%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET205-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 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 : n018.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 08:29:47 AM UTC 2026
% Result : Unsatisfiable 186.91s 46.32s
% Output : Refutation 177.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 113
% Syntax : Number of formulae : 710 ( 223 unt; 0 typ; 0 def)
% Number of atoms : 1723 ( 649 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 6695 ( 929 ~; 683 |; 0 &;5083 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 15 con; 0-3 aty)
% Number of variables : 1075 ( 0 ^;1075 !; 0 ?;1075 :)
% Comments :
%------------------------------------------------------------------------------
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(x_decl,type,
x: $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(omega_decl,type,
omega: $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(operation_decl,type,
operation: $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(function_decl,type,
function: $i > $o ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(choice_decl,type,
choice: $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(single_valued1_decl,type,
single_valued1: $i > $i ).
thf(single_valued2_decl,type,
single_valued2: $i > $i ).
thf(single_valued3_decl,type,
single_valued3: $i > $i ).
thf(complement_decl,type,
complement: $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(identity_relation_decl,type,
identity_relation: $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(application_function_decl,type,
application_function: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(successor_decl,type,
successor: $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(singleton_relation_decl,type,
singleton_relation: $i ).
thf(second_decl,type,
second: $i > $i ).
thf(compose_class_decl,type,
compose_class: $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(25,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_set) ).
thf(169,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(170,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[169]) ).
thf(49,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_in_universal) ).
thf(220,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(16,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(148,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(149,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[148]) ).
thf(7395,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,149]) ).
thf(7396,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[7395]) ).
thf(7397,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7396:[bind(A,$thf( omega ))]]) ).
thf(7500,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(simp,[status(thm)],[7397]) ).
thf(7547,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) ) )
| ( A @ singleton @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[170,7500]) ).
thf(7548,plain,
omega @ singleton @ ( omega @ member ),
inference(pattern_uni,[status(thm)],[7547:[bind(A,$thf( omega )),bind(B,$thf( omega ))]]) ).
thf(8,axiom,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation3) ).
thf(131,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(132,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[131]) ).
thf(1221,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,132]) ).
thf(1222,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[1221]) ).
thf(1223,plain,
domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ),
inference(pattern_uni,[status(thm)],[1222:[bind(A,$thf( omega ))]]) ).
thf(51,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',complement1) ).
thf(223,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(224,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[223]) ).
thf(1290,plain,
! [B: $i,A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1223,224]) ).
thf(1291,plain,
! [B: $i,A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1290]) ).
thf(1292,plain,
~ ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[1291:[bind(A,$thf( omega @ domain_of @ ( omega @ ordered_pair ) )),bind(B,$thf( domain_relation ))]]) ).
thf(7625,plain,
( ( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,1292]) ).
thf(7626,plain,
( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7625]) ).
thf(7691,plain,
( ( ( domain_relation @ complement )
!= ( omega @ singleton ) )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[7626]) ).
thf(111,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity2) ).
thf(347,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(63,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one2) ).
thf(248,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(249,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[248]) ).
thf(56,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',identity_relation) ).
thf(234,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(112,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',regularity1) ).
thf(350,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(351,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[350]) ).
thf(352,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[351]) ).
thf(360,plain,
! [B: $i,A: $i] :
( ~ $true
| ( ( B @ complement @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[224]) ).
thf(363,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ( A != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[360]) ).
thf(365,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= B )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[363]) ).
thf(11,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection1) ).
thf(137,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(138,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[137]) ).
thf(2130,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,220]) ).
thf(2131,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2130:[bind(A,$thf( omega ))]]) ).
thf(67,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(258,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(805,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ( universal_class @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[170,258]) ).
thf(806,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[805:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(1216,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( domain_relation @ ( B @ domain_of @ ( B @ ordered_pair ) @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[806,132]) ).
thf(1217,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( domain_relation @ ( B @ domain_of @ ( B @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[1216]) ).
thf(1218,plain,
! [A: $i] : ( domain_relation @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[1217:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(1268,plain,
! [A: $i] : ( domain_relation @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[1218]) ).
thf(1412,plain,
! [C: $i,B: $i,A: $i] :
( ( ( domain_relation @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1268,224]) ).
thf(1413,plain,
! [C: $i,B: $i,A: $i] :
( ( ( domain_relation @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1412]) ).
thf(1414,plain,
! [A: $i] :
~ ( domain_relation @ complement @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[1413:[bind(A,$thf( H )),bind(B,$thf( H @ singleton @ domain_of @ ( H @ singleton @ ordered_pair ) )),bind(C,$thf( domain_relation ))]]) ).
thf(1454,plain,
! [A: $i] :
~ ( domain_relation @ complement @ ( A @ singleton @ domain_of @ ( A @ singleton @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[1414]) ).
thf(13,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',image) ).
thf(142,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(143,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[142]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class_definition) ).
thf(167,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(168,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[167]) ).
thf(475,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[168,168]) ).
thf(476,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[475:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(490,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[476]) ).
thf(3300,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ power_class )
= ( C @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[143,490]) ).
thf(3301,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3300:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3317,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[3301]) ).
thf(1,negated_conjecture,
( ( null_class @ ( x @ cross_product ) )
!= null_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_cross_product_with_null_class1_1) ).
thf(114,plain,
( ( null_class @ ( x @ cross_product ) )
!= null_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(115,plain,
( ( null_class @ ( x @ cross_product ) )
!= null_class ),
inference(polarity_switch,[status(thm)],[114]) ).
thf(116,plain,
( ( null_class @ ( x @ cross_product ) )
!= null_class ),
inference(lifteq,[status(thm)],[115]) ).
thf(2100,plain,
! [A: $i] :
( ( A
!= ( null_class @ ( x @ cross_product ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,116]) ).
thf(2101,plain,
! [A: $i] :
( ( A
!= ( null_class @ ( x @ cross_product ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2100]) ).
thf(2102,plain,
null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[2101:[bind(A,$thf( null_class @ ( x @ cross_product ) ))]]) ).
thf(2642,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2102,224]) ).
thf(2643,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2642]) ).
thf(2644,plain,
~ ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[2643:[bind(A,$thf( null_class @ ( x @ cross_product ) @ regular )),bind(B,$thf( null_class @ ( x @ cross_product ) ))]]) ).
thf(35,axiom,
omega @ inductive,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(189,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(104,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive1) ).
thf(335,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(336,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[335]) ).
thf(922,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,336]) ).
thf(923,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[922]) ).
thf(924,plain,
omega @ ( null_class @ member ),
inference(pattern_uni,[status(thm)],[923:[bind(A,$thf( omega ))]]) ).
thf(1044,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[924,224]) ).
thf(1045,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1044]) ).
thf(1046,plain,
~ ( omega @ complement @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[1045:[bind(A,$thf( null_class )),bind(B,$thf( omega ))]]) ).
thf(2609,plain,
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,1046]) ).
thf(2610,plain,
( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2609]) ).
thf(2676,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= ( omega @ complement ) )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2610]) ).
thf(47,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(216,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(22,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(162,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(163,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[162]) ).
thf(164,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[163]) ).
thf(11178,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ ( x @ cross_product ) ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,116]) ).
thf(11179,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( null_class @ ( x @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11178:[bind(A,$thf( null_class @ ( x @ cross_product ) )),bind(B,$thf( B ))]]) ).
thf(13340,plain,
( ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) )
| ~ ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[11179]) ).
thf(14572,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,13340]) ).
thf(14573,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[14572]) ).
thf(14692,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( A
!= ( null_class @ ( x @ cross_product ) ) )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[14573]) ).
thf(14712,plain,
( ( universal_class != null_class )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[14692]) ).
thf(14745,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,14712]) ).
thf(14746,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[14745]) ).
thf(14810,plain,
! [A: $i] :
( ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( A != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[14746]) ).
thf(14818,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[14810]) ).
thf(16018,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,14818]) ).
thf(16019,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[16018]) ).
thf(16020,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16019:[bind(A,$thf( universal_class ))]]) ).
thf(18601,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( ( null_class @ ( x @ cross_product ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[16020,224]) ).
thf(18602,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( ( null_class @ ( x @ cross_product ) )
!= universal_class ) ),
inference(simp,[status(thm)],[18601]) ).
thf(18603,plain,
( ~ ( universal_class @ complement @ ( universal_class @ regular @ member ) )
| ( ( null_class @ ( x @ cross_product ) )
!= universal_class ) ),
inference(pattern_uni,[status(thm)],[18602:[bind(A,$thf( universal_class @ regular )),bind(B,$thf( universal_class ))]]) ).
thf(110,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor_relation1) ).
thf(346,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(357,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[220,224]) ).
thf(358,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[357]) ).
thf(359,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[358:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(366,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,359]) ).
thf(367,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[366]) ).
thf(368,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[367]) ).
thf(369,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[368]) ).
thf(12750,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,369]) ).
thf(12751,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[12750:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(13117,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[12751]) ).
thf(14316,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[13117,216]) ).
thf(14317,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[14316]) ).
thf(14375,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[346,14317]) ).
thf(14376,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14375]) ).
thf(14399,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[14376]) ).
thf(7,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(129,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(130,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[129]) ).
thf(14365,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[130,14317]) ).
thf(14366,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14365]) ).
thf(14403,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( universal_class != omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14366]) ).
thf(14414,plain,
( ( universal_class != omega )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[14403]) ).
thf(48,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ apply ) )
= C )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn3) ).
thf(217,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ apply ) )
= C )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(76,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function1) ).
thf(276,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(55,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',diagonalisation) ).
thf(232,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(233,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[232]) ).
thf(506,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[233,168]) ).
thf(507,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[506:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(533,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[507]) ).
thf(28,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(175,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)],[28]) ).
thf(176,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
inference(lifteq,[status(thm)],[175]) ).
thf(7388,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( C @ ( B @ unordered_pair ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[806,149]) ).
thf(7389,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( C @ ( B @ unordered_pair ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[7388]) ).
thf(7390,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[7389:[bind(A,$thf( D )),bind(B,$thf( D @ singleton ))]]) ).
thf(7496,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[7390]) ).
thf(20297,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( B @ ( A @ ordered_pair ) @ ( D @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[176,7496]) ).
thf(20298,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[20297:[bind(A,$thf( E )),bind(B,$thf( G )),bind(C,$thf( G @ singleton @ ( E @ unordered_pair ) )),bind(D,$thf( E ))]]) ).
thf(20381,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[20298]) ).
thf(18,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',range_of) ).
thf(153,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(154,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(lifteq,[status(thm)],[153]) ).
thf(3306,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ complement )
= ( C @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[143,168]) ).
thf(3307,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3306:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3320,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3307]) ).
thf(3392,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ complement @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[154,3320]) ).
thf(3393,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3392:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(3478,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3393]) ).
thf(3576,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3478,168]) ).
thf(3577,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3576:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(19,axiom,
choice @ function,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice1) ).
thf(155,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(1953,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,155]) ).
thf(1954,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1953:[bind(A,$thf( choice ))]]) ).
thf(107,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(340,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)],[107]) ).
thf(92,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse) ).
thf(309,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(310,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[309]) ).
thf(537,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( B @ range_of ) ) ),
inference(paramod_ordered,[status(thm)],[310,154]) ).
thf(538,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ range_of ) ),
inference(pattern_uni,[status(thm)],[537:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(27,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one3) ).
thf(173,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(174,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[173]) ).
thf(33,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps3) ).
thf(185,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(89,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(301,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)],[89]) ).
thf(7627,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ singleton @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,7548]) ).
thf(7628,plain,
( ( omega @ singleton @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7627:[bind(A,$thf( omega ))]]) ).
thf(32,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(183,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(184,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[183]) ).
thf(2653,plain,
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,359]) ).
thf(2654,plain,
( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[2653]) ).
thf(2667,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= ( universal_class @ complement ) )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= omega ) ),
inference(simp,[status(thm)],[2654]) ).
thf(1049,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,365]) ).
thf(1050,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[1049]) ).
thf(1051,plain,
( ( omega @ complement )
!= omega ),
inference(pattern_uni,[status(thm)],[1050:[bind(A,$thf( null_class )),bind(B,$thf( omega ))]]) ).
thf(11078,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ complement ) )
| ( B != omega )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,1051]) ).
thf(11079,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ complement @ ( A @ subclass ) )
| ~ ( A @ ( omega @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11078:[bind(A,$thf( omega @ complement )),bind(B,$thf( B ))]]) ).
thf(13304,plain,
( ~ ( omega @ complement @ ( omega @ subclass ) )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[11079]) ).
thf(20871,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( omega @ complement @ subclass ) ) )
| ~ ( omega @ complement @ ( omega @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,13304]) ).
thf(20872,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( omega @ complement @ subclass ) ) )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[20871]) ).
thf(21052,plain,
! [A: $i] :
( ( universal_class != omega )
| ( A
!= ( omega @ complement ) )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[20872]) ).
thf(21086,plain,
( ( universal_class != omega )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[21052]) ).
thf(1285,plain,
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1223,1046]) ).
thf(1286,plain,
( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1285]) ).
thf(1323,plain,
( ( ( omega @ complement )
!= domain_relation )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= null_class ) ),
inference(simp,[status(thm)],[1286]) ).
thf(7650,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7548,224]) ).
thf(7651,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[7650]) ).
thf(7652,plain,
~ ( omega @ singleton @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7651:[bind(A,$thf( omega )),bind(B,$thf( omega @ singleton ))]]) ).
thf(109,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(344,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)],[109]) ).
thf(355,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,224]) ).
thf(356,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[355]) ).
thf(361,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[356]) ).
thf(364,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[361]) ).
thf(2664,plain,
! [A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,364]) ).
thf(2665,plain,
! [A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2664]) ).
thf(2677,plain,
! [A: $i] :
( ( ( null_class @ ( x @ cross_product ) )
!= A )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= omega )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2665]) ).
thf(2682,plain,
( ( ( null_class @ ( x @ cross_product ) @ regular )
!= omega )
| ( ( null_class @ ( x @ cross_product ) @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2677]) ).
thf(20,axiom,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn3) ).
thf(156,plain,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(157,plain,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
inference(lifteq,[status(thm)],[156]) ).
thf(2106,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( A @ ( x @ cross_product ) )
!= null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,116]) ).
thf(2107,plain,
! [A: $i] :
( ( ( A @ ( x @ cross_product ) )
!= null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2106:[]]) ).
thf(6270,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2107,1046]) ).
thf(6271,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6270]) ).
thf(6340,plain,
! [A: $i] :
( ( A
!= ( omega @ complement ) )
| ( ( A @ regular )
!= null_class )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6271]) ).
thf(6350,plain,
( ( ( omega @ complement @ regular )
!= null_class )
| ( ( omega @ complement @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6340]) ).
thf(86,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(296,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(2216,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2217,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2216:[bind(A,$thf( universal_class ))]]) ).
thf(20347,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[176,258]) ).
thf(20348,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20347:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( F @ singleton )),bind(D,$thf( H @ singleton @ ( F @ unordered_pair ) ))]]) ).
thf(20408,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20348]) ).
thf(100,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(325,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)],[100]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps2) ).
thf(150,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(151,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[150]) ).
thf(152,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ maps ) ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[151]) ).
thf(69,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(261,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)],[69]) ).
thf(4,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(122,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)],[4]) ).
thf(123,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)],[122]) ).
thf(6,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(127,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)],[6]) ).
thf(128,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)],[127]) ).
thf(409,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(paramod_ordered,[status(thm)],[258,224]) ).
thf(410,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(simp,[status(thm)],[409]) ).
thf(411,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[410:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(424,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[411]) ).
thf(20324,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ~ ( universal_class @ complement @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[176,424]) ).
thf(20325,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20324:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( F @ singleton )),bind(D,$thf( H @ singleton @ ( F @ unordered_pair ) ))]]) ).
thf(20401,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20325]) ).
thf(20594,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( F @ ( E @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[128,20401]) ).
thf(20595,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( F @ ( E @ ordered_pair ) @ member ) ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[20594]) ).
thf(20596,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20595:[bind(A,$thf( H )),bind(B,$thf( G )),bind(C,$thf( C )),bind(D,$thf( universal_class @ complement )),bind(E,$thf( H @ ( G @ ordered_pair ) )),bind(F,$thf( C ))]]) ).
thf(20708,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20596]) ).
thf(21850,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ flip @ ( E @ ( F @ ( G @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[123,20708]) ).
thf(21851,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ flip @ ( E @ ( F @ ( G @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[21850]) ).
thf(21852,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21851:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( universal_class @ complement @ flip )),bind(E,$thf( A )),bind(F,$thf( C )),bind(G,$thf( B ))]]) ).
thf(30,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation3) ).
thf(179,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(98,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function1) ).
thf(321,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(44,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn1) ).
thf(210,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(211,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[210]) ).
thf(1774,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[490,168]) ).
thf(1775,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1774:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1858,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1775]) ).
thf(58,axiom,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_can_define_singleton) ).
thf(238,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(3,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(119,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)],[3]) ).
thf(120,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)],[119]) ).
thf(121,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)],[120]) ).
thf(113,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive3) ).
thf(353,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(7878,plain,
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,7652]) ).
thf(7879,plain,
( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7878]) ).
thf(7899,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= ( omega @ singleton @ complement ) )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= omega ) ),
inference(simp,[status(thm)],[7879]) ).
thf(15,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps1) ).
thf(146,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(147,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[146]) ).
thf(80,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(280,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)],[80]) ).
thf(7866,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2107,7652]) ).
thf(7867,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[7866]) ).
thf(7907,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton @ complement ) )
| ( ( A @ regular )
!= omega )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[7867]) ).
thf(7931,plain,
( ( ( omega @ singleton @ complement @ regular )
!= omega )
| ( ( omega @ singleton @ complement @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[7907]) ).
thf(14,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(144,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[144]) ).
thf(14370,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[145,14317]) ).
thf(14371,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[14370]) ).
thf(14372,plain,
~ ( universal_class @ complement @ ( universal_class @ complement @ ( universal_class @ not_subclass_element ) @ member ) ),
inference(pattern_uni,[status(thm)],[14371:[bind(A,$thf( universal_class )),bind(B,$thf( universal_class @ complement ))]]) ).
thf(50,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_class2) ).
thf(221,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(77,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',element_relation1) ).
thf(277,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(14382,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[277,14317]) ).
thf(14383,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14382]) ).
thf(14410,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[14383]) ).
thf(1058,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,359]) ).
thf(1059,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1058]) ).
thf(1072,plain,
( ( ( universal_class @ complement )
!= omega )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1059]) ).
thf(9,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(133,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)],[9]) ).
thf(1090,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[806,1046]) ).
thf(1091,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1090]) ).
thf(1104,plain,
! [A: $i] :
( ( ( omega @ complement )
!= universal_class )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[1091]) ).
thf(2423,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,277]) ).
thf(2424,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( element_relation @ ( element_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2423:[bind(A,$thf( element_relation ))]]) ).
thf(84,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(292,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)],[84]) ).
thf(73,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(270,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)],[73]) ).
thf(1319,plain,
! [A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1223,364]) ).
thf(1320,plain,
! [A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1319]) ).
thf(1330,plain,
! [A: $i] :
( ( domain_relation != A )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= omega )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1320]) ).
thf(1338,plain,
( ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= omega )
| ( ( domain_relation @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1330]) ).
thf(803,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( C @ ( B @ unordered_pair ) ) )
| ~ ( universal_class @ complement @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[170,424]) ).
thf(804,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[803:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(1056,plain,
! [A: $i] :
( ( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,804]) ).
thf(1057,plain,
! [A: $i] :
( ( universal_class @ complement @ ( A @ singleton @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1056]) ).
thf(1067,plain,
! [A: $i] :
( ( ( universal_class @ complement )
!= omega )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[1057]) ).
thf(2740,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,2644]) ).
thf(2741,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[2740]) ).
thf(2759,plain,
! [A: $i] :
( ( A
!= ( null_class @ ( x @ cross_product ) @ complement ) )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[2741]) ).
thf(2773,plain,
( ( ( null_class @ ( x @ cross_product ) @ regular )
!= null_class )
| ~ ( null_class @ ( x @ cross_product ) @ complement @ inductive ) ),
inference(simp,[status(thm)],[2759]) ).
thf(14785,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[130,14712]) ).
thf(14786,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14785]) ).
thf(14808,plain,
! [A: $i] :
( ( A
!= ( null_class @ ( x @ cross_product ) ) )
| ( omega != null_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14786]) ).
thf(14817,plain,
( ( omega != null_class )
| ( universal_class != null_class )
| ~ ( null_class @ ( x @ cross_product ) @ inductive ) ),
inference(simp,[status(thm)],[14808]) ).
thf(2,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism3) ).
thf(117,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(41,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(201,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(5016,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3317,490]) ).
thf(5017,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[5016:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(7671,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,359]) ).
thf(7672,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7671]) ).
thf(7695,plain,
( ( ( universal_class @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7672]) ).
thf(7703,plain,
( ( universal_class @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[7695]) ).
thf(68,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(259,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(260,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[259]) ).
thf(2132,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,220]) ).
thf(2133,plain,
( ( null_class @ ( omega @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2132:[bind(A,$thf( universal_class ))]]) ).
thf(87,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(297,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)],[87]) ).
thf(36,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(190,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)],[36]) ).
thf(191,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[190]) ).
thf(192,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A = C )
| ( A = B ) ),
inference(lifteq,[status(thm)],[191]) ).
thf(23,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection2) ).
thf(165,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(3361,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ complement @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[538,3320]) ).
thf(3362,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3361:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(3471,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3362]) ).
thf(940,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( B @ singleton @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,804]) ).
thf(941,plain,
! [B: $i,A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( B @ singleton @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[940]) ).
thf(949,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( ( B @ singleton )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[941]) ).
thf(957,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[949]) ).
thf(75,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',apply) ).
thf(274,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(275,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(lifteq,[status(thm)],[274]) ).
thf(65,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ compose ) )
= B )
| ~ ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition2) ).
thf(252,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ compose ) )
= B )
| ~ ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(39,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(197,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(1294,plain,
! [B: $i,A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1223,365]) ).
thf(1295,plain,
! [B: $i,A: $i] :
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[1294]) ).
thf(1296,plain,
( ( domain_relation @ complement )
!= domain_relation ),
inference(pattern_uni,[status(thm)],[1295:[bind(A,$thf( omega @ domain_of @ ( omega @ ordered_pair ) )),bind(B,$thf( domain_relation ))]]) ).
thf(2212,plain,
! [A: $i] :
( ( A != domain_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2213,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( domain_relation @ ( domain_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2212:[bind(A,$thf( domain_relation ))]]) ).
thf(96,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(317,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)],[96]) ).
thf(235,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[234]) ).
thf(2376,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,235]) ).
thf(2377,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2376:[bind(A,$thf( subset_relation ))]]) ).
thf(34,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',one_to_one1) ).
thf(187,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(188,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[187]) ).
thf(12,axiom,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_domain_relation2) ).
thf(139,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(140,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[139]) ).
thf(141,plain,
! [B: $i,A: $i] :
( ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ domain_of )
= B ) ),
inference(lifteq,[status(thm)],[140]) ).
thf(7559,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7500,224]) ).
thf(7560,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[7559]) ).
thf(7561,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7560:[bind(A,$thf( E )),bind(B,$thf( omega )),bind(C,$thf( E @ ( omega @ unordered_pair ) ))]]) ).
thf(7609,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(simp,[status(thm)],[7561]) ).
thf(9420,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7500,7609]) ).
thf(9421,plain,
! [B: $i,A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[9420]) ).
thf(9512,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[9421]) ).
thf(9530,plain,
! [B: $i,A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) @ complement ) ),
inference(simp,[status(thm)],[9512]) ).
thf(37,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(193,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(7640,plain,
( ( ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,2644]) ).
thf(7641,plain,
( ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7640]) ).
thf(7682,plain,
( ( ( null_class @ ( x @ cross_product ) @ complement )
!= ( omega @ singleton ) )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= omega ) ),
inference(simp,[status(thm)],[7641]) ).
thf(45,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum_class2) ).
thf(212,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(5,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function2) ).
thf(124,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(52,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(225,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(13835,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7496,224]) ).
thf(13836,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ complement @ ( C @ member ) ) ),
inference(simp,[status(thm)],[13835]) ).
thf(13837,plain,
! [B: $i,A: $i] :
~ ( A @ ( B @ singleton @ unordered_pair ) @ complement @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[13836:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( H @ singleton )),bind(D,$thf( G @ ( H @ singleton @ unordered_pair ) ))]]) ).
thf(13900,plain,
! [B: $i,A: $i] :
~ ( A @ ( B @ singleton @ unordered_pair ) @ complement @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[13837]) ).
thf(7580,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7500,359]) ).
thf(7581,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7580]) ).
thf(7602,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7581]) ).
thf(7619,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) ),
inference(simp,[status(thm)],[7602]) ).
thf(1097,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,1046]) ).
thf(1098,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1097]) ).
thf(1099,plain,
~ ( omega @ complement @ inductive ),
inference(pattern_uni,[status(thm)],[1098:[bind(A,$thf( omega @ complement ))]]) ).
thf(134,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)],[133]) ).
thf(1771,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[490,168]) ).
thf(1772,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1771:[bind(A,$thf( E )),bind(B,$thf( E @ power_class @ ( element_relation @ image ) ))]]) ).
thf(1857,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1772]) ).
thf(61,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(244,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compatible1) ).
thf(266,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(42,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(205,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)],[42]) ).
thf(13801,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ( A @ singleton @ ( C @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[170,7496]) ).
thf(13802,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[13801:[bind(A,$thf( E @ singleton )),bind(B,$thf( E @ singleton )),bind(C,$thf( E ))]]) ).
thf(13908,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[13802]) ).
thf(13967,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[13908,224]) ).
thf(13968,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[13967]) ).
thf(13969,plain,
! [A: $i] :
~ ( A @ singleton @ singleton @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[13968:[bind(A,$thf( F )),bind(B,$thf( F @ singleton )),bind(C,$thf( F @ singleton @ singleton ))]]) ).
thf(14035,plain,
! [A: $i] :
~ ( A @ singleton @ singleton @ complement @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[13969]) ).
thf(108,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose1) ).
thf(343,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(57,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(236,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(237,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[236]) ).
thf(93,axiom,
! [B: $i,A: $i] :
( ( composition_function @ ( B @ ( A @ compose ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_composition_function3) ).
thf(311,plain,
! [B: $i,A: $i] :
( ( composition_function @ ( B @ ( A @ compose ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(3298,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ power_class @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ complement )
= ( C @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[143,490]) ).
thf(3299,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3298:[bind(A,$thf( element_relation )),bind(B,$thf( D @ power_class )),bind(C,$thf( D ))]]) ).
thf(3316,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3299]) ).
thf(4764,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3316,490]) ).
thf(4765,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[4764:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(6314,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( B @ singleton @ member ) ) )
| ~ $true
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2107,804]) ).
thf(6315,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( B @ singleton @ member ) ) )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6314]) ).
thf(6333,plain,
! [B: $i,A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( ( A @ regular )
!= ( B @ singleton ) )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6315]) ).
thf(6368,plain,
! [A: $i] :
( ( ( universal_class @ complement @ regular )
!= ( A @ singleton ) )
| ( ( universal_class @ complement @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6333]) ).
thf(20614,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( F @ ( E @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[123,20401]) ).
thf(20615,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( F @ ( E @ ordered_pair ) @ member ) ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[20614]) ).
thf(20616,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20615:[bind(A,$thf( A )),bind(B,$thf( G )),bind(C,$thf( H )),bind(D,$thf( universal_class @ complement )),bind(E,$thf( H @ ( G @ ordered_pair ) )),bind(F,$thf( A ))]]) ).
thf(20688,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20616]) ).
thf(31,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',function2) ).
thf(181,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(182,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[181]) ).
thf(23254,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,182]) ).
thf(23255,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[23254]) ).
thf(23256,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[23255:[bind(A,$thf( choice ))]]) ).
thf(23364,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[310,23256]) ).
thf(23365,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[23364:[bind(A,$thf( choice ))]]) ).
thf(166,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[165]) ).
thf(70,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(264,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)],[70]) ).
thf(21,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(158,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(159,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[158]) ).
thf(160,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[159]) ).
thf(161,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[160]) ).
thf(79,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(279,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(180,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[179]) ).
thf(1153,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( B @ range_of ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( B @ inverse @ domain_of ) ) ),
inference(paramod_ordered,[status(thm)],[538,154]) ).
thf(1154,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[1153:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(82,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ ( C @ compose ) )
!= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition3) ).
thf(286,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ ( C @ compose ) )
!= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(9478,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,7609]) ).
thf(9479,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[9478]) ).
thf(9515,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[9479]) ).
thf(9531,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[9515]) ).
thf(118,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[117]) ).
thf(3403,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3320,168]) ).
thf(3404,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3403:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(7540,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7500,365]) ).
thf(7541,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C ) ),
inference(simp,[status(thm)],[7540]) ).
thf(7542,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( A @ ( omega @ unordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[7541:[bind(A,$thf( E )),bind(B,$thf( omega )),bind(C,$thf( E @ ( omega @ unordered_pair ) ))]]) ).
thf(7622,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( A @ ( omega @ unordered_pair ) ) ),
inference(simp,[status(thm)],[7542]) ).
thf(94,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(313,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(7623,plain,
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,1046]) ).
thf(7624,plain,
( ( omega @ singleton @ ( omega @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[7623]) ).
thf(7689,plain,
( ( ( omega @ complement )
!= ( omega @ singleton ) )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[7624]) ).
thf(26,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(171,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(172,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[171]) ).
thf(16143,plain,
( ( ( null_class @ ( x @ cross_product ) @ inductive )
!= ( omega @ inductive ) )
| ( omega != null_class )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,14817]) ).
thf(16144,plain,
( ( ( null_class @ ( x @ cross_product ) @ inductive )
!= ( omega @ inductive ) )
| ( omega != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16143]) ).
thf(16182,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= omega )
| ( omega != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16144]) ).
thf(90,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(303,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)],[90]) ).
thf(43,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inductive2) ).
thf(208,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(13928,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[13908,365]) ).
thf(13929,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C ) ),
inference(simp,[status(thm)],[13928]) ).
thf(13930,plain,
! [A: $i] :
( ( A @ singleton @ singleton @ complement )
!= ( A @ singleton @ singleton ) ),
inference(pattern_uni,[status(thm)],[13929:[bind(A,$thf( F )),bind(B,$thf( F @ singleton )),bind(C,$thf( F @ singleton @ singleton ))]]) ).
thf(14042,plain,
! [A: $i] :
( ( A @ singleton @ singleton @ complement )
!= ( A @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13930]) ).
thf(20714,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ( ( D @ complement )
!= D )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20381,365]) ).
thf(20715,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ( ( D @ complement )
!= D ) ),
inference(simp,[status(thm)],[20714]) ).
thf(20716,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ complement )
!= ( B @ ( A @ ordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[20715:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( F @ singleton )),bind(D,$thf( G @ ( F @ ordered_pair ) ))]]) ).
thf(20832,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ complement )
!= ( B @ ( A @ ordered_pair ) ) ),
inference(simp,[status(thm)],[20716]) ).
thf(2382,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2383,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2382:[bind(A,$thf( universal_class ))]]) ).
thf(54,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ domain_of @ ( A @ maps ) ) )
| ~ ( B @ ( A @ range_of @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',maps4) ).
thf(230,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ domain_of @ ( A @ maps ) ) )
| ~ ( B @ ( A @ range_of @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(14781,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[296,14712]) ).
thf(14782,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[14781]) ).
thf(14803,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( null_class @ ( x @ cross_product ) ) )
| ( domain_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[14782]) ).
thf(2811,plain,
! [B: $i,A: $i] :
( ( ( A @ diagonalise )
!= ( B @ diagonalise ) )
| ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[233,533]) ).
thf(2812,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[2811:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(99,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(323,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)],[99]) ).
thf(14574,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ~ $true
| ~ ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,13340]) ).
thf(14575,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[14574]) ).
thf(14679,plain,
! [A: $i] :
( ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ( A != null_class )
| ~ ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[14575]) ).
thf(14706,plain,
( ( ( null_class @ ( x @ cross_product ) )
!= universal_class )
| ~ ( null_class @ ( null_class @ ( x @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[14679]) ).
thf(2722,plain,
( ( ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,2644]) ).
thf(2723,plain,
( ( null_class @ ( x @ cross_product ) @ complement @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2722]) ).
thf(2757,plain,
( ( ( null_class @ ( x @ cross_product ) @ complement )
!= omega )
| ( ( null_class @ ( x @ cross_product ) @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2723]) ).
thf(1790,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[233,490]) ).
thf(1791,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[1790:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1862,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[1791]) ).
thf(7836,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7500,7652]) ).
thf(7837,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7836]) ).
thf(7898,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7837]) ).
thf(7928,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) ),
inference(simp,[status(thm)],[7898]) ).
thf(102,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(330,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)],[102]) ).
thf(2841,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( B @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[533,168]) ).
thf(2842,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2841:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(2911,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[2842]) ).
thf(59,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(240,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(64,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',single_valued_term_defn2) ).
thf(250,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(251,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[250]) ).
thf(3288,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( C @ diagonalise @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ complement )
= ( identity_relation @ ( C @ intersection ) @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[143,533]) ).
thf(3289,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3288:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(3328,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[3289]) ).
thf(5247,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3328,533]) ).
thf(5248,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[5247:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(91,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(306,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)],[91]) ).
thf(10,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
thf(135,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(88,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(299,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)],[88]) ).
thf(7629,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ singleton @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,7548]) ).
thf(7630,plain,
( ( null_class @ singleton @ ( omega @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7629:[bind(A,$thf( omega ))]]) ).
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(278,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(7872,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,7652]) ).
thf(7873,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7872]) ).
thf(7901,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton @ complement ) )
| ( omega != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7873]) ).
thf(7929,plain,
( ( omega != null_class )
| ~ ( omega @ singleton @ complement @ inductive ) ),
inference(simp,[status(thm)],[7901]) ).
thf(1064,plain,
! [A: $i] :
( ( ( A @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,364]) ).
thf(1065,plain,
! [A: $i] :
( ( ( A @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1064]) ).
thf(1070,plain,
! [A: $i] :
( ( A != omega )
| ( omega != null_class )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1065]) ).
thf(1076,plain,
( ( omega != null_class )
| ( ( omega @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1070]) ).
thf(66,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(255,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)],[66]) ).
thf(20336,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ~ ( B @ ( A @ ordered_pair ) @ complement @ ( D @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[176,13900]) ).
thf(20337,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ ordered_pair ) @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[20336:[bind(A,$thf( E )),bind(B,$thf( G )),bind(C,$thf( G @ singleton @ ( E @ unordered_pair ) )),bind(D,$thf( E ))]]) ).
thf(20406,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ ordered_pair ) @ complement @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[20337]) ).
thf(7633,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,365]) ).
thf(7634,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[7633]) ).
thf(7635,plain,
( ( omega @ singleton @ complement )
!= ( omega @ singleton ) ),
inference(pattern_uni,[status(thm)],[7634:[bind(A,$thf( omega )),bind(B,$thf( omega @ singleton ))]]) ).
thf(101,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(327,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)],[101]) ).
thf(1779,plain,
! [B: $i,A: $i] :
( ( ( A @ power_class )
!= ( B @ power_class ) )
| ( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[168,490]) ).
thf(1780,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1779:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(81,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(282,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)],[81]) ).
thf(2374,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,235]) ).
thf(2375,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2374:[bind(A,$thf( subset_relation ))]]) ).
thf(20873,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ~ $true
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,13304]) ).
thf(20874,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[20873]) ).
thf(21075,plain,
! [A: $i] :
( ( ( omega @ complement )
!= universal_class )
| ( A != omega )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[20874]) ).
thf(21090,plain,
( ( ( omega @ complement )
!= universal_class )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[21075]) ).
thf(40,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(199,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)],[40]) ).
thf(331,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)],[330]) ).
thf(543,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[310,235]) ).
thf(544,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[543:[bind(A,$thf( subset_relation ))]]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
thf(177,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(2218,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2219,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2218:[bind(A,$thf( universal_class ))]]) ).
thf(6320,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2107,359]) ).
thf(6321,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6320]) ).
thf(6341,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( ( A @ regular )
!= omega )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6321]) ).
thf(6352,plain,
( ( ( universal_class @ complement @ regular )
!= omega )
| ( ( universal_class @ complement @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6341]) ).
thf(106,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(338,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)],[106]) ).
thf(1933,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,189]) ).
thf(1934,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[1933:[bind(A,$thf( omega ))]]) ).
thf(2378,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2379,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2378:[bind(A,$thf( successor_relation ))]]) ).
thf(7588,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7500,364]) ).
thf(7589,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[7588]) ).
thf(7590,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[7589:[bind(A,$thf( D )),bind(B,$thf( D @ ( omega @ unordered_pair ) ))]]) ).
thf(7612,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(simp,[status(thm)],[7590]) ).
thf(95,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(314,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)],[95]) ).
thf(6282,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ~ $true
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2107,365]) ).
thf(6283,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6282]) ).
thf(6284,plain,
! [A: $i] :
( ( ( A @ complement )
!= A )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(pattern_uni,[status(thm)],[6283:[bind(A,$thf( D )),bind(B,$thf( D @ regular )),bind(C,$thf( D ))]]) ).
thf(6358,plain,
! [A: $i] :
( ( ( A @ complement )
!= A )
| ( ( A @ ( x @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[6284]) ).
thf(14730,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,14712]) ).
thf(14731,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[14730]) ).
thf(14732,plain,
( ~ ( null_class @ ( x @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[14731:[bind(A,$thf( universal_class ))]]) ).
thf(7838,plain,
( ( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,7652]) ).
thf(7839,plain,
( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[7838]) ).
thf(7904,plain,
( ( ( omega @ singleton @ complement )
!= omega )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[7839]) ).
thf(1309,plain,
( ( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1223,359]) ).
thf(1310,plain,
( ( domain_relation @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[1309]) ).
thf(1321,plain,
( ( ( universal_class @ complement )
!= domain_relation )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[1310]) ).
thf(1364,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,1292]) ).
thf(1365,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1364]) ).
thf(1379,plain,
! [A: $i] :
( ( A
!= ( domain_relation @ complement ) )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1365]) ).
thf(1390,plain,
( ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= null_class )
| ~ ( domain_relation @ complement @ inductive ) ),
inference(simp,[status(thm)],[1379]) ).
thf(942,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,359]) ).
thf(943,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[942]) ).
thf(955,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( omega != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[943]) ).
thf(963,plain,
( ( omega != null_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[955]) ).
thf(97,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(319,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)],[97]) ).
thf(186,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[185]) ).
thf(7678,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7548,364]) ).
thf(7679,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[7678]) ).
thf(7680,plain,
( ( omega @ singleton @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[7679:[bind(A,$thf( omega @ singleton ))]]) ).
thf(21616,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ rotate @ ( G @ ( F @ ( E @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[128,20688]) ).
thf(21617,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ rotate @ ( G @ ( F @ ( E @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[21616]) ).
thf(21618,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21617:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( universal_class @ complement @ rotate )),bind(E,$thf( B )),bind(F,$thf( A )),bind(G,$thf( C ))]]) ).
thf(74,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn2) ).
thf(272,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ domain_of @ ( B @ member ) )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(46,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',operation1) ).
thf(214,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(83,axiom,
! [C: $i,B: $i,A: $i] :
( ( application_function @ ( B @ ( A @ apply ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn4) ).
thf(290,plain,
! [C: $i,B: $i,A: $i] :
( ( application_function @ ( B @ ( A @ apply ) @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ~ ( A @ domain_of @ ( B @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(14427,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,14414]) ).
thf(14428,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[14427]) ).
thf(14472,plain,
( ( ( universal_class @ complement )
!= omega )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[14428]) ).
thf(62,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(246,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(2384,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2385,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2384:[bind(A,$thf( universal_class ))]]) ).
thf(1349,plain,
( ( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,1292]) ).
thf(1350,plain,
( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1349]) ).
thf(1380,plain,
( ( ( domain_relation @ complement )
!= omega )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= null_class ) ),
inference(simp,[status(thm)],[1350]) ).
thf(2049,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,924]) ).
thf(2050,plain,
( ( null_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2049:[bind(A,$thf( omega ))]]) ).
thf(21636,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ rotate @ ( G @ ( F @ ( E @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[123,20688]) ).
thf(21637,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ rotate @ ( G @ ( F @ ( E @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[21636]) ).
thf(21638,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21637:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( universal_class @ complement @ rotate )),bind(E,$thf( B )),bind(F,$thf( C )),bind(G,$thf( A ))]]) ).
thf(72,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(268,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)],[72]) ).
thf(38,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(195,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(239,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[238]) ).
thf(60,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(242,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)],[60]) ).
thf(85,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(294,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)],[85]) ).
thf(125,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[124]) ).
thf(126,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ compose ) )
= C ) ),
inference(lifteq,[status(thm)],[125]) ).
thf(136,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[135]) ).
thf(103,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',choice2) ).
thf(332,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(21119,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ( universal_class != omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,21086]) ).
thf(21120,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[21119]) ).
thf(21223,plain,
! [A: $i] :
( ( ( omega @ complement )
!= universal_class )
| ( A != omega )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[21120]) ).
thf(21236,plain,
( ( ( omega @ complement )
!= universal_class )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[21223]) ).
thf(178,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[177]) ).
thf(21830,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ flip @ ( E @ ( F @ ( G @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ $true
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[128,20708]) ).
thf(21831,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( universal_class @ complement @ flip @ ( E @ ( F @ ( G @ ordered_pair ) @ ordered_pair ) @ member ) ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[21830]) ).
thf(21832,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21831:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( universal_class @ complement @ flip )),bind(E,$thf( C )),bind(F,$thf( A )),bind(G,$thf( B ))]]) ).
thf(14359,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[296,14317]) ).
thf(14360,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14359]) ).
thf(14407,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[14360]) ).
thf(53,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(227,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(215,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[214]) ).
thf(2613,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,365]) ).
thf(2614,plain,
! [B: $i,A: $i] :
( ( ( null_class @ ( x @ cross_product ) @ ( null_class @ ( x @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[2613]) ).
thf(2615,plain,
( ( null_class @ ( x @ cross_product ) @ complement )
!= ( null_class @ ( x @ cross_product ) ) ),
inference(pattern_uni,[status(thm)],[2614:[bind(A,$thf( null_class @ ( x @ cross_product ) @ regular )),bind(B,$thf( null_class @ ( x @ cross_product ) ))]]) ).
thf(105,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',application_function_defn1) ).
thf(337,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(71838,plain,
$false,
inference(e,[status(thm)],[7691,347,249,234,352,365,138,2131,1454,3317,2644,2676,18603,14399,170,14414,217,276,533,142,20381,14818,3577,1954,340,538,153,174,185,1046,301,7628,7496,184,2667,21086,1323,7652,344,2682,157,6350,2217,20408,189,325,152,261,7500,21852,216,924,164,179,321,211,1858,238,121,806,353,1051,7899,147,280,7931,14372,221,14410,1072,133,20401,116,1104,132,2424,292,233,248,270,1338,220,1067,2773,14817,117,14712,201,5017,7703,260,2133,297,424,192,165,3471,957,1223,275,137,252,197,14317,1296,2213,224,317,2377,156,188,169,141,9530,1292,193,7682,212,124,225,13900,173,129,7619,13340,7609,1099,134,1857,244,266,205,14035,128,343,237,311,4765,6368,20688,23365,166,148,264,161,279,149,180,296,1154,176,286,9531,118,3404,259,7622,13908,313,7689,144,335,236,804,181,350,187,172,16182,303,208,14042,20832,274,2383,230,14803,2812,16020,323,14706,2757,1862,7928,13304,155,330,2911,240,251,130,5248,223,306,135,299,7630,167,278,7929,1076,490,2107,162,255,123,145,20406,7635,150,327,1780,310,282,2375,21090,199,331,359,544,177,182,127,2219,6352,338,1934,2379,154,175,7612,314,6358,143,23256,336,14732,7904,1321,250,3478,3320,1390,963,319,258,158,20708,186,114,171,139,7680,21618,272,346,214,290,119,235,14472,246,2102,2385,168,146,1380,190,183,2050,21638,210,268,195,3328,239,242,294,126,136,332,1268,21236,131,178,277,21832,14407,369,227,7548,309,215,2615,3316,364,232,337,122]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET205-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 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
% 0.25/0.51 % Computer : n018.cluster.edu
% 0.25/0.51 % Model : x86_64 x86_64
% 0.25/0.51 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.25/0.51 % Memory : 8046.5625MB
% 0.25/0.51 % OS : Linux 6.8.0-71-generic
% 0.25/0.51 % CPULimit : 300
% 0.25/0.51 % WCLimit : 300
% 0.25/0.51 % DateTime : Sat Sep 26 07:21:52 UTC 2026
% 0.25/0.52 % CPUTime :
% 0.25/0.52 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.97/1.00 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.69/1.24 % [INFO] Parsing done (243ms).
% 1.69/1.25 % [INFO] Running in sequential loop mode.
% 2.67/1.59 % [INFO] eprover registered as external prover.
% 2.95/1.60 % [INFO] Scanning for conjecture ...
% 3.22/1.71 % [INFO] Found a conjecture (or negated_conjecture) and 112 axioms. Running axiom selection ...
% 3.52/1.82 % [INFO] Axiom selection finished. Selected 112 axioms (removed 0 axioms).
% 3.94/1.93 % [INFO] Problem is propositional (TPTP CNF).
% 3.94/1.94 % [INFO] Type checking passed.
% 3.94/1.95 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 186.91/46.31 % External prover 'e' found a proof!
% 186.91/46.31 % [INFO] Killing All external provers ...
% 186.91/46.31 % Time passed: 45668ms (effective reasoning time: 45054ms)
% 186.91/46.32 % Axioms used in derivation (112): operation4, flip2, inverse, range, domain1, inductive3, equal_implies_subclass1, complement1, one_to_one2, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, maps4, range_of, function3, regularity1, singleton_set, identity_relation, union, flip3, element_relation2, unordered_pairs_in_universal, compose2, homomorphism3, compatible3, application_function_defn2, symmetric_difference, cartesian_product1, not_subclass_members1, compose_can_define_singleton, image, definition_of_domain_relation1, replacement, compose_class_definition3, definition_of_composition_function2, successor, ordered_pair, successor_relation1, domain, homomorphism1, one_to_one3, single_valued_term_defn3, single_valued_class1, compose3, omega_is_inductive2, rotate1, choice1, single_valued_term_defn1, power_class2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, intersection1, sum_class_definition, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, maps1, not_subclass_members2, single_valued_term_defn2, successor_relation3, function1, maps3, domain2, compatible4, intersection3, single_valued_class2, compose_class_definition1, unordered_pair_member, apply, element_relation1, operation1, application_function_defn1, restriction1, cartesian_product3, definition_of_domain_relation3, flip1, homomorphism5, maps2, compatible1, unordered_pair2, application_function_defn4, complement2, operation2, rotate3, compose_class_definition2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, intersection2, operation3, application_function_defn3, definition_of_domain_relation2, choice2, definition_of_composition_function1, homomorphism2, power_class_definition, definition_of_composition_function3, unordered_pair3, subset_relation, compose1
% 186.91/46.32 % No. of inferences in proof: 710
% 186.91/46.32 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 45668 ms resp. 45054 ms w/o parsing
% 177.10/46.52 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 177.10/46.53 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------