%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP134^2 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:06:53 AM UTC 2026
% Result : Theorem 112.92s 90.47s
% Output : Refutation 113.79s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 290
% Syntax : Number of formulae : 582 ( 139 unt; 0 typ; 0 def)
% Number of atoms : 1951 ( 434 equ; 8 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 6186 ( 112 ~; 30 |; 114 &;5064 @)
% ( 0 <=>; 866 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 199 ( 199 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 42 usr; 5 con; 0-3 aty)
% Number of variables : 1791 ( 159 ^;1604 !; 28 ?;1791 :)
% Comments :
%------------------------------------------------------------------------------
thf(nat_type,type,
nat: $tType ).
thf(one_decl,type,
one:
!>[TA: $tType] : $o ).
thf(zero_decl,type,
zero:
!>[TA: $tType] : $o ).
thf(ord_decl,type,
ord:
!>[TA: $tType] : $o ).
thf(neg_numeral_decl,type,
neg_numeral:
!>[TA: $tType] : $o ).
thf(order_decl,type,
order:
!>[TA: $tType] : $o ).
thf(group_add_decl,type,
group_add:
!>[TA: $tType] : $o ).
thf(finite_finite_decl,type,
finite_finite:
!>[TA: $tType] : $o ).
thf(monoid_add_decl,type,
monoid_add:
!>[TA: $tType] : $o ).
thf(linorder_decl,type,
linorder:
!>[TA: $tType] : $o ).
thf(preorder_decl,type,
preorder:
!>[TA: $tType] : $o ).
thf(zero_neq_one_decl,type,
zero_neq_one:
!>[TA: $tType] : $o ).
thf(wellorder_decl,type,
wellorder:
!>[TA: $tType] : $o ).
thf(semigroup_add_decl,type,
semigroup_add:
!>[TA: $tType] : $o ).
thf(linordered_idom_decl,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(comm_monoid_add_decl,type,
comm_monoid_add:
!>[TA: $tType] : $o ).
thf(ab_semigroup_add_decl,type,
ab_semigroup_add:
!>[TA: $tType] : $o ).
thf(dense_linorder_decl,type,
dense_linorder:
!>[TA: $tType] : $o ).
thf(cancel_semigroup_add_decl,type,
cancel_semigroup_add:
!>[TA: $tType] : $o ).
thf(cancel1352612707id_add_decl,type,
cancel1352612707id_add:
!>[TA: $tType] : $o ).
thf(linord219039673up_add_decl,type,
linord219039673up_add:
!>[TA: $tType] : $o ).
thf(ordere216010020id_add_decl,type,
ordere216010020id_add:
!>[TA: $tType] : $o ).
thf(ordere779506340up_add_decl,type,
ordere779506340up_add:
!>[TA: $tType] : $o ).
thf(linord1659791738miring_decl,type,
linord1659791738miring:
!>[TA: $tType] : $o ).
thf(canoni770627133id_add_decl,type,
canoni770627133id_add:
!>[TA: $tType] : $o ).
thf(strict797366125id_add_decl,type,
strict797366125id_add:
!>[TA: $tType] : $o ).
thf(ordere236663937imp_le_decl,type,
ordere236663937imp_le:
!>[TA: $tType] : $o ).
thf(ordere223160158up_add_decl,type,
ordere223160158up_add:
!>[TA: $tType] : $o ).
thf(ordere516151231imp_le_decl,type,
ordere516151231imp_le:
!>[TA: $tType] : $o ).
thf(semiri456707255roduct_decl,type,
semiri456707255roduct:
!>[TA: $tType] : $o ).
thf(finite_Fpow_decl,type,
finite_Fpow:
!>[TA: $tType] : ( ( set @ TA ) > ( set @ ( set @ TA ) ) ) ).
thf(finite_finite2_decl,type,
finite_finite2:
!>[TA: $tType] : ( ( set @ TA ) > $o ) ).
thf(one_one_decl,type,
one_one:
!>[TA: $tType] : TA ).
thf(plus_plus_decl,type,
plus_plus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(zero_zero_decl,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(neg_numeral_dbl_inc_decl,type,
neg_numeral_dbl_inc:
!>[TA: $tType] : ( TA > TA ) ).
thf(number2016821345itions_decl,type,
number2016821345itions: ( nat > nat ) > nat > $o ).
thf(ord_less_decl,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_decl,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_decl,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(member_decl,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(n_decl,type,
n: nat ).
thf(262,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_le0) ).
thf(1024,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(22,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_add__less__cancel__left) ).
thf(347,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(206,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( finite_finite2 @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_rev__finite__subset) ).
thf(893,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( finite_finite2 @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(289,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ C @ B )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ D @ C )
=> ( C = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_finite__has__minimal2) ).
thf(1110,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ C @ B )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ D @ C )
=> ( C = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(213,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_Collect__mono__iff) ).
thf(909,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(17,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_add__nonneg__eq__0__iff) ).
thf(333,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(278,axiom,
one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oone) ).
thf(1083,plain,
one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(65,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_less__imp__le) ).
thf(468,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(169,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_add__mono) ).
thf(777,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(94,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ B ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_zero__eq__add__iff__both__eq__0) ).
thf(555,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ B ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(136,axiom,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( neg_numeral_dbl_inc @ TA )
= ( ^ [A: TA] : ( plus_plus @ TA @ ( plus_plus @ TA @ A @ A ) @ ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_dbl__inc__def) ).
thf(683,plain,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( neg_numeral_dbl_inc @ TA )
= ( ^ [A: TA] : ( plus_plus @ TA @ ( plus_plus @ TA @ A @ A ) @ ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(9,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_ord__le__eq__trans) ).
thf(308,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(273,axiom,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_not__finite__existsD) ).
thf(1073,plain,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(198,axiom,
! [A: nat > $o,B: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] :
( ( A @ C )
& ( ord_less @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_finite__M__bounded__by__nat) ).
thf(873,plain,
! [A: nat > $o,B: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] :
( ( A @ C )
& ( ord_less @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(134,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_add__nonneg__nonneg) ).
thf(677,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(200,axiom,
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] :
( ! [B: nat] : ( ord_less_eq @ nat @ ( A @ B ) @ n )
& ! [B: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ B )
=> ( ( A @ B )
= ( zero_zero @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062) ).
thf(878,plain,
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] :
( ! [B: nat] : ( ord_less_eq @ nat @ ( A @ B ) @ n )
& ! [B: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ B )
=> ( ( A @ B )
= ( zero_zero @ nat ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(54,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( zero @ TB )
=> ! [A: ( TC > TB ) > TA,B: TA] :
( ( A
= ( ^ [C: TC > TB] : B ) )
=> ( ( A
@ ^ [C: TC] : ( zero_zero @ TB ) )
= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_fun__cong__unused__0) ).
thf(440,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( zero @ TB )
=> ! [A: ( TC > TB ) > TA,B: TA] :
( ( A
= ( ^ [C: TC > TB] : B ) )
=> ( ( A
@ ^ [C: TC] : ( zero_zero @ TB ) )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(276,axiom,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_finite__Collect__le__nat) ).
thf(1078,plain,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(85,axiom,
preorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_8) ).
thf(533,plain,
preorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(92,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_subsetD) ).
thf(550,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(73,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_le__cases) ).
thf(500,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(67,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_less__eq__set__def) ).
thf(472,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(222,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_bot__nat__0_Onot__eq__extremum) ).
thf(930,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(270,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_bot__nat__0_Oextremum__unique) ).
thf(1063,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(19,axiom,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_is__num__normalize_I1_J) ).
thf(340,plain,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(187,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_mem__Collect__eq) ).
thf(829,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(251,axiom,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( finite_finite2 @ ( set @ TA )
@ ( collect @ ( set @ TA )
@ ^ [B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_finite__Collect__subsets) ).
thf(1001,plain,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( finite_finite2 @ ( set @ TA )
@ ( collect @ ( set @ TA )
@ ^ [B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(46,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_ord__eq__le__trans) ).
thf(417,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(127,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
| ~ ( ord_less_eq @ TA @ A @ B )
| ~ ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_verit__la__disequality) ).
thf(651,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
| ~ ( ord_less_eq @ TA @ A @ B )
| ~ ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(13,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_linear) ).
thf(320,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(98,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_eq__refl) ).
thf(565,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(35,axiom,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( neg_numeral_dbl_inc @ TA @ ( zero_zero @ TA ) )
= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_dbl__inc__simps_I2_J) ).
thf(389,plain,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( neg_numeral_dbl_inc @ TA @ ( zero_zero @ TA ) )
= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(151,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_dual__order_Otrans) ).
thf(724,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(149,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ B @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_add__decreasing2) ).
thf(719,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ B @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(173,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_add_Ocommute) ).
thf(785,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(75,axiom,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_add_Oassoc) ).
thf(504,plain,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(26,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_add__mono__thms__linordered__semiring_I2_J) ).
thf(357,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(228,axiom,
ordere223160158up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(949,plain,
ordere223160158up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(89,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_subset__trans) ).
thf(542,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(104,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_le__imp__less__or__eq) ).
thf(587,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(116,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( A != B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_less__le) ).
thf(619,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( A != B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(47,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_double__zero__sym) ).
thf(421,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= ( plus_plus @ TA @ A @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(237,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_nat__le__linear) ).
thf(969,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(166,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_zero__le) ).
thf(770,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(202,axiom,
! [A: nat,B: nat] :
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [C: nat > nat] :
( ! [D: nat] : ( ord_less_eq @ nat @ ( C @ D ) @ A )
& ! [D: nat] :
( ( ord_less_eq @ nat @ B @ D )
=> ( ( C @ D )
= ( zero_zero @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_bound__domain__and__range__impl__finitely__many__functions) ).
thf(881,plain,
! [A: nat,B: nat] :
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [C: nat > nat] :
( ! [D: nat] : ( ord_less_eq @ nat @ ( C @ D ) @ A )
& ! [D: nat] :
( ( ord_less_eq @ nat @ B @ D )
=> ( ( C @ D )
= ( zero_zero @ nat ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(57,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_verit__comp__simplify1_I1_J) ).
thf(448,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(68,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_le__add__same__cancel1) ).
thf(475,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(161,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_subset__iff) ).
thf(756,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(264,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_trans__le__add1) ).
thf(1040,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(125,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_subset__refl) ).
thf(646,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(24,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_order__refl) ).
thf(353,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(183,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_le__less__linear) ).
thf(821,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(70,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_dual__order_Orefl) ).
thf(480,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(72,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less_eq @ TA @ B @ C ) )
=> ( ( ( ord_less_eq @ TA @ B @ A )
=> ~ ( ord_less_eq @ TA @ A @ C ) )
=> ( ( ( ord_less_eq @ TA @ A @ C )
=> ~ ( ord_less_eq @ TA @ C @ B ) )
=> ( ( ( ord_less_eq @ TA @ C @ B )
=> ~ ( ord_less_eq @ TA @ B @ A ) )
=> ( ( ( ord_less_eq @ TA @ B @ C )
=> ~ ( ord_less_eq @ TA @ C @ A ) )
=> ~ ( ( ord_less_eq @ TA @ C @ A )
=> ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_le__cases3) ).
thf(485,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less_eq @ TA @ B @ C ) )
=> ( ( ( ord_less_eq @ TA @ B @ A )
=> ~ ( ord_less_eq @ TA @ A @ C ) )
=> ( ( ( ord_less_eq @ TA @ A @ C )
=> ~ ( ord_less_eq @ TA @ C @ B ) )
=> ( ( ( ord_less_eq @ TA @ C @ B )
=> ~ ( ord_less_eq @ TA @ B @ A ) )
=> ( ( ( ord_less_eq @ TA @ B @ C )
=> ~ ( ord_less_eq @ TA @ C @ A ) )
=> ~ ( ( ord_less_eq @ TA @ C @ A )
=> ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(189,axiom,
ordere236663937imp_le @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(834,plain,
ordere236663937imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(121,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_not__one__le__zero) ).
thf(634,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(3,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( TA > $o ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_predicate1I) ).
thf(293,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( TA > $o ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(38,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
= ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_double__add__le__zero__iff__single__add__le__zero) ).
thf(396,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
= ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(49,axiom,
! [TA: $tType] :
( ( strict797366125id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_pos__add__strict) ).
thf(428,plain,
! [TA: $tType] :
( ( strict797366125id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(287,axiom,
! [A: nat > nat] :
( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
= ( A
= ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_partitions__zero) ).
thf(1104,plain,
! [A: nat > nat] :
( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
= ( A
= ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(76,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
= ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_add__le__same__cancel2) ).
thf(507,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
= ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(10,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ! [C: TA] :
( B
!= ( plus_plus @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_less__eqE) ).
thf(312,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ! [C: TA] :
( B
!= ( plus_plus @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(1,conjecture,
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] : ( number2016821345itions @ A @ n ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
~ ( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] : ( number2016821345itions @ A @ n ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(292,plain,
~ ( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] : ( number2016821345itions @ A @ n ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(59,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_add__left__cancel) ).
thf(452,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(217,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_add__leD1) ).
thf(920,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(33,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_add__nonpos__eq__0__iff) ).
thf(380,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(226,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA > $o,D: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
=> ( D @ E ) ) )
=> ( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E ) ) )
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ B )
& ( D @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_subset__CollectI) ).
thf(942,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA > $o,D: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ( C @ E )
=> ( D @ E ) ) )
=> ( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E ) ) )
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ B )
& ( D @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(163,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_le__less) ).
thf(761,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(86,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_add__nonpos__nonpos) ).
thf(534,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(178,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ B @ ( plus_plus @ TA @ C @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_add__increasing2) ).
thf(798,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ B @ ( plus_plus @ TA @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(281,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_le__refl) ).
thf(1091,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(230,axiom,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_Collect__restrict) ).
thf(952,plain,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(7,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_add__less__same__cancel2) ).
thf(302,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(108,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_antisym) ).
thf(598,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(36,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_add__le__imp__le__left) ).
thf(392,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(4,axiom,
order @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_11) ).
thf(297,plain,
order @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(48,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_equalityD2) ).
thf(424,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(174,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less_eq @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_not__le) ).
thf(788,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less_eq @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(153,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_add__mono__thms__linordered__semiring_I3_J) ).
thf(729,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(216,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ~ ( ( ord_less_eq @ nat @ A @ C )
=> ~ ( ord_less_eq @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_add__leE) ).
thf(916,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ~ ( ( ord_less_eq @ nat @ A @ C )
=> ~ ( ord_less_eq @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(27,axiom,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).
thf(361,plain,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(201,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_finite) ).
thf(879,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(52,axiom,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_one__reorient) ).
thf(435,plain,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(131,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_add__right__imp__eq) ).
thf(665,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(81,axiom,
! [TA: $tType] :
( ( semiri456707255roduct @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ A @ B ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_add__0__iff) ).
thf(522,plain,
! [TA: $tType] :
( ( semiri456707255roduct @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ A @ B ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(63,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
= ( B = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_antisym__conv) ).
thf(461,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
= ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(146,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( plus_plus @ TA @ A @ A )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_double__zero) ).
thf(710,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ( plus_plus @ TA @ A @ A )
= ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(258,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_add__leD2) ).
thf(1017,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(41,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_eq__iff) ).
thf(403,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(253,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_less__imp__le__nat) ).
thf(1006,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(106,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_add__left__mono) ).
thf(593,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(113,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ A @ B )
= ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_antisym__conv1) ).
thf(610,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ A @ B )
= ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(246,axiom,
zero_neq_one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ozero__neq__one) ).
thf(989,plain,
zero_neq_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(193,axiom,
ordere779506340up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(857,plain,
ordere779506340up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(80,axiom,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_zero__neq__one) ).
thf(519,plain,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(102,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_measure__induct__rule) ).
thf(578,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(204,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_pigeonhole__infinite__rel) ).
thf(884,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(223,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_Collect__mem__eq) ).
thf(933,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(105,axiom,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_set__eq__subset) ).
thf(590,plain,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(135,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_dense__le) ).
thf(679,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(91,axiom,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_add_Oleft__neutral) ).
thf(547,plain,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(145,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_order__le__less__subst1) ).
thf(706,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(210,axiom,
! [TA: $tType,A: set @ TA,B: TA > $o,C: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A
@ ( collect @ TA
@ ^ [D: TA] :
( ( B @ D )
& ( C @ D ) ) ) )
= ( ( ord_less_eq @ ( set @ TA ) @ A @ ( collect @ TA @ B ) )
& ( ord_less_eq @ ( set @ TA ) @ A @ ( collect @ TA @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_conj__subset__def) ).
thf(901,plain,
! [TA: $tType,A: set @ TA,B: TA > $o,C: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A
@ ( collect @ TA
@ ^ [D: TA] :
( ( B @ D )
& ( C @ D ) ) ) )
= ( ( ord_less_eq @ ( set @ TA ) @ A @ ( collect @ TA @ B ) )
& ( ord_less_eq @ ( set @ TA ) @ A @ ( collect @ TA @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(137,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_add__mono__thms__linordered__field_I3_J) ).
thf(686,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(274,axiom,
ordere516151231imp_le @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le) ).
thf(1075,plain,
ordere516151231imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(117,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_dual__order_Ostrict__implies__order) ).
thf(622,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(244,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_le__add1) ).
thf(985,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(155,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_order_Ostrict__implies__order) ).
thf(738,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(128,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_order__less__le__subst2) ).
thf(654,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(61,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_order_Ostrict__trans2) ).
thf(457,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(150,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( ord_less_eq @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB] :
! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_le__fun__def) ).
thf(721,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( ord_less_eq @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB] :
! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(99,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_verit__sum__simplify) ).
thf(569,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(170,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
| ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_add__less__zeroD) ).
thf(779,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
| ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(29,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_add__right__mono) ).
thf(371,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(20,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_add__less__le__mono) ).
thf(343,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(268,axiom,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( A @ D @ E )
= ( A @ E @ D ) )
=> ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ D @ ( plus_plus @ nat @ D @ E ) ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_Euclid__induct) ).
thf(1054,plain,
! [A: nat > nat > $o,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( A @ D @ E )
= ( A @ E @ D ) )
=> ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
=> ( ! [D: nat,E: nat] :
( ( A @ D @ E )
=> ( A @ D @ ( plus_plus @ nat @ D @ E ) ) )
=> ( A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(95,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_add__mono__thms__linordered__field_I4_J) ).
thf(558,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(263,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ~ ! [D: nat] :
( ( A @ D )
=> ~ ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_bounded__Max__nat) ).
thf(1026,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ~ ! [D: nat] :
( ( A @ D )
=> ~ ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(84,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_add__le__cancel__right) ).
thf(530,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(176,axiom,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Finite__Set_Ofinite) ).
thf(794,plain,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(130,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_not__gr__zero) ).
thf(662,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(220,axiom,
semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).
thf(926,plain,
semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(212,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_add__le__mono1) ).
thf(906,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(5,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ( ord_less_eq @ ( TA > $o ) @ A @ C )
=> ( C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_rev__predicate1D) ).
thf(298,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ( ord_less_eq @ ( TA > $o ) @ A @ C )
=> ( C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(44,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_add__neg__neg) ).
thf(412,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(159,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_add__le__imp__le__right) ).
thf(750,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(192,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(856,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(158,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ C @ D ) ) )
=> ( ord_less_eq @ TA @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_dense__ge__bounded) ).
thf(745,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ C @ D ) ) )
=> ( ord_less_eq @ TA @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(55,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Finite__Set_Ofinite_5) ).
thf(444,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(109,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_subset__eq) ).
thf(601,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(78,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_order__class_Oorder_Oantisym) ).
thf(513,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(50,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_add__le__cancel__left) ).
thf(430,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(31,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_comm__monoid__add__class_Oadd__0) ).
thf(375,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(154,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ D @ C ) ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_dense__le__bounded) ).
thf(733,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ D @ C ) ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(103,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_zero__le__double__add__iff__zero__le__single__add) ).
thf(584,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(140,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_less__le__trans) ).
thf(694,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(167,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_subset__antisym) ).
thf(772,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(157,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_le__funD) ).
thf(743,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(233,axiom,
( ord_less_eq @ ( set @ ( nat > nat ) )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] : ( number2016821345itions @ A @ n ) )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] :
( ! [B: nat] : ( ord_less_eq @ nat @ ( A @ B ) @ n )
& ! [B: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ B )
=> ( ( A @ B )
= ( zero_zero @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062) ).
thf(958,plain,
( ord_less_eq @ ( set @ ( nat > nat ) )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] : ( number2016821345itions @ A @ n ) )
@ ( collect @ ( nat > nat )
@ ^ [A: nat > nat] :
( ! [B: nat] : ( ord_less_eq @ nat @ ( A @ B ) @ n )
& ! [B: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ B )
=> ( ( A @ B )
= ( zero_zero @ nat ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(211,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ~ ( finite_finite2 @ TA @ A )
=> ~ ( finite_finite2 @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_infinite__super) ).
thf(904,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ~ ( finite_finite2 @ TA @ A )
=> ~ ( finite_finite2 @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(240,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_finite__code) ).
thf(975,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(199,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_neq0__conv) ).
thf(875,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(185,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(826,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(224,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_finite__Collect__disjI) ).
thf(936,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(100,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_not__less) ).
thf(572,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(147,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less_eq @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_verit__comp__simplify1_I3_J) ).
thf(713,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less_eq @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(288,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_less__nat__zero__code) ).
thf(1107,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(280,axiom,
strict797366125id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add) ).
thf(1090,plain,
strict797366125id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(227,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_le__eq__less__or__eq) ).
thf(947,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(23,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ B @ A )
& ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_dual__order_Oeq__iff) ).
thf(350,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ B @ A )
& ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(88,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_add__increasing) ).
thf(540,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(172,axiom,
ord @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_12) ).
thf(784,plain,
ord @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(255,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_trans__le__add2) ).
thf(1011,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(266,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_finite__Collect__conjI) ).
thf(1048,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(12,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_order_Otrans) ).
thf(318,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(265,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_finite__set__choice) ).
thf(1043,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(115,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_less__add__same__cancel2) ).
thf(616,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(195,axiom,
canoni770627133id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).
thf(863,plain,
canoni770627133id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(164,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= B )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_add__cancel__left__left) ).
thf(764,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= B )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(242,axiom,
! [A: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_plus__nat_Oadd__0) ).
thf(979,plain,
! [A: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(110,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
? [C: TA] :
( B
= ( plus_plus @ TA @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_le__iff__add) ).
thf(604,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
? [C: TA] :
( B
= ( plus_plus @ TA @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(209,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(900,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(42,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_less__add__same__cancel1) ).
thf(406,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(247,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_le__trans) ).
thf(990,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(249,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less @ nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_nat__add__left__cancel__less) ).
thf(996,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(205,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_le__Suc__ex) ).
thf(890,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(277,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ A @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_add__gr__0) ).
thf(1080,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ A @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(120,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ B @ A ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_add__cancel__right__left) ).
thf(631,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ B @ A ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(181,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_in__mono) ).
thf(816,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(152,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_order_Onot__eq__order__implies__strict) ).
thf(726,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(207,axiom,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_finite__Collect__less__nat) ).
thf(895,plain,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(260,axiom,
preorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).
thf(1022,plain,
preorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(132,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_add__left__imp__eq) ).
thf(668,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(16,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= A )
= ( B
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_add__cancel__left__right) ).
thf(330,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= A )
= ( B
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(269,axiom,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: TA > TB,C: TA > TB] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( B @ D )
!= ( zero_zero @ TB ) ) ) ) )
=> ( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( C @ D )
!= ( zero_zero @ TB ) ) ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( plus_plus @ TB @ ( B @ D ) @ ( C @ D ) )
!= ( zero_zero @ TB ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_sum_Ofinite__Collect__op) ).
thf(1061,plain,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: TA > TB,C: TA > TB] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( B @ D )
!= ( zero_zero @ TB ) ) ) ) )
=> ( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( C @ D )
!= ( zero_zero @ TB ) ) ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ A )
& ( ( plus_plus @ TB @ ( B @ D ) @ ( C @ D ) )
!= ( zero_zero @ TB ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(21,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_leD) ).
thf(345,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(194,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_less__mono__imp__le__mono) ).
thf(858,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(93,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( B = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_dual__order_Oantisym) ).
thf(552,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(214,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_finite__subset) ).
thf(912,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(39,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_dual__order_Ostrict__trans1) ).
thf(399,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(234,axiom,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_Collect__subset) ).
thf(959,plain,
! [TA: $tType,A: set @ TA,B: TA > $o] :
( ord_less_eq @ ( set @ TA )
@ ( collect @ TA
@ ^ [C: TA] :
( ( member @ TA @ C @ A )
& ( B @ C ) ) )
@ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(122,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_order_Oorder__iff__strict) ).
thf(636,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(271,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_Collect__mono) ).
thf(1066,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(257,axiom,
( ( finite_finite2 @ nat )
= ( ^ [A: set @ nat] :
? [B: nat] :
! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less_eq @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_finite__nat__set__iff__bounded__le) ).
thf(1015,plain,
( ( finite_finite2 @ nat )
= ( ^ [A: set @ nat] :
? [B: nat] :
! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less_eq @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(112,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_dual__order_Ostrict__trans2) ).
thf(608,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(37,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_less__numeral__extra_I3_J) ).
thf(394,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(97,axiom,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_4) ).
thf(563,plain,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(53,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_add__pos__pos) ).
thf(438,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(218,axiom,
linorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).
thf(922,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(129,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_group__cancel_Oadd1) ).
thf(658,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(60,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB] :
( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
=> ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_le__funI) ).
thf(455,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB] :
( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
=> ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(203,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).
thf(883,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(275,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TB] :
( ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( finite_finite2 @ ( TA > TB )
@ ( collect @ ( TA > TB )
@ ^ [D: TA > TB] :
! [E: TA] :
( ( ( member @ TA @ E @ A )
=> ( member @ TB @ ( D @ E ) @ B ) )
& ( ~ ( member @ TA @ E @ A )
=> ( ( D @ E )
= C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_finite__set__of__finite__funs) ).
thf(1076,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TB] :
( ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( finite_finite2 @ ( TA > TB )
@ ( collect @ ( TA > TB )
@ ^ [D: TA > TB] :
! [E: TA] :
( ( ( member @ TA @ E @ A )
=> ( member @ TB @ ( D @ E ) @ B ) )
& ( ~ ( member @ TA @ E @ A )
=> ( ( D @ E )
= C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(221,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_le__neq__implies__less) ).
thf(927,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(126,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ A @ B ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_add__cancel__right__right) ).
thf(648,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( plus_plus @ TA @ A @ B ) )
= ( B
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(79,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_add_Oleft__commute) ).
thf(516,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(225,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_partitions__bounds) ).
thf(939,plain,
! [A: nat > nat,B: nat,C: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(25,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_order__trans) ).
thf(355,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(184,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ~ ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_less__le__not__le) ).
thf(823,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ~ ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(82,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_add_Oleft__cancel) ).
thf(525,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(252,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_bot__nat__0_Oextremum__uniqueI) ).
thf(1003,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(272,axiom,
! [A: nat] :
( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_Nat_Oadd__0__right) ).
thf(1070,plain,
! [A: nat] :
( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(144,axiom,
linorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_9) ).
thf(705,plain,
linorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(231,axiom,
cancel1352612707id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).
thf(954,plain,
cancel1352612707id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(43,axiom,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_add_Oright__neutral) ).
thf(409,plain,
! [TA: $tType] :
( ( monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(285,axiom,
! [TA: $tType] :
( ( finite_Fpow @ TA )
= ( ^ [A: set @ TA] :
( collect @ ( set @ TA )
@ ^ [B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ A )
& ( finite_finite2 @ TA @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_Fpow__def) ).
thf(1098,plain,
! [TA: $tType] :
( ( finite_Fpow @ TA )
= ( ^ [A: set @ TA] :
( collect @ ( set @ TA )
@ ^ [B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ B @ A )
& ( finite_finite2 @ TA @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(248,axiom,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_less__one) ).
thf(993,plain,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(45,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_zero__less__double__add__iff__zero__less__single__add) ).
thf(414,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
= ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(238,axiom,
linord1659791738miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).
thf(971,plain,
linord1659791738miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(175,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_add_Oright__cancel) ).
thf(791,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(74,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_leI) ).
thf(502,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(191,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_Nat_Oex__has__greatest__nat) ).
thf(840,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(66,axiom,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_add__decreasing) ).
thf(470,plain,
! [TA: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(180,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_equalityE) ).
thf(808,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(124,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
= ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_add__less__same__cancel1) ).
thf(643,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
= ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(90,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_add__less__cancel__right) ).
thf(544,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(18,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_order__le__less__subst2) ).
thf(336,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(143,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_le__funE) ).
thf(703,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(261,axiom,
semiri456707255roduct @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).
thf(1023,plain,
semiri456707255roduct @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(114,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_le__zero__eq) ).
thf(613,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(267,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_nat__le__iff__add) ).
thf(1052,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(196,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus @ nat @ A @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_add__eq__self__zero) ).
thf(864,plain,
! [A: nat,B: nat] :
( ( ( plus_plus @ nat @ A @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(254,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_le__0__eq) ).
thf(1008,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(56,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_order_Ostrict__trans1) ).
thf(446,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(107,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
= ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_add__le__same__cancel1) ).
thf(595,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
= ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(283,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_Collect__cong) ).
thf(1094,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(241,axiom,
ord @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).
thf(978,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(119,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_order__subst2) ).
thf(627,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(28,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_ord__eq__le__subst) ).
thf(363,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(71,axiom,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_double__add__less__zero__iff__single__add__less__zero) ).
thf(482,plain,
! [TA: $tType] :
( ( linord219039673up_add @ TA )
=> ! [A: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
= ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(133,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ! [C: TA] :
( ( B
= ( plus_plus @ TA @ A @ C ) )
=> ( C
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_canonically__ordered__monoid__add__class_OlessE) ).
thf(671,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ! [C: TA] :
( ( B
= ( plus_plus @ TA @ A @ C ) )
=> ( C
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(138,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_add_Ocomm__neutral) ).
thf(688,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(162,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_le__numeral__extra_I4_J) ).
thf(759,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(58,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_add__mono__thms__linordered__semiring_I1_J) ).
thf(450,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(279,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_infinite__descent__measure) ).
thf(1084,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(142,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_add__mono__thms__linordered__semiring_I4_J) ).
thf(699,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(235,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ B @ C )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( C = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_finite__has__maximal2) ).
thf(961,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ B @ C )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( C = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(286,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_add__is__0) ).
thf(1101,plain,
! [A: nat,B: nat] :
( ( ( plus_plus @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(40,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_le__numeral__extra_I3_J) ).
thf(401,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(171,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_add__eq__0__iff__both__eq__0) ).
thf(781,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
& ( B
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(69,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_zero__le__one) ).
thf(478,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(34,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less_eq @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_linorder__wlog) ).
thf(383,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less_eq @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(188,axiom,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA,D: TA > $o] :
( ( member @ TA @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ C )
& ( D @ E ) ) ) )
=> ( D @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_prop__restrict) ).
thf(832,plain,
! [TA: $tType,A: TA,B: set @ TA,C: set @ TA,D: TA > $o] :
( ( member @ TA @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ C )
& ( D @ E ) ) ) )
=> ( D @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(87,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_equalityD1) ).
thf(536,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(215,axiom,
( ( finite_finite2 @ nat )
= ( ^ [A: set @ nat] :
? [B: nat] :
! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less @ nat @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_finite__nat__set__iff__bounded) ).
thf(914,plain,
( ( finite_finite2 @ nat )
= ( ^ [A: set @ nat] :
? [B: nat] :
! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less @ nat @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(32,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).
thf(378,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(51,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_less__numeral__extra_I4_J) ).
thf(433,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(77,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_zero__reorient) ).
thf(510,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(148,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( B = A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_dual__order_Oorder__iff__strict) ).
thf(716,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( B = A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(179,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_ord__le__eq__subst) ).
thf(800,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(290,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_eq__imp__le) ).
thf(1116,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(6,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o,C: TA] :
( ( ord_less_eq @ ( TA > $o ) @ A @ B )
=> ( ( A @ C )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_predicate1D) ).
thf(300,plain,
! [TA: $tType,A: TA > $o,B: TA > $o,C: TA] :
( ( ord_less_eq @ ( TA > $o ) @ A @ B )
=> ( ( A @ C )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(250,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_le__add2) ).
thf(999,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(160,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_dense__ge) ).
thf(752,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(11,axiom,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_le__add__same__cancel2) ).
thf(315,plain,
! [TA: $tType] :
( ( ordere516151231imp_le @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
= ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(236,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_bot__nat__0_Oextremum) ).
thf(967,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(182,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(818,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(197,axiom,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_less__or__eq__imp__le) ).
thf(867,plain,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(30,axiom,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_7) ).
thf(373,plain,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(229,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_less__eq__nat_Osimps_I1_J) ).
thf(950,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(284,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).
thf(1097,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(190,axiom,
! [A: set @ nat,B: nat] :
( ! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less @ nat @ C @ B ) )
=> ( finite_finite2 @ nat @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_bounded__nat__set__is__finite) ).
thf(835,plain,
! [A: set @ nat,B: nat] :
( ! [C: nat] :
( ( member @ nat @ C @ A )
=> ( ord_less @ nat @ C @ B ) )
=> ( finite_finite2 @ nat @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(8,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_add__right__cancel) ).
thf(305,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(15,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_order__subst1) ).
thf(326,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(186,axiom,
ordere216010020id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__comm__monoid__add) ).
thf(828,plain,
ordere216010020id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(141,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ A @ B ) )
= ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_antisym__conv2) ).
thf(696,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ A @ B ) )
= ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(232,axiom,
! [A: nat > nat,B: nat] :
( ! [C: nat] : ( ord_less_eq @ nat @ C @ ( A @ C ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less_eq @ nat @ ( A @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_finite__less__ub) ).
thf(955,plain,
! [A: nat > nat,B: nat] :
( ! [C: nat] : ( ord_less_eq @ nat @ C @ ( A @ C ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less_eq @ nat @ ( A @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(177,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_add__le__less__mono) ).
thf(796,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(208,axiom,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_partitions__imp__finite__elements) ).
thf(897,plain,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [C: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(165,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_le__neq__trans) ).
thf(767,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(156,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_order__class_Oorder_Oeq__iff) ).
thf(740,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(245,axiom,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_nat__less__le) ).
thf(987,plain,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(64,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_group__cancel_Oadd2) ).
thf(464,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(101,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_add_Ogroup__left__neutral) ).
thf(575,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(259,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ D )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_add__le__mono) ).
thf(1019,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ D )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(83,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_not__le__imp__less) ).
thf(528,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(62,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_6) ).
thf(459,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(243,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less_eq @ nat @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_nat__add__left__cancel__le) ).
thf(982,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less_eq @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(139,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_pred__subset__eq) ).
thf(691,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(291,axiom,
monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__add) ).
thf(1120,plain,
monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(111,axiom,
finite_finite @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Finite__Set_Ofinite_10) ).
thf(607,plain,
finite_finite @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(282,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(1093,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(118,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( A != B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_order_Ostrict__iff__order) ).
thf(624,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( A != B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(14,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_subsetI) ).
thf(322,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(219,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_le__antisym) ).
thf(923,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(123,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_order__less__le__subst1) ).
thf(639,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(168,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_le__less__trans) ).
thf(775,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(96,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( B != A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_dual__order_Ostrict__iff__order) ).
thf(560,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( B != A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(256,axiom,
wellorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).
thf(1014,plain,
wellorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(239,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ A
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ B )
& ( C @ D ) ) ) )
= ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_subset__Collect__iff) ).
thf(972,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ A
@ ( collect @ TA
@ ^ [D: TA] :
( ( member @ TA @ D @ B )
& ( C @ D ) ) ) )
= ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(1121,plain,
$false,
inference(e,[status(thm)],[1024,347,893,1110,909,333,1083,468,777,555,683,308,1073,873,677,878,440,1078,533,550,500,472,930,1063,340,829,1001,417,651,320,565,389,724,719,785,504,357,949,542,587,619,421,969,770,881,448,475,756,1040,646,353,821,480,485,834,634,293,396,428,1104,507,312,292,452,920,380,942,761,534,798,1091,952,302,598,392,297,424,788,729,916,361,879,435,665,522,461,710,1017,403,1006,593,610,989,857,519,578,884,933,590,679,547,706,901,686,1075,622,985,738,654,457,721,569,779,371,343,1054,558,1026,530,794,662,926,906,298,412,750,856,745,444,601,513,430,375,733,584,694,772,743,958,904,975,875,826,936,572,713,1107,1090,947,350,540,784,1011,1048,318,1043,616,863,764,979,604,900,406,990,996,890,1080,631,816,726,895,1022,668,330,1061,345,858,552,912,399,959,636,1066,1015,608,394,563,438,922,658,455,883,1076,927,648,516,939,355,823,525,1003,1070,705,954,409,1098,993,414,971,791,502,840,470,808,643,544,336,703,1023,613,1052,864,1008,446,595,1094,978,627,363,482,671,688,759,450,1084,699,961,1101,401,781,478,383,832,536,914,378,433,510,716,800,1116,300,999,752,315,967,818,867,373,950,1097,835,305,326,828,696,955,796,897,767,740,987,464,575,1019,528,459,982,691,1120,607,1093,624,322,923,639,775,560,1014,972]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : ITP134^2 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.09/0.20 % Computer : n010.cluster.edu
% 0.09/0.20 % Model : x86_64 x86_64
% 0.09/0.20 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.20 % Memory : 8046.5625MB
% 0.09/0.20 % OS : Linux 6.8.0-71-generic
% 0.09/0.20 % CPULimit : 300
% 0.09/0.20 % WCLimit : 300
% 0.09/0.20 % DateTime : Tue Sep 29 19:59:03 UTC 2026
% 0.09/0.20 % CPUTime :
% 0.09/0.20 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.69/0.52 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.83/0.89 % [INFO] Parsing done (375ms).
% 2.33/0.90 % [INFO] Running in sequential loop mode.
% 2.91/1.12 % [INFO] eprover registered as external prover.
% 2.91/1.12 % [INFO] Scanning for conjecture ...
% 3.55/1.20 % [INFO] Found a conjecture (or negated_conjecture) and 289 axioms. Running axiom selection ...
% 3.94/1.30 % [INFO] Axiom selection finished. Selected 289 axioms (removed 0 axioms).
% 4.36/1.49 % [INFO] Problem is higher-order (TPTP THF).
% 4.76/1.51 % [INFO] Type checking passed.
% 4.76/1.51 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 112.92/90.46 % External prover 'e' found a proof!
% 112.92/90.47 % [INFO] Killing All external provers ...
% 112.92/90.47 % Time passed: 90176ms (effective reasoning time: 89562ms)
% 112.92/90.47 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 112.92/90.47 % Axioms used in derivation (289): fact_123_ord__le__eq__subst, fact_128_le__cases, fact_139_dual__order_Otrans, fact_76_add__mono__thms__linordered__semiring_I1_J, fact_171_subset__Collect__iff, fact_147_equalityD2, fact_126_linear, fact_229_not__le, fact_150_Collect__mono, fact_203_order_Onot__eq__order__implies__strict, fact_233_order__le__less__subst1, fact_153_Collect__mono__iff, fact_86_add__eq__self__zero, fact_102_add__nonneg__nonneg, tcon_Nat_Onat___Orderings_Oorder_2, fact_241_le__eq__less__or__eq, fact_186_add__less__same__cancel1, fact_131_antisym__conv, tcon_fun___Finite__Set_Ofinite, fact_24_double__add__le__zero__iff__single__add__le__zero, fact_94_add__leD2, fact_138_linorder__wlog, fact_103_add__increasing2, tcon_Nat_Onat___Orderings_Olinorder, fact_12_finite, fact_225_antisym__conv2, fact_183_neq0__conv, fact_122_ord__eq__le__subst, fact_172_subset__CollectI, fact_130_le__cases3, fact_37_add__eq__0__iff__both__eq__0, fact_196_add__mono__thms__linordered__field_I3_J, fact_145_subset__eq, fact_61_one__reorient, fact_44_Collect__mem__eq, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_246_less__numeral__extra_I3_J, fact_16_finite__has__maximal2, fact_216_not__le__imp__less, fact_15_finite__has__minimal2, fact_194_add__less__le__mono, fact_36_zero__eq__add__iff__both__eq__0, tcon_Nat_Onat___Rings_Ozero__neq__one, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le, fact_124_eq__iff, fact_237_leD, fact_69_add__le__imp__le__right, fact_193_less__one, tcon_fun___Orderings_Oord, fact_80_add_Ocomm__neutral, fact_208_dense__le__bounded, tcon_HOL_Obool___Orderings_Olinorder_9, fact_55_add_Oleft__cancel, fact_217_less__le__not__le, fact_228_not__less, fact_111_subset__antisym, fact_8_not__finite__existsD, fact_30_bot__nat__0_Oextremum, fact_14_finite__Collect__le__nat, fact_185_finite__Collect__less__nat, fact_100_add__nonneg__eq__0__iff, fact_177_partitions__imp__finite__elements, fact_7_finite__set__of__finite__funs, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_143_subsetD, fact_200_add__pos__pos, fact_68_zero__le, fact_22_le__add__same__cancel1, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_251_infinite__descent__measure, fact_187_add__less__same__cancel2, fact_144_equalityE, fact_156_less__eq__set__def, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_106_add__decreasing, fact_2_finite__Collect__conjI, fact_136_order__trans, fact_13_bound__domain__and__range__impl__finitely__many__functions, fact_33_le__zero__eq, fact_121_order__subst2, tcon_Nat_Onat___Groups_Oone, fact_43_mem__Collect__eq, fact_180_add__less__cancel__left, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_101_add__nonpos__nonpos, fact_226_antisym__conv1, fact_152_set__eq__subset, fact_107_sum_Ofinite__Collect__op, fact_109_zero__le__one, fact_174_prop__restrict, fact_51_add__left__imp__eq, fact_209_dense__ge__bounded, fact_141_dual__order_Oantisym, fact_137_dual__order_Orefl, fact_85_less__eq__nat_Osimps_I1_J, fact_157_zero__neq__one, tcon_Nat_Onat___Groups_Omonoid__add, fact_40_add__cancel__left__right, tcon_HOL_Obool___Orderings_Oorder_11, fact_23_le__add__same__cancel2, fact_219_le__less__linear, fact_71_le__iff__add, fact_189_less__add__same__cancel2, fact_221_dense__ge, fact_218_le__imp__less__or__eq, fact_98_add__leE, fact_250_finite__M__bounded__by__nat, fact_245_verit__comp__simplify1_I1_J, fact_88_nat__le__iff__add, fact_160_add__0__iff, fact_108_not__one__le__zero, fact_4_finite__code, fact_74_add__left__mono, fact_66_le__trans, fact_235_le__less, fact_81_comm__monoid__add__class_Oadd__0, fact_249_finite__nat__set__iff__bounded, fact_169_conj__subset__def, fact_149_subset__refl, fact_197_add__mono__thms__linordered__field_I4_J, fact_75_add__mono, fact_63_nat__le__linear, fact_18_infinite__super, fact_38_add__cancel__right__right, tcon_Set_Oset___Orderings_Oorder_6, fact_162_predicate1I, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_25_zero__le__double__add__iff__zero__le__single__add, fact_112_subsetI, tcon_Nat_Onat___Groups_Ozero, fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062, fact_73_less__eqE, tcon_Nat_Onat___Orderings_Owellorder, fact_28_Nat_Oadd__0__right, tcon_Set_Oset___Orderings_Oord_7, fact_3_finite__Collect__disjI, fact_155_Collect__subset, fact_32_add__left__cancel, fact_72_add__right__mono, fact_206_dual__order_Oorder__iff__strict, fact_49_zero__reorient, fact_47_add_Oright__neutral, fact_140_dual__order_Oeq__iff, fact_192_add__gr__0, fact_97_le__add1, fact_84_le__0__eq, fact_120_order__subst1, fact_110_finite__less__ub, fact_62_Nat_Oex__has__greatest__nat, fact_64_le__antisym, fact_46_double__zero, fact_60_ab__semigroup__add__class_Oadd__ac_I1_J, fact_244_less__mono__imp__le__mono, fact_176_Fpow__def, fact_146_equalityD1, fact_6_partitions__bounds, fact_5_partitions__zero, fact_54_add_Oright__cancel, tcon_Nat_Onat___Orderings_Oord_3, fact_224_less__imp__le, fact_59_add__mono__thms__linordered__semiring_I4_J, fact_65_eq__imp__le, fact_163_predicate1D, fact_20_add__le__same__cancel1, fact_184_nat__add__left__cancel__less, fact_239_nat__less__le, fact_133_ord__eq__le__trans, fact_231_order__less__le__subst1, fact_179_add__less__cancel__right, fact_242_less__or__eq__imp__le, fact_39_add__cancel__right__left, fact_154_bounded__Max__nat, tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add, fact_10_finite__Collect__subsets, fact_26_nat__add__left__cancel__le, tcon_Set_Oset___Orderings_Opreorder_4, fact_119_le__fun__def, fact_243_le__neq__implies__less, fact_99_add__nonpos__eq__0__iff, fact_165_verit__la__disequality, fact_178_not__gr__zero, fact_92_add__le__mono, fact_227_le__neq__trans, fact_41_add__cancel__left__left, fact_105_add__increasing, fact_67_le__refl, fact_212_order_Ostrict__iff__order, fact_167_pred__subset__eq, fact_93_le__Suc__ex, tcon_Set_Oset___Finite__Set_Ofinite_5, fact_247_verit__comp__simplify1_I3_J, fact_252_measure__induct__rule, fact_77_add__mono__thms__linordered__semiring_I2_J, fact_57_group__cancel_Oadd2, fact_114_finite__nat__set__iff__bounded__le, fact_142_in__mono, fact_127_eq__refl, fact_118_le__funI, fact_78_add__mono__thms__linordered__semiring_I3_J, fact_205_dual__order_Ostrict__iff__order, fact_91_add__le__mono1, fact_232_order__le__less__subst2, fact_17_rev__finite__subset, fact_201_add__neg__neg, fact_132_order__class_Oorder_Oeq__iff, fact_170_dbl__inc__def, tcon_fun___Orderings_Opreorder, tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_116_le__funD, fact_53_add_Ocommute, fact_50_add__right__imp__eq, fact_202_add__less__zeroD, fact_191_zero__less__double__add__iff__zero__less__single__add, tcon_Nat_Onat___Orderings_Opreorder_1, fact_79_add_Ogroup__left__neutral, fact_222_less__le__trans, fact_159_verit__sum__simplify, fact_83_bot__nat__0_Oextremum__unique, fact_181_bot__nat__0_Onot__eq__extremum, fact_89_trans__le__add2, fact_210_dual__order_Ostrict__trans2, tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_148_subset__iff, fact_240_less__imp__le__nat, fact_27_add__is__0, fact_9_pigeonhole__infinite__rel, fact_70_add__le__imp__le__left, fact_31_add__right__cancel, fact_188_less__add__same__cancel1, fact_42_double__zero__sym, fact_35_add__le__cancel__left, fact_198_pos__add__strict, fact_115_Euclid__induct, fact_214_order_Ostrict__trans2, fact_175_fun__cong__unused__0, fact_45_Collect__cong, fact_164_rev__predicate1D, tcon_fun___Orderings_Oorder, fact_21_add__le__same__cancel2, fact_34_add__le__cancel__right, fact_11_finite__set__choice, fact_195_add__le__less__mono, tcon_Nat_Onat___Groups_Osemigroup__add, fact_234_less__le, fact_161_le__numeral__extra_I3_J, fact_230_order__less__le__subst2, fact_215_order_Ostrict__trans1, fact_151_subset__trans, fact_58_group__cancel_Oadd1, fact_19_finite__subset, tcon_HOL_Obool___Orderings_Opreorder_8, fact_104_add__decreasing2, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_158_le__numeral__extra_I4_J, tcon_HOL_Obool___Orderings_Oord_12, fact_204_dual__order_Ostrict__implies__order, fact_90_trans__le__add1, tcon_Nat_Onat___Groups_Oab__semigroup__add, fact_238_less__numeral__extra_I4_J, fact_87_plus__nat_Oadd__0, fact_248_bounded__nat__set__is__finite, fact_168_dbl__inc__simps_I2_J, fact_95_add__leD1, fact_135_order__class_Oorder_Oantisym, fact_173_Collect__restrict, fact_220_dense__le, fact_182_less__nat__zero__code, fact_207_order_Ostrict__implies__order, fact_56_add_Oassoc, fact_211_dual__order_Ostrict__trans1, fact_213_order_Oorder__iff__strict, fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062, fact_48_add_Oleft__neutral, fact_96_le__add2, fact_82_bot__nat__0_Oextremum__uniqueI, fact_236_leI, fact_117_le__funE, fact_223_le__less__trans, fact_29_le0, fact_199_canonically__ordered__monoid__add__class_OlessE, tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, fact_134_ord__le__eq__trans, fact_125_antisym, fact_190_double__add__less__zero__iff__single__add__less__zero, fact_52_add_Oleft__commute, fact_129_order_Otrans, fact_166_is__num__normalize_I1_J, fact_113_order__refl, tcon_HOL_Obool___Finite__Set_Ofinite_10
% 112.92/90.47 % No. of inferences in proof: 582
% 112.92/90.47 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 90176 ms resp. 89562 ms w/o parsing
% 113.79/90.71 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 113.79/90.72 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------