%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET206-6 : 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 : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:29:47 AM UTC 2026
% Result : Unsatisfiable 161.12s 39.36s
% Output : Refutation 159.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 113
% Syntax : Number of formulae : 817 ( 244 unt; 0 typ; 0 def)
% Number of atoms : 2053 ( 824 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 7883 (1126 ~; 862 |; 0 &;5895 @)
% ( 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 : 1178 ( 0 ^;1178 !; 0 ?;1178 :)
% Comments :
%------------------------------------------------------------------------------
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(y_decl,type,
y: $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(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(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(23827,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[155,182]) ).
thf(23828,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[23827]) ).
thf(23829,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[23828:[bind(A,$thf( choice ))]]) ).
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(194,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[193]) ).
thf(40305,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[23829,194]) ).
thf(40306,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[40305]) ).
thf(40307,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[40306:[bind(A,$thf( choice ))]]) ).
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(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(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(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(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(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(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(1,negated_conjecture,
( ( y @ ( null_class @ cross_product ) )
!= null_class ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_cross_product_with_null_class2_1) ).
thf(114,plain,
( ( y @ ( null_class @ cross_product ) )
!= null_class ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(115,plain,
( ( y @ ( null_class @ cross_product ) )
!= null_class ),
inference(polarity_switch,[status(thm)],[114]) ).
thf(116,plain,
( ( y @ ( null_class @ cross_product ) )
!= null_class ),
inference(lifteq,[status(thm)],[115]) ).
thf(2107,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( y @ ( A @ cross_product ) )
!= null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,116]) ).
thf(2108,plain,
! [A: $i] :
( ( ( y @ ( A @ cross_product ) )
!= null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2107:[]]) ).
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(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(10546,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2108,359]) ).
thf(10547,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10546]) ).
thf(10562,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( ( A @ regular )
!= omega )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10547]) ).
thf(10577,plain,
( ( ( universal_class @ complement @ regular )
!= omega )
| ( ( y @ ( universal_class @ complement @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10562]) ).
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(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(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(12965,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(12966,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( universal_class @ complement @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[12965:[bind(A,$thf( universal_class @ complement )),bind(B,$thf( B ))]]) ).
thf(13438,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( universal_class @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[12966]) ).
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(14584,plain,
( ~ ( universal_class @ complement @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[13438,216]) ).
thf(14585,plain,
~ ( universal_class @ complement @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[14584]) ).
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(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(17684,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1857,490]) ).
thf(17685,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[17684:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(17998,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[17685]) ).
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(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(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(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(11183,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ complement ) )
| ( B != omega )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,1051]) ).
thf(11184,plain,
! [A: $i] :
( ( A != omega )
| ~ ( omega @ complement @ ( A @ subclass ) )
| ~ ( A @ ( omega @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11183:[bind(A,$thf( omega @ complement )),bind(B,$thf( B ))]]) ).
thf(13590,plain,
( ~ ( omega @ complement @ ( omega @ subclass ) )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[11184]) ).
thf(21285,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( omega @ complement @ subclass ) ) )
| ~ ( omega @ complement @ ( omega @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,13590]) ).
thf(21286,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ ( omega @ complement @ subclass ) ) )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[21285]) ).
thf(21493,plain,
! [A: $i] :
( ( universal_class != omega )
| ( A
!= ( omega @ complement ) )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[21286]) ).
thf(21518,plain,
( ( universal_class != omega )
| ~ ( omega @ complement @ ( omega @ subclass ) ) ),
inference(simp,[status(thm)],[21493]) ).
thf(22401,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ( universal_class != omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,21518]) ).
thf(22402,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[22401]) ).
thf(22488,plain,
! [A: $i] :
( ( ( omega @ complement )
!= universal_class )
| ( A != omega )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[22402]) ).
thf(22521,plain,
( ( ( omega @ complement )
!= universal_class )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[22488]) ).
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(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(11251,plain,
! [B: $i,A: $i] :
( ( A
!= ( y @ ( null_class @ cross_product ) ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,116]) ).
thf(11252,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( y @ ( null_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11251:[bind(A,$thf( y @ ( null_class @ cross_product ) )),bind(B,$thf( B ))]]) ).
thf(13619,plain,
( ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) )
| ~ ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[11252]) ).
thf(16377,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,13619]) ).
thf(16378,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[16377]) ).
thf(16513,plain,
! [A: $i] :
( ( universal_class != null_class )
| ( A
!= ( y @ ( null_class @ cross_product ) ) )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[16378]) ).
thf(16517,plain,
( ( universal_class != null_class )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[16513]) ).
thf(16604,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[346,16517]) ).
thf(16605,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16604]) ).
thf(16618,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( y @ ( null_class @ cross_product ) ) )
| ( successor_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16605]) ).
thf(40158,plain,
( ( universal_class != y )
| ( universal_class != null_class )
| ( successor_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16618]) ).
thf(40261,plain,
( ( universal_class != y )
| ( successor_relation != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[40158]) ).
thf(41396,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class != y )
| ( null_class != null_class )
| ( universal_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,40261]) ).
thf(41397,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class != y )
| ( universal_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[41396]) ).
thf(41398,plain,
( ( universal_class != y )
| ( universal_class != null_class )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[41397:[bind(A,$thf( successor_relation ))]]) ).
thf(2100,plain,
! [A: $i] :
( ( A
!= ( y @ ( null_class @ cross_product ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,116]) ).
thf(2101,plain,
! [A: $i] :
( ( A
!= ( y @ ( null_class @ cross_product ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2100]) ).
thf(2102,plain,
y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[2101:[bind(A,$thf( y @ ( null_class @ cross_product ) ))]]) ).
thf(2656,plain,
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,359]) ).
thf(2657,plain,
( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[2656]) ).
thf(2671,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( ( y @ ( null_class @ cross_product ) @ regular )
!= omega ) ),
inference(simp,[status(thm)],[2657]) ).
thf(2379,plain,
! [A: $i] :
( ( A != successor_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2380,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2379:[bind(A,$thf( successor_relation ))]]) ).
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(7470,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(7471,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( B @ ( A @ unordered_pair ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[7470]) ).
thf(7472,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7471:[bind(A,$thf( omega ))]]) ).
thf(7601,plain,
! [A: $i] : ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) ),
inference(simp,[status(thm)],[7472]) ).
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(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(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(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(7630,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) ) )
| ( A @ singleton @ ( omega @ member ) ) ),
inference(paramod_ordered,[status(thm)],[170,7601]) ).
thf(7631,plain,
omega @ singleton @ ( omega @ member ),
inference(pattern_uni,[status(thm)],[7630:[bind(A,$thf( omega )),bind(B,$thf( omega ))]]) ).
thf(7715,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ singleton @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,7631]) ).
thf(7716,plain,
( ( omega @ singleton @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7715:[bind(A,$thf( omega ))]]) ).
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(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(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(143,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[142]) ).
thf(3362,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(3363,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3362:[bind(A,$thf( element_relation )),bind(B,$thf( D @ power_class )),bind(C,$thf( D ))]]) ).
thf(3380,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)],[3363]) ).
thf(4950,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)],[3380,490]) ).
thf(4951,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[4950:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(6649,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ power_class @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[154,4951]) ).
thf(6650,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[6649:[bind(A,$thf( universal_class @ ( F @ power_class @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(6887,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[6650]) ).
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(2105,plain,
! [A: $i] :
( ( A != y )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,116]) ).
thf(2106,plain,
( ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ( y @ ( y @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2105:[bind(A,$thf( y ))]]) ).
thf(3141,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2106,224]) ).
thf(3142,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3141]) ).
thf(3143,plain,
( ~ ( y @ complement @ ( y @ regular @ member ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(pattern_uni,[status(thm)],[3142:[bind(A,$thf( y @ regular )),bind(B,$thf( y ))]]) ).
thf(3908,plain,
( ( ( y @ complement @ ( y @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[924,3143]) ).
thf(3909,plain,
( ( ( y @ complement @ ( y @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3908]) ).
thf(3947,plain,
( ( ( y @ complement )
!= omega )
| ( ( y @ regular )
!= null_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3909]) ).
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(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(209,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[208]) ).
thf(53340,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,209]) ).
thf(53341,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[53340]) ).
thf(53342,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[53341:[bind(A,$thf( omega ))]]) ).
thf(53564,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( omega @ ( successor_relation @ image ) ) )
| ( omega @ ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[143,53342]) ).
thf(53565,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ range_of @ subclass ),
inference(pattern_uni,[status(thm)],[53564:[bind(A,$thf( successor_relation )),bind(B,$thf( omega ))]]) ).
thf(53635,plain,
! [A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ range_of ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[538,53565]) ).
thf(53636,plain,
omega @ ( universal_class @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[53635:[bind(A,$thf( universal_class @ ( omega @ ( successor_relation @ restrict ) ) ))]]) ).
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(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(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(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(40448,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ single_valued_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,40307]) ).
thf(40449,plain,
( ( null_class @ single_valued_class )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[40448:[bind(A,$thf( choice ))]]) ).
thf(7734,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[7631,224]) ).
thf(7735,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[7734]) ).
thf(7736,plain,
~ ( omega @ singleton @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7735:[bind(A,$thf( omega )),bind(B,$thf( omega @ singleton ))]]) ).
thf(7964,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,7736]) ).
thf(7965,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7964]) ).
thf(7985,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton @ complement ) )
| ( omega != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[7965]) ).
thf(8022,plain,
( ( omega != null_class )
| ~ ( omega @ singleton @ complement @ inductive ) ),
inference(simp,[status(thm)],[7985]) ).
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(196,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[195]) ).
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(7463,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(7464,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)],[7463]) ).
thf(7465,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[7464:[bind(A,$thf( D )),bind(B,$thf( D @ singleton ))]]) ).
thf(7599,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[7465]) ).
thf(14092,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)],[7599,224]) ).
thf(14093,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)],[14092]) ).
thf(14094,plain,
! [B: $i,A: $i] :
~ ( A @ ( B @ singleton @ unordered_pair ) @ complement @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[14093:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( H @ singleton )),bind(D,$thf( G @ ( H @ singleton @ unordered_pair ) ))]]) ).
thf(14164,plain,
! [B: $i,A: $i] :
~ ( A @ ( B @ singleton @ unordered_pair ) @ complement @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[14094]) ).
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(16557,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,16517]) ).
thf(16558,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16557]) ).
thf(16630,plain,
! [A: $i] :
( ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( A != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16558]) ).
thf(16640,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[16630]) ).
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(11427,plain,
! [B: $i,A: $i] :
( ( A
!= ( domain_relation @ complement ) )
| ( B != domain_relation )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[164,1296]) ).
thf(11428,plain,
! [A: $i] :
( ( A != domain_relation )
| ~ ( domain_relation @ complement @ ( A @ subclass ) )
| ~ ( A @ ( domain_relation @ complement @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[11427:[bind(A,$thf( domain_relation @ complement )),bind(B,$thf( B ))]]) ).
thf(13691,plain,
( ~ ( domain_relation @ complement @ ( domain_relation @ subclass ) )
| ~ ( domain_relation @ ( domain_relation @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[11428]) ).
thf(43422,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ ( domain_relation @ complement @ subclass ) ) )
| ~ ( domain_relation @ complement @ ( domain_relation @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,13691]) ).
thf(43423,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ ( domain_relation @ complement @ subclass ) ) )
| ~ ( domain_relation @ complement @ ( domain_relation @ subclass ) ) ),
inference(simp,[status(thm)],[43422]) ).
thf(43735,plain,
! [A: $i] :
( ( domain_relation != universal_class )
| ( A
!= ( domain_relation @ complement ) )
| ~ ( domain_relation @ complement @ ( domain_relation @ subclass ) ) ),
inference(simp,[status(thm)],[43423]) ).
thf(43799,plain,
( ( domain_relation != universal_class )
| ~ ( domain_relation @ complement @ ( domain_relation @ subclass ) ) ),
inference(simp,[status(thm)],[43735]) ).
thf(43833,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ complement @ ( domain_relation @ subclass ) ) )
| ( domain_relation != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[216,43799]) ).
thf(43834,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ complement @ ( domain_relation @ subclass ) ) )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[43833]) ).
thf(43983,plain,
! [A: $i] :
( ( ( domain_relation @ complement )
!= universal_class )
| ( A != domain_relation )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[43834]) ).
thf(44030,plain,
( ( ( domain_relation @ complement )
!= universal_class )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[43983]) ).
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(7761,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,359]) ).
thf(7762,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7761]) ).
thf(7773,plain,
( ( ( universal_class @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7762]) ).
thf(7792,plain,
( ( universal_class @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[7773]) ).
thf(3364,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(3365,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3364:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3381,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)],[3365]) ).
thf(5212,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)],[3381,490]) ).
thf(5213,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[5212:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(7021,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ complement @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[154,5213]) ).
thf(7022,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[7021:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(7144,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[7022]) ).
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(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(41393,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class != y )
| ( successor_relation != null_class )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,40261]) ).
thf(41394,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class != y )
| ( successor_relation != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[41393]) ).
thf(41395,plain,
( ( universal_class != y )
| ( successor_relation != null_class )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[41394:[bind(A,$thf( universal_class ))]]) ).
thf(41753,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != y )
| ( successor_relation != null_class ) ),
inference(paramod_ordered,[status(thm)],[41395,224]) ).
thf(41754,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != y )
| ( successor_relation != null_class ) ),
inference(simp,[status(thm)],[41753]) ).
thf(41755,plain,
( ~ ( universal_class @ complement @ ( universal_class @ regular @ member ) )
| ( universal_class != y )
| ( successor_relation != null_class ) ),
inference(pattern_uni,[status(thm)],[41754:[bind(A,$thf( universal_class @ regular )),bind(B,$thf( universal_class ))]]) ).
thf(2614,plain,
! [B: $i,A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,365]) ).
thf(2615,plain,
! [B: $i,A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[2614]) ).
thf(2616,plain,
( ( y @ ( null_class @ cross_product ) @ complement )
!= ( y @ ( null_class @ cross_product ) ) ),
inference(pattern_uni,[status(thm)],[2615:[bind(A,$thf( y @ ( null_class @ cross_product ) @ regular )),bind(B,$thf( y @ ( null_class @ cross_product ) ))]]) ).
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(3370,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(3371,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3370:[bind(A,$thf( element_relation )),bind(B,$thf( D @ complement )),bind(C,$thf( D ))]]) ).
thf(3384,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3371]) ).
thf(3460,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)],[154,3384]) ).
thf(3461,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3460:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(3525,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ power_class ) ),
inference(simp,[status(thm)],[3461]) ).
thf(10490,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ~ $true
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2108,365]) ).
thf(10491,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10490]) ).
thf(10492,plain,
! [A: $i] :
( ( ( A @ complement )
!= A )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(pattern_uni,[status(thm)],[10491:[bind(A,$thf( D )),bind(B,$thf( D @ regular )),bind(C,$thf( D ))]]) ).
thf(10602,plain,
! [A: $i] :
( ( ( A @ complement )
!= A )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10492]) ).
thf(7640,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)],[7601,224]) ).
thf(7641,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[7640]) ).
thf(7642,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[7641:[bind(A,$thf( E )),bind(B,$thf( omega )),bind(C,$thf( E @ ( omega @ unordered_pair ) ))]]) ).
thf(7707,plain,
! [A: $i] :
~ ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ),
inference(simp,[status(thm)],[7642]) ).
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(16379,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ~ $true
| ~ ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,13619]) ).
thf(16380,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ~ ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[16379]) ).
thf(16500,plain,
! [A: $i] :
( ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( A != null_class )
| ~ ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[16380]) ).
thf(16527,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ~ ( null_class @ ( y @ ( null_class @ cross_product ) @ subclass ) ) ),
inference(simp,[status(thm)],[16500]) ).
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(16542,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class != null_class )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,16517]) ).
thf(16543,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[16542]) ).
thf(16544,plain,
( ~ ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16543:[bind(A,$thf( universal_class ))]]) ).
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(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(3606,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[233,3525]) ).
thf(3607,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3606:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(3712,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[3607]) ).
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(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(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(3638,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[3525,168]) ).
thf(3639,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3638:[bind(A,$thf( H )),bind(B,$thf( universal_class @ ( H @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of ))]]) ).
thf(3720,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[3639]) ).
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(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(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(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(2217,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2218,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2217:[bind(A,$thf( universal_class ))]]) ).
thf(16288,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,2218]) ).
thf(16289,plain,
( ( null_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[16288:[bind(A,$thf( universal_class ))]]) ).
thf(16333,plain,
( ( null_class @ ( null_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(simp,[status(thm)],[16289]) ).
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(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(3352,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(3353,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[3352:[bind(A,$thf( element_relation )),bind(B,$thf( D @ diagonalise )),bind(C,$thf( D ))]]) ).
thf(3392,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)],[3353]) ).
thf(5406,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)],[3392,533]) ).
thf(5407,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[5406:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(15834,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( B @ diagonalise @ ( element_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[154,5407]) ).
thf(15835,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[15834:[bind(A,$thf( universal_class @ ( F @ diagonalise @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(16168,plain,
! [A: $i] :
( ( universal_class @ ( A @ diagonalise @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[15835]) ).
thf(7625,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)],[7601,365]) ).
thf(7626,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C ) ),
inference(simp,[status(thm)],[7625]) ).
thf(7627,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( A @ ( omega @ unordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[7626:[bind(A,$thf( E )),bind(B,$thf( omega )),bind(C,$thf( E @ ( omega @ unordered_pair ) ))]]) ).
thf(7703,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( A @ ( omega @ unordered_pair ) ) ),
inference(simp,[status(thm)],[7627]) ).
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(20746,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)],[176,7599]) ).
thf(20747,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[20746:[bind(A,$thf( E )),bind(B,$thf( G )),bind(C,$thf( G @ singleton @ ( E @ unordered_pair ) )),bind(D,$thf( E ))]]) ).
thf(20783,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[20747]) ).
thf(21131,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)],[20783,365]) ).
thf(21132,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)],[21131]) ).
thf(21133,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ complement )
!= ( B @ ( A @ ordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[21132:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( F @ singleton )),bind(D,$thf( G @ ( F @ ordered_pair ) ))]]) ).
thf(21238,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ complement )
!= ( B @ ( A @ ordered_pair ) ) ),
inference(simp,[status(thm)],[21133]) ).
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(44203,plain,
! [B: $i,A: $i] :
( ( ( successor_relation @ ( successor_relation @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(paramod_ordered,[status(thm)],[41398,365]) ).
thf(44204,plain,
! [B: $i,A: $i] :
( ( ( successor_relation @ ( successor_relation @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[44203]) ).
thf(44205,plain,
( ( ( successor_relation @ complement )
!= successor_relation )
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(pattern_uni,[status(thm)],[44204:[bind(A,$thf( successor_relation @ regular )),bind(B,$thf( successor_relation ))]]) ).
thf(44429,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( successor_relation @ complement )
!= successor_relation )
| ( universal_class != y )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,44205]) ).
thf(44430,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( successor_relation @ complement )
!= successor_relation )
| ( universal_class != y )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[44429]) ).
thf(44431,plain,
( ( ( successor_relation @ complement )
!= successor_relation )
| ( universal_class != y )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[44430:[bind(A,$thf( universal_class ))]]) ).
thf(7675,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7601,364]) ).
thf(7676,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[7675]) ).
thf(7677,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[7676:[bind(A,$thf( D )),bind(B,$thf( D @ ( omega @ unordered_pair ) ))]]) ).
thf(7710,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= universal_class ),
inference(simp,[status(thm)],[7677]) ).
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(2131,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,220]) ).
thf(2132,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2131:[bind(A,$thf( omega ))]]) ).
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(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(14633,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[130,14585]) ).
thf(14634,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14633]) ).
thf(14657,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( universal_class != omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14634]) ).
thf(14680,plain,
( ( universal_class != omega )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[14657]) ).
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(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(16597,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[130,16517]) ).
thf(16598,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( y @ ( null_class @ cross_product ) @ ( null_class @ subclass ) ) )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16597]) ).
thf(16623,plain,
! [A: $i] :
( ( A
!= ( y @ ( null_class @ cross_product ) ) )
| ( omega != null_class )
| ( universal_class != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[16598]) ).
thf(16634,plain,
( ( omega != null_class )
| ( universal_class != null_class )
| ~ ( y @ ( null_class @ cross_product ) @ inductive ) ),
inference(simp,[status(thm)],[16623]) ).
thf(18972,plain,
( ( ( y @ ( null_class @ cross_product ) @ inductive )
!= ( omega @ inductive ) )
| ( omega != null_class )
| ( universal_class != null_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,16634]) ).
thf(18973,plain,
( ( ( y @ ( null_class @ cross_product ) @ inductive )
!= ( omega @ inductive ) )
| ( omega != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[18972]) ).
thf(19011,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= omega )
| ( omega != null_class )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[18973]) ).
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(14651,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[277,14585]) ).
thf(14652,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14651]) ).
thf(14666,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[14652]) ).
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(53754,plain,
! [A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ range_of ) )
| ( omega @ ( A @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[154,53565]) ).
thf(53755,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[53754:[bind(A,$thf( universal_class @ ( omega @ ( successor_relation @ restrict ) ) ))]]) ).
thf(3154,plain,
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2106,359]) ).
thf(3155,plain,
( ( ( y @ ( y @ regular @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3154]) ).
thf(3179,plain,
( ( ( universal_class @ complement )
!= y )
| ( ( y @ regular )
!= omega )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3155]) ).
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(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(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(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(20736,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(20737,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20736:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( F @ singleton )),bind(D,$thf( H @ singleton @ ( F @ unordered_pair ) ))]]) ).
thf(20778,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20737]) ).
thf(21020,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)],[123,20778]) ).
thf(21021,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)],[21020]) ).
thf(21022,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21021:[bind(A,$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(21106,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[21022]) ).
thf(10496,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2108,1046]) ).
thf(10497,plain,
! [A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10496]) ).
thf(10557,plain,
! [A: $i] :
( ( A
!= ( omega @ complement ) )
| ( ( A @ regular )
!= null_class )
| ( ( y @ ( A @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10497]) ).
thf(10603,plain,
( ( ( omega @ complement @ regular )
!= null_class )
| ( ( y @ ( omega @ complement @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[10557]) ).
thf(21287,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ~ $true
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,13590]) ).
thf(21288,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( omega @ complement @ ( omega @ subclass ) ) )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[21287]) ).
thf(21451,plain,
! [A: $i] :
( ( ( omega @ complement )
!= universal_class )
| ( A != omega )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[21288]) ).
thf(21516,plain,
( ( ( omega @ complement )
!= universal_class )
| ~ ( omega @ ( omega @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[21451]) ).
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(19250,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class )
!= ( identity_relation @ ( B @ intersection ) @ domain_of @ power_class ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ diagonalise @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[533,1862]) ).
thf(19251,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ diagonalise @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[19250:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(19476,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
!= ( B @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( B @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1862,490]) ).
thf(19477,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[19476:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(19708,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ power_class )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[19477]) ).
thf(7711,plain,
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,1046]) ).
thf(7712,plain,
( ( omega @ singleton @ ( omega @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[7711]) ).
thf(7789,plain,
( ( ( omega @ complement )
!= ( omega @ singleton ) )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[7712]) ).
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(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(7768,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,364]) ).
thf(7769,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[7768]) ).
thf(7770,plain,
( ( omega @ singleton @ complement )
!= universal_class ),
inference(pattern_uni,[status(thm)],[7769:[bind(A,$thf( omega @ singleton ))]]) ).
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(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(18845,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,16640]) ).
thf(18846,plain,
! [A: $i] :
( ( A != universal_class )
| ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[18845]) ).
thf(18847,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= universal_class )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[18846:[bind(A,$thf( universal_class ))]]) ).
thf(19088,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( ( y @ ( null_class @ cross_product ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[18847,224]) ).
thf(19089,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( ( y @ ( null_class @ cross_product ) )
!= universal_class ) ),
inference(simp,[status(thm)],[19088]) ).
thf(19090,plain,
( ~ ( universal_class @ complement @ ( universal_class @ regular @ member ) )
| ( ( y @ ( null_class @ cross_product ) )
!= universal_class ) ),
inference(pattern_uni,[status(thm)],[19089:[bind(A,$thf( universal_class @ regular )),bind(B,$thf( universal_class ))]]) ).
thf(41663,plain,
! [A: $i] :
( ( A != universal_class )
| ( y != null_class )
| ( successor_relation != null_class )
| ( universal_class @ ( universal_class @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,41395]) ).
thf(41664,plain,
( ( y != null_class )
| ( successor_relation != null_class )
| ( universal_class @ ( universal_class @ regular @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[41663:[bind(A,$thf( universal_class ))]]) ).
thf(41864,plain,
( ( y != null_class )
| ( successor_relation != null_class )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(simp,[status(thm)],[41664]) ).
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(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(166,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[165]) ).
thf(23937,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[310,23829]) ).
thf(23938,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[23937:[bind(A,$thf( choice ))]]) ).
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(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(14694,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[189,14680]) ).
thf(14695,plain,
( ( ( universal_class @ complement @ inductive )
!= ( omega @ inductive ) )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[14694]) ).
thf(14744,plain,
( ( ( universal_class @ complement )
!= omega )
| ( universal_class != omega ) ),
inference(simp,[status(thm)],[14695]) ).
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(180,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[179]) ).
thf(235,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[234]) ).
thf(2375,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,235]) ).
thf(2376,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2375:[bind(A,$thf( subset_relation ))]]) ).
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(7667,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7601,359]) ).
thf(7668,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7667]) ).
thf(7691,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7668]) ).
thf(7704,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( universal_class @ complement ) ),
inference(simp,[status(thm)],[7691]) ).
thf(118,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[117]) ).
thf(9579,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,7707]) ).
thf(9580,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[9579]) ).
thf(9627,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[9580]) ).
thf(9663,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ complement )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[9627]) ).
thf(3427,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)],[538,3384]) ).
thf(3428,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ complement )
= ( A @ power_class ) ),
inference(pattern_uni,[status(thm)],[3427:[bind(A,$thf( universal_class @ ( F @ complement @ ( element_relation @ restrict ) ) )),bind(B,$thf( F ))]]) ).
thf(3547,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)],[3428]) ).
thf(2133,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,220]) ).
thf(2134,plain,
( ( null_class @ ( omega @ member ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2133:[bind(A,$thf( universal_class ))]]) ).
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(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(2219,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2220,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2219:[bind(A,$thf( universal_class ))]]) ).
thf(2424,plain,
! [A: $i] :
( ( A != element_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,277]) ).
thf(2425,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( element_relation @ ( element_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2424:[bind(A,$thf( element_relation ))]]) ).
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(20707,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)],[176,14164]) ).
thf(20708,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ ordered_pair ) @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[20707:[bind(A,$thf( E )),bind(B,$thf( G )),bind(C,$thf( G @ singleton @ ( E @ unordered_pair ) )),bind(D,$thf( E ))]]) ).
thf(20801,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ ordered_pair ) @ complement @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[20708]) ).
thf(21005,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)],[128,20778]) ).
thf(21006,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)],[21005]) ).
thf(21007,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[21006:[bind(A,$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(21105,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[21007]) ).
thf(22034,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)],[128,21105]) ).
thf(22035,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)],[22034]) ).
thf(22036,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[22035:[bind(A,$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(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(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(2213,plain,
! [A: $i] :
( ( A != domain_relation )
| ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,296]) ).
thf(2214,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( null_class @ subclass ) )
| ( domain_relation @ ( domain_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2213:[bind(A,$thf( domain_relation ))]]) ).
thf(2377,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,235]) ).
thf(2378,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2377:[bind(A,$thf( subset_relation ))]]) ).
thf(3164,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2106,364]) ).
thf(3165,plain,
! [A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3164]) ).
thf(3171,plain,
! [A: $i] :
( ( y != A )
| ( ( y @ regular )
!= omega )
| ( ( A @ complement )
!= universal_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3165]) ).
thf(3182,plain,
( ( ( y @ regular )
!= omega )
| ( ( y @ complement )
!= universal_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3171]) ).
thf(14060,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)],[170,7599]) ).
thf(14061,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[14060:[bind(A,$thf( E @ singleton )),bind(B,$thf( E @ singleton )),bind(C,$thf( E ))]]) ).
thf(14172,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[14061]) ).
thf(14192,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)],[14172,365]) ).
thf(14193,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ complement )
!= C ) ),
inference(simp,[status(thm)],[14192]) ).
thf(14194,plain,
! [A: $i] :
( ( A @ singleton @ singleton @ complement )
!= ( A @ singleton @ singleton ) ),
inference(pattern_uni,[status(thm)],[14193:[bind(A,$thf( F )),bind(B,$thf( F @ singleton )),bind(C,$thf( F @ singleton @ singleton ))]]) ).
thf(14311,plain,
! [A: $i] :
( ( A @ singleton @ singleton @ complement )
!= ( A @ singleton @ singleton ) ),
inference(simp,[status(thm)],[14194]) ).
thf(44293,plain,
! [B: $i,A: $i] :
( ( ( successor_relation @ ( successor_relation @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(paramod_ordered,[status(thm)],[41398,224]) ).
thf(44294,plain,
! [B: $i,A: $i] :
( ( ( successor_relation @ ( successor_relation @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[44293]) ).
thf(44295,plain,
( ~ ( successor_relation @ complement @ ( successor_relation @ regular @ member ) )
| ( universal_class != y )
| ( universal_class != null_class ) ),
inference(pattern_uni,[status(thm)],[44294:[bind(A,$thf( successor_relation @ regular )),bind(B,$thf( successor_relation ))]]) ).
thf(3929,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( y @ complement @ ( y @ regular @ member ) ) )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,3143]) ).
thf(3930,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( y @ complement @ ( y @ regular @ member ) ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[3929]) ).
thf(3957,plain,
! [A: $i] :
( ( A
!= ( y @ complement ) )
| ( ( y @ regular )
!= null_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[3930]) ).
thf(3971,plain,
( ( ( y @ regular )
!= null_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class )
| ~ ( y @ complement @ inductive ) ),
inference(simp,[status(thm)],[3957]) ).
thf(198,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[197]) ).
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(22251,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)],[128,21106]) ).
thf(22252,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)],[22251]) ).
thf(22253,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[22252:[bind(A,$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(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(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(2645,plain,
! [B: $i,A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2102,224]) ).
thf(2646,plain,
! [B: $i,A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2645]) ).
thf(2647,plain,
~ ( y @ ( null_class @ cross_product ) @ complement @ ( y @ ( null_class @ cross_product ) @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[2646:[bind(A,$thf( y @ ( null_class @ cross_product ) @ regular )),bind(B,$thf( y @ ( null_class @ cross_product ) ))]]) ).
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(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(14642,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[346,14585]) ).
thf(14643,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14642]) ).
thf(14656,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[14643]) ).
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(2667,plain,
! [A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,364]) ).
thf(2668,plain,
! [A: $i] :
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( A @ ( omega @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2667]) ).
thf(2679,plain,
! [A: $i] :
( ( ( y @ ( null_class @ cross_product ) )
!= A )
| ( ( y @ ( null_class @ cross_product ) @ regular )
!= omega )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2668]) ).
thf(2684,plain,
( ( ( y @ ( null_class @ cross_product ) @ regular )
!= omega )
| ( ( y @ ( null_class @ cross_product ) @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[2679]) ).
thf(7717,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ singleton @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,7631]) ).
thf(7718,plain,
( ( null_class @ singleton @ ( omega @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7717:[bind(A,$thf( omega ))]]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[144]) ).
thf(17695,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( B @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( B @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[490,1857]) ).
thf(17696,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[17695:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(14227,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)],[14172,224]) ).
thf(14228,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ complement @ ( B @ member ) ) ),
inference(simp,[status(thm)],[14227]) ).
thf(14229,plain,
! [A: $i] :
~ ( A @ singleton @ singleton @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[14228:[bind(A,$thf( F )),bind(B,$thf( F @ singleton )),bind(C,$thf( F @ singleton @ singleton ))]]) ).
thf(14304,plain,
! [A: $i] :
~ ( A @ singleton @ singleton @ complement @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[14229]) ).
thf(22051,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)],[123,21105]) ).
thf(22052,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)],[22051]) ).
thf(22053,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ flip @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[22052:[bind(A,$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(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(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(2610,plain,
( ( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2102,1046]) ).
thf(2611,plain,
( ( y @ ( null_class @ cross_product ) @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2610]) ).
thf(2678,plain,
( ( ( y @ ( null_class @ cross_product ) )
!= ( omega @ complement ) )
| ( ( y @ ( null_class @ cross_product ) @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2611]) ).
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(2385,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2386,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2385:[bind(A,$thf( universal_class ))]]) ).
thf(3641,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)],[3525,168]) ).
thf(3642,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3641:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
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(2428,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,277]) ).
thf(2429,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2428:[bind(A,$thf( universal_class ))]]) ).
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(9543,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7601,7707]) ).
thf(9544,plain,
! [B: $i,A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( B @ ( omega @ unordered_pair ) @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[9543]) ).
thf(9633,plain,
! [B: $i,A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[9544]) ).
thf(9666,plain,
! [B: $i,A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( B @ ( omega @ unordered_pair ) @ complement ) ),
inference(simp,[status(thm)],[9633]) ).
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(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(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(7719,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class @ ( omega @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,7631]) ).
thf(7720,plain,
( ( null_class @ ( omega @ member ) )
| ( omega @ singleton @ ( omega @ singleton @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[7719:[bind(A,$thf( omega @ singleton ))]]) ).
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(14626,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[296,14585]) ).
thf(14627,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[14626]) ).
thf(14673,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( universal_class @ complement ) )
| ( domain_relation != universal_class ) ),
inference(simp,[status(thm)],[14627]) ).
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(2728,plain,
( ( ( y @ ( null_class @ cross_product ) @ complement @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,2647]) ).
thf(2729,plain,
( ( y @ ( null_class @ cross_product ) @ complement @ ( y @ ( null_class @ cross_product ) @ regular @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[2728]) ).
thf(2775,plain,
( ( ( y @ ( null_class @ cross_product ) @ complement )
!= omega )
| ( ( y @ ( null_class @ cross_product ) @ regular )
!= null_class ) ),
inference(simp,[status(thm)],[2729]) ).
thf(2817,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(2818,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[2817:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(14639,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)],[145,14585]) ).
thf(14640,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subclass ) )
!= ( universal_class @ complement @ ( universal_class @ subclass ) ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(simp,[status(thm)],[14639]) ).
thf(14641,plain,
~ ( universal_class @ complement @ ( universal_class @ complement @ ( universal_class @ not_subclass_element ) @ member ) ),
inference(pattern_uni,[status(thm)],[14640:[bind(A,$thf( universal_class )),bind(B,$thf( universal_class @ complement ))]]) ).
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(20755,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(20756,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[20755:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( F @ singleton )),bind(D,$thf( H @ singleton @ ( F @ unordered_pair ) ))]]) ).
thf(20785,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[20756]) ).
thf(186,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(cnf,[status(esa)],[185]) ).
thf(2430,plain,
! [A: $i] :
( ( A != universal_class )
| ( null_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,277]) ).
thf(2431,plain,
( ( null_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2430:[bind(A,$thf( universal_class ))]]) ).
thf(2383,plain,
! [A: $i] :
( ( A != universal_class )
| ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[352,346]) ).
thf(2384,plain,
( ( universal_class @ ( null_class @ cross_product ) @ ( successor_relation @ subclass ) )
| ( universal_class @ ( universal_class @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2383:[bind(A,$thf( universal_class ))]]) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[205]) ).
thf(207,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[206]) ).
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(2848,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(2849,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[2848:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(2917,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(simp,[status(thm)],[2849]) ).
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(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(7927,plain,
( ( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[924,7736]) ).
thf(7928,plain,
( ( omega @ singleton @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[7927]) ).
thf(7991,plain,
( ( ( omega @ singleton @ complement )
!= omega )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[7928]) ).
thf(22268,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)],[123,21106]) ).
thf(22269,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)],[22268]) ).
thf(22270,plain,
! [C: $i,B: $i,A: $i] :
~ ( universal_class @ complement @ rotate @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[22269:[bind(A,$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(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(3468,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)],[3384,168]) ).
thf(3469,plain,
! [A: $i] :
( ( universal_class @ ( A @ complement @ ( element_relation @ restrict ) ) @ range_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[3468:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(43424,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ complement @ ( domain_relation @ subclass ) ) )
| ~ $true
| ~ ( domain_relation @ ( domain_relation @ complement @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[216,13691]) ).
thf(43425,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( domain_relation @ complement @ ( domain_relation @ subclass ) ) )
| ~ ( domain_relation @ ( domain_relation @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[43424]) ).
thf(43768,plain,
! [A: $i] :
( ( ( domain_relation @ complement )
!= universal_class )
| ( A != domain_relation )
| ~ ( domain_relation @ ( domain_relation @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[43425]) ).
thf(43811,plain,
( ( ( domain_relation @ complement )
!= universal_class )
| ~ ( domain_relation @ ( domain_relation @ complement @ subclass ) ) ),
inference(simp,[status(thm)],[43768]) ).
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(2746,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( y @ ( null_class @ cross_product ) @ complement @ ( y @ ( null_class @ cross_product ) @ regular @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[336,2647]) ).
thf(2747,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( y @ ( null_class @ cross_product ) @ complement @ ( y @ ( null_class @ cross_product ) @ regular @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[2746]) ).
thf(2771,plain,
! [A: $i] :
( ( A
!= ( y @ ( null_class @ cross_product ) @ complement ) )
| ( ( y @ ( null_class @ cross_product ) @ regular )
!= null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[2747]) ).
thf(2783,plain,
( ( ( y @ ( null_class @ cross_product ) @ regular )
!= null_class )
| ~ ( y @ ( null_class @ cross_product ) @ complement @ inductive ) ),
inference(simp,[status(thm)],[2771]) ).
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(7922,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7601,7736]) ).
thf(7923,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) @ ( omega @ member ) )
!= ( omega @ singleton @ complement @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7922]) ).
thf(8006,plain,
! [A: $i] :
( ( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) )
| ( omega != omega ) ),
inference(simp,[status(thm)],[7923]) ).
thf(8019,plain,
! [A: $i] :
( ( A @ ( omega @ unordered_pair ) )
!= ( omega @ singleton @ complement ) ),
inference(simp,[status(thm)],[8006]) ).
thf(3107,plain,
( ( ( y @ ( y @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2106,1046]) ).
thf(3108,plain,
( ( ( y @ ( y @ regular @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3107]) ).
thf(3177,plain,
( ( ( omega @ complement )
!= y )
| ( ( y @ regular )
!= null_class )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3108]) ).
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(200,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(cnf,[status(esa)],[199]) ).
thf(178,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[177]) ).
thf(3111,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[2106,365]) ).
thf(3112,plain,
! [B: $i,A: $i] :
( ( ( y @ ( y @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(simp,[status(thm)],[3111]) ).
thf(3113,plain,
( ( ( y @ complement )
!= y )
| ( ( null_class @ ( null_class @ cross_product ) )
!= null_class ) ),
inference(pattern_uni,[status(thm)],[3112:[bind(A,$thf( y @ regular )),bind(B,$thf( y ))]]) ).
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(7721,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,365]) ).
thf(7722,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ( ( B @ complement )
!= B ) ),
inference(simp,[status(thm)],[7721]) ).
thf(7723,plain,
( ( omega @ singleton @ complement )
!= ( omega @ singleton ) ),
inference(pattern_uni,[status(thm)],[7722:[bind(A,$thf( omega )),bind(B,$thf( omega @ singleton ))]]) ).
thf(215,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[214]) ).
thf(3643,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[168,3525]) ).
thf(3644,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[3643:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(3721,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[3644]) ).
thf(7745,plain,
( ( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[7631,1292]) ).
thf(7746,plain,
( ( domain_relation @ complement @ ( omega @ domain_of @ ( omega @ ordered_pair ) @ member ) )
!= ( omega @ singleton @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[7745]) ).
thf(7776,plain,
( ( ( domain_relation @ complement )
!= ( omega @ singleton ) )
| ( ( omega @ domain_of @ ( omega @ ordered_pair ) )
!= omega ) ),
inference(simp,[status(thm)],[7746]) ).
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(66343,plain,
$false,
inference(e,[status(thm)],[40307,1454,10577,347,249,14585,17998,234,22521,352,365,138,41398,2671,2380,7601,170,217,276,533,142,340,1954,7716,538,153,174,185,6887,1046,3947,301,53636,184,13619,1323,344,40449,8022,196,157,14164,189,325,152,16640,44030,261,216,924,7792,7144,179,13590,321,211,238,121,806,353,41755,2616,164,1858,3525,1051,10602,7707,1072,132,133,23829,116,40261,1104,16527,147,16544,280,221,3712,292,233,248,117,270,3720,201,1338,16333,220,1067,16168,7703,21238,260,44431,7710,297,424,192,165,2132,957,1223,275,137,14680,252,197,1296,224,44205,317,156,3381,19011,169,14666,188,141,53755,4951,3143,3179,7631,1292,193,5213,2108,212,124,225,1099,21106,10603,21516,173,19251,19708,7789,129,134,1857,244,7770,266,205,311,19090,41864,128,343,237,166,23938,148,279,264,14744,161,7599,180,296,43799,149,2376,1154,7736,286,7704,118,9663,16634,176,3547,259,2134,313,144,335,236,804,350,53565,2220,181,2425,187,172,20801,22036,230,5407,303,3384,274,2214,2378,3182,208,14311,44295,1862,3971,155,198,330,22253,323,240,251,41395,2647,130,278,223,306,135,299,14656,167,3380,1076,20783,490,162,255,209,2684,123,7718,194,145,17696,14304,22053,282,150,327,2678,1780,2386,310,3642,199,331,2429,127,338,9666,359,544,177,182,1934,336,16517,154,175,314,16618,143,7720,250,1321,14673,1390,2775,2818,14641,14172,963,319,20785,258,158,21518,186,114,21105,139,171,2431,2384,13691,207,272,346,2917,214,290,119,235,246,2218,2102,7991,168,146,22270,1380,190,183,2050,18847,3469,210,43811,268,195,2783,239,242,294,126,136,8019,3177,332,1268,200,178,369,3392,131,3113,277,227,7723,20778,309,215,53342,2106,3721,364,232,7776,337,122]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : SET206-6 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.12 % 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
% 0.23/0.49 % Computer : n020.cluster.edu
% 0.23/0.49 % Model : x86_64 x86_64
% 0.23/0.49 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.23/0.49 % Memory : 8046.5625MB
% 0.23/0.49 % OS : Linux 6.8.0-71-generic
% 0.23/0.49 % CPULimit : 300
% 0.23/0.49 % WCLimit : 300
% 0.23/0.49 % DateTime : Sat Sep 26 07:20:34 UTC 2026
% 0.23/0.50 % CPUTime :
% 0.23/0.50 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 1.38/1.32 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.33/1.72 % [INFO] Parsing done (394ms).
% 2.33/1.75 % [INFO] Running in sequential loop mode.
% 3.13/2.25 % [INFO] eprover registered as external prover.
% 3.13/2.26 % [INFO] Scanning for conjecture ...
% 3.62/2.43 % [INFO] Found a conjecture (or negated_conjecture) and 112 axioms. Running axiom selection ...
% 4.05/2.60 % [INFO] Axiom selection finished. Selected 112 axioms (removed 0 axioms).
% 4.41/2.78 % [INFO] Problem is propositional (TPTP CNF).
% 4.69/2.81 % [INFO] Type checking passed.
% 4.69/2.81 % [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 ...
% 161.12/39.35 % External prover 'e' found a proof!
% 161.12/39.35 % [INFO] Killing All external provers ...
% 161.12/39.35 % Time passed: 38673ms (effective reasoning time: 37594ms)
% 161.12/39.36 % 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
% 161.12/39.36 % No. of inferences in proof: 817
% 161.12/39.36 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : 38673 ms resp. 37594 ms w/o parsing
% 159.15/39.69 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 159.49/39.71 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------