%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET103+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% 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 : n004.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 08:29:26 AM UTC 2026
% Result : Theorem 17.33s 6.06s
% Output : Refutation 16.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 49
% Syntax : Number of formulae : 187 ( 75 unt; 0 typ; 5 def)
% Number of atoms : 544 ( 135 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 1718 ( 153 ~; 93 |; 115 &;1273 @)
% ( 19 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 36 usr; 11 con; 0-3 aty)
% Number of variables : 293 ( 1 ^; 274 !; 13 ?; 293 :)
% ( 0 !>; 0 ?*; 0 @-; 5 @+)
% Comments :
%------------------------------------------------------------------------------
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(disjoint_decl,type,
disjoint: $i > $i > $o ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i ).
thf(sk5_decl,type,
sk5: $i > $i ).
thf(sk7_decl,type,
sk7: $i ).
thf(sk8_decl,type,
sk8: $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ! [B: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: $i] :
~ ( ( universal_class @ ( A @ member ) )
| ( ( null_class @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )
= ( A @ ( sk1 @ ordered_pair ) ) ) ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( ^ [A: $i] :
@+[B: $i] :
( ( A @ ( B @ disjoint ) )
& ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ) ) ).
thf(sk7_def,definition,
( sk7
= ( @+[A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ) ) ) ).
thf(sk8_def,definition,
( sk8
= ( @+[A: $i] :
( ! [B: $i] :
( ( universal_class @ ( B @ member ) )
=> ( ( B @ ( B @ ( A @ apply ) @ member ) )
| ( B = null_class ) ) )
& ( A @ function ) ) ) ) ).
thf(26,axiom,
! [A: $i,B: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair) ).
thf(190,plain,
! [A: $i,B: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(191,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(cnf,[status(esa)],[190]) ).
thf(39,axiom,
! [A: $i,B: $i] :
( ( A @ complement @ ( B @ member ) )
<=> ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement) ).
thf(247,plain,
! [A: $i,B: $i] :
( ( ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) )
=> ( A @ complement @ ( B @ member ) ) )
& ( ( A @ complement @ ( B @ member ) )
=> ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(248,plain,
( ! [A: $i,B: $i] :
( ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) )
=> ( A @ complement @ ( B @ member ) ) )
& ! [A: $i,B: $i] :
( ( A @ complement @ ( B @ member ) )
=> ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[247]) ).
thf(249,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ complement @ ( D @ member ) )
| ( C @ ( D @ member ) )
| ~ ( universal_class @ ( D @ member ) ) )
& ( ~ ( A @ ( B @ member ) )
| ~ ( A @ complement @ ( B @ member ) ) )
& ( ( universal_class @ ( B @ member ) )
| ~ ( A @ complement @ ( B @ member ) ) ) ),
inference(cnf,[status(esa)],[248]) ).
thf(251,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( A @ complement @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[249]) ).
thf(727,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( C @ complement @ ( D @ member ) ) ),
inference(paramod_ordered,[status(thm)],[191,251]) ).
thf(728,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( C @ complement @ ( D @ member ) ) ),
inference(simp,[status(thm)],[727]) ).
thf(729,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[728:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( universal_class )),bind(D,$thf( F @ ( E @ unordered_pair ) ))]]) ).
thf(747,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[729]) ).
thf(29,axiom,
? [A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',infinity) ).
thf(205,plain,
? [A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(206,plain,
! [A: $i] :
( ( ( A @ ( sk7 @ subclass ) )
| ~ ( A @ inductive ) )
& ( sk7 @ inductive )
& ( universal_class @ ( sk7 @ member ) ) ),
inference(cnf,[status(esa)],[205]) ).
thf(207,plain,
universal_class @ ( sk7 @ member ),
inference(cnfConj,[status(thm)],[206]) ).
thf(286,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk7 @ member ) )
!= ( A @ ( B @ member ) ) )
| ~ $true
| ~ ( A @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[207,251]) ).
thf(287,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk7 @ member ) )
!= ( A @ ( B @ member ) ) )
| ~ ( A @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[286]) ).
thf(288,plain,
~ ( universal_class @ complement @ ( sk7 @ member ) ),
inference(pattern_uni,[status(thm)],[287:[bind(A,$thf( universal_class )),bind(B,$thf( sk7 ))]]) ).
thf(16,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) )
<=> ( ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
& ( universal_class @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_defn2) ).
thf(125,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
& ( universal_class @ ( C @ member ) ) )
=> ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) ) )
& ( ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) )
=> ( ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
& ( universal_class @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(24,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate) ).
thf(178,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(179,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(cnf,[status(esa)],[178]) ).
thf(3,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',extensionality) ).
thf(51,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(52,plain,
( ! [A: $i,B: $i] :
( ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) )
=> ( A = B ) )
& ! [A: $i,B: $i] :
( ( A = B )
=> ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) ) ) ),
inference(miniscope,[status(thm)],[51]) ).
thf(53,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C = D )
| ~ ( C @ ( D @ subclass ) )
| ~ ( D @ ( C @ subclass ) ) )
& ( ( A @ ( B @ subclass ) )
| ( A != B ) )
& ( ( B @ ( A @ subclass ) )
| ( A != B ) ) ),
inference(cnf,[status(esa)],[52]) ).
thf(56,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnfConj,[status(thm)],[53]) ).
thf(61,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[56]) ).
thf(62,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(simp,[status(thm)],[61]) ).
thf(1,conjecture,
! [A: $i,B: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(46,plain,
~ ! [A: $i,B: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(47,plain,
( ~ ( universal_class @ ( sk2 @ member ) )
& ( ( null_class @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )
!= ( sk2 @ ( sk1 @ ordered_pair ) ) ) ),
inference(cnf,[status(esa)],[46]) ).
thf(49,plain,
~ ( universal_class @ ( sk2 @ member ) ),
inference(cnfConj,[status(thm)],[47]) ).
thf(276,plain,
( ( ( universal_class @ ( sk7 @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[207,49]) ).
thf(277,plain,
( ( universal_class @ ( sk7 @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[276]) ).
thf(278,plain,
( ( universal_class != universal_class )
| ( sk7 != sk2 ) ),
inference(simp,[status(thm)],[277]) ).
thf(279,plain,
sk7 != sk2,
inference(simp,[status(thm)],[278]) ).
thf(424,plain,
! [B: $i,A: $i] :
( ( A != sk7 )
| ( B != sk2 )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[62,279]) ).
thf(425,plain,
! [A: $i] :
( ( A != sk2 )
| ~ ( sk7 @ ( A @ subclass ) )
| ~ ( A @ ( sk7 @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[424:[bind(A,$thf( sk7 ))]]) ).
thf(512,plain,
( ~ ( sk7 @ ( sk2 @ subclass ) )
| ~ ( sk2 @ ( sk7 @ subclass ) ) ),
inference(simp,[status(thm)],[425]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_defn) ).
thf(241,plain,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(6,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(69,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(12,axiom,
! [A: $i] :
( ( A @ function )
<=> ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function_defn) ).
thf(101,plain,
! [A: $i] :
( ( ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) )
=> ( A @ function ) )
& ( ( A @ function )
=> ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( B @ power_class @ ( A @ member ) )
<=> ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_defn) ).
thf(115,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) )
=> ( B @ power_class @ ( A @ member ) ) )
& ( ( B @ power_class @ ( A @ member ) )
=> ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(42,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip) ).
thf(260,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(261,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(cnf,[status(esa)],[260]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ cross_product ) @ ( C @ member ) )
=> ( C
= ( C @ second @ ( C @ first @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cross_product) ).
thf(147,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ cross_product ) @ ( C @ member ) )
=> ( C
= ( C @ second @ ( C @ first @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(17,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ intersection ) @ ( C @ member ) )
<=> ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection) ).
thf(132,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) )
=> ( B @ ( A @ intersection ) @ ( C @ member ) ) )
& ( ( B @ ( A @ intersection ) @ ( C @ member ) )
=> ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( B @ sum_class @ ( A @ member ) )
<=> ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_defn) ).
thf(74,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ sum_class @ ( A @ member ) ) )
& ( ( B @ sum_class @ ( A @ member ) )
=> ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(719,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[191,49]) ).
thf(720,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[719]) ).
thf(742,plain,
! [B: $i,A: $i] :
( ( universal_class != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= sk2 ) ),
inference(simp,[status(thm)],[720]) ).
thf(746,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= sk2 ),
inference(simp,[status(thm)],[742]) ).
thf(70,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(cnf,[status(esa)],[69]) ).
thf(27,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( ( A = C )
| ( A = B ) )
& ( universal_class @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_defn) ).
thf(192,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( ( A = C )
| ( A = B ) )
& ( universal_class @ ( A @ member ) ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( ( A = C )
| ( A = B ) )
& ( universal_class @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(22,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( A @ rotate @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
<=> ( ( A @ ( B @ ( D @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate_defn) ).
thf(165,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( A @ ( B @ ( D @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) ) )
=> ( A @ rotate @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) ) )
& ( ( A @ rotate @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
=> ( ( A @ ( B @ ( D @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( D @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ union ) @ ( C @ member ) )
<=> ( ( B @ ( C @ member ) )
| ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_defn) ).
thf(229,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( B @ ( C @ member ) )
| ( A @ ( C @ member ) ) )
=> ( B @ ( A @ union ) @ ( C @ member ) ) )
& ( ( B @ ( A @ union ) @ ( C @ member ) )
=> ( ( B @ ( C @ member ) )
| ( A @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restrict_defn) ).
thf(63,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(64,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
inference(cnf,[status(esa)],[63]) ).
thf(65,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
inference(lifteq,[status(thm)],[64]) ).
thf(40,axiom,
? [A: $i] :
( ! [B: $i] :
( ( universal_class @ ( B @ member ) )
=> ( ( B @ ( B @ ( A @ apply ) @ member ) )
| ( B = null_class ) ) )
& ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice) ).
thf(254,plain,
? [A: $i] :
( ! [B: $i] :
( ( universal_class @ ( B @ member ) )
=> ( ( B @ ( B @ ( A @ apply ) @ member ) )
| ( B = null_class ) ) )
& ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(255,plain,
! [A: $i] :
( ( ( A @ ( A @ ( sk8 @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) )
& ( sk8 @ function ) ),
inference(cnf,[status(esa)],[254]) ).
thf(256,plain,
sk8 @ function,
inference(cnfConj,[status(thm)],[255]) ).
thf(30,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_defn) ).
thf(210,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(211,plain,
~ ? [A: $i] : ( null_class @ ( A @ member ) ),
inference(miniscope,[status(thm)],[210]) ).
thf(212,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(cnf,[status(esa)],[211]) ).
thf(35,axiom,
! [A: $i,B: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( B @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_defn1) ).
thf(237,plain,
! [A: $i,B: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( B @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image_defn) ).
thf(244,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(1012,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( sk2 @ ( sk7 @ subclass ) ) )
| ~ ( sk7 @ ( sk2 @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[70,512]) ).
thf(1013,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( sk2 @ ( sk7 @ subclass ) ) )
| ~ ( sk7 @ ( sk2 @ subclass ) ) ),
inference(simp,[status(thm)],[1012]) ).
thf(1053,plain,
! [A: $i] :
( ( sk2 != universal_class )
| ( A != sk7 )
| ~ ( sk7 @ ( sk2 @ subclass ) ) ),
inference(simp,[status(thm)],[1013]) ).
thf(1071,plain,
( ( sk2 != universal_class )
| ~ ( sk7 @ ( sk2 @ subclass ) ) ),
inference(simp,[status(thm)],[1053]) ).
thf(295,plain,
( ( ( universal_class @ complement @ ( sk7 @ member ) )
!= ( universal_class @ ( sk7 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[207,288]) ).
thf(296,plain,
( ( universal_class @ complement @ ( sk7 @ member ) )
!= ( universal_class @ ( sk7 @ member ) ) ),
inference(simp,[status(thm)],[295]) ).
thf(297,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( sk7 != sk7 ) ),
inference(simp,[status(thm)],[296]) ).
thf(298,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[297]) ).
thf(208,plain,
sk7 @ inductive,
inference(cnfConj,[status(thm)],[206]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ inductive )
<=> ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive_defn) ).
thf(108,plain,
! [A: $i] :
( ( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) )
=> ( A @ inductive ) )
& ( ( A @ inductive )
=> ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(109,plain,
( ! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) )
=> ( A @ inductive ) )
& ! [A: $i] :
( ( A @ inductive )
=> ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[108]) ).
thf(110,plain,
! [B: $i,A: $i] :
( ( ( B @ inductive )
| ~ ( B @ ( B @ ( successor_relation @ image ) @ subclass ) )
| ~ ( B @ ( null_class @ member ) ) )
& ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) )
& ( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ) ),
inference(cnf,[status(esa)],[109]) ).
thf(111,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(cnfConj,[status(thm)],[110]) ).
thf(303,plain,
! [A: $i] :
( ( ( sk7 @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[208,111]) ).
thf(304,plain,
! [A: $i] :
( ( ( sk7 @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[303]) ).
thf(305,plain,
sk7 @ ( null_class @ member ),
inference(pattern_uni,[status(thm)],[304:[bind(A,$thf( sk7 ))]]) ).
thf(571,plain,
! [B: $i,A: $i] :
( ( ( sk7 @ ( null_class @ member ) )
!= ( A @ ( B @ member ) ) )
| ~ $true
| ~ ( A @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[305,251]) ).
thf(572,plain,
! [B: $i,A: $i] :
( ( ( sk7 @ ( null_class @ member ) )
!= ( A @ ( B @ member ) ) )
| ~ ( A @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[571]) ).
thf(573,plain,
~ ( sk7 @ complement @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[572:[bind(A,$thf( sk7 )),bind(B,$thf( null_class ))]]) ).
thf(25,axiom,
! [A: $i,B: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
<=> ( ( ( A @ successor )
= B )
& ( universal_class @ ( B @ member ) )
& ( universal_class @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation_defn2) ).
thf(180,plain,
! [A: $i,B: $i] :
( ( ( ( ( A @ successor )
= B )
& ( universal_class @ ( B @ member ) )
& ( universal_class @ ( A @ member ) ) )
=> ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) )
& ( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
=> ( ( ( A @ successor )
= B )
& ( universal_class @ ( B @ member ) )
& ( universal_class @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(28,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation_defn1) ).
thf(204,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(41,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation) ).
thf(259,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply_defn) ).
thf(71,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(18,axiom,
! [A: $i] :
( ( A != null_class )
=> ? [B: $i] :
( ( A @ ( B @ disjoint ) )
& ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity) ).
thf(139,plain,
! [A: $i] :
( ( A != null_class )
=> ? [B: $i] :
( ( A @ ( B @ disjoint ) )
& ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(140,plain,
! [A: $i] :
( ( ( A @ ( A @ sk5 @ disjoint ) )
| ( A = null_class ) )
& ( ( A @ ( A @ sk5 @ member ) )
| ( A = null_class ) )
& ( ( universal_class @ ( A @ sk5 @ member ) )
| ( A = null_class ) ) ),
inference(cnf,[status(esa)],[139]) ).
thf(141,plain,
! [A: $i] :
( ( universal_class @ ( A @ sk5 @ member ) )
| ( A = null_class ) ),
inference(cnfConj,[status(thm)],[140]) ).
thf(144,plain,
! [A: $i] :
( ( universal_class @ ( A @ sk5 @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[141]) ).
thf(23,axiom,
! [A: $i,B: $i] :
( ( ( universal_class @ ( B @ member ) )
& ( universal_class @ ( A @ member ) ) )
=> ( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
& ( ( B @ ( A @ ordered_pair ) @ first )
= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',first_second) ).
thf(172,plain,
! [A: $i,B: $i] :
( ( ( universal_class @ ( B @ member ) )
& ( universal_class @ ( A @ member ) ) )
=> ( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
& ( ( B @ ( A @ ordered_pair ) @ first )
= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(45,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ sum_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class) ).
thf(274,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ sum_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( A @ domain_of @ ( B @ member ) )
<=> ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain_of) ).
thf(81,plain,
! [A: $i,B: $i] :
( ( ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
& ( universal_class @ ( B @ member ) ) )
=> ( A @ domain_of @ ( B @ member ) ) )
& ( ( A @ domain_of @ ( B @ member ) )
=> ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(301,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[111,49]) ).
thf(302,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ ( sk2 @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[301]) ).
thf(315,plain,
! [A: $i] :
( ( A != universal_class )
| ( sk2 != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[302]) ).
thf(318,plain,
( ( sk2 != null_class )
| ~ ( universal_class @ inductive ) ),
inference(simp,[status(thm)],[315]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ inverse )
= ( universal_class @ ( A @ cross_product ) @ flip @ domain_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse_defn) ).
thf(98,plain,
! [A: $i] :
( ( A @ inverse )
= ( universal_class @ ( A @ cross_product ) @ flip @ domain_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_defn) ).
thf(271,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(592,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( sk7 @ complement @ ( null_class @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[111,573]) ).
thf(593,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( sk7 @ complement @ ( null_class @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[592]) ).
thf(594,plain,
~ ( sk7 @ complement @ inductive ),
inference(pattern_uni,[status(thm)],[593:[bind(A,$thf( sk7 @ complement ))]]) ).
thf(31,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( universal_class @ ( A @ member ) ) )
=> ( universal_class @ ( A @ ( B @ image ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(213,plain,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( universal_class @ ( A @ member ) ) )
=> ( universal_class @ ( A @ ( B @ image ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of_defn) ).
thf(66,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(312,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[111,212]) ).
thf(313,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[312]) ).
thf(314,plain,
~ ( null_class @ inductive ),
inference(pattern_uni,[status(thm)],[313:[bind(A,$thf( null_class )),bind(B,$thf( null_class ))]]) ).
thf(551,plain,
( ( ( sk7 @ inductive )
!= ( null_class @ inductive ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[208,314]) ).
thf(552,plain,
( ( sk7 @ inductive )
!= ( null_class @ inductive ) ),
inference(simp,[status(thm)],[551]) ).
thf(557,plain,
sk7 != null_class,
inference(simp,[status(thm)],[552]) ).
thf(20,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ! [C: $i] :
~ ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',disjoint_defn) ).
thf(150,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
~ ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) )
=> ( B @ ( A @ disjoint ) ) )
& ( ( B @ ( A @ disjoint ) )
=> ! [C: $i] :
~ ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(48,plain,
( ( null_class @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )
!= ( sk2 @ ( sk1 @ ordered_pair ) ) ),
inference(cnfConj,[status(thm)],[47]) ).
thf(50,plain,
( ( sk2 @ ( sk1 @ ordered_pair ) )
!= ( null_class @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[48]) ).
thf(567,plain,
( ( ( universal_class @ ( sk2 @ member ) )
!= ( sk7 @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[305,49]) ).
thf(568,plain,
( ( universal_class @ ( sk2 @ member ) )
!= ( sk7 @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[567]) ).
thf(585,plain,
( ( sk7 != universal_class )
| ( sk2 != null_class ) ),
inference(simp,[status(thm)],[568]) ).
thf(597,plain,
( ( ( sk7 @ complement @ ( null_class @ member ) )
!= ( sk7 @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[305,573]) ).
thf(598,plain,
( ( sk7 @ complement @ ( null_class @ member ) )
!= ( sk7 @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[597]) ).
thf(611,plain,
( ( ( sk7 @ complement )
!= sk7 )
| ( null_class != null_class ) ),
inference(simp,[status(thm)],[598]) ).
thf(613,plain,
( ( sk7 @ complement )
!= sk7 ),
inference(simp,[status(thm)],[611]) ).
thf(21,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
<=> ( ( D @ ( B @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cross_product_defn) ).
thf(158,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( B @ member ) )
& ( C @ ( A @ member ) ) )
=> ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) )
& ( ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
=> ( ( D @ ( B @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(54,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[53]) ).
thf(57,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[54]) ).
thf(58,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[57]) ).
thf(43,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ member ) )
<=> ? [B: $i] :
( ( A
= ( B @ ( B @ ordered_pair ) ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(262,plain,
! [A: $i] :
( ( ? [B: $i] :
( ( A
= ( B @ ( B @ ordered_pair ) ) )
& ( universal_class @ ( B @ member ) ) )
=> ( identity_relation @ ( A @ member ) ) )
& ( ( identity_relation @ ( A @ member ) )
=> ? [B: $i] :
( ( A
= ( B @ ( B @ ordered_pair ) ) )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(36,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ power_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class) ).
thf(239,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ power_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(280,plain,
! [A: $i] :
( ( ( universal_class @ ( sk7 @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[207,212]) ).
thf(281,plain,
! [A: $i] :
( ( universal_class @ ( sk7 @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[280]) ).
thf(282,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( sk7 != A ) ),
inference(simp,[status(thm)],[281]) ).
thf(283,plain,
universal_class != null_class,
inference(simp,[status(thm)],[282]) ).
thf(67,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(cnf,[status(esa)],[66]) ).
thf(68,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(lifteq,[status(thm)],[67]) ).
thf(15,axiom,
! [A: $i] :
( ( A @ singleton )
= ( A @ ( A @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set_defn) ).
thf(122,plain,
! [A: $i] :
( ( A @ singleton )
= ( A @ ( A @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(32,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) )
<=> ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip_defn) ).
thf(215,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) )
=> ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) )
& ( ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) )
=> ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
& ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
<=> ( ( B @ ( A @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation_defn) ).
thf(222,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( A @ member ) )
& ( universal_class @ ( B @ member ) ) )
=> ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) )
& ( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
=> ( ( B @ ( A @ member ) )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subclass ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_defn) ).
thf(90,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subclass ) ) )
& ( ( B @ ( A @ subclass ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(1259,plain,
$false,
inference(e,[status(thm)],[747,288,125,179,512,241,69,101,115,247,46,261,147,132,74,746,260,70,192,165,229,65,256,212,237,244,205,1071,298,279,573,180,191,204,259,71,144,49,172,274,81,318,98,271,208,594,213,66,251,108,557,63,150,50,585,613,314,158,139,58,207,262,51,190,305,210,239,283,68,62,178,254,122,215,222,90,111]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET103+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% 0.00/0.13 % 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
% 0.26/0.52 % Computer : n004.cluster.edu
% 0.26/0.52 % Model : x86_64 x86_64
% 0.26/0.52 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.26/0.52 % Memory : 8046.5625MB
% 0.26/0.52 % OS : Linux 6.8.0-71-generic
% 0.26/0.52 % CPULimit : 300
% 0.26/0.52 % WCLimit : 300
% 0.26/0.52 % DateTime : Sat Sep 26 07:07:51 UTC 2026
% 0.26/0.52 % CPUTime :
% 0.26/0.52 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
% 1.52/1.43 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.99/1.79 % [INFO] Parsing done (360ms).
% 2.41/1.81 % [INFO] Running in sequential loop mode.
% 3.37/2.40 % [INFO] eprover registered as external prover.
% 3.37/2.42 % [INFO] Scanning for conjecture ...
% 3.53/2.59 % [INFO] Found a conjecture (or negated_conjecture) and 43 axioms. Running axiom selection ...
% 3.88/2.71 % [INFO] Axiom selection finished. Selected 43 axioms (removed 0 axioms).
% 4.12/2.81 % [INFO] Problem is first-order (TPTP FOF).
% 4.12/2.82 % [INFO] Type checking passed.
% 4.12/2.83 % [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 ...
% 17.33/6.05 % External prover 'e' found a proof!
% 17.33/6.05 % [INFO] Killing All external provers ...
% 17.33/6.06 % Time passed: 5319ms (effective reasoning time: 4232ms)
% 17.33/6.06 % 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)>
% 17.33/6.06 % Axioms used in derivation (43): sum_class_defn, image_defn, cross_product_defn, extensionality, infinity, first_second, rotate, null_class_defn, sum_class, restrict_defn, compose_defn1, inductive_defn, unordered_pair, cross_product, flip_defn, function_defn, element_relation_defn, identity_relation, unordered_pair_defn, successor_relation_defn1, power_class_defn, intersection, inverse_defn, replacement, element_relation, singleton_set_defn, subclass_defn, domain_of, power_class, flip, compose_defn2, choice, complement, disjoint_defn, apply_defn, regularity, successor_defn, ordered_pair_defn, class_elements_are_sets, range_of_defn, union_defn, successor_relation_defn2, rotate_defn
% 17.33/6.06 % No. of inferences in proof: 182
% 17.33/6.06 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 5319 ms resp. 4232 ms w/o parsing
% 16.31/6.25 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 16.31/6.26 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------