.75/1.59 thf(fact_240_mult__mono, axiom, (! [A:$tType]: ((((ordered_semiring @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D)))))))))))).
% 2.75/1.59 thf(fact_241_mult__mono_H, axiom, (! [A:$tType]: ((((ordered_semiring @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D)))))))))))).
% 2.75/1.59 thf(fact_242_zero__le__square, axiom, (! [A:$tType]: ((((linordered_ring @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ A2)))))))).
% 2.75/1.59 thf(fact_243_split__mult__pos__le, axiom, (! [A:$tType]: ((((ordered_ring @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2))))))))).
% 2.75/1.59 thf(fact_244_mult__left__mono__neg, axiom, (! [A:$tType]: ((((ordered_ring @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 2.75/1.59 thf(fact_245_mult__nonpos__nonpos, axiom, (! [A:$tType]: ((((ordered_ring @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))).
% 2.75/1.59 thf(fact_246_mult__left__mono, axiom, (! [A:$tType]: ((((ordered_semiring @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 2.75/1.59 thf(fact_247_mult__right__mono__neg, axiom, (! [A:$tType]: ((((ordered_ring @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 2.75/1.59 thf(fact_248_mult__right__mono, axiom, (! [A:$tType]: ((((ordered_semiring @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 2.75/1.59 thf(fact_249_mult__le__0__iff, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 2.75/1.59 thf(fact_250_split__mult__neg__le, axiom, (! [A:$tType]: ((((ordered_semiring_0 @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2))) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A))))))))).
% 2.75/1.59 thf(fact_251_mult__nonneg__nonneg, axiom, (! [A:$tType]: ((((ordered_semiring_0 @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))).
% 2.75/1.59 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel, axiom, ((semiri1923998003cancel @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, axiom, ((semiri1193490041visors @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, ((cancel146912293up_add @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0, axiom, ((ordered_semiring_0 @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult, axiom, ((ab_semigroup_mult @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, ((comm_monoid_diff @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Oordered__semiring, axiom, ((ordered_semiring @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, ((comm_semiring_1 @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Osemigroup__mult, axiom, ((semigroup_mult @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Osemiring__1, axiom, ((semiring_1 @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Rings_Omult__zero, axiom, ((mult_zero @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Orderings_Oorder_1, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Nat_Onat___Groups_Ozero, axiom, ((zero @ nat) @ (type2 @ nat))).
% 2.75/1.59 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.75/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder_3, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.75/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_4, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_5, axiom, ((canoni770627133id_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_6, axiom, ((semiri1193490041visors @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_7, axiom, ((ab_semigroup_mult @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_8, axiom, ((ordered_semiring @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_9, axiom, ((comm_semiring_1 @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_10, axiom, ((semigroup_mult @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_11, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_12, axiom, ((semiring_1 @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_13, axiom, ((mult_zero @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_14, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_15, axiom, ((zero @ extended_enat) @ (type2 @ extended_enat))).
% 2.75/1.59 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.75/1.59 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.75/1.59 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.75/1.59 thf(conj_0, hypothesis, (! [X9:a,Xs6:(coindu1593790203_llist @ a)]: (((xsa = (((coindu1121789889_LCons @ a) @ X9) @ Xs6)) => thesis)))).
% 2.75/1.59 thf(conj_1, conjecture, thesis).
% 2.75/1.59 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2DTF10986.p
% 2.75/1.59 ------------------------
% 3.02/1.63 ---- Cleaned THF ---
% 3.02/1.63 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 3.02/1.63 coindu1593790203_llist: $tType > $tType ).
% 3.02/1.63
% 3.02/1.63 thf(ty_t_Extended__Nat_Oenat,type,
% 3.02/1.63 extended_enat: $tType ).
% 3.02/1.63
% 3.02/1.63 thf(ty_t_Set_Oset,type,
% 3.02/1.63 set: $tType > $tType ).
% 3.02/1.63
% 3.02/1.63 thf(ty_t_Nat_Onat,type,
% 3.02/1.63 nat: $tType ).
% 3.02/1.63
% 3.02/1.63 thf(ty_t_itself,type,
% 3.02/1.63 itself: $tType > $tType ).
% 3.02/1.63
% 3.02/1.63 thf(ty_tf_a,type,
% 3.02/1.63 a: $tType ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_HOL_Otype,type,
% 3.02/1.63 type:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ozero,type,
% 3.02/1.63 zero:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Orderings_Oorder,type,
% 3.02/1.63 order:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Omult__zero,type,
% 3.02/1.63 mult_zero:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ogroup__add,type,
% 3.02/1.63 group_add:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Osemiring__1,type,
% 3.02/1.63 semiring_1:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Orderings_Olinorder,type,
% 3.02/1.63 linorder:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Oordered__ring,type,
% 3.02/1.63 ordered_ring:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Osemigroup__mult,type,
% 3.02/1.63 semigroup_mult:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 3.02/1.63 comm_semiring_1:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Olinordered__ring,type,
% 3.02/1.63 linordered_ring:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Oordered__semiring,type,
% 3.02/1.63 ordered_semiring:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 3.02/1.63 comm_monoid_diff:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 3.02/1.63 ab_semigroup_mult:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 3.02/1.63 ordered_semiring_0:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.02/1.63 ordered_ab_group_add:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 3.02/1.63 linord581940658strict:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 3.02/1.63 cancel1352612707id_add:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 3.02/1.63 cancel146912293up_add:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 3.02/1.63 semiri1193490041visors:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.02/1.63 canoni770627133id_add:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 3.02/1.63 semiri1923998003cancel:
% 3.02/1.63 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
% 3.02/1.63 coindu1928975208n_lset:
% 3.02/1.63 !>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates,type,
% 3.02/1.63 coindu1529956631erates:
% 3.02/1.63 !>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
% 3.02/1.63 coindu268472904append:
% 3.02/1.63 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
% 3.02/1.63 coindu1213758845finite:
% 3.02/1.63 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
% 3.02/1.63 coindu1018505716length:
% 3.02/1.63 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
% 3.02/1.63 coindu300403952lexord:
% 3.02/1.63 !>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
% 3.02/1.63 coindu1121789889_LCons:
% 3.02/1.63 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
% 3.02/1.63 coindu1598213697e_LNil:
% 3.02/1.63 !>[A: $tType] : ( coindu1593790203_llist @ A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist,type,
% 3.02/1.63 coindu882539134_llist:
% 3.02/1.63 !>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
% 3.02/1.63 coindu1244876290_llist:
% 3.02/1.63 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist,type,
% 3.02/1.63 coindu44415597_llist:
% 3.02/1.63 !>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
% 3.02/1.63 coindu567634248member:
% 3.02/1.63 !>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olsetp,type,
% 3.02/1.63 coindu1055736764_lsetp:
% 3.02/1.63 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
% 3.02/1.63 coindu1599971794_llist:
% 3.02/1.63 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__Nat_Oenat__set,type,
% 3.02/1.63 coinductive_enat_set: set @ extended_enat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__Nat_Oenat__setp,type,
% 3.02/1.63 coindu530039314t_setp: extended_enat > $o ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 3.02/1.63 coindu1491768222unfold:
% 3.02/1.63 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_OeSuc,type,
% 3.02/1.63 extended_eSuc: extended_enat > extended_enat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Oenat,type,
% 3.02/1.63 extended_enat2: nat > extended_enat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 3.02/1.63 extended_case_enat:
% 3.02/1.63 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 3.02/1.63 extended_rec_enat:
% 3.02/1.63 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 3.02/1.63 extended_size_enat: extended_enat > nat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 3.02/1.63 extend1396239628finity:
% 3.02/1.63 !>[A: $tType] : A ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Extended__Nat_Othe__enat,type,
% 3.02/1.63 extended_the_enat: extended_enat > nat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.02/1.63 minus_minus:
% 3.02/1.63 !>[A: $tType] : ( A > A > A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 3.02/1.63 times_times:
% 3.02/1.63 !>[A: $tType] : ( A > A > A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.02/1.63 zero_zero:
% 3.02/1.63 !>[A: $tType] : A ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_If,type,
% 3.02/1.63 if:
% 3.02/1.63 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Nat_OSuc,type,
% 3.02/1.63 suc: nat > nat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 3.02/1.63 semiri532925092at_aux:
% 3.02/1.63 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Nat_Osize__class_Osize,type,
% 3.02/1.63 size_size:
% 3.02/1.63 !>[A: $tType] : ( A > nat ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Nat__Bijection_Otriangle,type,
% 3.02/1.63 nat_triangle: nat > nat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.02/1.63 ord_less_eq:
% 3.02/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Pure_Otype,type,
% 3.02/1.63 type2:
% 3.02/1.63 !>[A: $tType] : ( itself @ A ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_Set_OCollect,type,
% 3.02/1.63 collect:
% 3.02/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_c_member,type,
% 3.02/1.63 member:
% 3.02/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.02/1.63
% 3.02/1.63 thf(sy_v_na____,type,
% 3.02/1.63 na: nat ).
% 3.02/1.63
% 3.02/1.63 thf(sy_v_thesis____,type,
% 3.02/1.63 thesis: $o ).
% 3.02/1.63
% 3.02/1.63 thf(sy_v_xsa____,type,
% 3.02/1.63 xsa: coindu1593790203_llist @ a ).
% 3.02/1.63
% 3.02/1.63 thf(fact_0_len,axiom,
% 3.02/1.63 ( ( coindu1018505716length @ a @ xsa )
% 3.02/1.63 = ( extended_enat2 @ ( suc @ na ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_1_llist_Oinject,axiom,
% 3.02/1.63 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
% 3.02/1.63 = ( ( X21 = Y21 )
% 3.02/1.63 & ( X22 = Y22 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_2_lmember__code_I2_J,axiom,
% 3.02/1.63 ! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
% 3.02/1.63 = ( ( X = Y )
% 3.02/1.63 | ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_3_iterates_Ocode,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu1529956631erates @ A )
% 3.02/1.63 = ( ^ [F: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F @ ( F @ X2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_4_llist_Opred__inject_I2_J,axiom,
% 3.02/1.63 ! [A: $tType,P: A > $o,A2: A,Aa: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu44415597_llist @ A @ P @ ( coindu1121789889_LCons @ A @ A2 @ Aa ) )
% 3.02/1.63 = ( ( P @ A2 )
% 3.02/1.63 & ( coindu44415597_llist @ A @ P @ Aa ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_5_llexord__LCons__eq,axiom,
% 3.02/1.63 ! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
% 3.02/1.63 ( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
% 3.02/1.63 => ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_6_llexord__LCons__less,axiom,
% 3.02/1.63 ! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( R @ X @ Y )
% 3.02/1.63 => ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_7_unfold__llist__eq__LCons,axiom,
% 3.02/1.63 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X @ Xs ) )
% 3.02/1.63 = ( ~ ( IS_LNIL @ B2 )
% 3.02/1.63 & ( X
% 3.02/1.63 = ( LHD @ B2 ) )
% 3.02/1.63 & ( Xs
% 3.02/1.63 = ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_8_lsetp_Ointros_I2_J,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,X: A,X3: A] :
% 3.02/1.63 ( ( coindu1055736764_lsetp @ A @ Xs @ X )
% 3.02/1.63 => ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ X ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_9_lsetp_Ointros_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ X ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_10_lsetp_Ocases,axiom,
% 3.02/1.63 ! [A: $tType,A1: coindu1593790203_llist @ A,A22: A] :
% 3.02/1.63 ( ( coindu1055736764_lsetp @ A @ A1 @ A22 )
% 3.02/1.63 => ( ! [X4: A] :
% 3.02/1.63 ( ? [Xs2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( A1
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 3.02/1.63 => ( A22 != X4 ) )
% 3.02/1.63 => ~ ! [Xs2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ? [X5: A] :
% 3.02/1.63 ( A1
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
% 3.02/1.63 => ~ ( coindu1055736764_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_11_lsetp_Osimps,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu1055736764_lsetp @ A )
% 3.02/1.63 = ( ^ [A12: coindu1593790203_llist @ A,A23: A] :
% 3.02/1.63 ( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( A12
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 3.02/1.63 & ( A23 = X2 ) )
% 3.02/1.63 | ? [Xs3: coindu1593790203_llist @ A,X2: A,X6: A] :
% 3.02/1.63 ( ( A12
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X6 @ Xs3 ) )
% 3.02/1.63 & ( A23 = X2 )
% 3.02/1.63 & ( coindu1055736764_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_12_lsetp_Oinducts,axiom,
% 3.02/1.63 ! [A: $tType,X1: coindu1593790203_llist @ A,X23: A,P: ( coindu1593790203_llist @ A ) > A > $o] :
% 3.02/1.63 ( ( coindu1055736764_lsetp @ A @ X1 @ X23 )
% 3.02/1.63 => ( ! [X4: A,Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) @ X4 )
% 3.02/1.63 => ( ! [Xs2: coindu1593790203_llist @ A,X4: A,X5: A] :
% 3.02/1.63 ( ( coindu1055736764_lsetp @ A @ Xs2 @ X4 )
% 3.02/1.63 => ( ( P @ Xs2 @ X4 )
% 3.02/1.63 => ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) @ X4 ) ) )
% 3.02/1.63 => ( P @ X1 @ X23 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_13_llist_Osimps_I5_J,axiom,
% 3.02/1.63 ! [B: $tType,A: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
% 3.02/1.63 = ( F2 @ X21 @ X22 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_14_unfold__llist_Octr_I2_J,axiom,
% 3.02/1.63 ! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 3.02/1.63 ( ~ ( P2 @ A2 )
% 3.02/1.63 => ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 3.02/1.63 = ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_15_Suc_Ohyps,axiom,
% 3.02/1.63 ! [Xs: coindu1593790203_llist @ a] :
% 3.02/1.63 ( ( ( coindu1018505716length @ a @ Xs )
% 3.02/1.63 = ( extended_enat2 @ na ) )
% 3.02/1.63 => ( coindu1213758845finite @ a @ Xs ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_16_enat_Oinject,axiom,
% 3.02/1.63 ! [Nat: nat,Nat2: nat] :
% 3.02/1.63 ( ( ( extended_enat2 @ Nat )
% 3.02/1.63 = ( extended_enat2 @ Nat2 ) )
% 3.02/1.63 = ( Nat = Nat2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_17_nat_Oinject,axiom,
% 3.02/1.63 ! [X23: nat,Y2: nat] :
% 3.02/1.63 ( ( ( suc @ X23 )
% 3.02/1.63 = ( suc @ Y2 ) )
% 3.02/1.63 = ( X23 = Y2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_18_old_Onat_Oinject,axiom,
% 3.02/1.63 ! [Nat: nat,Nat2: nat] :
% 3.02/1.63 ( ( ( suc @ Nat )
% 3.02/1.63 = ( suc @ Nat2 ) )
% 3.02/1.63 = ( Nat = Nat2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_19_Suc__inject,axiom,
% 3.02/1.63 ! [X: nat,Y: nat] :
% 3.02/1.63 ( ( ( suc @ X )
% 3.02/1.63 = ( suc @ Y ) )
% 3.02/1.63 => ( X = Y ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_20_n__not__Suc__n,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( N
% 3.02/1.63 != ( suc @ N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_21_llength__LCons,axiom,
% 3.02/1.63 ! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
% 3.02/1.63 ( ( coindu1018505716length @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
% 3.02/1.63 = ( extended_eSuc @ ( coindu1018505716length @ B @ Xs ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_22_llexord_Ocoinduct,axiom,
% 3.02/1.63 ! [A: $tType,X7: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R: A > A > $o] :
% 3.02/1.63 ( ( X7 @ X @ Xa )
% 3.02/1.63 => ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( X7 @ X4 @ Xa2 )
% 3.02/1.63 => ( ? [Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Xb: A] :
% 3.02/1.63 ( ( X4
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
% 3.02/1.63 & ( Xa2
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Xb @ Ys2 ) )
% 3.02/1.63 & ( ( X7 @ Xs4 @ Ys2 )
% 3.02/1.63 | ( coindu300403952lexord @ A @ R @ Xs4 @ Ys2 ) ) )
% 3.02/1.63 | ? [Xb: A,Y3: A,Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( X4
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
% 3.02/1.63 & ( Xa2
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
% 3.02/1.63 & ( R @ Xb @ Y3 ) )
% 3.02/1.63 | ? [Ys2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( X4
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 & ( Xa2 = Ys2 ) ) ) )
% 3.02/1.63 => ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_23_llexord_Osimps,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu300403952lexord @ A )
% 3.02/1.63 = ( ^ [R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ? [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
% 3.02/1.63 ( ( A12
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 3.02/1.63 & ( A23
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
% 3.02/1.63 & ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys3 ) )
% 3.02/1.63 | ? [X2: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( A12
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 3.02/1.63 & ( A23
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Y4 @ Ys3 ) )
% 3.02/1.63 & ( R2 @ X2 @ Y4 ) )
% 3.02/1.63 | ? [Ys3: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( A12
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 & ( A23 = Ys3 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_24_llexord_Ocases,axiom,
% 3.02/1.63 ! [A: $tType,R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu300403952lexord @ A @ R @ A1 @ A22 )
% 3.02/1.63 => ( ! [Xs2: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X4: A] :
% 3.02/1.63 ( ( A1
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 3.02/1.63 => ( ( A22
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X4 @ Ys4 ) )
% 3.02/1.63 => ~ ( coindu300403952lexord @ A @ R @ Xs2 @ Ys4 ) ) )
% 3.02/1.63 => ( ! [X4: A] :
% 3.02/1.63 ( ? [Xs2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( A1
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 3.02/1.63 => ! [Y5: A] :
% 3.02/1.63 ( ? [Ys4: coindu1593790203_llist @ A] :
% 3.02/1.63 ( A22
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Y5 @ Ys4 ) )
% 3.02/1.63 => ~ ( R @ X4 @ Y5 ) ) )
% 3.02/1.63 => ~ ( ( A1
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 => ! [Ys4: coindu1593790203_llist @ A] : ( A22 != Ys4 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_25_eSuc__inject,axiom,
% 3.02/1.63 ! [M: extended_enat,N: extended_enat] :
% 3.02/1.63 ( ( ( extended_eSuc @ M )
% 3.02/1.63 = ( extended_eSuc @ N ) )
% 3.02/1.63 = ( M = N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_26_lfinite__code_I2_J,axiom,
% 3.02/1.63 ! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
% 3.02/1.63 ( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
% 3.02/1.63 = ( coindu1213758845finite @ B @ Xs ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_27_lfinite__LCons,axiom,
% 3.02/1.63 ! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
% 3.02/1.63 = ( coindu1213758845finite @ A @ Xs ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_28_lfinite__code_I1_J,axiom,
% 3.02/1.63 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_29_lfinite__LNil,axiom,
% 3.02/1.63 ! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_30_lfinite_Ocases,axiom,
% 3.02/1.63 ! [A: $tType,A2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ A2 )
% 3.02/1.63 => ( ( A2
% 3.02/1.63 != ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 => ~ ! [Xs2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ? [X4: A] :
% 3.02/1.63 ( A2
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
% 3.02/1.63 => ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_31_lfinite_Osimps,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu1213758845finite @ A )
% 3.02/1.63 = ( ^ [A3: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( A3
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 | ? [Xs3: coindu1593790203_llist @ A,X2: A] :
% 3.02/1.63 ( ( A3
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
% 3.02/1.63 & ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_32_lfinite_Oinducts,axiom,
% 3.02/1.63 ! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ X )
% 3.02/1.63 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 => ( ! [Xs2: coindu1593790203_llist @ A,X4: A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ Xs2 )
% 3.02/1.63 => ( ( P @ Xs2 )
% 3.02/1.63 => ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
% 3.02/1.63 => ( P @ X ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_33_llist_Odistinct_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu1598213697e_LNil @ A )
% 3.02/1.63 != ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_34_llist_Oexhaust,axiom,
% 3.02/1.63 ! [A: $tType,Y: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( Y
% 3.02/1.63 != ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 => ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
% 3.02/1.63 ( Y
% 3.02/1.63 != ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_35_neq__LNil__conv,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( Xs
% 3.02/1.63 != ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = ( ? [X2: A,Xs5: coindu1593790203_llist @ A] :
% 3.02/1.63 ( Xs
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_36_lfinite__LConsI,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ Xs )
% 3.02/1.63 => ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_37_llist_Osimps_I4_J,axiom,
% 3.02/1.63 ! [A: $tType,B: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B] :
% 3.02/1.63 ( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = F1 ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_38_enat__eSuc__iff,axiom,
% 3.02/1.63 ! [Y: nat,X: extended_enat] :
% 3.02/1.63 ( ( ( extended_enat2 @ Y )
% 3.02/1.63 = ( extended_eSuc @ X ) )
% 3.02/1.63 = ( ? [N2: nat] :
% 3.02/1.63 ( ( Y
% 3.02/1.63 = ( suc @ N2 ) )
% 3.02/1.63 & ( ( extended_enat2 @ N2 )
% 3.02/1.63 = X ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_39_eSuc__enat__iff,axiom,
% 3.02/1.63 ! [X: extended_enat,Y: nat] :
% 3.02/1.63 ( ( ( extended_eSuc @ X )
% 3.02/1.63 = ( extended_enat2 @ Y ) )
% 3.02/1.63 = ( ? [N2: nat] :
% 3.02/1.63 ( ( Y
% 3.02/1.63 = ( suc @ N2 ) )
% 3.02/1.63 & ( X
% 3.02/1.63 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_40_eSuc__enat,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 3.02/1.63 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_41_unfold__llist_Octr_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
% 3.02/1.63 ( ( P2 @ A2 )
% 3.02/1.63 => ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
% 3.02/1.63 = ( coindu1598213697e_LNil @ B ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_42_llist_Opred__inject_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,P: A > $o] : ( coindu44415597_llist @ A @ P @ ( coindu1598213697e_LNil @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_43_llexord__LNil,axiom,
% 3.02/1.63 ! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_44_mem__Collect__eq,axiom,
% 3.02/1.63 ! [A: $tType,A2: A,P: A > $o] :
% 3.02/1.63 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.02/1.63 = ( P @ A2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_45_Collect__mem__eq,axiom,
% 3.02/1.63 ! [A: $tType,A4: set @ A] :
% 3.02/1.63 ( ( collect @ A
% 3.02/1.63 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 3.02/1.63 = A4 ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_46_Collect__cong,axiom,
% 3.02/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.02/1.63 ( ! [X4: A] :
% 3.02/1.63 ( ( P @ X4 )
% 3.02/1.63 = ( Q @ X4 ) )
% 3.02/1.63 => ( ( collect @ A @ P )
% 3.02/1.63 = ( collect @ A @ Q ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_47_ext,axiom,
% 3.02/1.63 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 3.02/1.63 ( ! [X4: A] :
% 3.02/1.63 ( ( F3 @ X4 )
% 3.02/1.63 = ( G @ X4 ) )
% 3.02/1.63 => ( F3 = G ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_48_lmember__code_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,X: A] :
% 3.02/1.63 ~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_49_unfold__llist_Ocode,axiom,
% 3.02/1.63 ! [B: $tType,A: $tType] :
% 3.02/1.63 ( ( coindu1599971794_llist @ A @ B )
% 3.02/1.63 = ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_50_co_Oenat_Oinject,axiom,
% 3.02/1.63 ! [X23: extended_enat,Y2: extended_enat] :
% 3.02/1.63 ( ( ( extended_eSuc @ X23 )
% 3.02/1.63 = ( extended_eSuc @ Y2 ) )
% 3.02/1.63 = ( X23 = Y2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_51_enat_Osimps_I6_J,axiom,
% 3.02/1.63 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 3.02/1.63 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 3.02/1.63 = ( F1 @ Nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_52_enat_Osimps_I4_J,axiom,
% 3.02/1.63 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 3.02/1.63 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 3.02/1.63 = ( F1 @ Nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_53_lfinite__rev__induct,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ Xs )
% 3.02/1.63 => ( ( P @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 => ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ Xs2 )
% 3.02/1.63 => ( ( P @ Xs2 )
% 3.02/1.63 => ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X4 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
% 3.02/1.63 => ( P @ Xs ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_54_llength__LNil,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_55_gen__lset__code_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,A4: set @ A] :
% 3.02/1.63 ( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = A4 ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_56_of__nat__aux_Osimps_I2_J,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [Inc: A > A,N: nat,I: A] :
% 3.02/1.63 ( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
% 3.02/1.63 = ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_57_llist_Ocorec__code,axiom,
% 3.02/1.63 ! [A: $tType,C: $tType] :
% 3.02/1.63 ( ( coindu1244876290_llist @ C @ A )
% 3.02/1.63 = ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_58_the__enat_Osimps,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 3.02/1.63 = N ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_59_lappend__code_I2_J,axiom,
% 3.02/1.63 ! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_60_lappend__code_I1_J,axiom,
% 3.02/1.63 ! [A: $tType,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
% 3.02/1.63 = Ys ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_61_lappend__LNil2,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = Xs ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_62_lfinite__lappend,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
% 3.02/1.63 = ( ( coindu1213758845finite @ A @ Xs )
% 3.02/1.63 & ( coindu1213758845finite @ A @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_63_lappend__assoc,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
% 3.02/1.63 = ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_64_co_Oenat_Odistinct_I1_J,axiom,
% 3.02/1.63 ! [X23: extended_enat] :
% 3.02/1.63 ( ( zero_zero @ extended_enat )
% 3.02/1.63 != ( extended_eSuc @ X23 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_65_co_Oenat_OdiscI,axiom,
% 3.02/1.63 ! [Enat: extended_enat,X23: extended_enat] :
% 3.02/1.63 ( ( Enat
% 3.02/1.63 = ( extended_eSuc @ X23 ) )
% 3.02/1.63 => ( Enat
% 3.02/1.63 != ( zero_zero @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_66_enat__coexhaust,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( zero_zero @ extended_enat ) )
% 3.02/1.63 => ~ ! [N3: extended_enat] :
% 3.02/1.63 ( N
% 3.02/1.63 != ( extended_eSuc @ N3 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_67_co_Oenat_Oexhaust,axiom,
% 3.02/1.63 ! [Y: extended_enat] :
% 3.02/1.63 ( ( Y
% 3.02/1.63 != ( zero_zero @ extended_enat ) )
% 3.02/1.63 => ~ ! [X24: extended_enat] :
% 3.02/1.63 ( Y
% 3.02/1.63 != ( extended_eSuc @ X24 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_68_neq__zero__conv__eSuc,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( ? [N4: extended_enat] :
% 3.02/1.63 ( N
% 3.02/1.63 = ( extended_eSuc @ N4 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_69_lappend__LNil__LNil,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_70_LNil__eq__lappend__iff,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( ( coindu1598213697e_LNil @ A )
% 3.02/1.63 = ( coindu268472904append @ A @ Xs @ Ys ) )
% 3.02/1.63 = ( ( Xs
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 & ( Ys
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_71_lappend__eq__LNil__iff,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( ( coindu268472904append @ A @ Xs @ Ys )
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 = ( ( Xs
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) )
% 3.02/1.63 & ( Ys
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_72_lappend__inf,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ~ ( coindu1213758845finite @ A @ Xs )
% 3.02/1.63 => ( ( coindu268472904append @ A @ Xs @ Ys )
% 3.02/1.63 = Xs ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_73_zero__ne__eSuc,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( zero_zero @ extended_enat )
% 3.02/1.63 != ( extended_eSuc @ N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_74_llist_Ocorec_I2_J,axiom,
% 3.02/1.63 ! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
% 3.02/1.63 ( ~ ( P2 @ A2 )
% 3.02/1.63 => ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
% 3.02/1.63 = ( coindu1121789889_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_75_llist_Ocorec_I1_J,axiom,
% 3.02/1.63 ! [C: $tType,A: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
% 3.02/1.63 ( ( P2 @ A2 )
% 3.02/1.63 => ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
% 3.02/1.63 = ( coindu1598213697e_LNil @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_76_lappend__snocL1__conv__LCons2,axiom,
% 3.02/1.63 ! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
% 3.02/1.63 ( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
% 3.02/1.63 = ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_77_enat__set_Ocoinduct,axiom,
% 3.02/1.63 ! [X7: extended_enat > $o,X: extended_enat] :
% 3.02/1.63 ( ( X7 @ X )
% 3.02/1.63 => ( ! [X4: extended_enat] :
% 3.02/1.63 ( ( X7 @ X4 )
% 3.02/1.63 => ( ( X4
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 | ? [N5: extended_enat] :
% 3.02/1.63 ( ( X4
% 3.02/1.63 = ( extended_eSuc @ N5 ) )
% 3.02/1.63 & ( ( X7 @ N5 )
% 3.02/1.63 | ( member @ extended_enat @ N5 @ coinductive_enat_set ) ) ) ) )
% 3.02/1.63 => ( member @ extended_enat @ X @ coinductive_enat_set ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_78_enat__set_Osimps,axiom,
% 3.02/1.63 ! [A2: extended_enat] :
% 3.02/1.63 ( ( member @ extended_enat @ A2 @ coinductive_enat_set )
% 3.02/1.63 = ( ( A2
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 | ? [N2: extended_enat] :
% 3.02/1.63 ( ( A2
% 3.02/1.63 = ( extended_eSuc @ N2 ) )
% 3.02/1.63 & ( member @ extended_enat @ N2 @ coinductive_enat_set ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_79_enat__set_Ocases,axiom,
% 3.02/1.63 ! [A2: extended_enat] :
% 3.02/1.63 ( ( member @ extended_enat @ A2 @ coinductive_enat_set )
% 3.02/1.63 => ( ( A2
% 3.02/1.63 != ( zero_zero @ extended_enat ) )
% 3.02/1.63 => ~ ! [N6: extended_enat] :
% 3.02/1.63 ( ( A2
% 3.02/1.63 = ( extended_eSuc @ N6 ) )
% 3.02/1.63 => ~ ( member @ extended_enat @ N6 @ coinductive_enat_set ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_80_enat__setp_Ocoinduct,axiom,
% 3.02/1.63 ! [X7: extended_enat > $o,X: extended_enat] :
% 3.02/1.63 ( ( X7 @ X )
% 3.02/1.63 => ( ! [X4: extended_enat] :
% 3.02/1.63 ( ( X7 @ X4 )
% 3.02/1.63 => ( ( X4
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 | ? [N5: extended_enat] :
% 3.02/1.63 ( ( X4
% 3.02/1.63 = ( extended_eSuc @ N5 ) )
% 3.02/1.63 & ( ( X7 @ N5 )
% 3.02/1.63 | ( coindu530039314t_setp @ N5 ) ) ) ) )
% 3.02/1.63 => ( coindu530039314t_setp @ X ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_81_enat__setp_Osimps,axiom,
% 3.02/1.63 ( coindu530039314t_setp
% 3.02/1.63 = ( ^ [A3: extended_enat] :
% 3.02/1.63 ( ( A3
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 | ? [N2: extended_enat] :
% 3.02/1.63 ( ( A3
% 3.02/1.63 = ( extended_eSuc @ N2 ) )
% 3.02/1.63 & ( coindu530039314t_setp @ N2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_82_enat__setp_Ocases,axiom,
% 3.02/1.63 ! [A2: extended_enat] :
% 3.02/1.63 ( ( coindu530039314t_setp @ A2 )
% 3.02/1.63 => ( ( A2
% 3.02/1.63 != ( zero_zero @ extended_enat ) )
% 3.02/1.63 => ~ ! [N6: extended_enat] :
% 3.02/1.63 ( ( A2
% 3.02/1.63 = ( extended_eSuc @ N6 ) )
% 3.02/1.63 => ~ ( coindu530039314t_setp @ N6 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_83_case__enat__0,axiom,
% 3.02/1.63 ! [A: $tType,F3: nat > A,I: A] :
% 3.02/1.63 ( ( extended_case_enat @ A @ F3 @ I @ ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( F3 @ ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_84_the__enat__0,axiom,
% 3.02/1.63 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_85_enat__setp__enat__set__eq,axiom,
% 3.02/1.63 ( coindu530039314t_setp
% 3.02/1.63 = ( ^ [X2: extended_enat] : ( member @ extended_enat @ X2 @ coinductive_enat_set ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_86_not0__implies__Suc,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( zero_zero @ nat ) )
% 3.02/1.63 => ? [M2: nat] :
% 3.02/1.63 ( N
% 3.02/1.63 = ( suc @ M2 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_87_old_Onat_Oinducts,axiom,
% 3.02/1.63 ! [P: nat > $o,Nat: nat] :
% 3.02/1.63 ( ( P @ ( zero_zero @ nat ) )
% 3.02/1.63 => ( ! [Nat3: nat] :
% 3.02/1.63 ( ( P @ Nat3 )
% 3.02/1.63 => ( P @ ( suc @ Nat3 ) ) )
% 3.02/1.63 => ( P @ Nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_88_old_Onat_Oexhaust,axiom,
% 3.02/1.63 ! [Y: nat] :
% 3.02/1.63 ( ( Y
% 3.02/1.63 != ( zero_zero @ nat ) )
% 3.02/1.63 => ~ ! [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 != ( suc @ Nat3 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_89_Zero__not__Suc,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( zero_zero @ nat )
% 3.02/1.63 != ( suc @ M ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_90_Zero__neq__Suc,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( zero_zero @ nat )
% 3.02/1.63 != ( suc @ M ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_91_Suc__neq__Zero,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( suc @ M )
% 3.02/1.63 != ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_92_zero__induct,axiom,
% 3.02/1.63 ! [P: nat > $o,K: nat] :
% 3.02/1.63 ( ( P @ K )
% 3.02/1.63 => ( ! [N6: nat] :
% 3.02/1.63 ( ( P @ ( suc @ N6 ) )
% 3.02/1.63 => ( P @ N6 ) )
% 3.02/1.63 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_93_diff__induct,axiom,
% 3.02/1.63 ! [P: nat > nat > $o,M: nat,N: nat] :
% 3.02/1.63 ( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
% 3.02/1.63 => ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
% 3.02/1.63 => ( ! [X4: nat,Y5: nat] :
% 3.02/1.63 ( ( P @ X4 @ Y5 )
% 3.02/1.63 => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 3.02/1.63 => ( P @ M @ N ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_94_nat__induct,axiom,
% 3.02/1.63 ! [P: nat > $o,N: nat] :
% 3.02/1.63 ( ( P @ ( zero_zero @ nat ) )
% 3.02/1.63 => ( ! [N6: nat] :
% 3.02/1.63 ( ( P @ N6 )
% 3.02/1.63 => ( P @ ( suc @ N6 ) ) )
% 3.02/1.63 => ( P @ N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_95_nat_OdiscI,axiom,
% 3.02/1.63 ! [Nat: nat,X23: nat] :
% 3.02/1.63 ( ( Nat
% 3.02/1.63 = ( suc @ X23 ) )
% 3.02/1.63 => ( Nat
% 3.02/1.63 != ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_96_old_Onat_Odistinct_I1_J,axiom,
% 3.02/1.63 ! [Nat2: nat] :
% 3.02/1.63 ( ( zero_zero @ nat )
% 3.02/1.63 != ( suc @ Nat2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_97_old_Onat_Odistinct_I2_J,axiom,
% 3.02/1.63 ! [Nat4: nat] :
% 3.02/1.63 ( ( suc @ Nat4 )
% 3.02/1.63 != ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_98_nat_Odistinct_I1_J,axiom,
% 3.02/1.63 ! [X23: nat] :
% 3.02/1.63 ( ( zero_zero @ nat )
% 3.02/1.63 != ( suc @ X23 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_99_of__nat__aux_Osimps_I1_J,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [Inc: A > A,I: A] :
% 3.02/1.63 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 3.02/1.63 = I ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_100_enat__setp_Ointros_I1_J,axiom,
% 3.02/1.63 coindu530039314t_setp @ ( zero_zero @ extended_enat ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_101_enat__setp_Ointros_I2_J,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( coindu530039314t_setp @ N )
% 3.02/1.63 => ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_102_enat__set_Ointros_I1_J,axiom,
% 3.02/1.63 member @ extended_enat @ ( zero_zero @ extended_enat ) @ coinductive_enat_set ).
% 3.02/1.63
% 3.02/1.63 thf(fact_103_enat__set_Ointros_I2_J,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( member @ extended_enat @ N @ coinductive_enat_set )
% 3.02/1.63 => ( member @ extended_enat @ ( extended_eSuc @ N ) @ coinductive_enat_set ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_104_enat__0__iff_I2_J,axiom,
% 3.02/1.63 ! [X: nat] :
% 3.02/1.63 ( ( ( zero_zero @ extended_enat )
% 3.02/1.63 = ( extended_enat2 @ X ) )
% 3.02/1.63 = ( X
% 3.02/1.63 = ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_105_enat__0__iff_I1_J,axiom,
% 3.02/1.63 ! [X: nat] :
% 3.02/1.63 ( ( ( extended_enat2 @ X )
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( X
% 3.02/1.63 = ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_106_zero__enat__def,axiom,
% 3.02/1.63 ( ( zero_zero @ extended_enat )
% 3.02/1.63 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_107_zero__reorient,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( zero @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [X: A] :
% 3.02/1.63 ( ( ( zero_zero @ A )
% 3.02/1.63 = X )
% 3.02/1.63 = ( X
% 3.02/1.63 = ( zero_zero @ A ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_108_enat_Osize_I1_J,axiom,
% 3.02/1.63 ! [Nat: nat] :
% 3.02/1.63 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_109_enat_Osize_I3_J,axiom,
% 3.02/1.63 ! [Nat: nat] :
% 3.02/1.63 ( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_110_enat__unfold_Osimps,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( coindu1491768222unfold @ A )
% 3.02/1.63 = ( ^ [Stop: A > $o,Next: A > A,A3: A] : ( if @ extended_enat @ ( Stop @ A3 ) @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A3 ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_111_enat__unfold__unique,axiom,
% 3.02/1.63 ! [A: $tType,Stop2: A > $o,H: A > extended_enat,Next2: A > A,X: A] :
% 3.02/1.63 ( ! [X4: A] :
% 3.02/1.63 ( ( ( Stop2 @ X4 )
% 3.02/1.63 => ( ( H @ X4 )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) )
% 3.02/1.63 & ( ~ ( Stop2 @ X4 )
% 3.02/1.63 => ( ( H @ X4 )
% 3.02/1.63 = ( extended_eSuc @ ( H @ ( Next2 @ X4 ) ) ) ) ) )
% 3.02/1.63 => ( ( H @ X )
% 3.02/1.63 = ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ X ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_112_the__enat__eSuc,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ( extended_the_enat @ ( extended_eSuc @ N ) )
% 3.02/1.63 = ( suc @ ( extended_the_enat @ N ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_113_list__decode_Ocases,axiom,
% 3.02/1.63 ! [X: nat] :
% 3.02/1.63 ( ( X
% 3.02/1.63 != ( zero_zero @ nat ) )
% 3.02/1.63 => ~ ! [N6: nat] :
% 3.02/1.63 ( X
% 3.02/1.63 != ( suc @ N6 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_114_not__enat__eq,axiom,
% 3.02/1.63 ! [X: extended_enat] :
% 3.02/1.63 ( ( ! [Y4: nat] :
% 3.02/1.63 ( X
% 3.02/1.63 != ( extended_enat2 @ Y4 ) ) )
% 3.02/1.63 = ( X
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_115_not__infinity__eq,axiom,
% 3.02/1.63 ! [X: extended_enat] :
% 3.02/1.63 ( ( X
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( ? [I2: nat] :
% 3.02/1.63 ( X
% 3.02/1.63 = ( extended_enat2 @ I2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_116_eSuc__infinity,axiom,
% 3.02/1.63 ( ( extended_eSuc @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_117_enat_Osimps_I5_J,axiom,
% 3.02/1.63 ! [T: $tType,F1: nat > T,F2: T] :
% 3.02/1.63 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = F2 ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_118_enat__unfold__eq__0,axiom,
% 3.02/1.63 ! [A: $tType,Stop2: A > $o,Next2: A > A,A2: A] :
% 3.02/1.63 ( ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( Stop2 @ A2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_119_enat__unfold__stop,axiom,
% 3.02/1.63 ! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
% 3.02/1.63 ( ( Stop2 @ A2 )
% 3.02/1.63 => ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_120_enat_Osimps_I7_J,axiom,
% 3.02/1.63 ! [T: $tType,F1: nat > T,F2: T] :
% 3.02/1.63 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = F2 ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_121_enat_Osize_I2_J,axiom,
% 3.02/1.63 ( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_122_enat_Osize_I4_J,axiom,
% 3.02/1.63 ( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_123_infinity__ne__i0,axiom,
% 3.02/1.63 ( ( extend1396239628finity @ extended_enat )
% 3.02/1.63 != ( zero_zero @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_124_enat_Odistinct_I2_J,axiom,
% 3.02/1.63 ! [Nat5: nat] :
% 3.02/1.63 ( ( extend1396239628finity @ extended_enat )
% 3.02/1.63 != ( extended_enat2 @ Nat5 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_125_enat_Odistinct_I1_J,axiom,
% 3.02/1.63 ! [Nat: nat] :
% 3.02/1.63 ( ( extended_enat2 @ Nat )
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_126_enat2__cases,axiom,
% 3.02/1.63 ! [Y: extended_enat,Ya: extended_enat] :
% 3.02/1.63 ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ! [Nata: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 != ( extended_enat2 @ Nata ) ) )
% 3.02/1.63 => ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( Ya
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) )
% 3.02/1.63 => ( ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ! [Nat3: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 != ( extended_enat2 @ Nat3 ) ) )
% 3.02/1.63 => ~ ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( Ya
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_127_enat3__cases,axiom,
% 3.02/1.63 ! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 3.02/1.63 ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( ? [Nata: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 = ( extended_enat2 @ Nata ) )
% 3.02/1.63 => ! [Natb: nat] :
% 3.02/1.63 ( Yb
% 3.02/1.63 != ( extended_enat2 @ Natb ) ) ) )
% 3.02/1.63 => ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( ? [Nata: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 = ( extended_enat2 @ Nata ) )
% 3.02/1.63 => ( Yb
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ) )
% 3.02/1.63 => ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( ( Ya
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ! [Nata: nat] :
% 3.02/1.63 ( Yb
% 3.02/1.63 != ( extended_enat2 @ Nata ) ) ) )
% 3.02/1.63 => ( ( ? [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( ( Ya
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( Yb
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ) )
% 3.02/1.63 => ( ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ? [Nat3: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ! [Nata: nat] :
% 3.02/1.63 ( Yb
% 3.02/1.63 != ( extended_enat2 @ Nata ) ) ) )
% 3.02/1.63 => ( ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ? [Nat3: nat] :
% 3.02/1.63 ( Ya
% 3.02/1.63 = ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( Yb
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ) )
% 3.02/1.63 => ( ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ( Ya
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ! [Nat3: nat] :
% 3.02/1.63 ( Yb
% 3.02/1.63 != ( extended_enat2 @ Nat3 ) ) ) )
% 3.02/1.63 => ~ ( ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ( Ya
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( Yb
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_128_enat_Oexhaust,axiom,
% 3.02/1.63 ! [Y: extended_enat] :
% 3.02/1.63 ( ! [Nat3: nat] :
% 3.02/1.63 ( Y
% 3.02/1.63 != ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( Y
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_129_enat_Oinducts,axiom,
% 3.02/1.63 ! [P: extended_enat > $o,Enat: extended_enat] :
% 3.02/1.63 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 3.02/1.63 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( P @ Enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_130_enat__ex__split,axiom,
% 3.02/1.63 ( ( ^ [P4: extended_enat > $o] :
% 3.02/1.63 ? [X8: extended_enat] : ( P4 @ X8 ) )
% 3.02/1.63 = ( ^ [P5: extended_enat > $o] :
% 3.02/1.63 ( ( P5 @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 | ? [X2: nat] : ( P5 @ ( extended_enat2 @ X2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_131_eSuc__eq__infinity__iff,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( ( extended_eSuc @ N )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( N
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_132_infinity__eq__eSuc__iff,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( ( extend1396239628finity @ extended_enat )
% 3.02/1.63 = ( extended_eSuc @ N ) )
% 3.02/1.63 = ( N
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_133_enat__unfold__next,axiom,
% 3.02/1.63 ! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
% 3.02/1.63 ( ~ ( Stop2 @ A2 )
% 3.02/1.63 => ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
% 3.02/1.63 = ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ ( Next2 @ A2 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_134_enat__the__enat,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ( extended_enat2 @ ( extended_the_enat @ N ) )
% 3.02/1.63 = N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_135_exists__least__lemma,axiom,
% 3.02/1.63 ! [P: nat > $o] :
% 3.02/1.63 ( ~ ( P @ ( zero_zero @ nat ) )
% 3.02/1.63 => ( ? [X12: nat] : ( P @ X12 )
% 3.02/1.63 => ? [N6: nat] :
% 3.02/1.63 ( ~ ( P @ N6 )
% 3.02/1.63 & ( P @ ( suc @ N6 ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_136_dependent__nat__choice,axiom,
% 3.02/1.63 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 3.02/1.63 ( ? [X12: A] : ( P @ ( zero_zero @ nat ) @ X12 )
% 3.02/1.63 => ( ! [X4: A,N6: nat] :
% 3.02/1.63 ( ( P @ N6 @ X4 )
% 3.02/1.63 => ? [Y3: A] :
% 3.02/1.63 ( ( P @ ( suc @ N6 ) @ Y3 )
% 3.02/1.63 & ( Q @ N6 @ X4 @ Y3 ) ) )
% 3.02/1.63 => ? [F4: nat > A] :
% 3.02/1.63 ! [N5: nat] :
% 3.02/1.63 ( ( P @ N5 @ ( F4 @ N5 ) )
% 3.02/1.63 & ( Q @ N5 @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_137_times__enat__simps_I3_J,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( ( N
% 3.02/1.63 = ( zero_zero @ nat ) )
% 3.02/1.63 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) )
% 3.02/1.63 & ( ( N
% 3.02/1.63 != ( zero_zero @ nat ) )
% 3.02/1.63 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_138_times__enat__simps_I4_J,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( ( M
% 3.02/1.63 = ( zero_zero @ nat ) )
% 3.02/1.63 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) )
% 3.02/1.63 & ( ( M
% 3.02/1.63 != ( zero_zero @ nat ) )
% 3.02/1.63 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_139_times__enat__simps_I2_J,axiom,
% 3.02/1.63 ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_140_imult__is__0,axiom,
% 3.02/1.63 ! [M: extended_enat,N: extended_enat] :
% 3.02/1.63 ( ( ( times_times @ extended_enat @ M @ N )
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 = ( ( M
% 3.02/1.63 = ( zero_zero @ extended_enat ) )
% 3.02/1.63 | ( N
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.02/1.63 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 3.02/1.63 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_142_mult_Oassoc,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semigroup_mult @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.02/1.63 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 3.02/1.63 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_143_mult_Ocommute,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 3.02/1.63 => ( ( times_times @ A )
% 3.02/1.63 = ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_144_mult_Oleft__commute,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [B2: A,A2: A,C2: A] :
% 3.02/1.63 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 3.02/1.63 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_145_imult__is__infinity,axiom,
% 3.02/1.63 ! [A2: extended_enat,B2: extended_enat] :
% 3.02/1.63 ( ( ( times_times @ extended_enat @ A2 @ B2 )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( ( ( A2
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 & ( B2
% 3.02/1.63 != ( zero_zero @ extended_enat ) ) )
% 3.02/1.63 | ( ( B2
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 & ( A2
% 3.02/1.63 != ( zero_zero @ extended_enat ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_146_mult__zero__left,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( mult_zero @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_147_mult__zero__right,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( mult_zero @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_148_mult__eq__0__iff,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A] :
% 3.02/1.63 ( ( ( times_times @ A @ A2 @ B2 )
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 = ( ( A2
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 | ( B2
% 3.02/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_149_mult__cancel__left,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.02/1.63 ( ( ( times_times @ A @ C2 @ A2 )
% 3.02/1.63 = ( times_times @ A @ C2 @ B2 ) )
% 3.02/1.63 = ( ( C2
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 | ( A2 = B2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_150_mult__is__0,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ M @ N )
% 3.02/1.63 = ( zero_zero @ nat ) )
% 3.02/1.63 = ( ( M
% 3.02/1.63 = ( zero_zero @ nat ) )
% 3.02/1.63 | ( N
% 3.02/1.63 = ( zero_zero @ nat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_151_mult__0__right,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_152_mult__cancel1,axiom,
% 3.02/1.63 ! [K: nat,M: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ K @ M )
% 3.02/1.63 = ( times_times @ nat @ K @ N ) )
% 3.02/1.63 = ( ( M = N )
% 3.02/1.63 | ( K
% 3.02/1.63 = ( zero_zero @ nat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_153_mult__cancel2,axiom,
% 3.02/1.63 ! [M: nat,K: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ M @ K )
% 3.02/1.63 = ( times_times @ nat @ N @ K ) )
% 3.02/1.63 = ( ( M = N )
% 3.02/1.63 | ( K
% 3.02/1.63 = ( zero_zero @ nat ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_154_mult__cancel__right,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,C2: A,B2: A] :
% 3.02/1.63 ( ( ( times_times @ A @ A2 @ C2 )
% 3.02/1.63 = ( times_times @ A @ B2 @ C2 ) )
% 3.02/1.63 = ( ( C2
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 | ( A2 = B2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_155_mult__eq__1__iff,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ M @ N )
% 3.02/1.63 = ( suc @ ( zero_zero @ nat ) ) )
% 3.02/1.63 = ( ( M
% 3.02/1.63 = ( suc @ ( zero_zero @ nat ) ) )
% 3.02/1.63 & ( N
% 3.02/1.63 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_156_one__eq__mult__iff,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( ( suc @ ( zero_zero @ nat ) )
% 3.02/1.63 = ( times_times @ nat @ M @ N ) )
% 3.02/1.63 = ( ( M
% 3.02/1.63 = ( suc @ ( zero_zero @ nat ) ) )
% 3.02/1.63 & ( N
% 3.02/1.63 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_157_times__enat__simps_I1_J,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 3.02/1.63 = ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_158_mult__0,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_159_Suc__mult__cancel1,axiom,
% 3.02/1.63 ! [K: nat,M: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ ( suc @ K ) @ M )
% 3.02/1.63 = ( times_times @ nat @ ( suc @ K ) @ N ) )
% 3.02/1.63 = ( M = N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_160_mult__right__cancel,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.02/1.63 ( ( C2
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 => ( ( ( times_times @ A @ A2 @ C2 )
% 3.02/1.63 = ( times_times @ A @ B2 @ C2 ) )
% 3.02/1.63 = ( A2 = B2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_161_mult__left__cancel,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.02/1.63 ( ( C2
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 => ( ( ( times_times @ A @ C2 @ A2 )
% 3.02/1.63 = ( times_times @ A @ C2 @ B2 ) )
% 3.02/1.63 = ( A2 = B2 ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_162_no__zero__divisors,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A] :
% 3.02/1.63 ( ( A2
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 => ( ( B2
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 => ( ( times_times @ A @ A2 @ B2 )
% 3.02/1.63 != ( zero_zero @ A ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_163_divisors__zero,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( semiri1193490041visors @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A] :
% 3.02/1.63 ( ( ( times_times @ A @ A2 @ B2 )
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 => ( ( A2
% 3.02/1.63 = ( zero_zero @ A ) )
% 3.02/1.63 | ( B2
% 3.02/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_164_mult__not__zero,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( mult_zero @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A] :
% 3.02/1.63 ( ( ( times_times @ A @ A2 @ B2 )
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 => ( ( A2
% 3.02/1.63 != ( zero_zero @ A ) )
% 3.02/1.63 & ( B2
% 3.02/1.63 != ( zero_zero @ A ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_165_semiring__normalization__rules_I9_J,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_166_semiring__normalization__rules_I10_J,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_167_idiff__infinity__right,axiom,
% 3.02/1.63 ! [A2: nat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_168_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_169_diff__zero,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.63 = A2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_170_zero__diff,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_171_diff__0__right,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( group_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.63 = A2 ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_172_diff__self,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( group_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_173_idiff__0__right,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
% 3.02/1.63 = N ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_174_idiff__0,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_175_eSuc__minus__eSuc,axiom,
% 3.02/1.63 ! [N: extended_enat,M: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 3.02/1.63 = ( minus_minus @ extended_enat @ N @ M ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_176_idiff__infinity,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N )
% 3.02/1.63 = ( extend1396239628finity @ extended_enat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_177_idiff__enat__0__right,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
% 3.02/1.63 = N ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_178_idiff__enat__0,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
% 3.02/1.63 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_179_idiff__self,axiom,
% 3.02/1.63 ! [N: extended_enat] :
% 3.02/1.63 ( ( N
% 3.02/1.63 != ( extend1396239628finity @ extended_enat ) )
% 3.02/1.63 => ( ( minus_minus @ extended_enat @ N @ N )
% 3.02/1.63 = ( zero_zero @ extended_enat ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_180_eq__iff__diff__eq__0,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( group_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ( ( ^ [Y6: A,Z: A] : ( Y6 = Z ) )
% 3.02/1.63 = ( ^ [A3: A,B3: A] :
% 3.02/1.63 ( ( minus_minus @ A @ A3 @ B3 )
% 3.02/1.63 = ( zero_zero @ A ) ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_181_diff__eq__diff__eq,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( group_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,B2: A,C2: A,D: A] :
% 3.02/1.63 ( ( ( minus_minus @ A @ A2 @ B2 )
% 3.02/1.63 = ( minus_minus @ A @ C2 @ D ) )
% 3.02/1.63 => ( ( A2 = B2 )
% 3.02/1.63 = ( C2 = D ) ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_182_diff__right__commute,axiom,
% 3.02/1.63 ! [A: $tType] :
% 3.02/1.63 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 3.02/1.63 => ! [A2: A,C2: A,B2: A] :
% 3.02/1.63 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
% 3.02/1.63 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_183_nat__mult__eq__cancel__disj,axiom,
% 3.02/1.63 ! [K: nat,M: nat,N: nat] :
% 3.02/1.63 ( ( ( times_times @ nat @ K @ M )
% 3.02/1.63 = ( times_times @ nat @ K @ N ) )
% 3.02/1.63 = ( ( K
% 3.02/1.63 = ( zero_zero @ nat ) )
% 3.02/1.63 | ( M = N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_184_one__le__mult__iff,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
% 3.02/1.63 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
% 3.02/1.63 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_185_triangle__0,axiom,
% 3.02/1.63 ( ( nat_triangle @ ( zero_zero @ nat ) )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_186_diff__self__eq__0,axiom,
% 3.02/1.63 ! [M: nat] :
% 3.02/1.63 ( ( minus_minus @ nat @ M @ M )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_187_diff__0__eq__0,axiom,
% 3.02/1.63 ! [N: nat] :
% 3.02/1.63 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 3.02/1.63 = ( zero_zero @ nat ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_188_diff__Suc__Suc,axiom,
% 3.02/1.63 ! [M: nat,N: nat] :
% 3.02/1.63 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.02/1.63 = ( minus_minus @ nat @ M @ N ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_189_Suc__diff__diff,axiom,
% 3.02/1.63 ! [M: nat,N: nat,K: nat] :
% 3.02/1.63 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 3.02/1.63 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 3.02/1.63
% 3.02/1.63 thf(fact_190_diff__diff__cancel,axiom,
% 3.02/1.63 ! [I: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ I @ N )
% 3.02/1.64 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 3.02/1.64 = I ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_191_le__zero__eq,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [N: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.02/1.64 = ( N
% 3.02/1.64 = ( zero_zero @ A ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_192_diff__is__0__eq_H,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( ( minus_minus @ nat @ M @ N )
% 3.02/1.64 = ( zero_zero @ nat ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_193_diff__is__0__eq,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ( minus_minus @ nat @ M @ N )
% 3.02/1.64 = ( zero_zero @ nat ) )
% 3.02/1.64 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_194_le0,axiom,
% 3.02/1.64 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_195_Suc__le__mono,axiom,
% 3.02/1.64 ! [N: nat,M: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 3.02/1.64 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_196_idiff__enat__enat,axiom,
% 3.02/1.64 ! [A2: nat,B2: nat] :
% 3.02/1.64 ( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
% 3.02/1.64 = ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_197_diff__ge__0__iff__ge,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 3.02/1.64 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_198_le__cube,axiom,
% 3.02/1.64 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_199_le__square,axiom,
% 3.02/1.64 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_200_mult__le__mono,axiom,
% 3.02/1.64 ! [I: nat,J: nat,K: nat,L: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ I @ J )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ K @ L )
% 3.02/1.64 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_201_mult__le__mono1,axiom,
% 3.02/1.64 ! [I: nat,J: nat,K: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ I @ J )
% 3.02/1.64 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_202_mult__le__mono2,axiom,
% 3.02/1.64 ! [I: nat,J: nat,K: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ I @ J )
% 3.02/1.64 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_203_diff__mult__distrib,axiom,
% 3.02/1.64 ! [M: nat,N: nat,K: nat] :
% 3.02/1.64 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 3.02/1.64 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_204_diff__mult__distrib2,axiom,
% 3.02/1.64 ! [K: nat,M: nat,N: nat] :
% 3.02/1.64 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 3.02/1.64 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_205_minus__nat_Odiff__0,axiom,
% 3.02/1.64 ! [M: nat] :
% 3.02/1.64 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 3.02/1.64 = M ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_206_diffs0__imp__equal,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ( minus_minus @ nat @ M @ N )
% 3.02/1.64 = ( zero_zero @ nat ) )
% 3.02/1.64 => ( ( ( minus_minus @ nat @ N @ M )
% 3.02/1.64 = ( zero_zero @ nat ) )
% 3.02/1.64 => ( M = N ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_207_prod__decode__aux_Oinduct,axiom,
% 3.02/1.64 ! [P: nat > nat > $o,A0: nat,A1: nat] :
% 3.02/1.64 ( ! [K2: nat,M2: nat] :
% 3.02/1.64 ( ( ~ ( ord_less_eq @ nat @ M2 @ K2 )
% 3.02/1.64 => ( P @ ( suc @ K2 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K2 ) ) ) )
% 3.02/1.64 => ( P @ K2 @ M2 ) )
% 3.02/1.64 => ( P @ A0 @ A1 ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_208_zero__induct__lemma,axiom,
% 3.02/1.64 ! [P: nat > $o,K: nat,I: nat] :
% 3.02/1.64 ( ( P @ K )
% 3.02/1.64 => ( ! [N6: nat] :
% 3.02/1.64 ( ( P @ ( suc @ N6 ) )
% 3.02/1.64 => ( P @ N6 ) )
% 3.02/1.64 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_209_eq__diff__iff,axiom,
% 3.02/1.64 ! [K: nat,M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ K @ M )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ K @ N )
% 3.02/1.64 => ( ( ( minus_minus @ nat @ M @ K )
% 3.02/1.64 = ( minus_minus @ nat @ N @ K ) )
% 3.02/1.64 = ( M = N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_210_le__diff__iff,axiom,
% 3.02/1.64 ! [K: nat,M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ K @ M )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ K @ N )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 3.02/1.64 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_211_Nat_Odiff__diff__eq,axiom,
% 3.02/1.64 ! [K: nat,M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ K @ M )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ K @ N )
% 3.02/1.64 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 3.02/1.64 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_212_diff__le__mono,axiom,
% 3.02/1.64 ! [M: nat,N: nat,L: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_213_diff__le__self,axiom,
% 3.02/1.64 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_214_le__diff__iff_H,axiom,
% 3.02/1.64 ! [A2: nat,C2: nat,B2: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ A2 @ C2 )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 3.02/1.64 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_215_diff__le__mono2,axiom,
% 3.02/1.64 ! [M: nat,N: nat,L: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_216_Suc__diff__le,axiom,
% 3.02/1.64 ! [N: nat,M: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ N @ M )
% 3.02/1.64 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 3.02/1.64 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_217_diff__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,D: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ D @ C2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_218_diff__left__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [B2: A,A2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_219_diff__right__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_220_diff__eq__diff__less__eq,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A,D: A] :
% 3.02/1.64 ( ( ( minus_minus @ A @ A2 @ B2 )
% 3.02/1.64 = ( minus_minus @ A @ C2 @ D ) )
% 3.02/1.64 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_221_lift__Suc__mono__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( order @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [F3: nat > A,N: nat,N7: nat] :
% 3.02/1.64 ( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ N @ N7 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_222_lift__Suc__antimono__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( order @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [F3: nat > A,N: nat,N7: nat] :
% 3.02/1.64 ( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ N @ N7 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_223_nat__le__linear,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_224_le__antisym,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ N @ M )
% 3.02/1.64 => ( M = N ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_225_eq__imp__le,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( M = N )
% 3.02/1.64 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_226_le__trans,axiom,
% 3.02/1.64 ! [I: nat,J: nat,K: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ I @ J )
% 3.02/1.64 => ( ( ord_less_eq @ nat @ J @ K )
% 3.02/1.64 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_227_le__refl,axiom,
% 3.02/1.64 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_228_wlog__linorder__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( linorder @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [P: A > A > $o,B2: A,A2: A] :
% 3.02/1.64 ( ! [A5: A,B4: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A5 @ B4 )
% 3.02/1.64 => ( P @ A5 @ B4 ) )
% 3.02/1.64 => ( ( ( P @ B2 @ A2 )
% 3.02/1.64 => ( P @ A2 @ B2 ) )
% 3.02/1.64 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_229_full__nat__induct,axiom,
% 3.02/1.64 ! [P: nat > $o,N: nat] :
% 3.02/1.64 ( ! [N6: nat] :
% 3.02/1.64 ( ! [M3: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N6 )
% 3.02/1.64 => ( P @ M3 ) )
% 3.02/1.64 => ( P @ N6 ) )
% 3.02/1.64 => ( P @ N ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_230_not__less__eq__eq,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 3.02/1.64 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_231_Suc__n__not__le__n,axiom,
% 3.02/1.64 ! [N: nat] :
% 3.02/1.64 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_232_le__Suc__eq,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 3.02/1.64 = ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 | ( M
% 3.02/1.64 = ( suc @ N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_233_Suc__le__D,axiom,
% 3.02/1.64 ! [N: nat,M4: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
% 3.02/1.64 => ? [M2: nat] :
% 3.02/1.64 ( M4
% 3.02/1.64 = ( suc @ M2 ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_234_le__SucI,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_235_le__SucE,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 3.02/1.64 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 3.02/1.64 => ( M
% 3.02/1.64 = ( suc @ N ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_236_Suc__leD,axiom,
% 3.02/1.64 ! [M: nat,N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 3.02/1.64 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_237_zero__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_238_less__eq__nat_Osimps_I1_J,axiom,
% 3.02/1.64 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_239_le__0__eq,axiom,
% 3.02/1.64 ! [N: nat] :
% 3.02/1.64 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 3.02/1.64 = ( N
% 3.02/1.64 = ( zero_zero @ nat ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_240_mult__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A,D: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ C2 @ D )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_241_mult__mono_H,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A,D: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ C2 @ D )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_242_zero__le__square,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( linordered_ring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_243_split__mult__pos__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.64 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 3.02/1.64 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.64 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
% 3.02/1.64 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_244_mult__left__mono__neg,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [B2: A,A2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_245_mult__nonpos__nonpos,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.64 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 3.02/1.64 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_246_mult__left__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_247_mult__right__mono__neg,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_ring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [B2: A,A2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_248_mult__right__mono,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A,C2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_249_mult__le__0__iff,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( linord581940658strict @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 3.02/1.64 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.64 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 3.02/1.64 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.64 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_250_split__mult__neg__le,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.64 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 3.02/1.64 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 3.02/1.64 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
% 3.02/1.64 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(fact_251_mult__nonneg__nonneg,axiom,
% 3.02/1.64 ! [A: $tType] :
% 3.02/1.64 ( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
% 3.02/1.64 => ! [A2: A,B2: A] :
% 3.02/1.64 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.02/1.64 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 3.02/1.64 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.02/1.64 ! [A6: $tType,A7: $tType] :
% 3.02/1.64 ( ( order @ A7 @ ( type2 @ A7 ) )
% 3.02/1.64 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 3.02/1.64 semiri1923998003cancel @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.02/1.64 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 3.02/1.64 semiri1193490041visors @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 3.02/1.64 cancel146912293up_add @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 3.02/1.64 cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
% 3.02/1.64 ordered_semiring_0 @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 3.02/1.64 ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 3.02/1.64 comm_monoid_diff @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 3.02/1.64 ordered_semiring @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 3.02/1.64 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 3.02/1.64 semigroup_mult @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.02/1.64 linorder @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 3.02/1.64 semiring_1 @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 3.02/1.64 mult_zero @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 3.02/1.64 order @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 3.02/1.64 zero @ nat @ ( type2 @ nat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 3.02/1.64 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
% 3.02/1.64 linorder @ $o @ ( type2 @ $o ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
% 3.02/1.64 order @ $o @ ( type2 @ $o ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_5,axiom,
% 3.02/1.64 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_6,axiom,
% 3.02/1.64 semiri1193490041visors @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_7,axiom,
% 3.02/1.64 ab_semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_8,axiom,
% 3.02/1.64 ordered_semiring @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_9,axiom,
% 3.02/1.64 comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_10,axiom,
% 3.02/1.64 semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_11,axiom,
% 3.02/1.64 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_12,axiom,
% 3.02/1.64 semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_13,axiom,
% 3.02/1.64 mult_zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_14,axiom,
% 3.02/1.64 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_15,axiom,
% 3.02/1.64 zero @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.02/1.64
% 3.02/1.64 thf(help_If_3_1_T,axiom,
% 3.02/1.64 ! [P: $o] :
% 3.02/1.64 ( ( P = $true )
% 3.02/1.64 | ( P = $false ) ) ).
% 3.02/1.64
% 3.02/1.64 thf(help_If_2_1_T,axiom,
% 3.02/1.64 ! [A: $tType,X: A,Y: A] :
% 3.02/1.64 ( ( if @ A @ $false @ X @ Y )
% 3.02/1.64 = Y ) ).
% 3.02/1.64
% 3.02/1.64 thf(help_If_1_1_T,axiom,
% 3.02/1.64 ! [A: $tType,X: A,Y: A] :
% 3.02/1.64 ( ( if @ A @ $true @ X @ Y )
% 3.02/1.64 = X ) ).
% 3.02/1.64
% 3.02/1.64 thf(conj_0,hypothesis,
% 3.02/1.64 ! [X9: a,Xs6: coindu1593790203_llist @ a] :
% 3.02/1.64 ( ( xsa
% 3.02/1.64 = ( coindu1121789889_LCons @ a @ X9 @ Xs6 ) )
% 3.02/1.64 => thesis ) ).
% 3.02/1.64
% 3.02/1.64 thf(conj_1,conjecture,
% 3.02/1.64 thesis ).
% 3.02/1.64 ------------------------
% 3.02/1.67 % (11091)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/18Mi)
% 3.02/1.67 % (11085)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/183Mi)
% 3.02/1.67 % (11089)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_10986 for (2999ds/2Mi)
% 3.02/1.67 % (11088)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/2Mi)
% 3.02/1.67 % (11090)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_10986 for (2999ds/275Mi)
% 3.02/1.67 % (11087)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_10986 for (2999ds/27Mi)
% 3.02/1.67 % (11086)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_10986 for (2999ds/4Mi)
% 3.02/1.67 % (11088)Instruction limit reached!
% 3.02/1.67 % (11088)------------------------------
% 3.02/1.67 % (11088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (11088)Termination reason: Unknown
% 3.02/1.67 % (11088)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (11088)Memory used [KB]: 1407
% 3.02/1.67 % (11088)Time elapsed: 0.003 s
% 3.02/1.67 % (11088)Instructions burned: 2 (million)
% 3.02/1.67 % (11088)------------------------------
% 3.02/1.67 % (11088)------------------------------
% 3.02/1.67 % (11089)Instruction limit reached!
% 3.02/1.67 % (11089)------------------------------
% 3.02/1.67 % (11089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (11089)Termination reason: Unknown
% 3.02/1.67 % (11089)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (11089)Memory used [KB]: 1407
% 3.02/1.67 % (11089)Time elapsed: 0.003 s
% 3.02/1.67 % (11089)Instructions burned: 2 (million)
% 3.02/1.67 % (11089)------------------------------
% 3.02/1.67 % (11089)------------------------------
% 3.02/1.67 % (11086)Instruction limit reached!
% 3.02/1.67 % (11086)------------------------------
% 3.02/1.67 % (11086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.67 % (11086)Termination reason: Unknown
% 3.02/1.67 % (11086)Termination phase: shuffling
% 3.02/1.67
% 3.02/1.67 % (11086)Memory used [KB]: 1407
% 3.02/1.67 % (11086)Time elapsed: 0.004 s
% 3.02/1.67 % (11086)Instructions burned: 4 (million)
% 3.02/1.67 % (11086)------------------------------
% 3.02/1.67 % (11086)------------------------------
% 3.02/1.68 % (11091)Instruction limit reached!
% 3.02/1.68 % (11091)------------------------------
% 3.02/1.68 % (11091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.68 % (11091)Termination reason: Unknown
% 3.02/1.68 % (11091)Termination phase: Property scanning
% 3.02/1.68
% 3.02/1.68 % (11091)Memory used [KB]: 1663
% 3.02/1.68 % (11091)Time elapsed: 0.011 s
% 3.02/1.68 % (11091)Instructions burned: 18 (million)
% 3.02/1.68 % (11091)------------------------------
% 3.02/1.68 % (11091)------------------------------
% 3.02/1.68 % (11087)Instruction limit reached!
% 3.02/1.68 % (11087)------------------------------
% 3.02/1.68 % (11087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.68 % (11087)Termination reason: Unknown
% 3.02/1.68 % (11087)Termination phase: Property scanning
% 3.02/1.68
% 3.02/1.68 % (11087)Memory used [KB]: 1791
% 3.02/1.68 % (11087)Time elapsed: 0.015 s
% 3.02/1.68 % (11087)Instructions burned: 27 (million)
% 3.02/1.68 % (11087)------------------------------
% 3.02/1.68 % (11087)------------------------------
% 3.02/1.68 % (11092)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/3Mi)
% 3.02/1.68 % (11093)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_10986 for (2999ds/37Mi)
% 3.02/1.68 % (11094)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_10986 for (2999ds/15Mi)
% 3.02/1.69 % (11092)Instruction limit reached!
% 3.02/1.69 % (11092)------------------------------
% 3.02/1.69 % (11092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.69 % (11092)Termination reason: Unknown
% 3.02/1.69 % (11092)Termination phase: shuffling
% 3.02/1.69
% 3.02/1.69 % (11092)Memory used [KB]: 1407
% 3.02/1.69 % (11092)Time elapsed: 0.003 s
% 3.02/1.69 % (11092)Instructions burned: 3 (million)
% 3.02/1.69 % (11092)------------------------------
% 3.02/1.69 % (11092)------------------------------
% 3.02/1.69 % (11090)Refutation not found, incomplete strategy
% 3.02/1.69 % (11090)------------------------------
% 3.02/1.69 % (11090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.69 % (11090)Termination reason: Refutation not found, incomplete strategy
% 3.02/1.69
% 3.02/1.69
% 3.02/1.69 % (11090)Memory used [KB]: 6396
% 3.02/1.69 % (11090)Time elapsed: 0.022 s
% 3.02/1.69 % (11090)Instructions burned: 39 (million)
% 3.02/1.69 % (11090)------------------------------
% 3.02/1.69 % (11090)------------------------------
% 3.02/1.69 % (11095)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_10986 for (2999ds/3Mi)
% 3.02/1.69 % (11094)Instruction limit reached!
% 3.02/1.69 % (11094)------------------------------
% 3.02/1.69 % (11094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.69 % (11094)Termination reason: Unknown
% 3.02/1.69 % (11094)Termination phase: shuffling
% 3.02/1.69
% 3.02/1.69 % (11094)Memory used [KB]: 1663
% 3.02/1.69 % (11094)Time elapsed: 0.009 s
% 3.02/1.69 % (11094)Instructions burned: 15 (million)
% 3.02/1.69 % (11094)------------------------------
% 3.02/1.69 % (11094)------------------------------
% 3.02/1.69 % (11095)Instruction limit reached!
% 3.02/1.69 % (11095)------------------------------
% 3.02/1.69 % (11095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.69 % (11095)Termination reason: Unknown
% 3.02/1.69 % (11095)Termination phase: shuffling
% 3.02/1.69
% 3.02/1.69 % (11095)Memory used [KB]: 1407
% 3.02/1.69 % (11095)Time elapsed: 0.003 s
% 3.02/1.69 % (11095)Instructions burned: 3 (million)
% 3.02/1.69 % (11095)------------------------------
% 3.02/1.69 % (11095)------------------------------
% 3.06/1.70 % (11096)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_10986 for (2999ds/1041Mi)
% 3.06/1.70 % (11097)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/7Mi)
% 3.06/1.70 % (11093)Instruction limit reached!
% 3.06/1.70 % (11093)------------------------------
% 3.06/1.70 % (11093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.70 % (11093)Termination reason: Unknown
% 3.06/1.70 % (11093)Termination phase: Property scanning
% 3.06/1.70
% 3.06/1.70 % (11093)Memory used [KB]: 2174
% 3.06/1.70 % (11093)Time elapsed: 0.019 s
% 3.06/1.70 % (11093)Instructions burned: 38 (million)
% 3.06/1.70 % (11093)------------------------------
% 3.06/1.70 % (11093)------------------------------
% 3.06/1.70 % (11098)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/16Mi)
% 3.06/1.71 % (11097)Instruction limit reached!
% 3.06/1.71 % (11097)------------------------------
% 3.06/1.71 % (11097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.71 % (11097)Termination reason: Unknown
% 3.06/1.71 % (11097)Termination phase: shuffling
% 3.06/1.71
% 3.06/1.71 % (11097)Memory used [KB]: 1407
% 3.06/1.71 % (11097)Time elapsed: 0.005 s
% 3.06/1.71 % (11097)Instructions burned: 7 (million)
% 3.06/1.71 % (11097)------------------------------
% 3.06/1.71 % (11097)------------------------------
% 3.06/1.71 % (11099)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_10986 for (2999ds/3Mi)
% 3.06/1.71 % (11100)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_10986 for (2999ds/3Mi)
% 3.06/1.71 % (11099)Instruction limit reached!
% 3.06/1.71 % (11099)------------------------------
% 3.06/1.71 % (11099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.71 % (11099)Termination reason: Unknown
% 3.06/1.71 % (11099)Termination phase: shuffling
% 3.06/1.71
% 3.06/1.71 % (11099)Memory used [KB]: 1407
% 3.06/1.71 % (11099)Time elapsed: 0.003 s
% 3.06/1.71 % (11099)Instructions burned: 3 (million)
% 3.06/1.71 % (11099)------------------------------
% 3.06/1.71 % (11099)------------------------------
% 3.06/1.71 % (11100)Instruction limit reached!
% 3.06/1.71 % (11100)------------------------------
% 3.06/1.71 % (11100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.71 % (11100)Termination reason: Unknown
% 3.06/1.71 % (11100)Termination phase: shuffling
% 3.06/1.71
% 3.06/1.71 % (11100)Memory used [KB]: 1407
% 3.06/1.71 % (11100)Time elapsed: 0.003 s
% 3.06/1.71 % (11100)Instructions burned: 3 (million)
% 3.06/1.71 % (11100)------------------------------
% 3.06/1.71 % (11100)------------------------------
% 3.06/1.71 % (11098)Instruction limit reached!
% 3.06/1.71 % (11098)------------------------------
% 3.06/1.71 % (11098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.71 % (11098)Termination reason: Unknown
% 3.06/1.71 % (11098)Termination phase: shuffling
% 3.06/1.71
% 3.06/1.71 % (11098)Memory used [KB]: 1791
% 3.06/1.71 % (11098)Time elapsed: 0.010 s
% 3.06/1.71 % (11098)Instructions burned: 17 (million)
% 3.06/1.71 % (11098)------------------------------
% 3.06/1.71 % (11098)------------------------------
% 3.06/1.72 % (11101)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/7Mi)
% 3.06/1.72 % (11102)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/3Mi)
% 3.06/1.72 % (11101)Instruction limit reached!
% 3.06/1.72 % (11101)------------------------------
% 3.06/1.72 % (11101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.72 % (11101)Termination reason: Unknown
% 3.06/1.72 % (11101)Termination phase: shuffling
% 3.06/1.72
% 3.06/1.72 % (11101)Memory used [KB]: 1535
% 3.06/1.72 % (11101)Time elapsed: 0.005 s
% 3.06/1.72 % (11101)Instructions burned: 7 (million)
% 3.06/1.72 % (11101)------------------------------
% 3.06/1.72 % (11101)------------------------------
% 3.06/1.72 % (11102)Instruction limit reached!
% 3.06/1.72 % (11102)------------------------------
% 3.06/1.72 % (11102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.72 % (11102)Termination reason: Unknown
% 3.06/1.72 % (11102)Termination phase: shuffling
% 3.06/1.72
% 3.06/1.72 % (11102)Memory used [KB]: 1407
% 3.06/1.72 % (11102)Time elapsed: 0.003 s
% 3.06/1.72 % (11102)Instructions burned: 3 (million)
% 3.06/1.72 % (11102)------------------------------
% 3.06/1.72 % (11102)------------------------------
% 3.06/1.73 % (11103)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/4Mi)
% 3.06/1.73 % (11104)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/18Mi)
% 3.06/1.73 % (11103)Instruction limit reached!
% 3.06/1.73 % (11103)------------------------------
% 3.06/1.73 % (11103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.73 % (11103)Termination reason: Unknown
% 3.06/1.73 % (11103)Termination phase: shuffling
% 3.06/1.73 % (11105)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/710Mi)
% 3.06/1.73
% 3.06/1.73 % (11103)Memory used [KB]: 1407
% 3.06/1.73 % (11103)Time elapsed: 0.004 s
% 3.06/1.73 % (11103)Instructions burned: 5 (million)
% 3.06/1.73 % (11103)------------------------------
% 3.06/1.73 % (11103)------------------------------
% 3.06/1.74 % (11104)Instruction limit reached!
% 3.06/1.74 % (11104)------------------------------
% 3.06/1.74 % (11104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.74 % (11104)Termination reason: Unknown
% 3.06/1.74 % (11104)Termination phase: Property scanning
% 3.06/1.74
% 3.06/1.74 % (11104)Memory used [KB]: 1791
% 3.06/1.74 % (11104)Time elapsed: 0.011 s
% 3.06/1.74 % (11104)Instructions burned: 19 (million)
% 3.06/1.74 % (11104)------------------------------
% 3.06/1.74 % (11104)------------------------------
% 3.06/1.74 % (11106)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/6Mi)
% 3.06/1.74 % (11107)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_10986 for (2999ds/902Mi)
% 3.06/1.74 % (11106)Instruction limit reached!
% 3.06/1.74 % (11106)------------------------------
% 3.06/1.74 % (11106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.74 % (11106)Termination reason: Unknown
% 3.06/1.74 % (11106)Termination phase: shuffling
% 3.06/1.74
% 3.06/1.74 % (11106)Memory used [KB]: 1407
% 3.06/1.74 % (11106)Time elapsed: 0.005 s
% 3.06/1.74 % (11106)Instructions burned: 7 (million)
% 3.06/1.74 % (11106)------------------------------
% 3.06/1.74 % (11106)------------------------------
% 3.06/1.74 % (11108)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_10986 for (2999ds/21Mi)
% 3.06/1.75 % (11109)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_10986 for (2999ds/5Mi)
% 3.06/1.75 % (11108)Instruction limit reached!
% 3.06/1.75 % (11108)------------------------------
% 3.06/1.75 % (11108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.75 % (11108)Termination reason: Unknown
% 3.06/1.75 % (11108)Termination phase: Property scanning
% 3.06/1.75
% 3.06/1.75 % (11108)Memory used [KB]: 1918
% 3.06/1.75 % (11108)Time elapsed: 0.012 s
% 3.06/1.75 % (11108)Instructions burned: 21 (million)
% 3.06/1.75 % (11108)------------------------------
% 3.06/1.75 % (11108)------------------------------
% 3.06/1.76 % (11109)Instruction limit reached!
% 3.06/1.76 % (11109)------------------------------
% 3.06/1.76 % (11109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.76 % (11109)Termination reason: Unknown
% 3.06/1.76 % (11109)Termination phase: shuffling
% 3.06/1.76
% 3.06/1.76 % (11109)Memory used [KB]: 1407
% 3.06/1.76 % (11109)Time elapsed: 0.004 s
% 3.06/1.76 % (11109)Instructions burned: 6 (million)
% 3.06/1.76 % (11109)------------------------------
% 3.06/1.76 % (11109)------------------------------
% 3.06/1.76 % (11085)Instruction limit reached!
% 3.06/1.76 % (11085)------------------------------
% 3.06/1.76 % (11085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.76 % (11085)Termination reason: Unknown
% 3.06/1.76 % (11085)Termination phase: Saturation
% 3.06/1.76
% 3.06/1.76 % (11085)Memory used [KB]: 7164
% 3.06/1.76 % (11085)Time elapsed: 0.092 s
% 3.06/1.76 % (11085)Instructions burned: 183 (million)
% 3.06/1.76 % (11085)------------------------------
% 3.06/1.76 % (11085)------------------------------
% 3.06/1.76 % (11110)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_10986 for (2999ds/6Mi)
% 3.06/1.76 % (11110)Instruction limit reached!
% 3.06/1.76 % (11110)------------------------------
% 3.06/1.76 % (11110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.76 % (11110)Termination reason: Unknown
% 3.06/1.76 % (11110)Termination phase: shuffling
% 3.06/1.76
% 3.06/1.76 % (11110)Memory used [KB]: 1407
% 3.06/1.76 % (11110)Time elapsed: 0.004 s
% 3.06/1.76 % (11110)Instructions burned: 6 (million)
% 3.06/1.76 % (11110)------------------------------
% 3.06/1.76 % (11110)------------------------------
% 3.06/1.77 % (11111)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_10986 for (2998ds/377Mi)
% 3.06/1.77 % (11112)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_10986 for (2998ds/779Mi)
% 3.06/1.77 % (11113)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_10986 for (2998ds/19Mi)
% 3.06/1.78 % (11114)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_10986 for (2998ds/879Mi)
% 3.06/1.78 % (11113)Instruction limit reached!
% 3.06/1.78 % (11113)------------------------------
% 3.06/1.78 % (11113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.78 % (11113)Termination reason: Unknown
% 3.06/1.78 % (11113)Termination phase: shuffling
% 3.06/1.78
% 3.06/1.78 % (11113)Memory used [KB]: 1663
% 3.06/1.78 % (11113)Time elapsed: 0.010 s
% 3.06/1.78 % (11113)Instructions burned: 19 (million)
% 3.06/1.78 % (11113)------------------------------
% 3.06/1.78 % (11113)------------------------------
% 3.06/1.80 % (11115)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_10986 for (2998ds/17Mi)
% 3.06/1.81 % (11115)Instruction limit reached!
% 3.06/1.81 % (11115)------------------------------
% 3.06/1.81 % (11115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.81 % (11115)Termination reason: Unknown
% 3.06/1.81 % (11115)Termination phase: Property scanning
% 3.06/1.81
% 3.06/1.81 % (11115)Memory used [KB]: 1791
% 3.06/1.81 % (11115)Time elapsed: 0.011 s
% 3.06/1.81 % (11115)Instructions burned: 18 (million)
% 3.06/1.81 % (11115)------------------------------
% 3.06/1.81 % (11115)------------------------------
% 3.06/1.82 % (11116)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_10986 for (2998ds/3Mi)
% 3.06/1.82 % (11116)Instruction limit reached!
% 3.06/1.82 % (11116)------------------------------
% 3.06/1.82 % (11116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.82 % (11116)Termination reason: Unknown
% 3.06/1.82 % (11116)Termination phase: shuffling
% 3.06/1.82
% 3.06/1.82 % (11116)Memory used [KB]: 1407
% 3.06/1.82 % (11116)Time elapsed: 0.003 s
% 3.06/1.82 % (11116)Instructions burned: 3 (million)
% 3.06/1.82 % (11116)------------------------------
% 3.06/1.82 % (11116)------------------------------
% 3.06/1.84 % (11117)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_10986 for (2998ds/30Mi)
% 3.06/1.85 % (11117)Instruction limit reached!
% 3.06/1.85 % (11117)------------------------------
% 3.06/1.85 % (11117)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.86 % (11117)Termination reason: Unknown
% 3.06/1.86 % (11117)Termination phase: Property scanning
% 3.06/1.86
% 3.06/1.86 % (11117)Memory used [KB]: 1791
% 3.06/1.86 % (11117)Time elapsed: 0.016 s
% 3.06/1.86 % (11117)Instructions burned: 30 (million)
% 3.06/1.86 % (11117)------------------------------
% 3.06/1.86 % (11117)------------------------------
% 3.06/1.87 % (11118)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_10986 for (2997ds/127Mi)
% 3.06/1.93 % (11118)Instruction limit reached!
% 3.06/1.93 % (11118)------------------------------
% 3.06/1.93 % (11118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.93 % (11118)Termination reason: Unknown
% 3.06/1.93 % (11118)Termination phase: Saturation
% 3.06/1.93
% 3.06/1.93 % (11118)Memory used [KB]: 6908
% 3.06/1.93 % (11118)Time elapsed: 0.088 s
% 3.06/1.93 % (11118)Instructions burned: 127 (million)
% 3.06/1.93 % (11118)------------------------------
% 3.06/1.93 % (11118)------------------------------
% 3.06/1.95 % (11119)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_10986 for (2997ds/100Mi)
% 3.06/1.96 % (11111)Instruction limit reached!
% 3.06/1.96 % (11111)------------------------------
% 3.06/1.96 % (11111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.96 % (11111)Termination reason: Unknown
% 3.06/1.96 % (11111)Termination phase: Saturation
% 3.06/1.96
% 3.06/1.96 % (11111)Memory used [KB]: 7547
% 3.06/1.96 % (11111)Time elapsed: 0.193 s
% 3.06/1.96 % (11111)Instructions burned: 378 (million)
% 3.06/1.96 % (11111)------------------------------
% 3.06/1.96 % (11111)------------------------------
% 3.06/1.98 % (11120)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_10986 for (2996ds/3Mi)
% 3.06/1.98 % (11120)Instruction limit reached!
% 3.06/1.98 % (11120)------------------------------
% 3.06/1.98 % (11120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/1.98 % (11120)Termination reason: Unknown
% 3.06/1.98 % (11120)Termination phase: shuffling
% 3.06/1.98
% 3.06/1.98 % (11120)Memory used [KB]: 1407
% 3.06/1.98 % (11120)Time elapsed: 0.003 s
% 3.06/1.98 % (11120)Instructions burned: 3 (million)
% 3.06/1.98 % (11120)------------------------------
% 3.06/1.98 % (11120)------------------------------
% 3.06/2.00 % (11121)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_10986 for (2996ds/20Mi)
% 3.06/2.00 % (11119)Instruction limit reached!
% 3.06/2.00 % (11119)------------------------------
% 3.06/2.00 % (11119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/2.00 % (11119)Termination reason: Unknown
% 3.06/2.00 % (11119)Termination phase: Property scanning
% 3.06/2.00
% 3.06/2.00 % (11119)Memory used [KB]: 2302
% 3.06/2.00 % (11119)Time elapsed: 0.047 s
% 3.06/2.00 % (11119)Instructions burned: 101 (million)
% 3.06/2.00 % (11119)------------------------------
% 3.06/2.00 % (11119)------------------------------
% 3.06/2.01 % (11121)Instruction limit reached!
% 3.06/2.01 % (11121)------------------------------
% 3.06/2.01 % (11121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.06/2.01 % (11121)Termination reason: Unknown
% 3.06/2.01 % (11121)Termination phase: Property scanning
% 3.06/2.01
% 3.06/2.01 % (11121)Memory used [KB]: 1663
% 3.06/2.01 % (11121)Time elapsed: 0.012 s
% 3.06/2.01 % (11121)Instructions burned: 21 (million)
% 3.06/2.01 % (11121)------------------------------
% 3.06/2.01 % (11121)------------------------------
% 3.06/2.01 % (11122)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_10986 for (2996ds/86Mi)
% 3.06/2.02 % (11123)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_10986 for (2996ds/107Mi)
% 5.58/2.05 % (11122)Instruction limit reached!
% 5.58/2.05 % (11122)------------------------------
% 5.58/2.05 % (11122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.05 % (11122)Termination reason: Unknown
% 5.58/2.05 % (11122)Termination phase: Saturation
% 5.58/2.05
% 5.58/2.05 % (11122)Memory used [KB]: 6652
% 5.58/2.05 % (11122)Time elapsed: 0.043 s
% 5.58/2.05 % (11122)Instructions burned: 86 (million)
% 5.58/2.05 % (11122)------------------------------
% 5.58/2.05 % (11122)------------------------------
% 5.58/2.07 % (11105)Instruction limit reached!
% 5.58/2.07 % (11105)------------------------------
% 5.58/2.07 % (11105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.07 % (11105)Termination reason: Unknown
% 5.58/2.07 % (11105)Termination phase: Saturation
% 5.58/2.07
% 5.58/2.07 % (11105)Memory used [KB]: 7803
% 5.58/2.07 % (11105)Time elapsed: 0.340 s
% 5.58/2.07 % (11105)Instructions burned: 710 (million)
% 5.58/2.07 % (11105)------------------------------
% 5.58/2.07 % (11105)------------------------------
% 5.58/2.07 % (11124)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_10986 for (2995ds/39Mi)
% 5.58/2.08 % (11123)Instruction limit reached!
% 5.58/2.08 % (11123)------------------------------
% 5.58/2.08 % (11123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.08 % (11123)Termination reason: Unknown
% 5.58/2.08 % (11123)Termination phase: Saturation
% 5.58/2.08
% 5.58/2.08 % (11123)Memory used [KB]: 6780
% 5.58/2.08 % (11123)Time elapsed: 0.055 s
% 5.58/2.08 % (11123)Instructions burned: 108 (million)
% 5.58/2.08 % (11123)------------------------------
% 5.58/2.08 % (11123)------------------------------
% 5.58/2.08 % (11125)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_10986 for (2995ds/448Mi)
% 5.58/2.09 % (11124)Instruction limit reached!
% 5.58/2.09 % (11124)------------------------------
% 5.58/2.09 % (11124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.09 % (11124)Termination reason: Unknown
% 5.58/2.09 % (11124)Termination phase: Property scanning
% 5.58/2.09
% 5.58/2.09 % (11124)Memory used [KB]: 2174
% 5.58/2.09 % (11124)Time elapsed: 0.020 s
% 5.58/2.09 % (11124)Instructions burned: 39 (million)
% 5.58/2.09 % (11124)------------------------------
% 5.58/2.09 % (11124)------------------------------
% 5.58/2.09 % (11126)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_10986 for (2995ds/46Mi)
% 5.58/2.10 % (11127)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_10986 for (2995ds/98Mi)
% 5.58/2.11 % (11126)Instruction limit reached!
% 5.58/2.11 % (11126)------------------------------
% 5.58/2.11 % (11126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.11 % (11126)Termination reason: Unknown
% 5.58/2.11 % (11126)Termination phase: Property scanning
% 5.58/2.11
% 5.58/2.11 % (11126)Memory used [KB]: 2046
% 5.58/2.11 % (11126)Time elapsed: 0.024 s
% 5.58/2.11 % (11126)Instructions burned: 46 (million)
% 5.58/2.11 % (11126)------------------------------
% 5.58/2.11 % (11126)------------------------------
% 6.24/2.13 % (11128)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_10986 for (2995ds/507Mi)
% 6.24/2.14 % (11112)Instruction limit reached!
% 6.24/2.14 % (11112)------------------------------
% 6.24/2.14 % (11112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.14 % (11112)Termination reason: Unknown
% 6.24/2.14 % (11112)Termination phase: Saturation
% 6.24/2.14
% 6.24/2.14 % (11112)Memory used [KB]: 8571
% 6.24/2.14 % (11112)Time elapsed: 0.369 s
% 6.24/2.14 % (11112)Instructions burned: 780 (million)
% 6.24/2.14 % (11112)------------------------------
% 6.24/2.14 % (11112)------------------------------
% 6.24/2.15 % (11127)Instruction limit reached!
% 6.24/2.15 % (11127)------------------------------
% 6.24/2.15 % (11127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.15 % (11127)Termination reason: Unknown
% 6.24/2.15 % (11127)Termination phase: Property scanning
% 6.24/2.15
% 6.24/2.15 % (11127)Memory used [KB]: 2302
% 6.24/2.15 % (11127)Time elapsed: 0.046 s
% 6.24/2.15 % (11127)Instructions burned: 99 (million)
% 6.24/2.15 % (11127)------------------------------
% 6.24/2.15 % (11127)------------------------------
% 6.24/2.16 % (11129)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_10986 for (2994ds/149Mi)
% 6.24/2.17 % (11130)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_10986 for (2994ds/39Mi)
% 6.24/2.18 % (11107)Instruction limit reached!
% 6.24/2.18 % (11107)------------------------------
% 6.24/2.18 % (11107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.18 % (11107)Termination reason: Unknown
% 6.24/2.18 % (11107)Termination phase: Saturation
% 6.24/2.18
% 6.24/2.18 % (11107)Memory used [KB]: 9210
% 6.24/2.18 % (11107)Time elapsed: 0.440 s
% 6.24/2.18 % (11107)Instructions burned: 903 (million)
% 6.24/2.18 % (11107)------------------------------
% 6.24/2.18 % (11107)------------------------------
% 6.24/2.18 % (11130)Instruction limit reached!
% 6.24/2.18 % (11130)------------------------------
% 6.24/2.18 % (11130)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.18 % (11130)Termination reason: Unknown
% 6.24/2.18 % (11130)Termination phase: Preprocessing 3
% 6.24/2.18
% 6.24/2.18 % (11130)Memory used [KB]: 2046
% 6.24/2.18 % (11130)Time elapsed: 0.021 s
% 6.24/2.18 % (11130)Instructions burned: 40 (million)
% 6.24/2.18 % (11130)------------------------------
% 6.24/2.18 % (11130)------------------------------
% 6.24/2.19 % (11131)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_10986 for (2994ds/6Mi)
% 6.56/2.20 % (11131)Instruction limit reached!
% 6.56/2.20 % (11131)------------------------------
% 6.56/2.20 % (11131)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.20 % (11131)Termination reason: Unknown
% 6.56/2.20 % (11131)Termination phase: shuffling
% 6.56/2.20
% 6.56/2.20 % (11131)Memory used [KB]: 1407
% 6.56/2.20 % (11131)Time elapsed: 0.005 s
% 6.56/2.20 % (11131)Instructions burned: 7 (million)
% 6.56/2.20 % (11131)------------------------------
% 6.56/2.20 % (11131)------------------------------
% 6.56/2.20 % (11114)Instruction limit reached!
% 6.56/2.20 % (11114)------------------------------
% 6.56/2.20 % (11114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.20 % (11114)Termination reason: Unknown
% 6.56/2.20 % (11114)Termination phase: Saturation
% 6.56/2.20
% 6.56/2.20 % (11114)Memory used [KB]: 7931
% 6.56/2.20 % (11114)Time elapsed: 0.446 s
% 6.56/2.20 % (11114)Instructions burned: 880 (million)
% 6.56/2.20 % (11114)------------------------------
% 6.56/2.20 % (11114)------------------------------
% 6.56/2.20 % (11132)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_10986 for (2994ds/1087Mi)
% 6.56/2.21 % (11096)Instruction limit reached!
% 6.56/2.21 % (11096)------------------------------
% 6.56/2.21 % (11096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.21 % (11096)Termination reason: Unknown
% 6.56/2.21 % (11096)Termination phase: Saturation
% 6.56/2.21
% 6.56/2.21 % (11096)Memory used [KB]: 9722
% 6.56/2.21 % (11096)Time elapsed: 0.516 s
% 6.56/2.21 % (11096)Instructions burned: 1041 (million)
% 6.56/2.21 % (11096)------------------------------
% 6.56/2.21 % (11096)------------------------------
% 6.56/2.21 % (11133)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_10986 for (2994ds/47Mi)
% 6.56/2.22 % (11134)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_10986 for (2994ds/44Mi)
% 6.56/2.22 % (11129)Instruction limit reached!
% 6.56/2.22 % (11129)------------------------------
% 6.56/2.22 % (11129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.22 % (11129)Termination reason: Unknown
% 6.56/2.22 % (11129)Termination phase: Property scanning
% 6.56/2.22
% 6.56/2.22 % (11129)Memory used [KB]: 3454
% 6.56/2.22 % (11129)Time elapsed: 0.068 s
% 6.56/2.22 % (11129)Instructions burned: 149 (million)
% 6.56/2.22 % (11129)------------------------------
% 6.56/2.22 % (11129)------------------------------
% 6.56/2.23 % (11135)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_10986 for (2994ds/432Mi)
% 6.56/2.24 % (11134)Instruction limit reached!
% 6.56/2.24 % (11134)------------------------------
% 6.56/2.24 % (11134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.24 % (11134)Termination reason: Unknown
% 6.56/2.24 % (11134)Termination phase: Property scanning
% 6.56/2.24
% 6.56/2.24 % (11134)Memory used [KB]: 2174
% 6.56/2.24 % (11134)Time elapsed: 0.022 s
% 6.56/2.24 % (11134)Instructions burned: 45 (million)
% 6.56/2.24 % (11134)------------------------------
% 6.56/2.24 % (11134)------------------------------
% 6.56/2.24 % (11133)Instruction limit reached!
% 6.56/2.24 % (11133)------------------------------
% 6.56/2.24 % (11133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.24 % (11133)Termination reason: Unknown
% 6.56/2.24 % (11133)Termination phase: Property scanning
% 6.56/2.24
% 6.56/2.24 % (11133)Memory used [KB]: 2046
% 6.56/2.24 % (11133)Time elapsed: 0.024 s
% 6.56/2.24 % (11133)Instructions burned: 47 (million)
% 6.56/2.24 % (11133)------------------------------
% 6.56/2.24 % (11133)------------------------------
% 6.56/2.24 % (11136)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_10986 for (2994ds/62Mi)
% 6.56/2.25 % (11137)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_10986 for (2994ds/31Mi)
% 6.56/2.25 % (11138)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_10986 for (2994ds/96Mi)
% 6.56/2.27 % (11136)Instruction limit reached!
% 6.56/2.27 % (11136)------------------------------
% 6.56/2.27 % (11136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.27 % (11136)Termination reason: Unknown
% 6.56/2.27 % (11136)Termination phase: Preprocessing 3
% 6.56/2.27
% 6.56/2.27 % (11136)Memory used [KB]: 2174
% 6.56/2.27 % (11136)Time elapsed: 0.030 s
% 6.56/2.27 % (11136)Instructions burned: 63 (million)
% 6.56/2.27 % (11136)------------------------------
% 6.56/2.27 % (11136)------------------------------
% 6.56/2.27 % (11137)Instruction limit reached!
% 6.56/2.27 % (11137)------------------------------
% 6.56/2.27 % (11137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.56/2.27 % (11137)Termination reason: Unknown
% 6.56/2.27 % (11137)Termination phase: shuffling
% 6.56/2.27
% 6.56/2.27 % (11137)Memory used [KB]: 1918
% 6.56/2.27 % (11137)Time elapsed: 0.017 s
% 6.56/2.27 % (11137)Instructions burned: 32 (million)
% 6.56/2.27 % (11137)------------------------------
% 6.56/2.27 % (11137)------------------------------
% 6.56/2.28 % (11139)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_10986 for (2993ds/453Mi)
% 6.56/2.28 % (11140)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_10986 for (2993ds/71Mi)
% 6.85/2.30 % (11125)Instruction limit reached!
% 6.85/2.30 % (11125)------------------------------
% 6.85/2.30 % (11125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.30 % (11125)Termination reason: Unknown
% 6.85/2.30 % (11125)Termination phase: Saturation
% 6.85/2.30
% 6.85/2.30 % (11125)Memory used [KB]: 8059
% 6.85/2.30 % (11125)Time elapsed: 0.215 s
% 6.85/2.30 % (11125)Instructions burned: 448 (million)
% 6.85/2.30 % (11125)------------------------------
% 6.85/2.30 % (11125)------------------------------
% 6.85/2.30 % (11138)Instruction limit reached!
% 6.85/2.30 % (11138)------------------------------
% 6.85/2.30 % (11138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.30 % (11138)Termination reason: Unknown
% 6.85/2.30 % (11138)Termination phase: Saturation
% 6.85/2.30
% 6.85/2.30 % (11138)Memory used [KB]: 6652
% 6.85/2.30 % (11138)Time elapsed: 0.047 s
% 6.85/2.30 % (11138)Instructions burned: 96 (million)
% 6.85/2.30 % (11138)------------------------------
% 6.85/2.30 % (11138)------------------------------
% 6.85/2.31 % (11141)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_10986 for (2993ds/83Mi)
% 6.85/2.31 % (11142)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_10986 for (2993ds/11Mi)
% 6.85/2.32 % (11140)Instruction limit reached!
% 6.85/2.32 % (11140)------------------------------
% 6.85/2.32 % (11140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.32 % (11140)Termination reason: Unknown
% 6.85/2.32 % (11140)Termination phase: Property scanning
% 6.85/2.32
% 6.85/2.32 % (11140)Memory used [KB]: 2046
% 6.85/2.32 % (11140)Time elapsed: 0.035 s
% 6.85/2.32 % (11140)Instructions burned: 71 (million)
% 6.85/2.32 % (11140)------------------------------
% 6.85/2.32 % (11140)------------------------------
% 6.85/2.32 % (11142)Instruction limit reached!
% 6.85/2.32 % (11142)------------------------------
% 6.85/2.32 % (11142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.32 % (11142)Termination reason: Unknown
% 6.85/2.32 % (11142)Termination phase: shuffling
% 6.85/2.32
% 6.85/2.32 % (11142)Memory used [KB]: 1663
% 6.85/2.32 % (11142)Time elapsed: 0.007 s
% 6.85/2.32 % (11142)Instructions burned: 12 (million)
% 6.85/2.32 % (11142)------------------------------
% 6.85/2.32 % (11142)------------------------------
% 6.85/2.33 % (11143)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_10986 for (2993ds/41Mi)
% 6.85/2.33 % (11144)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10986 for (2993ds/6Mi)
% 6.85/2.34 % (11144)Instruction limit reached!
% 6.85/2.34 % (11144)------------------------------
% 6.85/2.34 % (11144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.34 % (11144)Termination reason: Unknown
% 6.85/2.34 % (11144)Termination phase: shuffling
% 6.85/2.34
% 6.85/2.34 % (11144)Memory used [KB]: 1407
% 6.85/2.34 % (11144)Time elapsed: 0.005 s
% 6.85/2.34 % (11144)Instructions burned: 7 (million)
% 6.85/2.34 % (11144)------------------------------
% 6.85/2.34 % (11144)------------------------------
% 6.85/2.35 % (11143)Instruction limit reached!
% 6.85/2.35 % (11143)------------------------------
% 6.85/2.35 % (11143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.35 % (11143)Termination reason: Unknown
% 6.85/2.35 % (11143)Termination phase: Property scanning
% 6.85/2.35
% 6.85/2.35 % (11143)Memory used [KB]: 2174
% 6.85/2.35 % (11143)Time elapsed: 0.021 s
% 6.85/2.35 % (11143)Instructions burned: 43 (million)
% 6.85/2.35 % (11143)------------------------------
% 6.85/2.35 % (11143)------------------------------
% 6.85/2.35 % (11145)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_10986 for (2993ds/150Mi)
% 6.85/2.36 % (11141)Instruction limit reached!
% 6.85/2.36 % (11141)------------------------------
% 6.85/2.36 % (11141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.36 % (11141)Termination reason: Unknown
% 6.85/2.36 % (11141)Termination phase: Saturation
% 6.85/2.36
% 6.85/2.36 % (11141)Memory used [KB]: 6524
% 6.85/2.36 % (11141)Time elapsed: 0.044 s
% 6.85/2.36 % (11141)Instructions burned: 83 (million)
% 6.85/2.36 % (11141)------------------------------
% 6.85/2.36 % (11141)------------------------------
% 6.85/2.37 % (11146)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_10986 for (2992ds/343Mi)
% 6.85/2.37 % (11147)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_10986 for (2992ds/109Mi)
% 6.85/2.38 % (11128)Instruction limit reached!
% 6.85/2.38 % (11128)------------------------------
% 6.85/2.38 % (11128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.85/2.38 % (11128)Termination reason: Unknown
% 6.85/2.38 % (11128)Termination phase: Saturation
% 6.85/2.38
% 6.85/2.38 % (11128)Memory used [KB]: 8315
% 6.85/2.38 % (11128)Time elapsed: 0.253 s
% 6.85/2.38 % (11128)Instructions burned: 508 (million)
% 6.85/2.38 % (11128)------------------------------
% 6.85/2.38 % (11128)------------------------------
% 7.60/2.40 % (11148)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_10986 for (2992ds/253Mi)
% 7.60/2.42 % (11147)Instruction limit reached!
% 7.60/2.42 % (11147)------------------------------
% 7.60/2.42 % (11147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.60/2.42 % (11147)Termination reason: Unknown
% 7.60/2.42 % (11147)Termination phase: Saturation
% 7.60/2.42
% 7.60/2.42 % (11147)Memory used [KB]: 6780
% 7.60/2.42 % (11147)Time elapsed: 0.054 s
% 7.60/2.42 % (11147)Instructions burned: 109 (million)
% 7.60/2.42 % (11147)------------------------------
% 7.60/2.42 % (11147)------------------------------
% 7.60/2.43 % (11145)Instruction limit reached!
% 7.60/2.43 % (11145)------------------------------
% 7.60/2.43 % (11145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.60/2.43 % (11145)Termination reason: Unknown
% 7.60/2.43 % (11145)Termination phase: Saturation
% 7.60/2.43
% 7.60/2.43 % (11145)Memory used [KB]: 7036
% 7.60/2.43 % (11145)Time elapsed: 0.077 s
% 7.60/2.43 % (11145)Instructions burned: 151 (million)
% 7.60/2.43 % (11145)------------------------------
% 7.60/2.43 % (11145)------------------------------
% 7.60/2.44 % (11149)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_10986 for (2992ds/140Mi)
% 7.60/2.44 % (11135)Instruction limit reached!
% 7.60/2.44 % (11135)------------------------------
% 7.60/2.44 % (11135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.60/2.44 % (11135)Termination reason: Unknown
% 7.60/2.44 % (11135)Termination phase: Saturation
% 7.60/2.44
% 7.60/2.44 % (11135)Memory used [KB]: 3454
% 7.60/2.44 % (11135)Time elapsed: 0.216 s
% 7.60/2.44 % (11135)Instructions burned: 433 (million)
% 7.60/2.44 % (11135)------------------------------
% 7.60/2.44 % (11135)------------------------------
% 7.60/2.44 % (11148)First to succeed.
% 7.60/2.45 % (11148)Refutation found. Thanks to Tanya!
% 7.60/2.45 % SZS status Theorem for DTF2THF_10986
% 7.60/2.45 % SZS output start Proof for DTF2THF_10986
% 7.60/2.45 thf(type_def_5, type, itself: ($tType) > $tType).
% 7.60/2.45 thf(type_def_7, type, set: ($tType) > $tType).
% 7.60/2.45 thf(type_def_8, type, coindu1593790203_llist: ($tType) > $tType).
% 7.60/2.45 thf(type_def_9, type, extended_enat: $tType).
% 7.60/2.45 thf(type_def_10, type, nat: $tType).
% 7.60/2.45 thf(type_def_11, type, a: $tType).
% 7.60/2.45 thf(func_def_0, type, coindu1593790203_llist: ($tType) > $tType).
% 7.60/2.45 thf(func_def_1, type, extended_enat: $tType).
% 7.60/2.45 thf(func_def_2, type, set: ($tType) > $tType).
% 7.60/2.45 thf(func_def_3, type, nat: $tType).
% 7.60/2.45 thf(func_def_4, type, itself: ($tType) > $tType).
% 7.60/2.45 thf(func_def_5, type, a: $tType).
% 7.60/2.45 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_7, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_9, type, mult_zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_10, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_11, type, semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_12, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_13, type, ordered_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_14, type, semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_15, type, comm_semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_16, type, linordered_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_17, type, ordered_semiring: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_18, type, comm_monoid_diff: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_19, type, ab_semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_20, type, ordered_semiring_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_21, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_22, type, linord581940658strict: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_23, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_24, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_25, type, semiri1193490041visors: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_27, type, semiri1923998003cancel: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.60/2.45 thf(func_def_28, type, coindu1928975208n_lset: !>[X0: $tType]:(set @ X0 > coindu1593790203_llist @ X0 > set @ X0)).
% 7.60/2.45 thf(func_def_29, type, coindu1529956631erates: !>[X0: $tType]:((X0 > X0) > X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_30, type, coindu268472904append: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_31, type, coindu1213758845finite: !>[X0: $tType]:(coindu1593790203_llist @ X0 > $o)).
% 7.60/2.45 thf(func_def_32, type, coindu1018505716length: !>[X0: $tType]:(coindu1593790203_llist @ X0 > extended_enat)).
% 7.60/2.45 thf(func_def_33, type, coindu300403952lexord: !>[X0: $tType]:((X0 > X0 > $o) > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0 > $o)).
% 7.60/2.45 thf(func_def_34, type, coindu1121789889_LCons: !>[X0: $tType]:(X0 > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_35, type, coindu1598213697e_LNil: !>[X0: $tType]:(coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_36, type, coindu882539134_llist: !>[X0: $tType, X1: $tType]:(X0 > (X1 > coindu1593790203_llist @ X1 > X0) > coindu1593790203_llist @ X1 > X0)).
% 7.60/2.45 thf(func_def_37, type, coindu1244876290_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > $o) > (X0 > coindu1593790203_llist @ X1) > (X0 > X0) > X0 > coindu1593790203_llist @ X1)).
% 7.60/2.45 thf(func_def_38, type, coindu44415597_llist: !>[X0: $tType]:((X0 > $o) > coindu1593790203_llist @ X0 > $o)).
% 7.60/2.45 thf(func_def_39, type, coindu567634248member: !>[X0: $tType]:(X0 > coindu1593790203_llist @ X0 > $o)).
% 7.60/2.45 thf(func_def_40, type, coindu1055736764_lsetp: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0 > $o)).
% 7.60/2.45 thf(func_def_41, type, coindu1599971794_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coindu1593790203_llist @ X1)).
% 7.60/2.45 thf(func_def_42, type, coinductive_enat_set: set @ extended_enat).
% 7.60/2.45 thf(func_def_43, type, coindu530039314t_setp: extended_enat > $o).
% 7.60/2.45 thf(func_def_44, type, coindu1491768222unfold: !>[X0: $tType]:((X0 > $o) > (X0 > X0) > X0 > extended_enat)).
% 7.60/2.45 thf(func_def_45, type, extended_eSuc: extended_enat > extended_enat).
% 7.60/2.45 thf(func_def_46, type, extended_enat2: nat > extended_enat).
% 7.60/2.45 thf(func_def_47, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 7.60/2.45 thf(func_def_48, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 7.60/2.45 thf(func_def_49, type, extended_size_enat: extended_enat > nat).
% 7.60/2.45 thf(func_def_50, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 7.60/2.45 thf(func_def_51, type, extended_the_enat: extended_enat > nat).
% 7.60/2.45 thf(func_def_52, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 7.60/2.45 thf(func_def_53, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 7.60/2.45 thf(func_def_54, type, zero_zero: !>[X0: $tType]:(X0)).
% 7.60/2.45 thf(func_def_55, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 7.60/2.45 thf(func_def_56, type, suc: nat > nat).
% 7.60/2.45 thf(func_def_57, type, semiri532925092at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 7.60/2.45 thf(func_def_58, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 7.60/2.45 thf(func_def_59, type, nat_triangle: nat > nat).
% 7.60/2.45 thf(func_def_60, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 7.60/2.45 thf(func_def_61, type, type2: !>[X0: $tType]:(itself @ X0)).
% 7.60/2.45 thf(func_def_62, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 7.60/2.45 thf(func_def_63, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 7.60/2.45 thf(func_def_64, type, na: nat).
% 7.60/2.45 thf(func_def_66, type, xsa: coindu1593790203_llist @ a).
% 7.60/2.45 thf(func_def_82, type, sK0: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_83, type, sK1: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_84, type, sK2: !>[X0: $tType]:((coindu1593790203_llist @ X0 > $o) > X0)).
% 7.60/2.45 thf(func_def_85, type, sK3: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_86, type, sK4: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_87, type, sK5: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_88, type, sK6: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_89, type, sK7: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_90, type, sK8: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 7.60/2.45 thf(func_def_91, type, sK9: nat > nat).
% 7.60/2.45 thf(func_def_92, type, sK10: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 7.60/2.45 thf(func_def_93, type, sK11: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_94, type, sK12: nat > nat).
% 7.60/2.45 thf(func_def_95, type, sK13: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat > X0)).
% 7.60/2.45 thf(func_def_96, type, sK14: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > X0)).
% 7.60/2.45 thf(func_def_97, type, sK15: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat)).
% 7.60/2.45 thf(func_def_98, type, sK16: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_99, type, sK17: extended_enat > nat > nat).
% 7.60/2.45 thf(func_def_100, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 7.60/2.45 thf(func_def_101, type, sK19: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_102, type, sK20: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_103, type, sK21: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_104, type, sK22: extended_enat > extended_enat).
% 7.60/2.45 thf(func_def_105, type, sK23: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_106, type, sK24: (nat > nat > $o) > nat).
% 7.60/2.45 thf(func_def_107, type, sK25: extended_enat > extended_enat).
% 7.60/2.45 thf(func_def_108, type, sK26: (nat > $o) > nat).
% 7.60/2.45 thf(func_def_109, type, sK27: nat > nat).
% 7.60/2.45 thf(func_def_110, type, sK28: !>[X0: $tType]:(coindu1593790203_llist @ X0 > X0)).
% 7.60/2.45 thf(func_def_111, type, sK29: !>[X0: $tType]:(coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 7.60/2.45 thf(func_def_112, type, sK30: extended_enat > extended_enat).
% 7.60/2.45 thf(func_def_113, type, sK31: nat > extended_enat > nat).
% 7.60/2.45 thf(func_def_114, type, sK32: nat > nat).
% 7.60/2.45 thf(func_def_116, type, ph34: !>[X0: $tType]:(X0)).
% 7.60/2.45 thf(f1260,plain,(
% 7.60/2.45 $false),
% 7.60/2.45 inference(avatar_sat_refutation,[status(thm)],[f1118,f1119,f1247])).
% 7.60/2.45 thf(f1247,plain,(
% 7.60/2.45 ~spl33_2),
% 7.60/2.45 inference(avatar_contradiction_clause,[status(thm)],[f1246])).
% 7.60/2.45 thf(f1246,plain,(
% 7.60/2.45 $false | ~spl33_2),
% 7.60/2.45 inference(trivial_inequality_removal,[status(thm)],[f1212])).
% 7.60/2.45 thf(f1212,plain,(
% 7.60/2.45 (zero_zero @ nat != zero_zero @ nat) | ~spl33_2),
% 7.60/2.45 inference(superposition,[status(thm)],[f1024,f1181])).
% 7.60/2.45 thf(f1181,plain,(
% 7.60/2.45 ((suc @ na) = zero_zero @ nat) | ~spl33_2),
% 7.60/2.45 inference(trivial_inequality_removal,[status(thm)],[f1174])).
% 7.60/2.45 thf(f1174,plain,(
% 7.60/2.45 ((suc @ na) = zero_zero @ nat) | (zero_zero @ extended_enat != zero_zero @ extended_enat) | ~spl33_2),
% 7.60/2.45 inference(superposition,[status(thm)],[f1005,f1159])).
% 7.60/2.45 thf(f1159,plain,(
% 7.60/2.45 ((extended_enat2 @ (suc @ na)) = zero_zero @ extended_enat) | ~spl33_2),
% 7.60/2.45 inference(backward_demodulation,[status(thm)],[f969,f1138])).
% 7.60/2.45 thf(f1138,plain,(
% 7.60/2.45 ((coindu1018505716length @ a @ xsa) = zero_zero @ extended_enat) | ~spl33_2),
% 7.60/2.45 inference(superposition,[status(thm)],[f930,f1128])).
% 7.60/2.45 thf(f1128,plain,(
% 7.60/2.45 (xsa = coindu1598213697e_LNil @ a) | ~spl33_2),
% 7.60/2.45 inference(equality_resolution,[status(thm)],[f1123])).
% 7.60/2.45 thf(f1123,plain,(
% 7.60/2.45 ( ! [X0 : coindu1593790203_llist @ a] : ((xsa != X0) | (coindu1598213697e_LNil @ a = X0)) ) | ~spl33_2),
% 7.60/2.45 inference(superposition,[status(thm)],[f1117,f947])).
% 7.60/2.45 thf(f947,plain,(
% 7.60/2.45 ( ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : ((coindu1598213697e_LNil @ X0 = X1) | ((coindu1121789889_LCons @ X0 @ (sK10 @ X0 @ X1) @ (sK11 @ X0 @ X1)) = X1)) )),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f841])).
% 7.60/2.45 thf(f841,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : (((coindu1598213697e_LNil @ X0 != X1) | ! [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) != X1)) & (((coindu1121789889_LCons @ X0 @ (sK10 @ X0 @ X1) @ (sK11 @ X0 @ X1)) = X1) | (coindu1598213697e_LNil @ X0 = X1)))),
% 7.60/2.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f839,f840])).
% 7.60/2.45 thf(f840,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : (? [X4 : X0,X5 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X4 @ X5) = X1) => ((coindu1121789889_LCons @ X0 @ (sK10 @ X0 @ X1) @ (sK11 @ X0 @ X1)) = X1))),
% 7.60/2.45 introduced(definition,[],[choice_axiom])).
% 7.60/2.45 thf(f839,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : (((coindu1598213697e_LNil @ X0 != X1) | ! [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) != X1)) & (? [X4 : X0,X5 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X4 @ X5) = X1) | (coindu1598213697e_LNil @ X0 = X1)))),
% 7.60/2.45 inference(rectify,[status(thm)],[f838])).
% 7.60/2.45 thf(f838,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : (((coindu1598213697e_LNil @ X0 != X1) | ! [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) != X1)) & (? [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) = X1) | (coindu1598213697e_LNil @ X0 = X1)))),
% 7.60/2.45 inference(nnf_transformation,[status(thm)],[f600])).
% 7.60/2.45 thf(f600,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : ((coindu1598213697e_LNil @ X0 != X1) <=> ? [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) = X1))),
% 7.60/2.45 inference(fool_elimination,[status(thm)],[f599])).
% 7.60/2.45 thf(f599,plain,(
% 7.60/2.45 ! [X0 : $tType,X1 : coindu1593790203_llist @ X0] : (? [X2 : X0,X3 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X2 @ X3) = X1) = (coindu1598213697e_LNil @ X0 != X1))),
% 7.60/2.45 inference(rectify,[status(thm)],[f36])).
% 7.60/2.45 thf(f36,axiom,(
% 7.60/2.45 ! [X0 : $tType,X17 : coindu1593790203_llist @ X0] : (? [X12 : X0,X59 : coindu1593790203_llist @ X0] : ((coindu1121789889_LCons @ X0 @ X12 @ X59) = X17) = (coindu1598213697e_LNil @ X0 != X17))),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',fact_35_neq__LNil__conv)).
% 7.60/2.45 thf(f1117,plain,(
% 7.60/2.45 ( ! [X0 : a,X1 : coindu1593790203_llist @ a] : ((xsa != (coindu1121789889_LCons @ a @ X0 @ X1))) ) | ~spl33_2),
% 7.60/2.45 inference(avatar_component_clause,[status(thm)],[f1116])).
% 7.60/2.45 thf(f1116,definition,(
% 7.60/2.45 spl33_2 <=> ! [X0 : a,X1 : coindu1593790203_llist @ a] : (xsa != (coindu1121789889_LCons @ a @ X0 @ X1))),
% 7.60/2.45 introduced(definition,[new_symbols(naming,[spl33_2])],[avatar_definition])).
% 7.60/2.45 thf(f930,plain,(
% 7.60/2.45 ( ! [X0 : $tType] : (((coindu1018505716length @ X0 @ coindu1598213697e_LNil @ X0) = zero_zero @ extended_enat)) )),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f55])).
% 7.60/2.45 thf(f55,axiom,(
% 7.60/2.45 ! [X0 : $tType] : ((coindu1018505716length @ X0 @ coindu1598213697e_LNil @ X0) = zero_zero @ extended_enat)),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',fact_54_llength__LNil)).
% 7.60/2.45 thf(f969,plain,(
% 7.60/2.45 ((coindu1018505716length @ a @ xsa) = (extended_enat2 @ (suc @ na)))),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f1])).
% 7.60/2.45 thf(f1,axiom,(
% 7.60/2.45 ((coindu1018505716length @ a @ xsa) = (extended_enat2 @ (suc @ na)))),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',fact_0_len)).
% 7.60/2.45 thf(f1005,plain,(
% 7.60/2.45 ( ! [X0 : nat] : ((zero_zero @ extended_enat != (extended_enat2 @ X0)) | (zero_zero @ nat = X0)) )),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f867])).
% 7.60/2.45 thf(f867,plain,(
% 7.60/2.45 ! [X0 : nat] : (((zero_zero @ nat = X0) | (zero_zero @ extended_enat != (extended_enat2 @ X0))) & ((zero_zero @ extended_enat = (extended_enat2 @ X0)) | (zero_zero @ nat != X0)))),
% 7.60/2.45 inference(nnf_transformation,[status(thm)],[f472])).
% 7.60/2.45 thf(f472,plain,(
% 7.60/2.45 ! [X0 : nat] : ((zero_zero @ nat = X0) <=> (zero_zero @ extended_enat = (extended_enat2 @ X0)))),
% 7.60/2.45 inference(fool_elimination,[status(thm)],[f471])).
% 7.60/2.45 thf(f471,plain,(
% 7.60/2.45 ! [X0 : nat] : ((zero_zero @ nat = X0) = (zero_zero @ extended_enat = (extended_enat2 @ X0)))),
% 7.60/2.45 inference(rectify,[status(thm)],[f105])).
% 7.60/2.45 thf(f105,axiom,(
% 7.60/2.45 ! [X8 : nat] : ((zero_zero @ nat = X8) = (zero_zero @ extended_enat = (extended_enat2 @ X8)))),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',fact_104_enat__0__iff_I2_J)).
% 7.60/2.45 thf(f1024,plain,(
% 7.60/2.45 ( ! [X0 : nat] : ((zero_zero @ nat != (suc @ X0))) )),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f722])).
% 7.60/2.45 thf(f722,plain,(
% 7.60/2.45 ! [X0 : nat] : (zero_zero @ nat != (suc @ X0))),
% 7.60/2.45 inference(rectify,[status(thm)],[f90])).
% 7.60/2.45 thf(f90,axiom,(
% 7.60/2.45 ! [X55 : nat] : (zero_zero @ nat != (suc @ X55))),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',fact_89_Zero__not__Suc)).
% 7.60/2.45 thf(f1119,plain,(
% 7.60/2.45 ~spl33_1),
% 7.60/2.45 inference(avatar_split_clause,[status(thm)],[f962,f1112])).
% 7.60/2.45 thf(f1112,definition,(
% 7.60/2.45 spl33_1 <=> (thesis = $true)),
% 7.60/2.45 introduced(definition,[new_symbols(naming,[spl33_1])],[avatar_definition])).
% 7.60/2.45 thf(f962,plain,(
% 7.60/2.45 (thesis != $true)),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f701])).
% 7.60/2.45 thf(f701,plain,(
% 7.60/2.45 (thesis != $true)),
% 7.60/2.45 inference(flattening,[status(thm)],[f500])).
% 7.60/2.45 thf(f500,plain,(
% 7.60/2.45 ~(thesis = $true)),
% 7.60/2.45 inference(fool_elimination,[status(thm)],[f499])).
% 7.60/2.45 thf(f499,plain,(
% 7.60/2.45 ~thesis),
% 7.60/2.45 inference(rectify,[status(thm)],[f289])).
% 7.60/2.45 thf(f289,negated_conjecture,(
% 7.60/2.45 ~thesis),
% 7.60/2.45 inference(negated_conjecture,[status(cth)],[f288])).
% 7.60/2.45 thf(f288,conjecture,(
% 7.60/2.45 thesis),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',conj_1)).
% 7.60/2.45 thf(f1118,plain,(
% 7.60/2.45 spl33_1 | spl33_2),
% 7.60/2.45 inference(avatar_split_clause,[status(thm)],[f997,f1116,f1112])).
% 7.60/2.45 thf(f997,plain,(
% 7.60/2.45 ( ! [X0 : a,X1 : coindu1593790203_llist @ a] : ((thesis = $true) | (xsa != (coindu1121789889_LCons @ a @ X0 @ X1))) )),
% 7.60/2.45 inference(cnf_transformation,[status(thm)],[f753])).
% 7.60/2.45 thf(f753,plain,(
% 7.60/2.45 ! [X0 : a,X1 : coindu1593790203_llist @ a] : ((xsa != (coindu1121789889_LCons @ a @ X0 @ X1)) | (thesis = $true))),
% 7.60/2.45 inference(ennf_transformation,[status(thm)],[f342])).
% 7.60/2.45 thf(f342,plain,(
% 7.60/2.45 ! [X0 : a,X1 : coindu1593790203_llist @ a] : ((xsa = (coindu1121789889_LCons @ a @ X0 @ X1)) => (thesis = $true))),
% 7.60/2.45 inference(fool_elimination,[status(thm)],[f341])).
% 7.60/2.45 thf(f341,plain,(
% 7.60/2.45 ! [X0 : a,X1 : coindu1593790203_llist @ a] : ((xsa = (coindu1121789889_LCons @ a @ X0 @ X1)) => thesis)),
% 7.60/2.45 inference(rectify,[status(thm)],[f287])).
% 7.60/2.45 thf(f287,axiom,(
% 7.60/2.45 ! [X119 : a,X120 : coindu1593790203_llist @ a] : ((xsa = (coindu1121789889_LCons @ a @ X119 @ X120)) => thesis)),
% 7.60/2.45 file('/export/starexec/sandbox/tmp/tmp.oI3INOMot7/DTF2THF_10986.p',conj_0)).
% 7.60/2.45 % SZS output end Proof for DTF2THF_10986
% 7.60/2.45 % (11148)------------------------------
% 7.60/2.45 % (11148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.60/2.45 % (11148)Termination reason: Refutation
% 7.60/2.45
% 7.60/2.45 % (11148)Memory used [KB]: 6780
% 7.60/2.45 % (11148)Time elapsed: 0.047 s
% 7.60/2.45 % (11148)Instructions burned: 86 (million)
% 7.60/2.45 % (11148)------------------------------
% 7.60/2.45 % (11148)------------------------------
% 7.60/2.45 % (11084)Success in time 0.799 s
%------------------------------------------------------------------------------