%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET809+4 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n005.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 : Sun Sep 27 08:30:56 AM UTC 2026
% Result : Theorem 12.39s 5.36s
% Output : Refutation 12.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 22
% Syntax : Number of formulae : 171 ( 58 unt; 0 typ; 2 def)
% Number of atoms : 552 ( 143 equ; 0 cnn)
% Maximal formula atoms : 22 ( 3 avg)
% Number of connectives : 1617 ( 201 ~; 103 |; 84 &;1133 @)
% ( 17 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 26 ( 23 usr; 6 con; 0-3 aty)
% Number of variables : 416 ( 2 ^; 402 !; 10 ?; 416 :)
% ( 0 !>; 0 ?*; 0 @-; 2 @+)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(on_decl,type,
on: $i ).
thf(equal_set_decl,type,
equal_set: $i > $i > $o ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(least_decl,type,
least: $i > $i > $i > $o ).
thf(apply_decl,type,
apply: $i > $i > $i > $o ).
thf(initial_segment_decl,type,
initial_segment: $i > $i > $i > $i ).
thf(power_set_decl,type,
power_set: $i > $i ).
thf(set_decl,type,
set: $i > $o ).
thf(strict_well_order_decl,type,
strict_well_order: $i > $i > $o ).
thf(member_predicate_decl,type,
member_predicate: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(product_decl,type,
product: $i > $i ).
thf(strict_order_decl,type,
strict_order: $i > $i > $o ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(suc_decl,type,
suc: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(difference_decl,type,
difference: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(sum_decl,type,
sum: $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk13_decl,type,
sk13: $i > $i > $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ( ( on @ ( A @ member ) )
=> ~ ( A @ ( A @ member ) ) ) ) ) ).
thf(sk13_def,definition,
( sk13
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( B @ ( C @ member ) )
=> ( A @ ( C @ member ) ) ) ) ) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton) ).
thf(69,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( B @ singleton @ ( A @ member ) ) )
& ( ( B @ singleton @ ( A @ member ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_member) ).
thf(181,plain,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(70,plain,
( ! [A: $i,B: $i] :
( ( A = B )
=> ( B @ singleton @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
=> ( A = B ) ) ),
inference(miniscope,[status(thm)],[69]) ).
thf(71,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ singleton @ ( C @ member ) )
| ( C != D ) )
& ( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[70]) ).
thf(73,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ member ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[71]) ).
thf(75,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ member ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[73]) ).
thf(76,plain,
! [A: $i] : ( A @ singleton @ ( A @ member ) ),
inference(simp,[status(thm)],[75]) ).
thf(21,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( member_predicate @ apply ) ) )
<=> ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rel_member) ).
thf(209,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ member ) )
=> ( B @ ( A @ ( member_predicate @ apply ) ) ) )
& ( ( B @ ( A @ ( member_predicate @ apply ) ) )
=> ( B @ ( A @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(1,conjecture,
! [A: $i] :
( ( on @ ( A @ member ) )
=> ~ ( A @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thV2) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ~ ( A @ ( A @ member ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(22,plain,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ~ ( A @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(23,plain,
( ( sk1 @ ( sk1 @ member ) )
& ( on @ ( sk1 @ member ) ) ),
inference(cnf,[status(esa)],[22]) ).
thf(24,plain,
on @ ( sk1 @ member ),
inference(cnfConj,[status(thm)],[23]) ).
thf(25,plain,
sk1 @ ( sk1 @ member ),
inference(cnfConj,[status(thm)],[23]) ).
thf(15,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
<=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',difference) ).
thf(167,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ( ( B @ ( C @ difference ) @ ( A @ member ) )
=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(168,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[167]) ).
thf(169,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( F @ difference ) @ ( D @ member ) )
| ( E @ ( D @ member ) )
| ~ ( F @ ( D @ member ) ) )
& ( ~ ( B @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[168]) ).
thf(171,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[169]) ).
thf(230,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[25,171]) ).
thf(231,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[230]) ).
thf(232,plain,
! [A: $i] :
~ ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[231:[bind(A,$thf( sk1 )),bind(B,$thf( sk1 ))]]) ).
thf(239,plain,
! [A: $i] :
~ ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[232]) ).
thf(269,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24,239]) ).
thf(270,plain,
! [A: $i] :
( ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[269]) ).
thf(274,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ difference ) )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[270]) ).
thf(277,plain,
! [A: $i] :
( ( sk1 @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[274]) ).
thf(13,axiom,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).
thf(158,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(159,plain,
~ ? [A: $i] : ( empty_set @ ( A @ member ) ),
inference(miniscope,[status(thm)],[158]) ).
thf(160,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(cnf,[status(esa)],[159]) ).
thf(217,plain,
! [A: $i] :
( ( ( sk1 @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,160]) ).
thf(218,plain,
! [A: $i] :
( ( sk1 @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[217]) ).
thf(220,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( sk1 != A ) ),
inference(simp,[status(thm)],[218]) ).
thf(222,plain,
sk1 != empty_set,
inference(simp,[status(thm)],[220]) ).
thf(267,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( B @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[76,239]) ).
thf(268,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( B @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[267]) ).
thf(273,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ ( B @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[268]) ).
thf(276,plain,
! [A: $i] :
( ( sk1 @ ( A @ difference ) )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[273]) ).
thf(20,axiom,
! [A: $i,B: $i] :
( ( B @ sum @ ( A @ member ) )
<=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum) ).
thf(202,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) )
=> ( B @ sum @ ( A @ member ) ) )
& ( ( B @ sum @ ( A @ member ) )
=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(16,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection) ).
thf(174,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) )
=> ( C @ ( B @ intersection ) @ ( A @ member ) ) )
& ( ( C @ ( B @ intersection ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(5,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
<=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',initial_segment) ).
thf(46,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) )
=> ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
& ( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( A = C )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair) ).
thf(191,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( A = C )
| ( A = B ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( A = C )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(192,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ( A = C )
| ( A = B ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( A = C )
| ( A = B ) ) ) ),
inference(miniscope,[status(thm)],[191]) ).
thf(193,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ unordered_pair ) @ ( D @ member ) )
| ( D != F ) )
& ( ( F @ ( E @ unordered_pair ) @ ( D @ member ) )
| ( D != E ) )
& ( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[192]) ).
thf(196,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(cnfConj,[status(thm)],[193]) ).
thf(200,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(lifteq,[status(thm)],[196]) ).
thf(201,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( B @ member ) ),
inference(simp,[status(thm)],[200]) ).
thf(225,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[24,171]) ).
thf(226,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[225]) ).
thf(227,plain,
! [A: $i] :
~ ( on @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[226:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(238,plain,
! [A: $i] :
~ ( on @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[227]) ).
thf(304,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[201,238]) ).
thf(305,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[304]) ).
thf(312,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ ( C @ difference ) ) )
| ( B != sk1 ) ),
inference(simp,[status(thm)],[305]) ).
thf(316,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( A @ unordered_pair ) )
!= ( on @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[312]) ).
thf(3,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_set) ).
thf(26,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ equal_set ) ) )
& ( ( B @ ( A @ equal_set ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(27,plain,
( ! [A: $i,B: $i] :
( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ equal_set ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(miniscope,[status(thm)],[26]) ).
thf(28,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ equal_set ) )
| ~ ( C @ ( D @ subset ) )
| ~ ( D @ ( C @ subset ) ) )
& ( ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) )
& ( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ) ),
inference(cnf,[status(esa)],[27]) ).
thf(29,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[28]) ).
thf(243,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[76,160]) ).
thf(244,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[243]) ).
thf(250,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[244]) ).
thf(253,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[250]) ).
thf(215,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24,160]) ).
thf(216,plain,
! [A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[215]) ).
thf(219,plain,
! [A: $i] :
( ( empty_set != on )
| ( sk1 != A ) ),
inference(simp,[status(thm)],[216]) ).
thf(221,plain,
empty_set != on,
inference(simp,[status(thm)],[219]) ).
thf(257,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24,238]) ).
thf(258,plain,
! [A: $i] :
( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[257]) ).
thf(262,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[258]) ).
thf(265,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[262]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_well_order ) )
<=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_well_order) ).
thf(85,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) )
=> ( B @ ( A @ strict_well_order ) ) )
& ( ( B @ ( A @ strict_well_order ) )
=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(195,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[193]) ).
thf(198,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(199,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ member ) ),
inference(simp,[status(thm)],[198]) ).
thf(286,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[199,239]) ).
thf(287,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[286]) ).
thf(293,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ ( C @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[287]) ).
thf(297,plain,
! [B: $i,A: $i] :
( ( A @ ( sk1 @ unordered_pair ) )
!= ( sk1 @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[293]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ least ) ) )
<=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',least) ).
thf(33,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) )
=> ( B @ ( A @ ( C @ least ) ) ) )
& ( ( B @ ( A @ ( C @ least ) ) )
=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(247,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[76,171]) ).
thf(248,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[247]) ).
thf(249,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(pattern_uni,[status(thm)],[248:[bind(A,$thf( E )),bind(B,$thf( E )),bind(C,$thf( E @ singleton ))]]) ).
thf(252,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(simp,[status(thm)],[249]) ).
thf(228,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,171]) ).
thf(229,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[228]) ).
thf(236,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= sk1 )
| ( sk1 != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[229]) ).
thf(241,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ difference ) )
!= sk1 )
| ~ ( A @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[236]) ).
thf(345,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[199,241]) ).
thf(346,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= sk1 ) ),
inference(simp,[status(thm)],[345]) ).
thf(347,plain,
! [B: $i,A: $i] :
( ( B @ ( sk1 @ unordered_pair ) @ ( A @ difference ) )
!= sk1 ),
inference(pattern_uni,[status(thm)],[346:[bind(A,$thf( sk1 )),bind(B,$thf( F )),bind(C,$thf( F @ ( sk1 @ unordered_pair ) )),bind(D,$thf( D ))]]) ).
thf(361,plain,
! [B: $i,A: $i] :
( ( B @ ( sk1 @ unordered_pair ) @ ( A @ difference ) )
!= sk1 ),
inference(simp,[status(thm)],[347]) ).
thf(308,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[201,239]) ).
thf(309,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[308]) ).
thf(313,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ ( C @ difference ) ) )
| ( B != sk1 ) ),
inference(simp,[status(thm)],[309]) ).
thf(317,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( A @ unordered_pair ) )
!= ( sk1 @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[313]) ).
thf(12,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset) ).
thf(150,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subset ) ) )
& ( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(151,plain,
( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subset ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[150]) ).
thf(152,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ subset ) )
| ~ ( E @ ( D @ ( E @ sk13 ) @ member ) ) )
& ( ( E @ ( D @ subset ) )
| ( D @ ( D @ ( E @ sk13 ) @ member ) ) )
& ( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ) ),
inference(cnf,[status(esa)],[151]) ).
thf(154,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[152]) ).
thf(156,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[154]) ).
thf(6,axiom,
! [A: $i,B: $i] :
( ( B @ power_set @ ( A @ member ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_set) ).
thf(53,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( B @ power_set @ ( A @ member ) ) )
& ( ( B @ power_set @ ( A @ member ) )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( B @ product @ ( A @ member ) )
<=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',product) ).
thf(77,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) )
=> ( B @ product @ ( A @ member ) ) )
& ( ( B @ product @ ( A @ member ) )
=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(11,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_order ) )
<=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_order) ).
thf(96,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) )
=> ( B @ ( A @ strict_order ) ) )
& ( ( B @ ( A @ strict_order ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(259,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( sk1 @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,238]) ).
thf(260,plain,
! [A: $i] :
( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( sk1 @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[259]) ).
thf(263,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) )
!= sk1 )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[260]) ).
thf(266,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= sk1 ),
inference(simp,[status(thm)],[263]) ).
thf(342,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( sk1 @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[76,241]) ).
thf(343,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( sk1 @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= sk1 ) ),
inference(simp,[status(thm)],[342]) ).
thf(344,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( A @ difference ) )
!= sk1 ),
inference(pattern_uni,[status(thm)],[343:[bind(A,$thf( sk1 )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( C ))]]) ).
thf(360,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( A @ difference ) )
!= sk1 ),
inference(simp,[status(thm)],[344]) ).
thf(31,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[28]) ).
thf(32,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[31]) ).
thf(255,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[76,238]) ).
thf(256,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[255]) ).
thf(261,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( on @ ( B @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[256]) ).
thf(264,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[261]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( A @ suc @ ( B @ member ) )
<=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(161,plain,
! [A: $i,B: $i] :
( ( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
=> ( A @ suc @ ( B @ member ) ) )
& ( ( A @ suc @ ( B @ member ) )
=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(7,axiom,
! [A: $i] :
( ( on @ ( A @ member ) )
<=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_number) ).
thf(59,plain,
! [A: $i] :
( ( ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) )
=> ( on @ ( A @ member ) ) )
& ( ( on @ ( A @ member ) )
=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(394,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24,252]) ).
thf(395,plain,
! [B: $i,A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[394]) ).
thf(407,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ difference ) )
!= on )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[395]) ).
thf(413,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[407]) ).
thf(271,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( sk1 @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,239]) ).
thf(272,plain,
! [A: $i] :
( ( sk1 @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( sk1 @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[271]) ).
thf(275,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ difference ) )
!= sk1 )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[272]) ).
thf(278,plain,
! [A: $i] :
( ( sk1 @ ( A @ difference ) )
!= sk1 ),
inference(simp,[status(thm)],[275]) ).
thf(279,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[199,160]) ).
thf(280,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[279]) ).
thf(291,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[280]) ).
thf(295,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[291]) ).
thf(339,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[201,241]) ).
thf(340,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= sk1 ) ),
inference(simp,[status(thm)],[339]) ).
thf(341,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= sk1 ),
inference(pattern_uni,[status(thm)],[340:[bind(A,$thf( E )),bind(B,$thf( sk1 )),bind(C,$thf( sk1 @ ( E @ unordered_pair ) )),bind(D,$thf( D ))]]) ).
thf(359,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= sk1 ),
inference(simp,[status(thm)],[341]) ).
thf(327,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[156,160]) ).
thf(328,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[327]) ).
thf(329,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(pattern_uni,[status(thm)],[328:[bind(A,$thf( empty_set )),bind(B,$thf( D )),bind(C,$thf( empty_set @ ( D @ sk13 ) ))]]) ).
thf(336,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(simp,[status(thm)],[329]) ).
thf(306,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[201,160]) ).
thf(307,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[306]) ).
thf(310,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( B != C ) ),
inference(simp,[status(thm)],[307]) ).
thf(314,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[310]) ).
thf(30,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[28]) ).
thf(18,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(183,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) )
=> ( C @ ( B @ union ) @ ( A @ member ) ) )
& ( ( C @ ( B @ union ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(288,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[199,238]) ).
thf(289,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[288]) ).
thf(290,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ ( C @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[289]) ).
thf(294,plain,
! [B: $i,A: $i] :
( ( A @ ( sk1 @ unordered_pair ) )
!= ( on @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[290]) ).
thf(505,plain,
$false,
inference(e,[status(thm)],[69,181,76,209,277,222,276,202,174,24,25,46,316,29,253,238,221,265,85,201,160,297,33,252,361,317,156,53,77,96,266,360,32,264,161,191,22,59,413,278,241,167,295,150,199,359,336,314,26,158,171,30,183,239,294]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET809+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.13 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.24/0.51 % Computer : n005.cluster.edu
% 0.24/0.51 % Model : x86_64 x86_64
% 0.24/0.51 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.24/0.51 % Memory : 8046.5625MB
% 0.24/0.51 % OS : Linux 6.8.0-71-generic
% 0.24/0.51 % CPULimit : 300
% 0.24/0.51 % WCLimit : 300
% 0.24/0.51 % DateTime : Sat Sep 26 08:17:17 UTC 2026
% 0.24/0.51 % CPUTime :
% 0.24/0.51 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.57/1.42 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.13/1.74 % [INFO] Parsing done (320ms).
% 2.13/1.76 % [INFO] Running in sequential loop mode.
% 3.41/2.43 % [INFO] eprover registered as external prover.
% 3.41/2.44 % [INFO] Scanning for conjecture ...
% 3.74/2.60 % [INFO] Found a conjecture (or negated_conjecture) and 19 axioms. Running axiom selection ...
% 3.94/2.70 % [INFO] Axiom selection finished. Selected 19 axioms (removed 0 axioms).
% 4.18/2.80 % [INFO] Problem is first-order (TPTP FOF).
% 4.18/2.81 % [INFO] Type checking passed.
% 4.18/2.82 % [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 ...
% 12.39/5.35 % External prover 'e' found a proof!
% 12.39/5.35 % [INFO] Killing All external provers ...
% 12.39/5.36 % Time passed: 4632ms (effective reasoning time: 3581ms)
% 12.39/5.36 % 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)>
% 12.39/5.36 % Axioms used in derivation (19): power_set, unordered_pair, rel_member, union, strict_well_order, intersection, set_member, initial_segment, least, subset, equal_set, singleton, ordinal_number, difference, strict_order, sum, product, successor, empty_set
% 12.39/5.36 % No. of inferences in proof: 169
% 12.39/5.36 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 4632 ms resp. 3581 ms w/o parsing
% 12.22/5.53 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 12.22/5.53 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------