%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET097+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% 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 : n019.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 08:29:24 AM UTC 2026
% Result : Theorem 213.67s 58.93s
% Output : Refutation 214.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 49
% Syntax : Number of formulae : 384 ( 105 unt; 0 typ; 5 def)
% Number of atoms : 1156 ( 354 equ; 0 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 3753 ( 478 ~; 364 |; 141 &;2678 @)
% ( 19 <=>; 73 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 36 usr; 10 con; 0-3 aty)
% Number of variables : 550 ( 3 ^; 515 !; 27 ?; 550 :)
% ( 0 !>; 0 ?*; 0 @-; 5 @+)
% Comments :
%------------------------------------------------------------------------------
thf(null_class_decl,type,
null_class: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $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(successor_decl,type,
successor: $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i > $i > $i ).
thf(sk3_decl,type,
sk3: $i > $i ).
thf(sk5_decl,type,
sk5: $i ).
thf(sk7_decl,type,
sk7: $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ( ? [B: $i] :
( ? [C: $i] : ( A @ ( B @ singleton @ complement @ intersection ) @ ( C @ member ) )
& ( A @ ( B @ member ) ) )
| ? [B: $i] :
( ( B @ singleton )
= A )
| ( A = null_class ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ( A @ ( C @ member ) )
& ( C @ ( B @ member ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( ^ [A: $i] :
@+[B: $i] :
( ( A @ ( B @ disjoint ) )
& ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( @+[A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ) ) ) ).
thf(sk7_def,definition,
( sk7
= ( @+[A: $i] :
( ! [B: $i] :
( ( universal_class @ ( B @ member ) )
=> ( ( B @ ( B @ ( A @ apply ) @ member ) )
| ( B = null_class ) ) )
& ( A @ function ) ) ) ) ).
thf(1,conjecture,
! [A: $i] :
( ? [B: $i] :
( ? [C: $i] : ( A @ ( B @ singleton @ complement @ intersection ) @ ( C @ member ) )
& ( A @ ( B @ member ) ) )
| ? [B: $i] :
( ( B @ singleton )
= A )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ? [B: $i] :
( ? [C: $i] : ( A @ ( B @ singleton @ complement @ intersection ) @ ( C @ member ) )
& ( A @ ( B @ member ) ) )
| ? [B: $i] :
( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(46,plain,
~ ! [A: $i] :
( ? [B: $i] :
( ? [C: $i] : ( A @ ( B @ singleton @ complement @ intersection ) @ ( C @ member ) )
& ( A @ ( B @ member ) ) )
| ? [B: $i] :
( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(47,plain,
! [C: $i,B: $i,A: $i] :
( ( ~ ( sk1 @ ( B @ singleton @ complement @ intersection ) @ ( C @ member ) )
| ~ ( sk1 @ ( B @ member ) ) )
& ( ( A @ singleton )
!= sk1 )
& ( sk1 != null_class ) ),
inference(cnf,[status(esa)],[46]) ).
thf(50,plain,
! [B: $i,A: $i] :
( ~ ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[47]) ).
thf(53,plain,
! [B: $i,A: $i] :
( ~ ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[50]) ).
thf(35,axiom,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_defn) ).
thf(214,plain,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(5,axiom,
! [A: $i,B: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( B @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_defn1) ).
thf(69,plain,
! [A: $i,B: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( B @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(7,axiom,
! [A: $i] :
( ( A @ function )
<=> ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function_defn) ).
thf(73,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)],[7]) ).
thf(74,plain,
( ! [A: $i] :
( ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) )
=> ( A @ function ) )
& ! [A: $i] :
( ( A @ function )
=> ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
& ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ) ) ),
inference(miniscope,[status(thm)],[73]) ).
thf(75,plain,
! [B: $i,A: $i] :
( ( ( B @ function )
| ~ ( identity_relation @ ( B @ inverse @ ( B @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ subclass ) ) )
& ( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) )
& ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ) ),
inference(cnf,[status(esa)],[74]) ).
thf(76,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(6,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(71,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(72,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(cnf,[status(esa)],[71]) ).
thf(3,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( A @ ( B @ subclass ) )
& ( B @ ( A @ subclass ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',extensionality) ).
thf(54,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(55,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)],[54]) ).
thf(56,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)],[55]) ).
thf(59,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnfConj,[status(thm)],[56]) ).
thf(64,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[59]) ).
thf(65,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(simp,[status(thm)],[64]) ).
thf(48,plain,
sk1 != null_class,
inference(cnfConj,[status(thm)],[47]) ).
thf(51,plain,
sk1 != null_class,
inference(lifteq,[status(thm)],[48]) ).
thf(371,plain,
! [B: $i,A: $i] :
( ( A != sk1 )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[65,51]) ).
thf(372,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( sk1 @ ( A @ subclass ) )
| ~ ( A @ ( sk1 @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[371:[bind(A,$thf( sk1 ))]]) ).
thf(599,plain,
( ~ ( sk1 @ ( null_class @ subclass ) )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[372]) ).
thf(1124,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ $true
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[72,599]) ).
thf(1125,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[1124]) ).
thf(1141,plain,
! [A: $i] :
( ( sk1 != universal_class )
| ( A != null_class )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[1125]) ).
thf(1157,plain,
( ( sk1 != universal_class )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[1141]) ).
thf(1441,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ $true
| ~ ( A @ function ) ),
inference(paramod_ordered,[status(thm)],[76,1157]) ).
thf(1442,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ ( A @ function ) ),
inference(simp,[status(thm)],[1441]) ).
thf(1461,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( A != sk1 )
| ( sk1 != universal_class )
| ~ ( A @ function ) ),
inference(simp,[status(thm)],[1442]) ).
thf(1469,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk1 != universal_class )
| ~ ( sk1 @ function ) ),
inference(simp,[status(thm)],[1461]) ).
thf(20,axiom,
! [A: $i] :
( ( A != null_class )
=> ? [B: $i] :
( ( A @ ( B @ disjoint ) )
& ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity) ).
thf(132,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)],[20]) ).
thf(133,plain,
! [A: $i] :
( ( ( A @ ( A @ sk3 @ disjoint ) )
| ( A = null_class ) )
& ( ( A @ ( A @ sk3 @ member ) )
| ( A = null_class ) )
& ( ( universal_class @ ( A @ sk3 @ member ) )
| ( A = null_class ) ) ),
inference(cnf,[status(esa)],[132]) ).
thf(134,plain,
! [A: $i] :
( ( universal_class @ ( A @ sk3 @ member ) )
| ( A = null_class ) ),
inference(cnfConj,[status(thm)],[133]) ).
thf(137,plain,
! [A: $i] :
( ( universal_class @ ( A @ sk3 @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[134]) ).
thf(27,axiom,
? [A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infinity) ).
thf(177,plain,
? [A: $i] :
( ! [B: $i] :
( ( B @ inductive )
=> ( B @ ( A @ subclass ) ) )
& ( A @ inductive )
& ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(178,plain,
! [A: $i] :
( ( ( A @ ( sk5 @ subclass ) )
| ~ ( A @ inductive ) )
& ( sk5 @ inductive )
& ( universal_class @ ( sk5 @ member ) ) ),
inference(cnf,[status(esa)],[177]) ).
thf(179,plain,
universal_class @ ( sk5 @ member ),
inference(cnfConj,[status(thm)],[178]) ).
thf(814,plain,
! [A: $i] :
( ( A != sk5 )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,179]) ).
thf(815,plain,
( ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( sk5 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[814:[bind(A,$thf( sk5 ))]]) ).
thf(1838,plain,
! [A: $i] :
( ( A
!= ( sk5 @ sk3 ) )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,815]) ).
thf(1839,plain,
! [A: $i] :
( ( A
!= ( sk5 @ sk3 ) )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(simp,[status(thm)],[1838]) ).
thf(1840,plain,
( ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( sk5 @ sk3 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[1839:[bind(A,$thf( sk5 @ sk3 ))]]) ).
thf(12,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip) ).
thf(87,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(88,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(cnf,[status(esa)],[87]) ).
thf(9,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation) ).
thf(83,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(1349,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,1157]) ).
thf(1350,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1349]) ).
thf(1369,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk1 != element_relation )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1350]) ).
thf(135,plain,
! [A: $i] :
( ( A @ ( A @ sk3 @ member ) )
| ( A = null_class ) ),
inference(cnfConj,[status(thm)],[133]) ).
thf(138,plain,
! [A: $i] :
( ( A @ ( A @ sk3 @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[135]) ).
thf(39,axiom,
? [A: $i] :
( ! [B: $i] :
( ( universal_class @ ( B @ member ) )
=> ( ( B @ ( B @ ( A @ apply ) @ member ) )
| ( B = null_class ) ) )
& ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice) ).
thf(239,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)],[39]) ).
thf(240,plain,
! [A: $i] :
( ( ( A @ ( A @ ( sk7 @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) )
& ( sk7 @ function ) ),
inference(cnf,[status(esa)],[239]) ).
thf(241,plain,
sk7 @ function,
inference(cnfConj,[status(thm)],[240]) ).
thf(976,plain,
! [A: $i] :
( ( A != sk7 )
| ( null_class @ function )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,241]) ).
thf(977,plain,
( ( null_class @ function )
| ( universal_class @ ( sk7 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[976:[bind(A,$thf( sk7 ))]]) ).
thf(18,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/sandbox2/benchmark/theBenchmark.p',compose_defn2) ).
thf(118,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)],[18]) ).
thf(119,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: $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 ) ) ) ) ),
inference(miniscope,[status(thm)],[118]) ).
thf(120,plain,
! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( F @ compose ) @ ( H @ ( G @ ordered_pair ) @ member ) )
| ~ ( G @ singleton @ ( E @ image ) @ ( F @ image ) @ ( H @ member ) )
| ~ ( universal_class @ ( G @ member ) ) )
& ( ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) ) )
& ( ( universal_class @ ( C @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) ) ) ),
inference(cnf,[status(esa)],[119]) ).
thf(122,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) ) ),
inference(cnfConj,[status(thm)],[120]) ).
thf(44,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_defn) ).
thf(274,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(275,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
inference(cnf,[status(esa)],[274]) ).
thf(276,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
= ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[275]) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image_defn) ).
thf(80,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(81,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
inference(cnf,[status(esa)],[80]) ).
thf(82,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ range_of ) ),
inference(lifteq,[status(thm)],[81]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of_defn) ).
thf(89,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(90,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(cnf,[status(esa)],[89]) ).
thf(91,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[90]) ).
thf(1521,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ ( B @ restrict ) ) @ inverse @ domain_of ) ),
inference(rewrite,[status(thm)],[82,91]) ).
thf(42,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restrict_defn) ).
thf(262,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)],[42]) ).
thf(263,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
inference(cnf,[status(esa)],[262]) ).
thf(264,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ ( B @ restrict ) ) )
= ( C @ ( A @ cross_product ) @ ( B @ intersection ) ) ),
inference(lifteq,[status(thm)],[263]) ).
thf(1737,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ image ) )
= ( universal_class @ ( A @ cross_product ) @ ( B @ intersection ) @ inverse @ domain_of ) ),
inference(rewrite,[status(thm)],[1521,264]) ).
thf(6022,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( C @ singleton @ ( A @ image ) @ cross_product ) @ ( B @ intersection ) @ inverse @ domain_of @ ( D @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ singleton @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ member ) ) ),
inference(rewrite,[status(thm)],[122,276,1737]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ singleton )
= ( A @ ( A @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set_defn) ).
thf(115,plain,
! [A: $i] :
( ( A @ singleton )
= ( A @ ( A @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( B @ sum_class @ ( A @ member ) )
<=> ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_defn) ).
thf(92,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)],[14]) ).
thf(93,plain,
( ! [A: $i,B: $i] :
( ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ sum_class @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ sum_class @ ( A @ member ) )
=> ? [C: $i] :
( ( B @ ( C @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[92]) ).
thf(94,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ sum_class @ ( C @ member ) )
| ~ ( D @ ( E @ member ) )
| ~ ( E @ ( C @ member ) ) )
& ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
| ~ ( B @ sum_class @ ( A @ member ) ) )
& ( ( A @ ( B @ sk2 ) @ ( A @ member ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[93]) ).
thf(96,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ sk2 ) @ member ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(28,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_defn) ).
thf(182,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(183,plain,
~ ? [A: $i] : ( null_class @ ( A @ member ) ),
inference(miniscope,[status(thm)],[182]) ).
thf(184,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(cnf,[status(esa)],[183]) ).
thf(2479,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[96,184]) ).
thf(2480,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2479]) ).
thf(2481,plain,
! [A: $i] :
~ ( null_class @ sum_class @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2480:[bind(A,$thf( E )),bind(B,$thf( null_class )),bind(C,$thf( E @ ( null_class @ sk2 ) ))]]) ).
thf(2563,plain,
! [A: $i] :
~ ( null_class @ sum_class @ ( A @ member ) ),
inference(simp,[status(thm)],[2481]) ).
thf(10709,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ sk3 @ member ) )
!= ( null_class @ sum_class @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[138,2563]) ).
thf(10710,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ sk3 @ member ) )
!= ( null_class @ sum_class @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[10709]) ).
thf(10711,plain,
( ( null_class @ sum_class )
= null_class ),
inference(pattern_uni,[status(thm)],[10710:[bind(A,$thf( null_class @ sum_class )),bind(B,$thf( null_class @ sum_class @ sk3 ))]]) ).
thf(1126,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,599]) ).
thf(1127,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1126]) ).
thf(1142,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk1 != element_relation )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1127]) ).
thf(16,axiom,
! [A: $i,B: $i] :
( ( B @ power_class @ ( A @ member ) )
<=> ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_defn) ).
thf(108,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)],[16]) ).
thf(109,plain,
( ! [A: $i,B: $i] :
( ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) )
=> ( B @ power_class @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ power_class @ ( A @ member ) )
=> ( ( B @ ( A @ subclass ) )
& ( universal_class @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[108]) ).
thf(110,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ power_class @ ( C @ member ) )
| ~ ( D @ ( C @ subclass ) )
| ~ ( universal_class @ ( C @ member ) ) )
& ( ( B @ ( A @ subclass ) )
| ~ ( B @ power_class @ ( A @ member ) ) )
& ( ( universal_class @ ( A @ member ) )
| ~ ( B @ power_class @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[109]) ).
thf(112,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[110]) ).
thf(279,plain,
! [A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,184]) ).
thf(280,plain,
! [A: $i] :
( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[279]) ).
thf(281,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( sk5 != A ) ),
inference(simp,[status(thm)],[280]) ).
thf(282,plain,
universal_class != null_class,
inference(simp,[status(thm)],[281]) ).
thf(357,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[65,282]) ).
thf(358,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[357:[bind(A,$thf( universal_class ))]]) ).
thf(595,plain,
( ~ ( universal_class @ ( null_class @ subclass ) )
| ~ ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[358]) ).
thf(768,plain,
( ~ $true
| ~ ( null_class @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[595,72]) ).
thf(769,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[768]) ).
thf(4469,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[112,769]) ).
thf(4470,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4469]) ).
thf(4471,plain,
~ ( null_class @ power_class @ ( universal_class @ member ) ),
inference(pattern_uni,[status(thm)],[4470:[bind(A,$thf( universal_class )),bind(B,$thf( null_class ))]]) ).
thf(4536,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,4471]) ).
thf(4537,plain,
( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) ),
inference(simp,[status(thm)],[4536]) ).
thf(4592,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( sk5 != universal_class ) ),
inference(simp,[status(thm)],[4537]) ).
thf(36,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subclass ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_defn) ).
thf(217,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)],[36]) ).
thf(865,plain,
! [A: $i] :
( ( A != sk1 )
| ( null_class != null_class )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,51]) ).
thf(866,plain,
! [A: $i] :
( ( A != sk1 )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(simp,[status(thm)],[865]) ).
thf(867,plain,
universal_class @ ( sk1 @ sk3 @ member ),
inference(pattern_uni,[status(thm)],[866:[bind(A,$thf( sk1 ))]]) ).
thf(287,plain,
! [B: $i,A: $i] :
( ~ $true
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[53]) ).
thf(291,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 )
| ( B != A )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[287]) ).
thf(294,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[291]) ).
thf(1063,plain,
! [A: $i] :
( ( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[867,294]) ).
thf(1064,plain,
! [A: $i] :
( ( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[1063]) ).
thf(1072,plain,
! [A: $i] :
( ( sk1 != universal_class )
| ( ( sk1 @ sk3 )
!= A )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[1064]) ).
thf(1078,plain,
( ( sk1 != universal_class )
| ( ( sk1 @ ( sk1 @ sk3 @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[1072]) ).
thf(1859,plain,
( ~ $true
| ( ( universal_class @ ( sk5 @ sk3 @ member ) )
!= ( universal_class @ ( null_class @ member ) ) )
| ( universal_class @ ( null_class @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[815]) ).
thf(1865,plain,
( ( universal_class != universal_class )
| ( ( sk5 @ sk3 )
!= null_class )
| ( universal_class @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1859]) ).
thf(1872,plain,
( ( ( sk5 @ sk3 )
!= null_class )
| ( universal_class @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1865]) ).
thf(29,axiom,
! [A: $i,B: $i] :
( ( ( B @ function )
& ( universal_class @ ( A @ member ) ) )
=> ( universal_class @ ( A @ ( B @ image ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(185,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)],[29]) ).
thf(11,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation_defn1) ).
thf(86,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(1345,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[86,1157]) ).
thf(1346,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1345]) ).
thf(1366,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk1 != successor_relation )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1346]) ).
thf(116,plain,
! [A: $i] :
( ( A @ singleton )
= ( A @ ( A @ unordered_pair ) ) ),
inference(cnf,[status(esa)],[115]) ).
thf(117,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[116]) ).
thf(26,axiom,
! [A: $i,B: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair) ).
thf(175,plain,
! [A: $i,B: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(176,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(cnf,[status(esa)],[175]) ).
thf(1264,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( universal_class @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[117,176]) ).
thf(1265,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1264:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(49,plain,
! [A: $i] :
( ( A @ singleton )
!= sk1 ),
inference(cnfConj,[status(thm)],[47]) ).
thf(52,plain,
! [A: $i] :
( ( A @ singleton )
!= sk1 ),
inference(lifteq,[status(thm)],[49]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ intersection ) @ ( C @ member ) )
<=> ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection) ).
thf(125,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)],[19]) ).
thf(852,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,86]) ).
thf(853,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( successor_relation @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[852:[bind(A,$thf( successor_relation ))]]) ).
thf(1455,plain,
! [A: $i] :
( ( ( sk7 @ function )
!= ( A @ function ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[241,76]) ).
thf(1456,plain,
! [A: $i] :
( ( ( sk7 @ function )
!= ( A @ function ) )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[1455]) ).
thf(1457,plain,
universal_class @ ( universal_class @ cross_product ) @ ( sk7 @ subclass ),
inference(pattern_uni,[status(thm)],[1456:[bind(A,$thf( sk7 ))]]) ).
thf(1475,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( sk7 @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1457,1157]) ).
thf(1476,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( sk7 @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1475]) ).
thf(1514,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk7 != sk1 )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[1476]) ).
thf(215,plain,
! [A: $i] :
( ( A @ successor )
= ( A @ singleton @ ( A @ union ) ) ),
inference(cnf,[status(esa)],[214]) ).
thf(216,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[215]) ).
thf(4576,plain,
( ( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[867,4471]) ).
thf(4577,plain,
( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) ),
inference(simp,[status(thm)],[4576]) ).
thf(4583,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( ( sk1 @ sk3 )
!= universal_class ) ),
inference(simp,[status(thm)],[4577]) ).
thf(10,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate) ).
thf(84,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(4463,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) )
| ~ $true
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[112,599]) ).
thf(4464,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4463]) ).
thf(4465,plain,
( ~ ( sk1 @ ( null_class @ subclass ) )
| ~ ( null_class @ power_class @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[4464:[bind(A,$thf( sk1 )),bind(B,$thf( null_class ))]]) ).
thf(5082,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ $true
| ~ ( null_class @ power_class @ ( sk1 @ member ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[112,4465]) ).
thf(5083,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ ( null_class @ power_class @ ( sk1 @ member ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[5082]) ).
thf(5084,plain,
( ~ ( null_class @ power_class @ ( sk1 @ member ) )
| ~ ( sk1 @ power_class @ ( null_class @ member ) ) ),
inference(pattern_uni,[status(thm)],[5083:[bind(A,$thf( null_class )),bind(B,$thf( sk1 ))]]) ).
thf(57,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[56]) ).
thf(60,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[57]) ).
thf(61,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[60]) ).
thf(10549,plain,
! [A: $i] :
( ( A != sk1 )
| ( null_class != null_class )
| ( A @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[138,51]) ).
thf(10550,plain,
! [A: $i] :
( ( A != sk1 )
| ( A @ ( A @ sk3 @ member ) ) ),
inference(simp,[status(thm)],[10549]) ).
thf(10551,plain,
sk1 @ ( sk1 @ sk3 @ member ),
inference(pattern_uni,[status(thm)],[10550:[bind(A,$thf( sk1 ))]]) ).
thf(285,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true
| ~ ( sk1 @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[179,53]) ).
thf(286,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[285]) ).
thf(289,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= universal_class )
| ( sk5 != B )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[286]) ).
thf(293,plain,
! [A: $i] :
( ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= universal_class )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[289]) ).
thf(43,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ member ) )
<=> ? [B: $i] :
( ( A
= ( B @ ( B @ ordered_pair ) ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(265,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(283,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ~ ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,53]) ).
thf(284,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ~ ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[283]) ).
thf(288,plain,
! [B: $i,A: $i] :
( ( sk1 != universal_class )
| ( sk5 != A )
| ~ ( sk1 @ ( A @ singleton @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[284]) ).
thf(292,plain,
! [A: $i] :
( ( sk1 != universal_class )
| ~ ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[288]) ).
thf(15,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/sandbox2/benchmark/theBenchmark.p',domain_of) ).
thf(99,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)],[15]) ).
thf(100,plain,
( ! [A: $i,B: $i] :
( ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
& ( universal_class @ ( B @ member ) ) )
=> ( A @ domain_of @ ( B @ member ) ) )
& ! [A: $i,B: $i] :
( ( A @ domain_of @ ( B @ member ) )
=> ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
& ( universal_class @ ( B @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[99]) ).
thf(101,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ domain_of @ ( D @ member ) )
| ( ( universal_class @ ( D @ singleton @ ( C @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( D @ member ) ) )
& ( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
| ~ ( A @ domain_of @ ( B @ member ) ) )
& ( ( universal_class @ ( B @ member ) )
| ~ ( A @ domain_of @ ( B @ member ) ) ) ),
inference(cnf,[status(esa)],[100]) ).
thf(102,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( A @ domain_of @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[101]) ).
thf(85,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(cnf,[status(esa)],[84]) ).
thf(32,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ union ) @ ( C @ member ) )
<=> ( ( B @ ( C @ member ) )
| ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_defn) ).
thf(201,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)],[32]) ).
thf(1041,plain,
! [A: $i] :
( ( A
!= ( sk1 @ sk3 ) )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,867]) ).
thf(1042,plain,
( ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( sk1 @ sk3 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[1041:[bind(A,$thf( sk1 @ sk3 ))]]) ).
thf(860,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,282]) ).
thf(861,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(simp,[status(thm)],[860]) ).
thf(862,plain,
universal_class @ ( universal_class @ sk3 @ member ),
inference(pattern_uni,[status(thm)],[861:[bind(A,$thf( universal_class ))]]) ).
thf(70,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( B @ compose ) @ subclass ) ),
inference(cnf,[status(esa)],[69]) ).
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/sandbox2/benchmark/theBenchmark.p',successor_relation_defn2) ).
thf(165,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(888,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,83]) ).
thf(889,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( element_relation @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[888:[bind(A,$thf( element_relation ))]]) ).
thf(4466,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ $true
| ~ ( null_class @ ( sk1 @ subclass ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[112,599]) ).
thf(4467,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( sk1 @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( sk1 @ subclass ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4466]) ).
thf(4468,plain,
( ~ ( null_class @ ( sk1 @ subclass ) )
| ~ ( sk1 @ power_class @ ( null_class @ member ) ) ),
inference(pattern_uni,[status(thm)],[4467:[bind(A,$thf( null_class )),bind(B,$thf( sk1 ))]]) ).
thf(5555,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ power_class @ ( null_class @ member ) ) )
| ~ ( null_class @ ( sk1 @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,4468]) ).
thf(5556,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ power_class @ ( null_class @ member ) ) )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[5555]) ).
thf(5658,plain,
( ( ( sk1 @ power_class )
!= universal_class )
| ( sk5 != null_class )
| ~ ( null_class @ ( sk1 @ subclass ) ) ),
inference(simp,[status(thm)],[5556]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ inductive )
<=> ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
& ( A @ ( null_class @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive_defn) ).
thf(225,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)],[37]) ).
thf(77,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(4423,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ $true
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[112,1157]) ).
thf(4424,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ( sk1 != universal_class )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4423]) ).
thf(4425,plain,
( ( sk1 != universal_class )
| ~ ( null_class @ power_class @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[4424:[bind(A,$thf( sk1 )),bind(B,$thf( null_class ))]]) ).
thf(34,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ power_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class) ).
thf(212,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ power_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(1477,plain,
! [A: $i] :
( ( A != sk7 )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,1457]) ).
thf(1478,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( sk7 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[1477:[bind(A,$thf( sk7 ))]]) ).
thf(1666,plain,
! [A: $i] :
( ( ( sk7 @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[241,77]) ).
thf(1667,plain,
! [A: $i] :
( ( ( sk7 @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[1666]) ).
thf(1668,plain,
identity_relation @ ( sk7 @ inverse @ ( sk7 @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[1667:[bind(A,$thf( sk7 ))]]) ).
thf(126,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) )
=> ( B @ ( A @ intersection ) @ ( C @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ intersection ) @ ( C @ member ) )
=> ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[125]) ).
thf(127,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ intersection ) @ ( F @ member ) )
| ~ ( E @ ( F @ member ) )
| ~ ( D @ ( F @ member ) ) )
& ( ( B @ ( C @ member ) )
| ~ ( B @ ( A @ intersection ) @ ( C @ member ) ) )
& ( ( A @ ( C @ member ) )
| ~ ( B @ ( A @ intersection ) @ ( C @ member ) ) ) ),
inference(cnf,[status(esa)],[126]) ).
thf(129,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( B @ ( A @ intersection ) @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[127]) ).
thf(128,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ member ) )
| ~ ( B @ ( A @ intersection ) @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[127]) ).
thf(40,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/sandbox2/benchmark/theBenchmark.p',first_second) ).
thf(244,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)],[40]) ).
thf(1453,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ function ) ),
inference(paramod_ordered,[status(thm)],[76,769]) ).
thf(1454,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ function ) ),
inference(simp,[status(thm)],[1453]) ).
thf(1458,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( A != universal_class )
| ~ ( A @ function ) ),
inference(simp,[status(thm)],[1454]) ).
thf(1466,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ~ ( universal_class @ function ) ),
inference(simp,[status(thm)],[1458]) ).
thf(2599,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ sum_class @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[96,2563]) ).
thf(2600,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ sum_class @ ( C @ member ) ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2599]) ).
thf(2601,plain,
! [A: $i] :
~ ( null_class @ sum_class @ sum_class @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2600:[bind(A,$thf( E )),bind(B,$thf( null_class @ sum_class )),bind(C,$thf( E @ ( null_class @ sum_class @ sk2 ) ))]]) ).
thf(2627,plain,
! [A: $i] :
~ ( null_class @ sum_class @ sum_class @ ( A @ member ) ),
inference(simp,[status(thm)],[2601]) ).
thf(2931,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ sum_class @ sum_class @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[96,2627]) ).
thf(2932,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk2 ) @ member ) )
!= ( null_class @ sum_class @ sum_class @ ( C @ member ) ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2931]) ).
thf(2933,plain,
! [A: $i] :
~ ( null_class @ sum_class @ sum_class @ sum_class @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2932:[bind(A,$thf( E )),bind(B,$thf( null_class @ sum_class @ sum_class )),bind(C,$thf( E @ ( null_class @ sum_class @ sum_class @ sk2 ) ))]]) ).
thf(2984,plain,
! [A: $i] :
~ ( null_class @ sum_class @ sum_class @ sum_class @ ( A @ member ) ),
inference(simp,[status(thm)],[2933]) ).
thf(3134,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( null_class @ sum_class @ sum_class @ sum_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1265,2984]) ).
thf(3135,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( null_class @ sum_class @ sum_class @ sum_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3134]) ).
thf(3188,plain,
! [B: $i,A: $i] :
( ( ( null_class @ sum_class @ sum_class @ sum_class )
!= universal_class )
| ( ( A @ singleton )
!= B ) ),
inference(simp,[status(thm)],[3135]) ).
thf(3218,plain,
( ( null_class @ sum_class @ sum_class @ sum_class )
!= universal_class ),
inference(simp,[status(thm)],[3188]) ).
thf(11005,plain,
( ( null_class @ sum_class @ sum_class )
!= universal_class ),
inference(rewrite,[status(thm)],[3218,10711]) ).
thf(11196,plain,
( ( null_class @ sum_class )
!= universal_class ),
inference(rewrite,[status(thm)],[11005,10711]) ).
thf(11082,plain,
! [A: $i] :
( ( ( sk1 @ ( sk1 @ sk3 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[10551,293]) ).
thf(11083,plain,
! [A: $i] :
( ( ( sk1 @ ( sk1 @ sk3 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= universal_class ) ),
inference(simp,[status(thm)],[11082]) ).
thf(11084,plain,
( ( sk1 @ ( sk1 @ sk3 @ singleton @ complement @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[11083:[bind(A,$thf( sk1 @ sk3 ))]]) ).
thf(11076,plain,
( ( ( sk1 @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[10551,4471]) ).
thf(11077,plain,
( ( sk1 @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) ),
inference(simp,[status(thm)],[11076]) ).
thf(11153,plain,
( ( ( null_class @ power_class )
!= sk1 )
| ( ( sk1 @ sk3 )
!= universal_class ) ),
inference(simp,[status(thm)],[11077]) ).
thf(295,plain,
! [A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) @ ( A @ member ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,292]) ).
thf(296,plain,
! [A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) @ ( A @ member ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[295]) ).
thf(297,plain,
! [A: $i] :
( ( ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) )
!= universal_class )
| ( sk5 != A )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[296]) ).
thf(298,plain,
( ( ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) )
!= universal_class )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[297]) ).
thf(4540,plain,
( ( ( universal_class @ ( universal_class @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[862,4471]) ).
thf(4541,plain,
( ( universal_class @ ( universal_class @ sk3 @ member ) )
!= ( null_class @ power_class @ ( universal_class @ member ) ) ),
inference(simp,[status(thm)],[4540]) ).
thf(4594,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( ( universal_class @ sk3 )
!= universal_class ) ),
inference(simp,[status(thm)],[4541]) ).
thf(33,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply_defn) ).
thf(209,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(210,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(cnf,[status(esa)],[209]) ).
thf(211,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[210]) ).
thf(1747,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( universal_class @ ( B @ singleton @ cross_product ) @ ( A @ intersection ) @ inverse @ domain_of @ sum_class ) ),
inference(rewrite,[status(thm)],[211,1737]) ).
thf(180,plain,
sk5 @ inductive,
inference(cnfConj,[status(thm)],[178]) ).
thf(103,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class )
| ~ ( A @ domain_of @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[101]) ).
thf(105,plain,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
inference(lifteq,[status(thm)],[103]) ).
thf(3302,plain,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ cross_product ) @ ( A @ intersection ) )
!= null_class ) ),
inference(rewrite,[status(thm)],[105,264]) ).
thf(30,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/sandbox2/benchmark/theBenchmark.p',flip_defn) ).
thf(187,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)],[30]) ).
thf(1915,plain,
( ~ $true
| ( ( universal_class @ ( sk1 @ sk3 @ sk3 @ member ) )
!= ( universal_class @ ( null_class @ member ) ) )
| ( universal_class @ ( null_class @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[1042]) ).
thf(1918,plain,
( ( universal_class != universal_class )
| ( ( sk1 @ sk3 @ sk3 )
!= null_class )
| ( universal_class @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1915]) ).
thf(1933,plain,
( ( ( sk1 @ sk3 @ sk3 )
!= null_class )
| ( universal_class @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1918]) ).
thf(97,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ sum_class @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(98,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ sum_class @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(simp,[status(thm)],[97]) ).
thf(4,axiom,
! [A: $i] :
( ( A @ inverse )
= ( universal_class @ ( A @ cross_product ) @ flip @ domain_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse_defn) ).
thf(66,plain,
! [A: $i] :
( ( A @ inverse )
= ( universal_class @ ( A @ cross_product ) @ flip @ domain_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(1726,plain,
( ( ( identity_relation @ ( sk7 @ inverse @ ( sk7 @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1668,769]) ).
thf(1727,plain,
( ( identity_relation @ ( sk7 @ inverse @ ( sk7 @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1726]) ).
thf(1731,plain,
( ( identity_relation != null_class )
| ( ( sk7 @ inverse @ ( sk7 @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[1727]) ).
thf(21,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ cross_product ) @ ( C @ member ) )
=> ( C
= ( C @ second @ ( C @ first @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cross_product) ).
thf(140,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)],[21]) ).
thf(303,plain,
! [A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,294]) ).
thf(304,plain,
! [A: $i] :
( ( ( universal_class @ ( sk5 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[303]) ).
thf(305,plain,
! [A: $i] :
( ( sk1 != universal_class )
| ( sk5 != A )
| ( ( sk1 @ ( A @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[304]) ).
thf(306,plain,
( ( sk1 != universal_class )
| ( ( sk1 @ ( sk5 @ singleton @ complement @ intersection ) )
!= sk1 ) ),
inference(simp,[status(thm)],[305]) ).
thf(780,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[86,769]) ).
thf(781,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[780]) ).
thf(795,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[781]) ).
thf(31,axiom,
! [A: $i,B: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
<=> ( ( B @ ( A @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation_defn) ).
thf(194,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)],[31]) ).
thf(854,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( successor_relation @ subclass ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,86]) ).
thf(855,plain,
( ( null_class @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[854:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(1511,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( sk7 @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1457,769]) ).
thf(1512,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( sk7 @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1511]) ).
thf(1516,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk7 != universal_class ) ),
inference(simp,[status(thm)],[1512]) ).
thf(95,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk2 ) @ ( A @ member ) )
| ~ ( B @ sum_class @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(121,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( C @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) ) ),
inference(cnfConj,[status(thm)],[120]) ).
thf(5685,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( C @ member ) )
| ~ ( A @ ( B @ compose ) @ ( D @ singleton @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ member ) ) ),
inference(rewrite,[status(thm)],[121,276]) ).
thf(22,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ disjoint ) )
<=> ! [C: $i] :
~ ( ( B @ ( C @ member ) )
& ( A @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',disjoint_defn) ).
thf(143,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)],[22]) ).
thf(1239,plain,
! [A: $i] :
( ( A
!= ( universal_class @ sk3 ) )
| ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,862]) ).
thf(1240,plain,
( ( universal_class @ ( null_class @ member ) )
| ( universal_class @ ( universal_class @ sk3 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[1239:[bind(A,$thf( universal_class @ sk3 ))]]) ).
thf(1118,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[86,599]) ).
thf(1119,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( sk1 @ subclass ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1118]) ).
thf(1140,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( sk1 != successor_relation )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1119]) ).
thf(41,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( ( A = C )
| ( A = B ) )
& ( universal_class @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_defn) ).
thf(250,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)],[41]) ).
thf(890,plain,
! [A: $i] :
( ( A
!= ( universal_class @ ( universal_class @ cross_product ) ) )
| ( null_class @ ( element_relation @ subclass ) )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,83]) ).
thf(891,plain,
( ( null_class @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[890:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) ))]]) ).
thf(24,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/sandbox2/benchmark/theBenchmark.p',rotate_defn) ).
thf(158,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)],[24]) ).
thf(113,plain,
! [B: $i,A: $i] :
( ( B @ power_class @ ( A @ member ) )
| ~ ( B @ ( A @ subclass ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[110]) ).
thf(114,plain,
! [B: $i,A: $i] :
( ( B @ power_class @ ( A @ member ) )
| ~ ( B @ ( A @ subclass ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[113]) ).
thf(10224,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ sk3 @ member ) )
!= ( null_class @ sum_class @ sum_class @ sum_class @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[138,2984]) ).
thf(10225,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ sk3 @ member ) )
!= ( null_class @ sum_class @ sum_class @ sum_class @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[10224]) ).
thf(10226,plain,
( ( null_class @ sum_class @ sum_class @ sum_class )
= null_class ),
inference(pattern_uni,[status(thm)],[10225:[bind(A,$thf( null_class @ sum_class @ sum_class @ sum_class )),bind(B,$thf( null_class @ sum_class @ sum_class @ sum_class @ sk3 ))]]) ).
thf(11259,plain,
( ( null_class @ sum_class @ sum_class )
= null_class ),
inference(rewrite,[status(thm)],[10226,10711]) ).
thf(136,plain,
! [A: $i] :
( ( A @ ( A @ sk3 @ disjoint ) )
| ( A = null_class ) ),
inference(cnfConj,[status(thm)],[133]) ).
thf(139,plain,
! [A: $i] :
( ( A @ ( A @ sk3 @ disjoint ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[136]) ).
thf(5016,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,4465]) ).
thf(5017,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[5016]) ).
thf(5104,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( sk5 != sk1 )
| ~ ( sk1 @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[5017]) ).
thf(104,plain,
! [B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[101]) ).
thf(106,plain,
! [B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
= null_class ) ),
inference(lifteq,[status(thm)],[104]) ).
thf(107,plain,
! [B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
= null_class ) ),
inference(simp,[status(thm)],[106]) ).
thf(3662,plain,
! [B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ cross_product ) @ ( A @ intersection ) )
= null_class ) ),
inference(rewrite,[status(thm)],[107,264]) ).
thf(23,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/sandbox2/benchmark/theBenchmark.p',cross_product_defn) ).
thf(151,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)],[23]) ).
thf(4722,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[179,4425]) ).
thf(4723,plain,
( ( ( universal_class @ ( sk5 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4722]) ).
thf(4787,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( sk5 != sk1 )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4723]) ).
thf(784,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,769]) ).
thf(785,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[784]) ).
thf(790,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[785]) ).
thf(123,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) )
| ~ ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
| ~ ( universal_class @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[120]) ).
thf(124,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( B @ compose ) @ ( D @ ( C @ ordered_pair ) @ member ) )
| ~ ( C @ singleton @ ( A @ image ) @ ( B @ image ) @ ( D @ member ) )
| ~ ( universal_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[123]) ).
thf(6600,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( B @ compose ) @ ( D @ singleton @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ member ) )
| ~ ( universal_class @ ( C @ singleton @ ( A @ image ) @ cross_product ) @ ( B @ intersection ) @ inverse @ domain_of @ ( D @ member ) )
| ~ ( universal_class @ ( C @ member ) ) ),
inference(rewrite,[status(thm)],[124,276,1737]) ).
thf(78,plain,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(79,plain,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[78]) ).
thf(12022,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ( A @ ( A @ sk3 @ disjoint ) ) ),
inference(paramod_ordered,[status(thm)],[139,282]) ).
thf(12023,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ sk3 @ disjoint ) ) ),
inference(simp,[status(thm)],[12022]) ).
thf(12024,plain,
universal_class @ ( universal_class @ sk3 @ disjoint ),
inference(pattern_uni,[status(thm)],[12023:[bind(A,$thf( universal_class ))]]) ).
thf(954,plain,
! [A: $i] :
( ( A != sk5 )
| ( null_class @ inductive )
| ( universal_class @ ( A @ sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,180]) ).
thf(955,plain,
( ( null_class @ inductive )
| ( universal_class @ ( sk5 @ sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[954:[bind(A,$thf( sk5 ))]]) ).
thf(4768,plain,
( ( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[867,4425]) ).
thf(4769,plain,
( ( ( universal_class @ ( sk1 @ sk3 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4768]) ).
thf(4777,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( ( sk1 @ sk3 )
!= sk1 )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4769]) ).
thf(130,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ intersection ) @ ( C @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[127]) ).
thf(131,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ intersection ) @ ( C @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) ) ),
inference(simp,[status(thm)],[130]) ).
thf(67,plain,
! [A: $i] :
( ( A @ inverse )
= ( universal_class @ ( A @ cross_product ) @ flip @ domain_of ) ),
inference(cnf,[status(esa)],[66]) ).
thf(68,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[67]) ).
thf(45,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ sum_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class) ).
thf(277,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
=> ( universal_class @ ( A @ sum_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(111,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( B @ power_class @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[110]) ).
thf(38,axiom,
! [A: $i,B: $i] :
( ( A @ complement @ ( B @ member ) )
<=> ( ~ ( A @ ( B @ member ) )
& ( universal_class @ ( B @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement) ).
thf(232,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)],[38]) ).
thf(4726,plain,
( ( ( universal_class @ ( universal_class @ sk3 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[862,4425]) ).
thf(4727,plain,
( ( ( universal_class @ ( universal_class @ sk3 @ member ) )
!= ( null_class @ power_class @ ( sk1 @ member ) ) )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4726]) ).
thf(4784,plain,
( ( ( null_class @ power_class )
!= universal_class )
| ( ( universal_class @ sk3 )
!= sk1 )
| ( sk1 != universal_class ) ),
inference(simp,[status(thm)],[4727]) ).
thf(189060,plain,
$false,
inference(e,[status(thm)],[53,214,69,1469,1840,88,1369,138,977,6022,115,10711,1142,4592,217,276,1078,1872,185,1366,1265,52,184,125,853,1514,46,4471,216,4583,84,5084,179,61,10551,293,132,89,265,292,117,102,85,201,1042,862,70,137,165,815,92,65,1457,889,5658,225,77,4425,212,96,1478,1668,129,73,128,244,1466,11196,11084,11153,298,4594,1747,264,180,176,4465,118,71,54,3302,86,187,274,1933,76,98,599,91,66,769,108,1731,140,306,241,80,795,209,112,194,855,1516,282,95,177,182,72,5685,143,1240,175,1140,250,99,87,1157,891,4468,158,114,11259,139,5104,3662,151,262,51,4787,790,867,6600,239,79,12024,955,294,4777,131,68,277,111,83,232,4784,1737]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET097+1 : TPTP v9.3.1. Bugfixed v5.4.0.
% 0.00/0.08 % 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
% 0.13/0.39 % Computer : n019.cluster.edu
% 0.13/0.39 % Model : x86_64 x86_64
% 0.13/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.39 % Memory : 8046.5625MB
% 0.13/0.39 % OS : Linux 6.8.0-71-generic
% 0.13/0.39 % CPULimit : 300
% 0.13/0.39 % WCLimit : 300
% 0.13/0.39 % DateTime : Sat Sep 26 07:06:48 UTC 2026
% 0.13/0.39 % CPUTime :
% 0.13/0.40 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
% 0.87/0.97 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.49/1.22 % [INFO] Parsing done (246ms).
% 1.49/1.24 % [INFO] Running in sequential loop mode.
% 2.15/1.64 % [INFO] eprover registered as external prover.
% 2.15/1.65 % [INFO] Scanning for conjecture ...
% 2.36/1.77 % [INFO] Found a conjecture (or negated_conjecture) and 43 axioms. Running axiom selection ...
% 2.54/1.86 % [INFO] Axiom selection finished. Selected 43 axioms (removed 0 axioms).
% 2.77/1.93 % [INFO] Problem is first-order (TPTP FOF).
% 2.77/1.94 % [INFO] Type checking passed.
% 2.77/1.95 % [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 ...
% 213.67/58.92 % External prover 'e' found a proof!
% 213.67/58.92 % [INFO] Killing All external provers ...
% 213.67/58.92 % Time passed: 58386ms (effective reasoning time: 57680ms)
% 213.67/58.92 % 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)>
% 213.67/58.93 % 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
% 213.67/58.93 % No. of inferences in proof: 379
% 213.67/58.93 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 58386 ms resp. 57680 ms w/o parsing
% 214.27/59.08 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 214.27/59.08 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------