%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET071-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 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 : n020.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 08:29:13 AM UTC 2026
% Result : Unsatisfiable 92.07s 29.38s
% Output : Refutation 93.88s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 94
% Syntax : Number of formulae : 602 ( 180 unt; 0 typ; 0 def)
% Number of atoms : 1445 ( 607 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 5693 ( 777 ~; 633 |; 0 &;4283 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 49 usr; 12 con; 0-3 aty)
% Number of variables : 749 ( 0 ^; 749 !; 0 ?; 749 :)
% Comments :
%------------------------------------------------------------------------------
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(x_decl,type,
x: $i ).
thf(y_decl,type,
y: $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(complement_decl,type,
complement: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(first_decl,type,
first: $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(3,negated_conjecture,
~ ( universal_class @ ( y @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_null_unordered_pair_3) ).
thf(97,plain,
~ ( universal_class @ ( y @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(101,plain,
~ ( universal_class @ ( y @ member ) ),
inference(polarity_switch,[status(thm)],[97]) ).
thf(35,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(173,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(139,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(140,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[139]) ).
thf(8782,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[173,140]) ).
thf(8783,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[8782]) ).
thf(8784,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[8783:[bind(A,$thf( omega ))]]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(169,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(170,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[169]) ).
thf(11,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(118,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(119,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[118]) ).
thf(120,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[119]) ).
thf(22,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(146,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(147,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[146]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(150,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(151,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[150]) ).
thf(1806,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ ( element_relation @ inverse @ compose ) @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( B @ domain_of @ intersection ) )
= ( B @ cantor ) ) ),
inference(paramod_ordered,[status(thm)],[147,151]) ).
thf(1807,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[1806:[bind(A,$thf( D @ ( element_relation @ inverse @ compose ) )),bind(B,$thf( D ))]]) ).
thf(1855,plain,
! [A: $i] :
( ( identity_relation @ ( A @ ( element_relation @ inverse @ compose ) @ intersection ) @ domain_of @ complement @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(simp,[status(thm)],[1807]) ).
thf(94,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(292,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(293,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[292]) ).
thf(294,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[293]) ).
thf(76,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(255,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(299,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[255,101]) ).
thf(300,plain,
( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[299]) ).
thf(308,plain,
( ( universal_class != universal_class )
| ( omega != y ) ),
inference(simp,[status(thm)],[300]) ).
thf(311,plain,
omega != y,
inference(simp,[status(thm)],[308]) ).
thf(2076,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,311]) ).
thf(2077,plain,
( ( null_class != y )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2076:[bind(A,$thf( omega ))]]) ).
thf(9,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(114,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(115,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[114]) ).
thf(89,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(281,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(282,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[281]) ).
thf(4,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(102,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(103,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[102]) ).
thf(2592,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[282,103]) ).
thf(8,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(112,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(113,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[112]) ).
thf(734,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[113,113]) ).
thf(735,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[734:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(771,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[735]) ).
thf(2617,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( C @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2592,771]) ).
thf(2618,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2617:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2683,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[2618]) ).
thf(10,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(116,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(117,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[116]) ).
thf(6806,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[2683,117]) ).
thf(6943,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[6806,771]) ).
thf(6944,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[6943:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(862,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[147,113]) ).
thf(863,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[862:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(905,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[863]) ).
thf(1707,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[905,771]) ).
thf(1708,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1707:[bind(A,$thf( E )),bind(B,$thf( E @ diagonalise @ ( element_relation @ image ) ))]]) ).
thf(1788,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1708]) ).
thf(18,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(136,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(137,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[136]) ).
thf(138,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[137]) ).
thf(7524,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[138,103]) ).
thf(42,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(187,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(2140,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,187]) ).
thf(2141,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2140:[bind(A,$thf( universal_class ))]]) ).
thf(7,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(109,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(110,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[109]) ).
thf(111,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[110]) ).
thf(2,negated_conjecture,
~ ( universal_class @ ( x @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_null_unordered_pair_2) ).
thf(96,plain,
~ ( universal_class @ ( x @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(100,plain,
~ ( universal_class @ ( x @ member ) ),
inference(polarity_switch,[status(thm)],[96]) ).
thf(306,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[255,100]) ).
thf(307,plain,
( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[306]) ).
thf(310,plain,
( ( universal_class != universal_class )
| ( omega != x ) ),
inference(simp,[status(thm)],[307]) ).
thf(313,plain,
omega != x,
inference(simp,[status(thm)],[310]) ).
thf(378,plain,
! [B: $i,A: $i] :
( ( A != omega )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[111,313]) ).
thf(379,plain,
! [A: $i] :
( ( A != x )
| ~ ( omega @ ( A @ subclass ) )
| ~ ( A @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[378:[bind(A,$thf( omega ))]]) ).
thf(538,plain,
( ~ ( omega @ ( x @ subclass ) )
| ~ ( x @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[379]) ).
thf(25,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(152,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(153,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[152]) ).
thf(36,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(174,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(41,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(185,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(1946,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,255]) ).
thf(1947,plain,
( ( null_class @ ( omega @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1946:[bind(A,$thf( universal_class ))]]) ).
thf(5907,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( null_class @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1947,101]) ).
thf(5908,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5907]) ).
thf(5953,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= y )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5908]) ).
thf(5961,plain,
( ( ( universal_class @ regular )
!= y )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5953]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(261,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(262,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[261]) ).
thf(295,plain,
! [B: $i,A: $i] :
( ~ $true
| ( ( B @ complement @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[262]) ).
thf(297,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ( A != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[295]) ).
thf(298,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[297]) ).
thf(5975,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( ( universal_class @ regular )
!= y ) ),
inference(paramod_ordered,[status(thm)],[5961,298]) ).
thf(5976,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( ( universal_class @ regular )
!= y ) ),
inference(simp,[status(thm)],[5975]) ).
thf(5977,plain,
( ( ( null_class @ complement )
!= null_class )
| ( ( universal_class @ regular )
!= y ) ),
inference(pattern_uni,[status(thm)],[5976:[bind(A,$thf( omega )),bind(B,$thf( null_class ))]]) ).
thf(92,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(288,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(303,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[255,262]) ).
thf(304,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[303]) ).
thf(305,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[304:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(314,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[255,305]) ).
thf(315,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[314]) ).
thf(316,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[315]) ).
thf(317,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[316]) ).
thf(340,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[111,317]) ).
thf(341,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[340:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(522,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[341]) ).
thf(75,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(254,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(3893,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[522,254]) ).
thf(3894,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[3893]) ).
thf(3911,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[187,3894]) ).
thf(3912,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[3911]) ).
thf(3947,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[3912]) ).
thf(62,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(227,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(228,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[227]) ).
thf(5,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(104,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(14,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(125,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(126,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[125]) ).
thf(4138,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[104,126]) ).
thf(4139,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4138]) ).
thf(4140,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4139:[bind(A,$thf( choice ))]]) ).
thf(4197,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[228,4140]) ).
thf(4198,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4197:[bind(A,$thf( choice ))]]) ).
thf(78,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(257,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(1601,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[113,771]) ).
thf(1602,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1601:[bind(A,$thf( E @ complement @ ( element_relation @ image ) )),bind(B,$thf( E ))]]) ).
thf(1650,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1602]) ).
thf(1920,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( y @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,101]) ).
thf(1921,plain,
( ~ ( null_class @ ( y @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1920:[bind(A,$thf( universal_class ))]]) ).
thf(81,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(263,plain,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(264,plain,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(lifteq,[status(thm)],[263]) ).
thf(1416,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[228,264]) ).
thf(1417,plain,
( ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[1416:[bind(A,$thf( element_relation ))]]) ).
thf(2593,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( C @ restrict ) ) )
!= ( universal_class @ ( E @ ( D @ restrict ) ) ) )
| ( ( C @ ( B @ ( A @ cross_product ) @ intersection ) @ inverse @ domain_of )
= ( E @ ( D @ image ) ) ) ),
inference(paramod_ordered,[status(thm)],[117,2592]) ).
thf(2594,plain,
! [B: $i,A: $i] :
( ( B @ ( universal_class @ ( A @ cross_product ) @ intersection ) @ inverse @ domain_of )
= ( A @ ( B @ image ) ) ),
inference(pattern_uni,[status(thm)],[2593:[bind(A,$thf( A )),bind(B,$thf( universal_class )),bind(C,$thf( C )),bind(D,$thf( C )),bind(E,$thf( A ))]]) ).
thf(2675,plain,
! [B: $i,A: $i] :
( ( B @ ( universal_class @ ( A @ cross_product ) @ intersection ) @ inverse @ domain_of )
= ( A @ ( B @ image ) ) ),
inference(simp,[status(thm)],[2594]) ).
thf(16250,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( B @ ( universal_class @ ( A @ cross_product ) @ intersection ) ) )
| ( ( A @ ( B @ image ) )
= ( subset_relation @ inverse @ domain_of ) ) ),
inference(paramod_ordered,[status(thm)],[1417,2675]) ).
thf(16251,plain,
( ( universal_class @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ image ) )
= ( subset_relation @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[16250:[bind(A,$thf( universal_class )),bind(B,$thf( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) ))]]) ).
thf(4062,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[2077,100]) ).
thf(4063,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4062]) ).
thf(4070,plain,
( ( omega != universal_class )
| ( ( omega @ regular )
!= x )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4063]) ).
thf(5947,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ( null_class @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1947,100]) ).
thf(5948,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5947]) ).
thf(5959,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= x )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5948]) ).
thf(5964,plain,
( ( ( universal_class @ regular )
!= x )
| ( null_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[5959]) ).
thf(6055,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( null_class @ ( omega @ member ) ) )
| ~ $true
| ( ( universal_class @ regular )
!= x ) ),
inference(paramod_ordered,[status(thm)],[5964,305]) ).
thf(6056,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( null_class @ ( omega @ member ) ) )
| ( ( universal_class @ regular )
!= x ) ),
inference(simp,[status(thm)],[6055]) ).
thf(6090,plain,
( ( ( universal_class @ complement )
!= null_class )
| ( omega != omega )
| ( ( universal_class @ regular )
!= x ) ),
inference(simp,[status(thm)],[6056]) ).
thf(6095,plain,
( ( ( universal_class @ complement )
!= null_class )
| ( ( universal_class @ regular )
!= x ) ),
inference(simp,[status(thm)],[6090]) ).
thf(47,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(196,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(27,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(156,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(157,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[156]) ).
thf(374,plain,
! [B: $i,A: $i] :
( ( A != omega )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[111,311]) ).
thf(375,plain,
! [A: $i] :
( ( A != y )
| ~ ( omega @ ( A @ subclass ) )
| ~ ( A @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[374:[bind(A,$thf( omega ))]]) ).
thf(536,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[375]) ).
thf(621,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ $true
| ~ ( y @ ( omega @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[254,536]) ).
thf(622,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[621]) ).
thf(628,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A != y )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[622]) ).
thf(638,plain,
( ( omega != universal_class )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[628]) ).
thf(1,negated_conjecture,
( ( y @ ( x @ unordered_pair ) )
!= null_class ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_null_unordered_pair_1) ).
thf(95,plain,
( ( y @ ( x @ unordered_pair ) )
!= null_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(98,plain,
( ( y @ ( x @ unordered_pair ) )
!= null_class ),
inference(polarity_switch,[status(thm)],[95]) ).
thf(99,plain,
( ( y @ ( x @ unordered_pair ) )
!= null_class ),
inference(lifteq,[status(thm)],[98]) ).
thf(2156,plain,
! [A: $i] :
( ( A
!= ( y @ ( x @ unordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,99]) ).
thf(2157,plain,
! [A: $i] :
( ( A
!= ( y @ ( x @ unordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2156]) ).
thf(2158,plain,
y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[2157:[bind(A,$thf( y @ ( x @ unordered_pair ) ))]]) ).
thf(1592,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[771,113]) ).
thf(1593,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1592:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1673,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1593]) ).
thf(4470,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[771,1673]) ).
thf(4471,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[4470:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(301,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[255,262]) ).
thf(302,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[301]) ).
thf(309,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[302]) ).
thf(312,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[309]) ).
thf(5972,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true
| ( ( universal_class @ regular )
!= y ) ),
inference(paramod_ordered,[status(thm)],[5961,312]) ).
thf(5973,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ( ( universal_class @ regular )
!= y ) ),
inference(simp,[status(thm)],[5972]) ).
thf(5974,plain,
( ( ( null_class @ complement )
!= universal_class )
| ( ( universal_class @ regular )
!= y ) ),
inference(pattern_uni,[status(thm)],[5973:[bind(A,$thf( null_class ))]]) ).
thf(57,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(216,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(1384,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ~ ( omega @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[254,538]) ).
thf(1385,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ~ ( omega @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1384]) ).
thf(1405,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != omega )
| ~ ( omega @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1385]) ).
thf(1415,plain,
( ( universal_class != x )
| ~ ( omega @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1405]) ).
thf(54,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(211,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(12,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(121,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(1942,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,173]) ).
thf(1943,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1942:[bind(A,$thf( omega ))]]) ).
thf(2771,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1943,100]) ).
thf(2772,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[2771]) ).
thf(2777,plain,
( ( omega != universal_class )
| ( ( omega @ regular )
!= x )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[2772]) ).
thf(2134,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,313]) ).
thf(2135,plain,
( ( null_class != x )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2134:[bind(A,$thf( omega ))]]) ).
thf(4781,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2135,298]) ).
thf(4782,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( null_class != x ) ),
inference(simp,[status(thm)],[4781]) ).
thf(4783,plain,
( ( ( omega @ complement )
!= omega )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[4782:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(4031,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[2077,262]) ).
thf(4032,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4031]) ).
thf(4033,plain,
( ~ ( omega @ complement @ ( omega @ regular @ member ) )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[4032:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(17,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(132,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(2136,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,187]) ).
thf(2137,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( element_relation @ ( element_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2136:[bind(A,$thf( element_relation ))]]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(160,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(161,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[160]) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) ) ),
inference(lifteq,[status(thm)],[161]) ).
thf(15289,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ singleton @ ( B @ image ) @ sum_class )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ singleton @ ( C @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[162,153]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(265,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(69,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(243,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(8864,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( omega @ ( successor_relation @ image ) ) )
| ( omega @ ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[2592,8784]) ).
thf(8865,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[8864:[bind(A,$thf( successor_relation )),bind(B,$thf( omega ))]]) ).
thf(8925,plain,
omega @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ),
inference(rewrite,[status(thm)],[8865,117]) ).
thf(8938,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[228,8925]) ).
thf(8939,plain,
omega @ ( universal_class @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[8938:[bind(A,$thf( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) ))]]) ).
thf(1599,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[113,771]) ).
thf(1600,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1599:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1603,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[771,771]) ).
thf(1604,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1603:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1651,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1604]) ).
thf(65,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(233,plain,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(72,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(248,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(12579,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[157,638]) ).
thf(12580,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12579]) ).
thf(12581,plain,
( ( omega != universal_class )
| ~ ( y @ inductive ) ),
inference(pattern_uni,[status(thm)],[12580:[bind(A,$thf( y ))]]) ).
thf(2163,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,104]) ).
thf(2164,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2163:[bind(A,$thf( choice ))]]) ).
thf(59,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(220,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(1582,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[147,771]) ).
thf(1583,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1582:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1670,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1583]) ).
thf(3713,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1670,771]) ).
thf(3714,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3713:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(3852,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3714]) ).
thf(50,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(201,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(12574,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ~ ( omega @ ( x @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[157,538]) ).
thf(12575,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ~ ( omega @ ( x @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12574]) ).
thf(12576,plain,
( ~ ( omega @ ( x @ subclass ) )
| ~ ( x @ inductive ) ),
inference(pattern_uni,[status(thm)],[12575:[bind(A,$thf( x ))]]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(192,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(275,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(31,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(165,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(3663,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[905,1670]) ).
thf(3664,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3663:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(77,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(256,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(3909,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[256,3894]) ).
thf(3910,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[3909]) ).
thf(3948,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[3910]) ).
thf(63,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(229,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(74,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(252,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(6041,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( ( universal_class @ regular )
!= x ) ),
inference(paramod_ordered,[status(thm)],[5964,298]) ).
thf(6042,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( ( universal_class @ regular )
!= x ) ),
inference(simp,[status(thm)],[6041]) ).
thf(6043,plain,
( ( ( null_class @ complement )
!= null_class )
| ( ( universal_class @ regular )
!= x ) ),
inference(pattern_uni,[status(thm)],[6042:[bind(A,$thf( omega )),bind(B,$thf( null_class ))]]) ).
thf(56,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(214,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(215,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[214]) ).
thf(541,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[211,101]) ).
thf(542,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[541]) ).
thf(565,plain,
! [B: $i,A: $i] :
( ( universal_class != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= y ) ),
inference(simp,[status(thm)],[542]) ).
thf(573,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= y ),
inference(simp,[status(thm)],[565]) ).
thf(1158,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= y ) ),
inference(paramod_ordered,[status(thm)],[215,573]) ).
thf(1159,plain,
! [A: $i] :
( ( A @ singleton )
!= y ),
inference(pattern_uni,[status(thm)],[1158:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(8856,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[8784,1415]) ).
thf(8857,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[8856]) ).
thf(8914,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[8857]) ).
thf(8920,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[8914]) ).
thf(48,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(197,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(1686,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[147,905]) ).
thf(1687,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1686:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(30,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(163,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(164,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[163]) ).
thf(21,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(144,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[144]) ).
thf(10765,plain,
! [A: $i] :
( ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ complement )
= ( A @ successor ) ),
inference(rewrite,[status(thm)],[164,145]) ).
thf(91,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(285,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(61,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(224,plain,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(1922,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[294,101]) ).
thf(1923,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1922]) ).
thf(2270,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( A @ regular )
!= y )
| ( A = null_class ) ),
inference(simp,[status(thm)],[1923]) ).
thf(2359,plain,
( ( ( universal_class @ regular )
!= y )
| ( universal_class = null_class ) ),
inference(simp,[status(thm)],[2270]) ).
thf(43,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(188,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(32,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(168,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[167]) ).
thf(17819,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(rewrite,[status(thm)],[168,103]) ).
thf(20,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(141,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(15031,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ inductive ) ),
inference(paramod_ordered,[status(thm)],[8784,12576]) ).
thf(15032,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[15031]) ).
thf(15098,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= x )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[15032]) ).
thf(15114,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= x )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[15098]) ).
thf(2615,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ power_class @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( C @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2592,771]) ).
thf(2616,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[2615:[bind(A,$thf( element_relation )),bind(B,$thf( D @ power_class )),bind(C,$thf( D ))]]) ).
thf(2682,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[2616]) ).
thf(6428,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(rewrite,[status(thm)],[2682,117]) ).
thf(2261,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( x @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,100]) ).
thf(2262,plain,
( ~ ( null_class @ ( x @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2261:[bind(A,$thf( universal_class ))]]) ).
thf(10721,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ~ ( null_class @ ( x @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2262,100]) ).
thf(10722,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10721]) ).
thf(10734,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= x )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10722]) ).
thf(10761,plain,
( ( ( universal_class @ regular )
!= x )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10734]) ).
thf(1386,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( omega @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[254,538]) ).
thf(1387,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( x @ subclass ) ) )
| ~ ( x @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1386]) ).
thf(1402,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A != x )
| ~ ( x @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1387]) ).
thf(1414,plain,
( ( omega != universal_class )
| ~ ( x @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1402]) ).
thf(1154,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)],[215,211]) ).
thf(1155,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1154:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(55,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(212,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(23,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(148,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(149,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[148]) ).
thf(2124,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,149]) ).
thf(2125,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2124:[bind(A,$thf( subset_relation ))]]) ).
thf(8644,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ~ ( null_class @ ( y @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1921,101]) ).
thf(8645,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8644]) ).
thf(8745,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= y )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8645]) ).
thf(8764,plain,
( ( ( universal_class @ regular )
!= y )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8745]) ).
thf(2051,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,256]) ).
thf(2052,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2051:[bind(A,$thf( universal_class ))]]) ).
thf(64,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(231,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(232,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[231]) ).
thf(1241,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[232,117]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(205,plain,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(1162,plain,
! [A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( y @ ( x @ unordered_pair ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[215,99]) ).
thf(1167,plain,
! [A: $i] :
( ( A != y )
| ( A != x )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[1162]) ).
thf(1170,plain,
( ( y != x )
| ( ( x @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[1167]) ).
thf(12594,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[157,536]) ).
thf(12595,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12594]) ).
thf(12596,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( y @ inductive ) ),
inference(pattern_uni,[status(thm)],[12595:[bind(A,$thf( y ))]]) ).
thf(6,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(105,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(70,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(244,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(166,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[165]) ).
thf(88,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(279,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(39,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(180,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(10867,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ complement )
!= ( B @ complement ) )
| ( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[10765,113]) ).
thf(10868,plain,
! [A: $i] :
( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[10867:[bind(A,$thf( G )),bind(B,$thf( G @ singleton @ complement @ ( G @ complement @ intersection ) ))]]) ).
thf(11063,plain,
! [A: $i] :
( ( A @ successor @ ( element_relation @ image ) @ complement )
= ( A @ singleton @ complement @ ( A @ complement @ intersection ) @ power_class ) ),
inference(simp,[status(thm)],[10868]) ).
thf(37,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(176,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(2047,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,256]) ).
thf(2048,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2047:[bind(A,$thf( successor_relation ))]]) ).
thf(1605,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[771,771]) ).
thf(1606,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1605:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1652,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1606]) ).
thf(79,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',function3) ).
thf(259,plain,
! [A: $i] :
( ( 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)],[79]) ).
thf(8729,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ~ ( null_class @ ( y @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1921,100]) ).
thf(8730,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8729]) ).
thf(8754,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= x )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8730]) ).
thf(8767,plain,
( ( ( universal_class @ regular )
!= x )
| ~ ( null_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[8754]) ).
thf(1701,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[905,113]) ).
thf(1702,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1701:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1786,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1702]) ).
thf(547,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(paramod_ordered,[status(thm)],[211,262]) ).
thf(548,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(simp,[status(thm)],[547]) ).
thf(549,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[548:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(572,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[549]) ).
thf(66,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(236,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(15,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(127,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(128,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[127]) ).
thf(129,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[128]) ).
thf(4887,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[129,103]) ).
thf(12584,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[157,3894]) ).
thf(12585,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12584]) ).
thf(12613,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12585]) ).
thf(12624,plain,
( ( omega != universal_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[12613]) ).
thf(28,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(158,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(159,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[158]) ).
thf(2733,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[1943,101]) ).
thf(2734,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[2733]) ).
thf(2774,plain,
( ( omega != universal_class )
| ( ( omega @ regular )
!= y )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[2734]) ).
thf(84,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(271,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(4805,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,2135]) ).
thf(4806,plain,
! [A: $i] :
( ( A != x )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4805:[]]) ).
thf(4833,plain,
( ( omega @ ( omega @ regular @ member ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[4806]) ).
thf(26,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(154,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(155,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[154]) ).
thf(49,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(198,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(106,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[105]) ).
thf(107,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[106]) ).
thf(108,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[107]) ).
thf(16,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(130,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(5989,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( null_class @ ( omega @ member ) ) )
| ~ $true
| ( ( universal_class @ regular )
!= y ) ),
inference(paramod_ordered,[status(thm)],[5961,305]) ).
thf(5990,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( null_class @ ( omega @ member ) ) )
| ( ( universal_class @ regular )
!= y ) ),
inference(simp,[status(thm)],[5989]) ).
thf(6022,plain,
( ( ( universal_class @ complement )
!= null_class )
| ( omega != omega )
| ( ( universal_class @ regular )
!= y ) ),
inference(simp,[status(thm)],[5990]) ).
thf(6027,plain,
( ( ( universal_class @ complement )
!= null_class )
| ( ( universal_class @ regular )
!= y ) ),
inference(simp,[status(thm)],[6022]) ).
thf(68,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(241,plain,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(10628,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ~ ( null_class @ ( x @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2262,101]) ).
thf(10629,plain,
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10628]) ).
thf(10740,plain,
( ( universal_class != universal_class )
| ( ( universal_class @ regular )
!= y )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10629]) ).
thf(10762,plain,
( ( ( universal_class @ regular )
!= y )
| ~ ( null_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[10740]) ).
thf(2053,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,256]) ).
thf(2054,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2053:[bind(A,$thf( universal_class ))]]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(123,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(124,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[123]) ).
thf(2816,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[124,103]) ).
thf(53,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(209,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(46,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(194,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(2395,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2158,298]) ).
thf(2396,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[2395]) ).
thf(2397,plain,
( ( y @ ( x @ unordered_pair ) @ complement )
!= ( y @ ( x @ unordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[2396:[bind(A,$thf( y @ ( x @ unordered_pair ) @ regular )),bind(B,$thf( y @ ( x @ unordered_pair ) ))]]) ).
thf(17048,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[173,12624]) ).
thf(17049,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[17048]) ).
thf(17087,plain,
( ( ( universal_class @ complement )
!= omega )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[17049]) ).
thf(2142,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,187]) ).
thf(2143,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2142:[bind(A,$thf( universal_class ))]]) ).
thf(40,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(182,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(8872,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,8784]) ).
thf(8873,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8872:[bind(A,$thf( successor_relation ))]]) ).
thf(563,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[211,100]) ).
thf(564,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( x @ member ) ) ),
inference(simp,[status(thm)],[563]) ).
thf(569,plain,
! [B: $i,A: $i] :
( ( universal_class != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= x ) ),
inference(simp,[status(thm)],[564]) ).
thf(576,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= x ),
inference(simp,[status(thm)],[569]) ).
thf(142,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[141]) ).
thf(143,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[142]) ).
thf(51,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(203,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(58,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(218,plain,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(16116,plain,
! [B: $i,A: $i] :
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( B @ ( universal_class @ ( A @ cross_product ) @ intersection ) ) )
| ( ( A @ ( B @ image ) )
= ( subset_relation @ inverse @ domain_of ) ) ),
inference(paramod_ordered,[status(thm)],[264,2675]) ).
thf(16117,plain,
( ( universal_class @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ image ) )
= ( subset_relation @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[16116:[bind(A,$thf( universal_class )),bind(B,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) ))]]) ).
thf(16432,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ image ) )
!= ( B @ ( A @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( subset_relation @ inverse @ domain_of ) ) ),
inference(paramod_ordered,[status(thm)],[16117,2592]) ).
thf(16433,plain,
( ( universal_class @ ( universal_class @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ restrict ) ) @ inverse @ domain_of )
= ( subset_relation @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[16432:[bind(A,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )),bind(B,$thf( universal_class ))]]) ).
thf(17220,plain,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ inverse @ domain_of )
= ( subset_relation @ inverse @ domain_of ) ),
inference(rewrite,[status(thm)],[16433,117]) ).
thf(2613,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( C @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2592,113]) ).
thf(2614,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2613:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(2681,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2614]) ).
thf(2924,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(rewrite,[status(thm)],[2681,117]) ).
thf(73,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(250,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(1156,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( ( A @ singleton )
!= x ) ),
inference(paramod_ordered,[status(thm)],[215,576]) ).
thf(1157,plain,
! [A: $i] :
( ( A @ singleton )
!= x ),
inference(pattern_uni,[status(thm)],[1156:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(3108,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( C @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2592,1600]) ).
thf(3109,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3108:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3300,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3109]) ).
thf(21606,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(rewrite,[status(thm)],[3300,117]) ).
thf(2932,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ ( universal_class @ ( B @ complement @ cross_product ) @ intersection ) @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[228,2924]) ).
thf(2933,plain,
! [A: $i] :
( ( universal_class @ ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[2932:[bind(A,$thf( element_relation @ ( universal_class @ ( G @ complement @ cross_product ) @ intersection ) )),bind(B,$thf( G ))]]) ).
thf(3104,plain,
! [A: $i] :
( ( universal_class @ ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[2933]) ).
thf(1802,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( element_relation @ inverse ) )
| ( ( B @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( B @ domain_of @ intersection ) )
= ( B @ cantor ) ) ),
inference(paramod_ordered,[status(thm)],[228,151]) ).
thf(1803,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(pattern_uni,[status(thm)],[1802:[bind(A,$thf( element_relation ))]]) ).
thf(1854,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ cross_product ) @ flip @ domain_of @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(simp,[status(thm)],[1803]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(171,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(2404,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2158,262]) ).
thf(2405,plain,
! [B: $i,A: $i] :
( ( ( y @ ( x @ unordered_pair ) @ ( y @ ( x @ unordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2404]) ).
thf(2406,plain,
~ ( y @ ( x @ unordered_pair ) @ complement @ ( y @ ( x @ unordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[2405:[bind(A,$thf( y @ ( x @ unordered_pair ) @ regular )),bind(B,$thf( y @ ( x @ unordered_pair ) ))]]) ).
thf(93,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(290,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(246,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(6553,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( element_relation @ ( universal_class @ ( A @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[6428,771]) ).
thf(6554,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ power_class @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[6553:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(4044,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,2077]) ).
thf(4045,plain,
! [A: $i] :
( ( A != y )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4044:[]]) ).
thf(4074,plain,
( ( omega @ ( omega @ regular @ member ) )
| ( y @ ( y @ regular @ member ) ) ),
inference(simp,[status(thm)],[4045]) ).
thf(12586,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[157,1414]) ).
thf(12587,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[12586]) ).
thf(12588,plain,
( ( omega != universal_class )
| ~ ( x @ inductive ) ),
inference(pattern_uni,[status(thm)],[12587:[bind(A,$thf( x ))]]) ).
thf(4459,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1673,771]) ).
thf(4460,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[4459:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(4619,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[4460]) ).
thf(1160,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ~ ( universal_class @ complement @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[215,572]) ).
thf(1161,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1160:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(4792,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2135,262]) ).
thf(4793,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[4792]) ).
thf(4794,plain,
( ~ ( omega @ complement @ ( omega @ regular @ member ) )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[4793:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(44,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(190,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(85,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(273,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(2122,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,149]) ).
thf(2123,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2122:[bind(A,$thf( subset_relation ))]]) ).
thf(67,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(239,plain,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(83,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(268,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(4022,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[2077,298]) ).
thf(4023,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4022]) ).
thf(4024,plain,
( ( ( omega @ complement )
!= omega )
| ( null_class != y ) ),
inference(pattern_uni,[status(thm)],[4023:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(1709,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[905,771]) ).
thf(1710,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1709:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1789,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1710]) ).
thf(2960,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[2924,113]) ).
thf(2961,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2960:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(6038,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true
| ( ( universal_class @ regular )
!= x ) ),
inference(paramod_ordered,[status(thm)],[5964,312]) ).
thf(6039,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ( ( universal_class @ regular )
!= x ) ),
inference(simp,[status(thm)],[6038]) ).
thf(6040,plain,
( ( ( null_class @ complement )
!= universal_class )
| ( ( universal_class @ regular )
!= x ) ),
inference(pattern_uni,[status(thm)],[6039:[bind(A,$thf( null_class ))]]) ).
thf(90,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(283,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(2263,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[294,100]) ).
thf(2264,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( x @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[2263]) ).
thf(2266,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( A @ regular )
!= x )
| ( A = null_class ) ),
inference(simp,[status(thm)],[2264]) ).
thf(2353,plain,
( ( ( universal_class @ regular )
!= x )
| ( universal_class = null_class ) ),
inference(simp,[status(thm)],[2266]) ).
thf(4016,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( null_class != y ) ),
inference(paramod_ordered,[status(thm)],[2077,101]) ).
thf(4017,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4016]) ).
thf(4066,plain,
( ( omega != universal_class )
| ( ( omega @ regular )
!= y )
| ( null_class != y ) ),
inference(simp,[status(thm)],[4017]) ).
thf(907,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[228,149]) ).
thf(908,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[907:[bind(A,$thf( subset_relation ))]]) ).
thf(713,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[254,638]) ).
thf(714,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[713]) ).
thf(716,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != omega )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[714]) ).
thf(723,plain,
( ( universal_class != y )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[716]) ).
thf(131,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[130]) ).
thf(87,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(277,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(2608,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ diagonalise @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( C @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[2592,905]) ).
thf(2609,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[2608:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(2680,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[2609]) ).
thf(6109,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[2680,117]) ).
thf(38,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(178,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(619,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[254,536]) ).
thf(620,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[619]) ).
thf(630,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != omega )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[620]) ).
thf(639,plain,
( ( universal_class != y )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[630]) ).
thf(8900,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( universal_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[8784,639]) ).
thf(8901,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[8900]) ).
thf(8909,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[8901]) ).
thf(8918,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[8909]) ).
thf(15199,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ $true
| ~ ( y @ inductive ) ),
inference(paramod_ordered,[status(thm)],[8784,12596]) ).
thf(15200,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ ( y @ inductive ) ),
inference(simp,[status(thm)],[15199]) ).
thf(15273,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( y @ inductive ) ),
inference(simp,[status(thm)],[15200]) ).
thf(15287,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( y @ inductive ) ),
inference(simp,[status(thm)],[15273]) ).
thf(122,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[121]) ).
thf(1944,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[294,255]) ).
thf(1945,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1944:[bind(A,$thf( omega ))]]) ).
thf(1903,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[254,1414]) ).
thf(1904,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( omega @ subclass ) ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1903]) ).
thf(1906,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != omega )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1904]) ).
thf(1917,plain,
( ( universal_class != x )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1906]) ).
thf(6140,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[6109,905]) ).
thf(6141,plain,
! [A: $i] :
( ( element_relation @ ( universal_class @ ( A @ diagonalise @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[6140:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(60,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(222,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(23642,plain,
$false,
inference(e,[status(thm)],[101,8784,170,120,1855,2077,115,6944,1788,7524,2141,538,153,174,185,5977,288,3947,4198,257,1650,1921,16251,4070,6095,125,196,157,638,152,228,2158,4471,261,5974,216,1415,211,121,905,2777,4783,147,4033,132,2137,1943,15289,265,312,116,243,8939,1600,1651,5961,292,233,248,117,12581,2164,220,102,3852,201,160,12576,192,275,165,3664,3948,229,252,6043,97,3894,1159,8920,4140,197,1687,10765,285,224,317,156,2359,188,17819,141,169,15114,6428,10761,1414,1155,212,96,109,256,2125,8764,173,1947,2052,1241,205,311,1170,5964,12596,105,244,298,166,148,264,279,180,149,573,11063,176,2048,1652,118,281,259,8767,1786,572,313,144,236,4887,12624,159,187,113,2774,271,4833,103,140,155,198,108,130,2592,6027,241,167,10762,2054,2816,255,209,112,123,6806,194,145,2397,17087,150,95,2143,263,182,8873,127,576,2675,154,1670,143,99,203,218,104,17220,771,2924,250,231,1157,21606,158,3104,114,1854,171,139,2262,2135,2406,290,246,6554,16117,4074,8925,12588,536,4619,214,1161,4794,262,146,190,1673,273,2123,151,305,239,268,4024,1789,2961,6040,283,294,126,136,2353,4066,908,723,131,163,277,6109,178,8918,15287,111,122,1945,215,639,1917,6141,222,100,1417,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET071-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.08 % 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
% 0.15/10.59 % Computer : n020.cluster.edu
% 0.15/10.59 % Model : x86_64 x86_64
% 0.15/10.59 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/10.59 % Memory : 8046.5625MB
% 0.15/10.59 % OS : Linux 6.8.0-71-generic
% 0.15/10.59 % CPULimit : 300
% 0.15/10.59 % WCLimit : 300
% 0.15/10.59 % DateTime : Sat Sep 26 07:03:14 UTC 2026
% 0.18/10.60 % CPUTime :
% 0.18/10.60 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.18/10.90 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.13/11.03 % [INFO] Parsing done (136ms).
% 1.13/11.04 % [INFO] Running in sequential loop mode.
% 1.55/11.23 % [INFO] eprover registered as external prover.
% 1.55/11.23 % [INFO] Scanning for conjecture ...
% 1.82/11.33 % [INFO] Found a conjecture (or negated_conjecture) and 91 axioms. Running axiom selection ...
% 2.03/11.43 % [INFO] Axiom selection finished. Selected 91 axioms (removed 0 axioms).
% 2.34/11.55 % [INFO] Problem is propositional (TPTP CNF).
% 2.34/11.56 % [INFO] Type checking passed.
% 2.34/11.56 % [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 ...
% 92.07/29.37 % External prover 'e' found a proof!
% 92.07/29.37 % [INFO] Killing All external provers ...
% 92.07/29.37 % Time passed: 18677ms (effective reasoning time: 18328ms)
% 92.07/29.38 % Axioms used in derivation (91): operation4, flip2, inverse, range, domain1, inductive3, operation3, equal_implies_subclass1, complement1, one_to_one2, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, function3, regularity1, singleton_set, identity_relation, union, flip3, element_relation2, unordered_pairs_in_universal, compose2, homomorphism3, compatible3, symmetric_difference, cartesian_product1, not_subclass_members1, image, replacement, successor, ordered_pair, successor_relation1, domain, homomorphism1, one_to_one3, single_valued_class1, compose3, omega_is_inductive2, rotate1, choice1, power_class2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, not_subclass_members2, successor_relation3, domain2, function1, compatible4, intersection3, single_valued_class2, unordered_pair_member, apply, element_relation1, operation1, restriction1, cartesian_product3, flip1, homomorphism5, compatible1, unordered_pair2, complement2, operation2, rotate3, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, intersection2, choice2, homomorphism2, power_class_definition, unordered_pair3, subset_relation, compose1
% 92.07/29.38 % No. of inferences in proof: 602
% 92.07/29.38 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 18677 ms resp. 18328 ms w/o parsing
% 93.88/29.66 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 93.88/29.67 % [INFO] Killing All external provers ...
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