%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET122-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n002.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 08:29:31 AM UTC 2026
% Result : Unsatisfiable 116.12s 29.13s
% Output : Refutation 116.91s
% Verified :
% SZS Type : Refutation
% Derivation depth : 29
% Number of leaves : 169
% Syntax : Number of formulae : 951 ( 273 unt; 0 typ; 0 def)
% Number of atoms : 2262 ( 970 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 7730 (1256 ~; 979 |; 0 &;5495 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 12 con; 0-3 aty)
% Number of variables : 1322 ( 0 ^;1322 !; 0 ?;1322 :)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(y_decl,type,
y: $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(second_decl,type,
second: $i > $i ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(x_decl,type,
x: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(member_of_decl,type,
member_of: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $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(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(null_class_decl,type,
null_class: $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(one_to_one_decl,type,
one_to_one: $i > $o ).
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(first_decl,type,
first: $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
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(regular_decl,type,
regular: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(member_of1_decl,type,
member_of1: $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(flip_decl,type,
flip: $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(range_decl,type,
range: $i > $i > $i > $i ).
thf(87,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(365,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(136,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_2) ).
thf(479,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(36,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation2) ).
thf(247,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)],[36]) ).
thf(248,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[247]) ).
thf(249,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[248]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair3) ).
thf(518,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(131,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(468,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(31,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(234,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(347,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(150,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(510,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(511,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[510]) ).
thf(512,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[511]) ).
thf(102,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(400,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(1,negated_conjecture,
~ ( universal_class @ ( y @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_OP_determines_components2_1) ).
thf(170,plain,
~ ( universal_class @ ( y @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(172,plain,
~ ( universal_class @ ( y @ member ) ),
inference(polarity_switch,[status(thm)],[170]) ).
thf(559,plain,
( ( ( universal_class @ ( null_class @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[400,172]) ).
thf(560,plain,
( ( universal_class @ ( null_class @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[559]) ).
thf(561,plain,
( ( universal_class != universal_class )
| ( null_class != y ) ),
inference(simp,[status(thm)],[560]) ).
thf(562,plain,
null_class != y,
inference(simp,[status(thm)],[561]) ).
thf(2994,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,562]) ).
thf(2995,plain,
! [A: $i] :
( ( A != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2994:[]]) ).
thf(3311,plain,
y @ ( y @ regular @ member ),
inference(simp,[status(thm)],[2995]) ).
thf(3349,plain,
! [A: $i] :
( ( A
!= ( y @ regular ) )
| ( y @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,3311]) ).
thf(3350,plain,
( ( y @ ( null_class @ member ) )
| ( y @ regular @ ( y @ regular @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3349:[bind(A,$thf( y @ regular ))]]) ).
thf(92,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(375,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(376,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[375]) ).
thf(11887,plain,
! [A: $i] :
( ( ( y @ regular @ ( y @ regular @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true
| ( y @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3350,376]) ).
thf(11888,plain,
! [A: $i] :
( ( ( y @ regular @ ( y @ regular @ regular @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ( y @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[11887]) ).
thf(11915,plain,
! [A: $i] :
( ( ( y @ regular )
!= null_class )
| ( ( y @ regular @ regular )
!= A )
| ( y @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[11888]) ).
thf(11922,plain,
( ( ( y @ regular )
!= null_class )
| ( y @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[11915]) ).
thf(23,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(217,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(440,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(46,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(269,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(15,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction2) ).
thf(202,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)],[15]) ).
thf(155,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(523,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(39,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(254,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(255,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[254]) ).
thf(90,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(372,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(128,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(462,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(463,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[462]) ).
thf(2234,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( D @ complement @ intersection ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[372,463]) ).
thf(2235,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( D @ complement @ intersection ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2234]) ).
thf(2272,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( D @ complement @ intersection ) )
!= universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= C ) ),
inference(simp,[status(thm)],[2235]) ).
thf(2275,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2272]) ).
thf(2303,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[255,2275]) ).
thf(2306,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[2303]) ).
thf(2309,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[2306]) ).
thf(138,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(483,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(573,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[483,376]) ).
thf(574,plain,
! [A: $i] :
( ( universal_class @ ( omega @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[573]) ).
thf(579,plain,
! [A: $i] :
( ( null_class != universal_class )
| ( omega != A ) ),
inference(simp,[status(thm)],[574]) ).
thf(582,plain,
null_class != universal_class,
inference(simp,[status(thm)],[579]) ).
thf(96,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_4) ).
thf(385,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(137,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(482,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(30,axiom,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(231,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(232,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[231]) ).
thf(233,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ subclass ) )
| ( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[232]) ).
thf(73,axiom,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(331,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(590,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[331,172]) ).
thf(591,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[590]) ).
thf(595,plain,
! [A: $i] :
( ( universal_class != universal_class )
| ( ( A @ singleton )
!= y ) ),
inference(simp,[status(thm)],[591]) ).
thf(598,plain,
! [A: $i] :
( ( A @ singleton )
!= y ),
inference(simp,[status(thm)],[595]) ).
thf(6132,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ singleton )
!= ( C @ singleton ) )
| ( A != y )
| ~ ( B @ singleton @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[233,598]) ).
thf(6133,plain,
! [B: $i,A: $i] :
( ( A != y )
| ~ ( B @ singleton @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(pattern_uni,[status(thm)],[6132:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B ))]]) ).
thf(6923,plain,
! [A: $i] :
( ~ ( A @ singleton @ ( y @ subclass ) )
| ( null_class = y ) ),
inference(simp,[status(thm)],[6133]) ).
thf(8472,plain,
! [A: $i] :
~ ( A @ singleton @ ( y @ subclass ) ),
inference(simplifyReflect,[status(thm)],[6923,562]) ).
thf(8518,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( B @ singleton @ ( y @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,8472]) ).
thf(8519,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( A @ subclass ) )
!= ( B @ singleton @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[8518]) ).
thf(8537,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
!= universal_class )
| ( A != y ) ),
inference(simp,[status(thm)],[8519]) ).
thf(8551,plain,
! [A: $i] :
( ( A @ singleton )
!= universal_class ),
inference(simp,[status(thm)],[8537]) ).
thf(8,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(187,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(188,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[187]) ).
thf(189,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[188]) ).
thf(22,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(216,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(563,plain,
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[483,172]) ).
thf(564,plain,
( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[563]) ).
thf(565,plain,
( ( universal_class != universal_class )
| ( omega != y ) ),
inference(simp,[status(thm)],[564]) ).
thf(566,plain,
omega != y,
inference(simp,[status(thm)],[565]) ).
thf(775,plain,
! [B: $i,A: $i] :
( ( A != omega )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,566]) ).
thf(776,plain,
! [A: $i] :
( ( A != y )
| ~ ( omega @ ( A @ subclass ) )
| ~ ( A @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[775:[bind(A,$thf( omega ))]]) ).
thf(945,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[776]) ).
thf(1108,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,945]) ).
thf(1109,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1108]) ).
thf(1131,plain,
! [A: $i] :
( ( A != y )
| ( omega != null_class )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1109]) ).
thf(1147,plain,
( ( omega != null_class )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1131]) ).
thf(1174,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,1147]) ).
thf(1175,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1174]) ).
thf(1181,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A != y )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1175]) ).
thf(1189,plain,
( ( omega != universal_class )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1181]) ).
thf(1248,plain,
! [B: $i,A: $i] :
( ( B != omega )
| ( omega != universal_class )
| ( A != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,1189]) ).
thf(1249,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A != null_class )
| ~ ( A @ ( omega @ subclass ) )
| ~ ( omega @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1248:[bind(A,$thf( A )),bind(B,$thf( omega ))]]) ).
thf(1252,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ subclass ) )
| ~ ( omega @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[1249]) ).
thf(10868,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[1252,216]) ).
thf(10869,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[10868]) ).
thf(68,axiom,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(316,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(317,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[316]) ).
thf(318,plain,
! [A: $i] :
( ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(lifteq,[status(thm)],[317]) ).
thf(3355,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3311,318]) ).
thf(3356,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3355]) ).
thf(3362,plain,
! [A: $i] :
( ( universal_class != y )
| ( ( y @ regular )
!= A )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3356]) ).
thf(3365,plain,
( ( universal_class != y )
| ( ( y @ regular @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3362]) ).
thf(133,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(472,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(6,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(182,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(21,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(214,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(215,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[214]) ).
thf(4198,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[182,215]) ).
thf(4199,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4198]) ).
thf(4200,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4199:[bind(A,$thf( choice ))]]) ).
thf(659,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,582]) ).
thf(660,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[659:[bind(A,$thf( null_class ))]]) ).
thf(998,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[660]) ).
thf(2318,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[998,482]) ).
thf(2319,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[2318]) ).
thf(4267,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4200,2319]) ).
thf(4268,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[4267]) ).
thf(4279,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4268]) ).
thf(162,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality1) ).
thf(538,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(48,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(274,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(241,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(242,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[241]) ).
thf(7732,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,242]) ).
thf(7733,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[7732]) ).
thf(7734,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[7733:[bind(A,$thf( omega ))]]) ).
thf(54,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/sandbox2/benchmark/theBenchmark.p',rotate2) ).
thf(287,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)],[54]) ).
thf(288,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(cnf,[status(esa)],[287]) ).
thf(825,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != y )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,562]) ).
thf(826,plain,
! [A: $i] :
( ( A != y )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[825:[bind(A,$thf( null_class ))]]) ).
thf(960,plain,
( ~ ( null_class @ ( y @ subclass ) )
| ~ ( y @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[826]) ).
thf(2080,plain,
( ~ ( null_class @ ( y @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[960,216]) ).
thf(2081,plain,
~ ( null_class @ ( y @ subclass ) ),
inference(simp,[status(thm)],[2080]) ).
thf(7737,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[242,2081]) ).
thf(7738,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7737]) ).
thf(7748,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= y )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7738]) ).
thf(7760,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= y )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7748]) ).
thf(7991,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,7760]) ).
thf(7992,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= y ) ),
inference(simp,[status(thm)],[7991]) ).
thf(7993,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= y ) ),
inference(simp,[status(thm)],[7992]) ).
thf(40,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cantor_class) ).
thf(256,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(257,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[256]) ).
thf(2,negated_conjecture,
( ( y @ ( x @ ordered_pair ) @ second )
!= ( y @ ( x @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_OP_determines_components2_2) ).
thf(171,plain,
( ( y @ ( x @ ordered_pair ) @ second )
!= ( y @ ( x @ ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(173,plain,
( ( y @ ( x @ ordered_pair ) @ second )
!= ( y @ ( x @ ordered_pair ) ) ),
inference(polarity_switch,[status(thm)],[171]) ).
thf(174,plain,
( ( y @ ( x @ ordered_pair ) @ second )
!= ( y @ ( x @ ordered_pair ) ) ),
inference(lifteq,[status(thm)],[173]) ).
thf(104,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(404,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(111,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_is_set) ).
thf(422,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(1048,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[422,318]) ).
thf(1049,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[1048]) ).
thf(1050,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[1049:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ ordered_pair ) ))]]) ).
thf(1055,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[1050]) ).
thf(7824,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,7734]) ).
thf(7825,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7824:[bind(A,$thf( successor_relation ))]]) ).
thf(148,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation2) ).
thf(504,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(49,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',restriction1) ).
thf(275,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)],[49]) ).
thf(276,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ restrict ) ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(lifteq,[status(thm)],[275]) ).
thf(203,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[202]) ).
thf(11966,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(rewrite,[status(thm)],[276,203]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_ordered_pair_determines_components1) ).
thf(344,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(1120,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,945]) ).
thf(1121,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1120]) ).
thf(1141,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != omega )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1121]) ).
thf(1152,plain,
( ( universal_class != y )
| ~ ( omega @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[1141]) ).
thf(7842,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( universal_class != y )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7734,1152]) ).
thf(7843,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[7842]) ).
thf(7856,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[7843]) ).
thf(7867,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= y )
| ( universal_class != y ) ),
inference(simp,[status(thm)],[7856]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(195,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(196,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(106,axiom,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(409,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(410,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[409]) ).
thf(411,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
= null_class ) ),
inference(lifteq,[status(thm)],[410]) ).
thf(1464,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( ( A @ singleton )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[411,172]) ).
thf(1465,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( ( A @ singleton )
= null_class ) ),
inference(simp,[status(thm)],[1464]) ).
thf(1466,plain,
( ( y @ singleton )
= null_class ),
inference(pattern_uni,[status(thm)],[1465:[bind(A,$thf( y ))]]) ).
thf(47,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(272,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(273,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[272]) ).
thf(2008,plain,
! [A: $i] :
( ( ( y @ singleton )
!= ( A @ singleton ) )
| ( ( null_class @ ( A @ union ) )
= ( A @ successor ) ) ),
inference(paramod_ordered,[status(thm)],[1466,273]) ).
thf(2009,plain,
( ( null_class @ ( y @ union ) )
= ( y @ successor ) ),
inference(pattern_uni,[status(thm)],[2008:[bind(A,$thf( y ))]]) ).
thf(37,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(250,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(251,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[250]) ).
thf(8957,plain,
( ( null_class @ complement @ ( y @ complement @ intersection ) @ complement )
= ( y @ successor ) ),
inference(rewrite,[status(thm)],[2009,251]) ).
thf(10,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(193,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(194,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[193]) ).
thf(9055,plain,
! [A: $i] :
( ( ( null_class @ complement @ ( y @ complement @ intersection ) @ complement )
!= ( A @ complement ) )
| ( ( y @ successor @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8957,194]) ).
thf(9056,plain,
( ( y @ successor @ ( element_relation @ image ) @ complement )
= ( null_class @ complement @ ( y @ complement @ intersection ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[9055:[bind(A,$thf( null_class @ complement @ ( y @ complement @ intersection ) ))]]) ).
thf(3357,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3311,463]) ).
thf(3358,plain,
! [B: $i,A: $i] :
( ( y @ ( y @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3357]) ).
thf(3360,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= y )
| ( ( y @ regular )
!= A ) ),
inference(simp,[status(thm)],[3358]) ).
thf(3363,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= y ),
inference(simp,[status(thm)],[3360]) ).
thf(4530,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != y ) ),
inference(paramod_ordered,[status(thm)],[255,3363]) ).
thf(4535,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != y ) ),
inference(simp,[status(thm)],[4530]) ).
thf(4546,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != y ) ),
inference(simp,[status(thm)],[4535]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_and_singletons) ).
thf(282,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(283,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ singleton @ union ) )
= ( B @ ( A @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[282]) ).
thf(15388,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
= ( B @ ( A @ unordered_pair ) ) ),
inference(rewrite,[status(thm)],[283,251]) ).
thf(4,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(178,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(179,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(112,axiom,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(423,plain,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(6975,plain,
! [B: $i,A: $i] :
( ( ( y @ singleton )
!= ( A @ singleton ) )
| ( null_class @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1466,423]) ).
thf(6976,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( y @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6975:[bind(A,$thf( y ))]]) ).
thf(6991,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( y @ singleton @ member ) ),
inference(simp,[status(thm)],[6976]) ).
thf(7017,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( null_class @ member ) ),
inference(rewrite,[status(thm)],[6991,1466]) ).
thf(7036,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( A @ unordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7017,376]) ).
thf(7037,plain,
! [B: $i,A: $i] :
( ( null_class @ ( A @ unordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[7036]) ).
thf(7060,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( A @ unordered_pair ) )
!= null_class )
| ( null_class != B ) ),
inference(simp,[status(thm)],[7037]) ).
thf(7071,plain,
! [A: $i] :
( ( null_class @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7060]) ).
thf(7086,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,7071]) ).
thf(7087,plain,
! [A: $i] :
( ( A @ ( null_class @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[7086:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(15566,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( null_class @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[15388,7087]) ).
thf(15567,plain,
! [A: $i] :
( ( A @ singleton @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[15566:[bind(A,$thf( null_class )),bind(B,$thf( B )),bind(C,$thf( B ))]]) ).
thf(15725,plain,
! [A: $i] :
( ( A @ singleton @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(simp,[status(thm)],[15567]) ).
thf(18379,plain,
! [A: $i] :
( ( ( y @ singleton )
!= ( A @ singleton ) )
| ( ( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[1466,15725]) ).
thf(18380,plain,
( ( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[18379:[bind(A,$thf( y ))]]) ).
thf(50,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(277,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(278,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[277]) ).
thf(279,plain,
! [A: $i] :
( ~ ( null_class @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[278]) ).
thf(13590,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( null_class @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,7087]) ).
thf(13591,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( null_class @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13590]) ).
thf(13592,plain,
! [A: $i] :
~ ( null_class @ ( A @ ( null_class @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[13591:[bind(A,$thf( D @ ( null_class @ unordered_pair ) )),bind(B,$thf( D ))]]) ).
thf(14103,plain,
! [A: $i] :
~ ( null_class @ ( A @ ( null_class @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[13592]) ).
thf(604,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[483,318]) ).
thf(605,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[604]) ).
thf(606,plain,
( ( omega @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[605:[bind(A,$thf( omega ))]]) ).
thf(53,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/sandbox2/benchmark/theBenchmark.p',homomorphism4) ).
thf(284,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)],[53]) ).
thf(109,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',domain1) ).
thf(416,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)],[109]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(325,plain,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(28,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetric_difference) ).
thf(227,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)],[28]) ).
thf(228,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[227]) ).
thf(58,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(293,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(294,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[293]) ).
thf(19280,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3311,294]) ).
thf(19281,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[19280]) ).
thf(19282,plain,
y @ ( y @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[19281:[bind(A,$thf( y @ regular )),bind(B,$thf( y ))]]) ).
thf(55,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(289,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(164,axiom,
null_class @ singleton @ ( null_class @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(544,plain,
null_class @ singleton @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(577,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[544,376]) ).
thf(578,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[577]) ).
thf(581,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[578]) ).
thf(584,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[581]) ).
thf(817,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,584]) ).
thf(818,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( null_class @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[817:[bind(A,$thf( null_class @ singleton )),bind(B,$thf( B ))]]) ).
thf(957,plain,
( ~ ( null_class @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[818]) ).
thf(5662,plain,
( ~ $true
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[957,216]) ).
thf(5663,plain,
~ ( null_class @ ( null_class @ singleton @ subclass ) ),
inference(simp,[status(thm)],[5662]) ).
thf(5676,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[289,5663]) ).
thf(5677,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5676]) ).
thf(5727,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[5677]) ).
thf(123,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality2) ).
thf(448,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(3,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists4) ).
thf(175,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(176,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(cnf,[status(esa)],[175]) ).
thf(177,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[176]) ).
thf(8736,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of @ singleton )
!= ( B @ singleton ) )
| ( A != universal_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[177,8551]) ).
thf(8737,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( A @ member_of )
= A ) ),
inference(pattern_uni,[status(thm)],[8736:[bind(A,$thf( C )),bind(B,$thf( C @ member_of ))]]) ).
thf(8738,plain,
( ( universal_class @ member_of )
= universal_class ),
inference(simp,[status(thm)],[8737]) ).
thf(2850,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,1147]) ).
thf(2851,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( y @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2850]) ).
thf(2852,plain,
( ~ ( omega @ ( y @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2851:[bind(A,$thf( omega ))]]) ).
thf(42,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(261,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(121,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_determines_components1) ).
thf(443,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(140,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(485,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(7834,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7734,2319]) ).
thf(7835,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[7834]) ).
thf(7845,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[7835]) ).
thf(32,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/sandbox2/benchmark/theBenchmark.p',compatible4) ).
thf(238,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)],[32]) ).
thf(1122,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ $true
| ~ ( y @ ( omega @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[482,945]) ).
thf(1123,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1122]) ).
thf(1140,plain,
! [A: $i] :
( ( omega != universal_class )
| ( A != y )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1123]) ).
thf(1151,plain,
( ( omega != universal_class )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[1140]) ).
thf(57,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_member_of_ordered_pair) ).
thf(292,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(2048,plain,
! [B: $i,A: $i] :
( ( ( y @ singleton )
!= ( A @ singleton ) )
| ( B @ ( A @ ordered_pair ) @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1466,292]) ).
thf(2049,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[2048:[bind(A,$thf( y ))]]) ).
thf(2071,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) ),
inference(simp,[status(thm)],[2049]) ).
thf(2258,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2071,463]) ).
thf(2259,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2258]) ).
thf(2271,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( null_class != B ) ),
inference(simp,[status(thm)],[2259]) ).
thf(2291,plain,
! [B: $i,A: $i] :
( ( A @ ( y @ ordered_pair ) )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[2271]) ).
thf(83,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product1) ).
thf(353,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)],[83]) ).
thf(139,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(484,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(1170,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,1147]) ).
thf(1171,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1170]) ).
thf(1183,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ( successor_relation != y )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1171]) ).
thf(6951,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[423,376]) ).
thf(6952,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[6951]) ).
thf(6987,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[6952]) ).
thf(6994,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6987]) ).
thf(7592,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,6994]) ).
thf(7593,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[7592:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(7617,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7593]) ).
thf(13973,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,7617]) ).
thf(13974,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13973]) ).
thf(13975,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[13974:[bind(A,$thf( E @ ( F @ singleton @ unordered_pair ) )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(14209,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[13975]) ).
thf(19,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(211,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[210]) ).
thf(38,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(252,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(253,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[252]) ).
thf(101,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product4) ).
thf(397,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)],[101]) ).
thf(44,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(265,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(266,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[265]) ).
thf(723,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,606]) ).
thf(724,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( omega @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( omega @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[723:[bind(A,$thf( omega @ singleton )),bind(B,$thf( B ))]]) ).
thf(1025,plain,
( ~ ( omega @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[724]) ).
thf(7110,plain,
( ~ $true
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1025,216]) ).
thf(7111,plain,
~ ( null_class @ ( omega @ singleton @ subclass ) ),
inference(simp,[status(thm)],[7110]) ).
thf(11262,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,7111]) ).
thf(11263,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11262]) ).
thf(11295,plain,
! [A: $i] :
( ( A != null_class )
| ( ( omega @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11263]) ).
thf(11312,plain,
( ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11295]) ).
thf(51,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(280,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(25,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(221,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(125,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets3) ).
thf(453,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(3088,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[512,463]) ).
thf(3089,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[3088]) ).
thf(3090,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3089:[bind(A,$thf( F @ ( F @ complement @ intersection ) )),bind(B,$thf( F @ ( F @ complement @ intersection ) @ regular )),bind(C,$thf( F ))]]) ).
thf(3243,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3090]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(243,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(115,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement2) ).
thf(428,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(507,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(2827,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,582]) ).
thf(2828,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2827:[]]) ).
thf(3238,plain,
universal_class @ ( universal_class @ regular @ member ),
inference(simp,[status(thm)],[2828]) ).
thf(3404,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3238,318]) ).
thf(3405,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3404]) ).
thf(3406,plain,
( ( universal_class @ regular @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[3405:[bind(A,$thf( universal_class @ regular ))]]) ).
thf(16,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible2) ).
thf(204,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)],[16]) ).
thf(205,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[204]) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[205]) ).
thf(610,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ~ $true
| ( ( B @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[331,318]) ).
thf(611,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( ( B @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[610]) ).
thf(612,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[611:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(618,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(simp,[status(thm)],[612]) ).
thf(13318,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,618]) ).
thf(13319,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13318]) ).
thf(13320,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[13319:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(14046,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(simp,[status(thm)],[13320]) ).
thf(3433,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,3406]) ).
thf(3434,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3433]) ).
thf(3435,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3434:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(64,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(307,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(7832,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7734,7111]) ).
thf(7833,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7832]) ).
thf(7849,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[7833]) ).
thf(94,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(380,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(7146,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,7111]) ).
thf(7147,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7146]) ).
thf(7179,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[7147]) ).
thf(11251,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,1151]) ).
thf(11252,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11251]) ).
thf(11253,plain,
( ( omega != universal_class )
| ~ ( y @ inductive ) ),
inference(pattern_uni,[status(thm)],[11252:[bind(A,$thf( y ))]]) ).
thf(11249,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,5663]) ).
thf(11250,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11249]) ).
thf(11290,plain,
! [A: $i] :
( ( A != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11250]) ).
thf(11307,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[11290]) ).
thf(7811,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7734,5663]) ).
thf(7812,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7811]) ).
thf(7859,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[7812]) ).
thf(24,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(220,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[219]) ).
thf(160,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(534,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(4029,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ regular @ singleton ) )
| ( universal_class != y )
| ( null_class != null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[411,3365]) ).
thf(4030,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ regular @ singleton ) )
| ( universal_class != y )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4029]) ).
thf(4031,plain,
( ( universal_class != y )
| ( universal_class @ ( y @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4030:[bind(A,$thf( y @ regular ))]]) ).
thf(29,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class2) ).
thf(229,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(230,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[229]) ).
thf(5781,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4200,230]) ).
thf(5782,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[5781]) ).
thf(5783,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[5782:[bind(A,$thf( choice ))]]) ).
thf(7828,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,7734]) ).
thf(7829,plain,
( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7828:[bind(A,$thf( omega ))]]) ).
thf(16089,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ $true
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7829,279]) ).
thf(16090,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[16089]) ).
thf(16091,plain,
( ( ( omega @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16090:[bind(A,$thf( omega @ ( successor_relation @ image ) ))]]) ).
thf(14,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(201,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(3814,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3435,463]) ).
thf(3815,plain,
! [B: $i,A: $i] :
( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3814]) ).
thf(3824,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) )
| ( ( universal_class @ regular @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3815]) ).
thf(3827,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) ),
inference(simp,[status(thm)],[3824]) ).
thf(7826,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,7734]) ).
thf(7827,plain,
( ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7826:[bind(A,$thf( omega ))]]) ).
thf(41,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(258,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(259,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[258]) ).
thf(260,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[259]) ).
thf(239,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)],[238]) ).
thf(240,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)],[239]) ).
thf(5,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(180,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(181,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[180]) ).
thf(7229,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)],[240,181]) ).
thf(59,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets1) ).
thf(295,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(296,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[295]) ).
thf(297,plain,
! [B: $i,A: $i] :
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ ordered_pair ) @ first )
= A ) ),
inference(lifteq,[status(thm)],[296]) ).
thf(113,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(424,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)],[113]) ).
thf(9,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_number_of_elements_in_class) ).
thf(190,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(191,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(cnf,[status(esa)],[190]) ).
thf(192,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(lifteq,[status(thm)],[191]) ).
thf(15605,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[15388,7071]) ).
thf(15606,plain,
! [A: $i] :
( ( null_class @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[15605:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(17262,plain,
! [A: $i] :
( ( ( y @ singleton )
!= ( A @ singleton ) )
| ( ( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[1466,15606]) ).
thf(17263,plain,
( ( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[17262:[bind(A,$thf( y ))]]) ).
thf(17465,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,17263]) ).
thf(17466,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[17465]) ).
thf(17467,plain,
~ ( null_class @ ( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement @ subclass ) ),
inference(pattern_uni,[status(thm)],[17466:[bind(A,$thf( null_class @ singleton @ complement @ ( null_class @ complement @ intersection ) @ complement ))]]) ).
thf(2140,plain,
! [B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2071,376]) ).
thf(2141,plain,
! [B: $i,A: $i] :
( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( null_class @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2140]) ).
thf(2146,plain,
! [B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) )
!= null_class )
| ( null_class != B ) ),
inference(simp,[status(thm)],[2141]) ).
thf(2150,plain,
! [A: $i] :
( ( A @ ( y @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2146]) ).
thf(7124,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[289,7111]) ).
thf(7125,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[7124]) ).
thf(7185,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[7125]) ).
thf(3064,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,255]) ).
thf(3065,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3064:[bind(A,$thf( subset_relation ))]]) ).
thf(167,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(551,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(126,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_3) ).
thf(456,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(405,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[404]) ).
thf(1601,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[405,255]) ).
thf(1602,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[1601:[bind(A,$thf( subset_relation ))]]) ).
thf(3140,plain,
! [A: $i] :
( ( A != omega )
| ( omega != universal_class )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,1189]) ).
thf(3141,plain,
! [A: $i] :
( ( A != omega )
| ( omega != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3140]) ).
thf(3142,plain,
( ( omega != universal_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3141:[bind(A,$thf( omega ))]]) ).
thf(4123,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(paramod_ordered,[status(thm)],[3142,318]) ).
thf(4124,plain,
! [A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4123]) ).
thf(4127,plain,
! [A: $i] :
( ( omega != universal_class )
| ( ( omega @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4124]) ).
thf(4140,plain,
( ( ( omega @ regular @ singleton )
!= null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4127]) ).
thf(12,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_class1) ).
thf(197,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(11807,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,11307]) ).
thf(11808,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[11807]) ).
thf(11813,plain,
( ( omega != null_class )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[11808]) ).
thf(118,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality4) ).
thf(435,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(12957,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,3406]) ).
thf(12958,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[12957]) ).
thf(12959,plain,
~ ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[12958:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(3862,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[194,3243]) ).
thf(3863,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3862:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(3892,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3863]) ).
thf(9072,plain,
! [A: $i] :
( ( ( null_class @ complement @ ( y @ complement @ intersection ) @ complement )
!= ( A @ complement ) )
| ( ( y @ successor @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[8957,3892]) ).
thf(9073,plain,
( ( y @ successor @ ( element_relation @ image ) @ ( null_class @ complement @ ( y @ complement @ intersection ) @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[9072:[bind(A,$thf( null_class @ complement @ ( y @ complement @ intersection ) ))]]) ).
thf(2328,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[289,2319]) ).
thf(2329,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[2328]) ).
thf(2359,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[2329]) ).
thf(7726,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[242,7111]) ).
thf(7727,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7726]) ).
thf(7751,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7727]) ).
thf(7763,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7751]) ).
thf(9326,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,7763]) ).
thf(9327,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9326]) ).
thf(9328,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9327]) ).
thf(97,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists3) ).
thf(388,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(2256,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[544,463]) ).
thf(2257,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2256]) ).
thf(2265,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[2257]) ).
thf(2283,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[2265]) ).
thf(2626,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[255,2283]) ).
thf(2632,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[2626]) ).
thf(2638,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( ( null_class @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[2632]) ).
thf(7214,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( null_class @ unordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,7087]) ).
thf(7215,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( null_class @ unordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[7214]) ).
thf(7216,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( A @ ( null_class @ unordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[7215:[bind(A,$thf( D @ ( null_class @ unordered_pair ) )),bind(B,$thf( D ))]]) ).
thf(7222,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( A @ ( null_class @ unordered_pair ) @ regular @ member ) ),
inference(simp,[status(thm)],[7216]) ).
thf(3845,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise )
= ( null_class @ domain_of @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3243,253]) ).
thf(3846,plain,
( ( identity_relation @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[3845:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(168,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_2_of_ordered_pair) ).
thf(554,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(2853,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,606]) ).
thf(2854,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2853]) ).
thf(2855,plain,
omega @ singleton @ ( omega @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[2854:[bind(A,$thf( omega @ singleton ))]]) ).
thf(19371,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2855,294]) ).
thf(19372,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[19371]) ).
thf(19373,plain,
omega @ singleton @ ( omega @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[19372:[bind(A,$thf( omega @ singleton @ regular )),bind(B,$thf( omega @ singleton ))]]) ).
thf(156,axiom,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists1) ).
thf(526,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(142,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(489,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(20,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(212,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(72,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_3_of_ordered_pair) ).
thf(328,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(7093,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ ( B @ unordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,7071]) ).
thf(7094,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ ( B @ unordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[7093]) ).
thf(7095,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( null_class @ ( A @ unordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[7094:[bind(A,$thf( null_class @ ( C @ unordered_pair ) )),bind(B,$thf( C ))]]) ).
thf(7103,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( null_class @ ( A @ unordered_pair ) @ regular @ member ) ),
inference(simp,[status(thm)],[7095]) ).
thf(8791,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ( null_class @ ( C @ unordered_pair ) @ ( A @ ( B @ unordered_pair ) @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[179,7103]) ).
thf(8792,plain,
! [A: $i] : ( null_class @ ( A @ unordered_pair ) @ ( A @ ( null_class @ unordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[8791:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(132,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_member_of_ordered_pair) ).
thf(471,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(27,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/sandbox2/benchmark/theBenchmark.p',domain) ).
thf(225,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)],[27]) ).
thf(4269,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4200,2081]) ).
thf(4270,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[4269]) ).
thf(4276,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= y ) ),
inference(simp,[status(thm)],[4270]) ).
thf(99,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair) ).
thf(393,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)],[99]) ).
thf(152,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_ordered_pair_determines_components1) ).
thf(515,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(244,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[243]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ second @ ( B @ ( A @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_1) ).
thf(311,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ second @ ( B @ ( A @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(163,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(541,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(2973,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,618]) ).
thf(2974,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2973]) ).
thf(2975,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[2974:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(3306,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(simp,[status(thm)],[2975]) ).
thf(2058,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[292,376]) ).
thf(2059,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2058]) ).
thf(2070,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[2059]) ).
thf(2076,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2070]) ).
thf(13135,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,11813]) ).
thf(13136,plain,
! [A: $i] :
( ( A != omega )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13135]) ).
thf(13137,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[13136:[bind(A,$thf( omega ))]]) ).
thf(7783,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ $true
| ~ ( y @ ( omega @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[7734,945]) ).
thf(7784,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[7783]) ).
thf(7850,plain,
( ( omega != omega )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[7784]) ).
thf(7863,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( y @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[7850]) ).
thf(12821,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( y @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,1147]) ).
thf(12822,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( y @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[12821]) ).
thf(12823,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[12822:[bind(A,$thf( omega ))]]) ).
thf(158,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(530,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(3026,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,274]) ).
thf(3027,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3026:[bind(A,$thf( omega ))]]) ).
thf(19331,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[3027,294]) ).
thf(19332,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[19331]) ).
thf(19333,plain,
( ( omega @ ( omega @ regular @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[19332:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(108,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class_definition) ).
thf(414,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(415,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[414]) ).
thf(2171,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[415,203]) ).
thf(12056,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ ( D @ ( universal_class @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ domain_of )
= ( D @ sum_class ) ) ),
inference(paramod_ordered,[status(thm)],[11966,2171]) ).
thf(12057,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(pattern_uni,[status(thm)],[12056:[bind(A,$thf( element_relation )),bind(B,$thf( universal_class )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(12173,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(simp,[status(thm)],[12057]) ).
thf(2090,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[289,2081]) ).
thf(2091,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[2090]) ).
thf(2121,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != y ) ),
inference(simp,[status(thm)],[2091]) ).
thf(60,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/sandbox2/benchmark/theBenchmark.p',compose2) ).
thf(298,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)],[60]) ).
thf(107,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/sandbox2/benchmark/theBenchmark.p',flip3) ).
thf(412,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)],[107]) ).
thf(3847,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[253,3243]) ).
thf(3848,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3847:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(3890,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3848]) ).
thf(8728,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ singleton )
!= ( C @ singleton ) )
| ( A != universal_class )
| ~ ( B @ singleton @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[233,8551]) ).
thf(8729,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ~ ( B @ singleton @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(pattern_uni,[status(thm)],[8728:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B ))]]) ).
thf(8741,plain,
! [A: $i] :
( ~ ( A @ singleton @ ( universal_class @ subclass ) )
| ( null_class = universal_class ) ),
inference(simp,[status(thm)],[8729]) ).
thf(9802,plain,
! [A: $i] :
~ ( A @ singleton @ ( universal_class @ subclass ) ),
inference(simplifyReflect,[status(thm)],[8741,582]) ).
thf(11282,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( B @ singleton @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,9802]) ).
thf(11283,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( B @ singleton @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11282]) ).
thf(11294,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11283]) ).
thf(11311,plain,
! [A: $i] :
( ( omega != universal_class )
| ~ ( A @ singleton @ inductive ) ),
inference(simp,[status(thm)],[11294]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(263,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(264,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[263]) ).
thf(599,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of @ singleton )
!= ( B @ singleton ) )
| ( A != y )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[177,598]) ).
thf(600,plain,
! [A: $i] :
( ( A != y )
| ( ( A @ member_of )
= A ) ),
inference(pattern_uni,[status(thm)],[599:[bind(A,$thf( C )),bind(B,$thf( C @ member_of ))]]) ).
thf(601,plain,
( ( y @ member_of )
= y ),
inference(simp,[status(thm)],[600]) ).
thf(151,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(513,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(116,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(430,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(17852,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,12823]) ).
thf(17853,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[17852]) ).
thf(17854,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[17853:[bind(A,$thf( null_class ))]]) ).
thf(56,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/sandbox2/benchmark/theBenchmark.p',flip2) ).
thf(290,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)],[56]) ).
thf(291,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(cnf,[status(esa)],[290]) ).
thf(165,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(545,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(281,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[280]) ).
thf(98,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/sandbox2/benchmark/theBenchmark.p',element_relation3) ).
thf(391,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)],[98]) ).
thf(95,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists2) ).
thf(382,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(67,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(313,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(146,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_number_of_elements_in_class) ).
thf(498,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(130,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/sandbox2/benchmark/theBenchmark.p',cartesian_product3) ).
thf(466,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)],[130]) ).
thf(75,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rotate1) ).
thf(335,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(82,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets4) ).
thf(350,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(105,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(406,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(2985,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,584]) ).
thf(2986,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2985]) ).
thf(2987,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[2986:[bind(A,$thf( null_class @ singleton ))]]) ).
thf(110,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_ordered_pairs_are_sets) ).
thf(419,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(93,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets1) ).
thf(377,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(641,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[372,172]) ).
thf(642,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[641]) ).
thf(646,plain,
! [B: $i,A: $i] :
( ( universal_class != universal_class )
| ( ( B @ ( A @ unordered_pair ) )
!= y ) ),
inference(simp,[status(thm)],[642]) ).
thf(649,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= y ),
inference(simp,[status(thm)],[646]) ).
thf(62,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_ordered_pairs_are_sets) ).
thf(303,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(8793,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ( A @ ( B @ unordered_pair ) @ ( null_class @ ( C @ unordered_pair ) @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[179,7103]) ).
thf(8794,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( null_class @ ( A @ unordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[8793:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(86,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(362,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(124,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection3) ).
thf(451,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)],[124]) ).
thf(35,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(245,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(2096,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,2081]) ).
thf(2097,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[2096]) ).
thf(2116,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != y ) ),
inference(simp,[status(thm)],[2097]) ).
thf(270,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(cnf,[status(esa)],[269]) ).
thf(271,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[270]) ).
thf(18,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible3) ).
thf(208,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)],[18]) ).
thf(144,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/sandbox2/benchmark/theBenchmark.p',subset_relation) ).
thf(494,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)],[144]) ).
thf(88,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets2) ).
thf(367,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(7032,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ( A @ ( B @ unordered_pair ) @ ( null_class @ member ) ) ),
inference(paramod_ordered,[status(thm)],[179,7017]) ).
thf(7033,plain,
! [A: $i] : ( A @ ( null_class @ unordered_pair ) @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[7032:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(154,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',right_cancellation) ).
thf(520,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(198,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[197]) ).
thf(135,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(477,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(19368,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2071,294]) ).
thf(19369,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( y @ ordered_pair ) @ ( null_class @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[19368]) ).
thf(19370,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( null_class @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[19369:[bind(A,$thf( E )),bind(B,$thf( null_class )),bind(C,$thf( E @ ( y @ ordered_pair ) ))]]) ).
thf(19448,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( null_class @ singleton @ subclass ) ),
inference(simp,[status(thm)],[19370]) ).
thf(9010,plain,
! [A: $i] :
( ( ( null_class @ complement @ ( y @ complement @ intersection ) @ complement )
!= ( A @ complement ) )
| ( ( A @ ( y @ successor @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[8957,3243]) ).
thf(9011,plain,
( ( null_class @ complement @ ( y @ complement @ intersection ) @ ( y @ successor @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[9010:[bind(A,$thf( null_class @ complement @ ( y @ complement @ intersection ) ))]]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(267,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(119,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/sandbox2/benchmark/theBenchmark.p',homomorphism5) ).
thf(438,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)],[119]) ).
thf(285,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(cnf,[status(esa)],[284]) ).
thf(286,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) ) ),
inference(lifteq,[status(thm)],[285]) ).
thf(262,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[261]) ).
thf(16181,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ singleton @ ( C @ image ) @ sum_class )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ singleton @ ( A @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[286,262]) ).
thf(226,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first ) ),
inference(lifteq,[status(thm)],[225]) ).
thf(5197,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( A @ ( C @ singleton @ ( B @ cross_product ) @ intersection ) @ not_subclass_element ) @ first ) ),
inference(rewrite,[status(thm)],[226,203]) ).
thf(13966,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ ( B @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,7071]) ).
thf(13967,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ ( B @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13966]) ).
thf(13968,plain,
! [A: $i] :
~ ( null_class @ ( null_class @ ( A @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[13967:[bind(A,$thf( null_class @ ( C @ unordered_pair ) )),bind(B,$thf( C ))]]) ).
thf(14207,plain,
! [A: $i] :
~ ( null_class @ ( null_class @ ( A @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[13968]) ).
thf(299,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(cnf,[status(esa)],[298]) ).
thf(11270,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,7863]) ).
thf(11271,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11270]) ).
thf(11272,plain,
( ( ( omega @ ( successor_relation @ image ) )
!= y )
| ~ ( y @ inductive ) ),
inference(pattern_uni,[status(thm)],[11271:[bind(A,$thf( y ))]]) ).
thf(7728,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[242,2319]) ).
thf(7729,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7728]) ).
thf(7752,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7729]) ).
thf(7764,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7752]) ).
thf(8033,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,7764]) ).
thf(8034,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[8033]) ).
thf(8035,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[8034]) ).
thf(643,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[372,318]) ).
thf(644,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[643]) ).
thf(645,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[644:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ unordered_pair ) ))]]) ).
thf(648,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[645]) ).
thf(1075,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,648]) ).
thf(1076,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[1075:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(209,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[208]) ).
thf(2614,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)],[209,181]) ).
thf(166,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(548,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(79,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair2) ).
thf(342,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(2934,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( y @ ordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,2150]) ).
thf(2935,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ ( y @ ordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2934]) ).
thf(2936,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( A @ ( y @ ordered_pair ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[2935:[bind(A,$thf( D @ ( y @ ordered_pair ) )),bind(B,$thf( D ))]]) ).
thf(3292,plain,
! [A: $i] : ( A @ ( y @ ordered_pair ) @ ( A @ ( y @ ordered_pair ) @ regular @ member ) ),
inference(simp,[status(thm)],[2936]) ).
thf(141,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function3) ).
thf(487,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)],[141]) ).
thf(3491,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2855,463]) ).
thf(3492,plain,
! [B: $i,A: $i] :
( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3491]) ).
thf(3497,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( omega @ singleton ) )
| ( ( omega @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3492]) ).
thf(3506,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[3497]) ).
thf(4566,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( ( omega @ singleton )
!= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[255,3506]) ).
thf(4570,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( ( omega @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[4566]) ).
thf(4581,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( ( omega @ singleton )
!= identity_relation ) ),
inference(simp,[status(thm)],[4570]) ).
thf(84,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(355,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(11228,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,945]) ).
thf(11229,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ~ ( omega @ ( y @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11228]) ).
thf(11230,plain,
( ~ ( omega @ ( y @ subclass ) )
| ~ ( y @ inductive ) ),
inference(pattern_uni,[status(thm)],[11229:[bind(A,$thf( y ))]]) ).
thf(13284,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,2076]) ).
thf(13285,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13284]) ).
thf(13286,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[13285:[bind(A,$thf( E @ ( D @ ordered_pair ) )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(14282,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[13286]) ).
thf(4271,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[405,4200]) ).
thf(4272,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4271:[bind(A,$thf( choice ))]]) ).
thf(19318,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2987,294]) ).
thf(19319,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[19318]) ).
thf(19320,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[19319:[bind(A,$thf( null_class @ singleton @ regular )),bind(B,$thf( null_class @ singleton ))]]) ).
thf(1250,plain,
! [B: $i,A: $i] :
( ( B != omega )
| ( A != universal_class )
| ( omega != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[189,1189]) ).
thf(1251,plain,
! [A: $i] :
( ( A != universal_class )
| ( omega != null_class )
| ~ ( A @ ( omega @ subclass ) )
| ~ ( omega @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1250:[bind(A,$thf( A )),bind(B,$thf( omega ))]]) ).
thf(1253,plain,
( ( omega != null_class )
| ~ ( universal_class @ ( omega @ subclass ) )
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1251]) ).
thf(11136,plain,
( ( omega != null_class )
| ~ $true
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[1253,482]) ).
thf(11137,plain,
( ( omega != null_class )
| ~ ( omega @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[11136]) ).
thf(169,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/sandbox2/benchmark/theBenchmark.p',range) ).
thf(557,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)],[169]) ).
thf(13542,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ unordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,1076]) ).
thf(13543,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ unordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13542]) ).
thf(13544,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[13543:[bind(A,$thf( F @ ( E @ unordered_pair ) @ singleton )),bind(B,$thf( F )),bind(C,$thf( E ))]]) ).
thf(14091,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[13544]) ).
thf(5055,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3306,463]) ).
thf(5056,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[5055]) ).
thf(5064,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( ( A @ singleton @ singleton @ regular )
!= B ) ),
inference(simp,[status(thm)],[5056]) ).
thf(5074,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ singleton )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[5064]) ).
thf(11852,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,11312]) ).
thf(11853,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[11852]) ).
thf(11855,plain,
( ( omega != null_class )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[11853]) ).
thf(1766,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,1151]) ).
thf(1767,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( omega @ subclass ) ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1766]) ).
thf(1772,plain,
! [A: $i] :
( ( universal_class != y )
| ( A != omega )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1767]) ).
thf(1780,plain,
( ( universal_class != y )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[1772]) ).
thf(5694,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,5663]) ).
thf(5695,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5694]) ).
thf(5731,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[5695]) ).
thf(89,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cartesian_product2) ).
thf(370,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)],[89]) ).
thf(77,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equality3) ).
thf(338,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(4121,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( omega != universal_class ) ),
inference(paramod_ordered,[status(thm)],[3142,172]) ).
thf(4122,plain,
( ( ( omega @ ( omega @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4121]) ).
thf(4128,plain,
( ( omega != universal_class )
| ( ( omega @ regular )
!= y )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4122]) ).
thf(4132,plain,
( ( ( omega @ regular )
!= y )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[4128]) ).
thf(3353,plain,
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3311,172]) ).
thf(3354,plain,
( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[3353]) ).
thf(3359,plain,
( ( universal_class != y )
| ( ( y @ regular )
!= y ) ),
inference(simp,[status(thm)],[3354]) ).
thf(4396,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( omega @ regular @ singleton ) )
| ( null_class != null_class )
| ( omega != universal_class )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[411,4140]) ).
thf(4397,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( omega @ regular @ singleton ) )
| ( omega != universal_class )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4396]) ).
thf(4398,plain,
( ( omega != universal_class )
| ( universal_class @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4397:[bind(A,$thf( omega @ regular ))]]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(199,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(3062,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[512,255]) ).
thf(3063,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3062:[bind(A,$thf( subset_relation ))]]) ).
thf(76,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(336,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(7710,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[242,5663]) ).
thf(7711,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7710]) ).
thf(7742,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7711]) ).
thf(7779,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[7742]) ).
thf(122,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(446,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(7882,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[179,7617]) ).
thf(7883,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[7882:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(7908,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[7883]) ).
thf(1166,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[289,1147]) ).
thf(1167,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( omega @ ( y @ subclass ) ) )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1166]) ).
thf(1180,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= omega )
| ( element_relation != y )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1167]) ).
thf(218,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[217]) ).
thf(213,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[212]) ).
thf(3665,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[213,181]) ).
thf(495,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)],[494]) ).
thf(5233,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[3243,3890]) ).
thf(5234,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[5233:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(5332,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
= null_class ),
inference(rewrite,[status(thm)],[5234,3846]) ).
thf(651,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= y ) ),
inference(paramod_ordered,[status(thm)],[179,649]) ).
thf(652,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= y ),
inference(pattern_uni,[status(thm)],[651:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(3024,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[512,172]) ).
thf(3025,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[3024]) ).
thf(3231,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( A @ regular )
!= y )
| ( A = null_class ) ),
inference(simp,[status(thm)],[3025]) ).
thf(3307,plain,
( ( ( universal_class @ regular )
!= y )
| ( null_class = universal_class ) ),
inference(simp,[status(thm)],[3231]) ).
thf(3369,plain,
( ( universal_class @ regular )
!= y ),
inference(simplifyReflect,[status(thm)],[3307,582]) ).
thf(7,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(183,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(184,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[183]) ).
thf(185,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[184]) ).
thf(186,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[185]) ).
thf(103,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/sandbox2/benchmark/theBenchmark.p',domain2) ).
thf(401,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)],[103]) ).
thf(69,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/sandbox2/benchmark/theBenchmark.p',homomorphism6) ).
thf(319,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)],[69]) ).
thf(26,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/sandbox2/benchmark/theBenchmark.p',operation4) ).
thf(222,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)],[26]) ).
thf(223,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)],[222]) ).
thf(224,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)],[223]) ).
thf(4745,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)],[224,181]) ).
thf(7840,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7734,2081]) ).
thf(7841,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( y @ subclass ) ) ),
inference(simp,[status(thm)],[7840]) ).
thf(7855,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= y ) ),
inference(simp,[status(thm)],[7841]) ).
thf(246,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[245]) ).
thf(17,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(207,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(11225,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,10869]) ).
thf(11226,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[11225]) ).
thf(11227,plain,
( ( omega != universal_class )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[11226:[bind(A,$thf( null_class ))]]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(536,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(12757,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,1055]) ).
thf(12758,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[12757]) ).
thf(12759,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[12758:[bind(A,$thf( F @ ( E @ ordered_pair ) @ singleton )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(14138,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[12759]) ).
thf(117,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_members) ).
thf(433,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(61,axiom,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',left_cancellation) ).
thf(300,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(6933,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ singleton @ ( C @ unordered_pair ) ) )
| ( A @ singleton @ ( B @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[196,423]) ).
thf(6934,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6933:[bind(A,$thf( D @ singleton )),bind(B,$thf( D )),bind(C,$thf( D @ singleton ))]]) ).
thf(6997,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[6934]) ).
thf(91,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/sandbox2/benchmark/theBenchmark.p',compose3) ).
thf(373,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)],[91]) ).
thf(63,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(305,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(78,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',flip1) ).
thf(341,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(268,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[267]) ).
thf(85,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/sandbox2/benchmark/theBenchmark.p',successor_relation3) ).
thf(358,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)],[85]) ).
thf(134,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets2) ).
thf(474,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(1046,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[422,172]) ).
thf(1047,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[1046]) ).
thf(1051,plain,
! [B: $i,A: $i] :
( ( universal_class != universal_class )
| ( ( B @ ( A @ ordered_pair ) )
!= y ) ),
inference(simp,[status(thm)],[1047]) ).
thf(1056,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= y ),
inference(simp,[status(thm)],[1051]) ).
thf(159,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation2) ).
thf(532,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(129,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(464,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(74,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(332,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(13618,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,11137]) ).
thf(13619,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13618]) ).
thf(13620,plain,
( ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[13619:[bind(A,$thf( omega ))]]) ).
thf(20513,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[266,13620]) ).
thf(20514,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ subclass ) ) )
| ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[20513]) ).
thf(20515,plain,
( ~ ( omega @ ( universal_class @ subclass ) )
| ~ ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[20514:[bind(A,$thf( null_class ))]]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',replacement) ).
thf(528,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)],[157]) ).
thf(200,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[199]) ).
thf(18585,plain,
! [A: $i] :
( ( A
!= ( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,18380]) ).
thf(18586,plain,
! [A: $i] :
( ( A
!= ( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18585]) ).
thf(18587,plain,
~ ( null_class @ ( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement @ subclass ) ),
inference(pattern_uni,[status(thm)],[18586:[bind(A,$thf( null_class @ complement @ ( null_class @ singleton @ complement @ intersection ) @ complement ))]]) ).
thf(147,axiom,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(501,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(100,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/sandbox2/benchmark/theBenchmark.p',rotate3) ).
thf(395,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)],[100]) ).
thf(15552,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( null_class @ ( C @ unordered_pair ) ) )
| ~ ( null_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[15388,14207]) ).
thf(15553,plain,
! [A: $i] :
~ ( null_class @ ( null_class @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ subclass ) ),
inference(pattern_uni,[status(thm)],[15552:[bind(A,$thf( A )),bind(B,$thf( null_class )),bind(C,$thf( A ))]]) ).
thf(114,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(427,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(127,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_of_1st_and_2nd_5) ).
thf(459,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(9368,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[274,7779]) ).
thf(9369,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[9368]) ).
thf(9370,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[9369]) ).
thf(70,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pair_determines_components2) ).
thf(322,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(65,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(309,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(13674,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ singleton @ ( C @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[279,6994]) ).
thf(13675,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ singleton @ ( C @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[13674]) ).
thf(13676,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[13675:[bind(A,$thf( F @ singleton @ ( D @ unordered_pair ) )),bind(B,$thf( F )),bind(C,$thf( D ))]]) ).
thf(14112,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[13676]) ).
thf(145,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(496,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(2334,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[484,2319]) ).
thf(2335,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[2334]) ).
thf(2351,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[2335]) ).
thf(143,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(491,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(33977,plain,
$false,
inference(e,[status(thm)],[365,479,249,518,468,234,347,11922,170,217,247,440,269,202,523,2309,582,385,8551,10869,3365,472,4279,945,538,3350,7734,288,8472,7993,257,174,404,1055,7825,372,504,11966,344,7867,196,9056,4546,18380,189,1147,14103,606,284,416,325,228,19282,5727,289,448,316,8738,2852,261,216,5663,179,443,376,485,7845,238,1151,2291,353,1183,14209,211,253,397,11312,280,221,293,453,3243,411,243,428,265,7760,507,3406,206,14046,3435,307,292,233,7849,380,7179,11253,11307,7859,220,534,4031,512,5783,16091,201,3827,7827,260,7229,598,297,424,192,17467,275,2150,7185,3065,551,456,229,566,484,252,1602,4140,197,11813,435,12959,9073,15606,2359,9328,388,2638,7222,3311,3846,554,7111,193,19373,526,489,15388,212,328,8792,471,256,225,4276,393,515,244,266,311,541,3306,2076,1466,13137,7863,12823,530,19333,12173,2121,298,412,618,2275,3890,11311,264,279,180,375,601,513,430,584,562,423,17854,291,7763,545,281,204,391,382,313,498,466,335,181,2319,350,187,406,172,219,2987,419,377,230,318,649,303,8794,274,3238,362,451,245,2116,271,208,494,3027,367,7033,483,520,251,198,462,477,17263,2171,19448,3142,9011,267,438,16181,241,15725,5197,14207,3363,299,11272,8035,1076,2614,548,255,342,3292,194,648,487,4581,355,11230,14282,295,282,4272,414,19320,11137,409,263,557,7087,14091,4200,5074,11855,1780,5731,331,370,463,338,4132,3359,6994,544,4398,199,177,3063,182,336,7779,175,446,7908,1180,7764,218,1189,3665,250,231,482,203,495,8957,5332,2071,2855,9802,258,652,3369,186,401,171,2283,319,4745,2081,7855,246,207,11227,272,7017,536,290,14138,433,510,214,287,300,6997,262,3892,405,190,183,422,373,305,210,242,294,341,268,358,195,474,1056,400,532,3506,464,332,20515,528,7829,7617,200,277,18587,501,13620,395,15553,1152,427,459,7071,9370,178,322,309,215,14112,222,496,7103,254,227,2351,491]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET122-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% 0.00/0.13 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.27/5.53 % Computer : n002.cluster.edu
% 0.27/5.53 % Model : x86_64 x86_64
% 0.27/5.53 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.27/5.53 % Memory : 8046.5625MB
% 0.27/5.53 % OS : Linux 6.8.0-71-generic
% 0.27/5.53 % CPULimit : 300
% 0.27/5.53 % WCLimit : 300
% 0.27/5.53 % DateTime : Sat Sep 26 07:13:58 UTC 2026
% 0.27/5.54 % CPUTime :
% 0.27/5.54 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.36/6.45 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.89/6.91 % [INFO] Parsing done (455ms).
% 2.89/6.93 % [INFO] Running in sequential loop mode.
% 3.90/7.56 % [INFO] eprover registered as external prover.
% 3.90/7.57 % [INFO] Scanning for conjecture ...
% 4.32/7.78 % [INFO] Found a conjecture (or negated_conjecture) and 167 axioms. Running axiom selection ...
% 4.70/7.97 % [INFO] Axiom selection finished. Selected 167 axioms (removed 0 axioms).
% 5.49/8.18 % [INFO] Problem is propositional (TPTP CNF).
% 5.60/8.20 % [INFO] Type checking passed.
% 5.60/8.21 % [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 ...
% 117.86/29.12 % External prover 'e' found a proof!
% 117.86/29.13 % [INFO] Killing All external provers ...
% 117.86/29.13 % Time passed: 23380ms (effective reasoning time: 22188ms)
% 116.12/29.13 % Axioms used in derivation (167): operation4, equal_implies_subclass1, complement1, one_to_one2, null_unordered_pair, property_3_of_ordered_pair, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, singleton_in_unordered_pair3, function3, corollary_to_set_in_its_singleton, regularity1, singleton_set, identity_relation, commutativity_of_unordered_pair, union, flip3, element_relation2, unordered_pairs_in_universal, corollary_of_null_class_is_subclass, equality1, compose2, homomorphism3, compatible3, symmetric_difference, cartesian_product1, image, unique_1st_and_2nd_in_pair_of_non_sets1, replacement, singletons_are_sets, not_subclass_members1, existence_of_null_class, ordered_pair, successor_relation1, property_2_of_ordered_pair, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, corollary_2_to_ordered_pair_determines_components1, singleton_in_unordered_pair1, singleton_identified_by_element2, member_of_non_singleton_unique1, null_class_is_a_set, member_exists2, one_to_one3, ordered_pair_determines_components1, single_valued_class1, existence_of_1st_and_2nd_5, compose3, omega_is_inductive2, rotate1, corollary_1_to_ordered_pairs_are_sets, left_cancellation, choice1, property_1_of_ordered_pair, power_class2, equality2, regularity2, singleton_member_of_ordered_pair, equal_implies_subclass2, element_relation3, homomorphism6, existence_of_1st_and_2nd_1, intersection1, existence_of_1st_and_2nd_4, flip2, two_subsets_of_singleton, ordered_pair_is_set, corollary_1_to_singletons_are_sets, inverse, corollary_1_to_number_of_elements_in_class, unique_1st_and_2nd_in_pair_of_non_sets4, corollary_to_unordered_pair_axiom3, sum_class_definition, corollary_1_to_ordered_pair_determines_components1, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, only_member_in_singleton, not_subclass_members2, ordered_pair_determines_components2, successor_relation3, member_exists3, domain2, function1, right_cancellation, compatible4, property_of_singletons2, intersection3, singleton_identified_by_element1, corollary_to_unordered_pair_axiom1, member_of_non_singleton_unique2, corollary_2_to_singletons_are_sets, single_valued_class2, unordered_pair_member, apply, corollary_2_to_number_of_elements_in_class, element_relation1, transitivity_of_subclass, operation1, member_of_singleton_is_unique, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, existence_of_1st_and_2nd_2, flip1, homomorphism5, unique_1st_and_2nd_in_pair_of_sets1, compatible1, unordered_pair2, unique_1st_and_2nd_in_pair_of_non_sets3, corollary_2_to_ordered_pairs_are_sets, number_of_elements_in_class, unordered_pair_member_of_ordered_pair, operation2, member_exists4, rotate3, property_of_singletons1, singleton_in_unordered_pair2, complement2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, unique_1st_and_2nd_in_pair_of_sets2, set_in_its_singleton, domain1, unordered_pair_is_subset, null_class_in_its_singleton, corollary_1_to_cartesian_product, null_class_is_unique, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, unordered_pairs_and_singletons, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, singleton_is_null_class, existence_of_1st_and_2nd_3, operation3, subclass_is_reflexive, homomorphism2, power_class_definition, unordered_pair_equals_singleton1, corollary_2_to_cartesian_product, choice2, unordered_pair3, special_classes_lemma, subset_relation, unique_1st_and_2nd_in_pair_of_non_sets2, compose1, member_exists1
% 116.12/29.13 % No. of inferences in proof: 951
% 116.12/29.13 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 23380 ms resp. 22188 ms w/o parsing
% 116.91/29.32 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 116.91/29.33 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------