%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : NUM142-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n003.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 08:08:16 AM UTC 2026
% Result : Unsatisfiable 170.34s 39.98s
% Output : Refutation 170.96s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 159
% Syntax : Number of formulae : 513 ( 155 unt; 0 typ; 0 def)
% Number of atoms : 1148 ( 384 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 4610 ( 583 ~; 507 |; 0 &;3520 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 21 con; 0-3 aty)
% Number of variables : 804 ( 0 ^; 804 !; 0 ?; 804 :)
% Comments :
%------------------------------------------------------------------------------
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(x_decl,type,
x: $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(homomorphism_decl,type,
homomorphism: $i > $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(recursion_equation_functions_decl,type,
recursion_equation_functions: $i > $i ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(rest_of_decl,type,
rest_of: $i > $i ).
thf(composition_function_decl,type,
composition_function: $i ).
thf(ordinal_numbers_decl,type,
ordinal_numbers: $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(operation_decl,type,
operation: $i > $o ).
thf(domain_relation_decl,type,
domain_relation: $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(union_of_range_map_decl,type,
union_of_range_map: $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(kind_1_ordinals_decl,type,
kind_1_ordinals: $i ).
thf(maps_decl,type,
maps: $i > $i > $i > $o ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(cross_product_decl,type,
cross_product: $i > $i > $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(compose_class_decl,type,
compose_class: $i > $i ).
thf(application_function_decl,type,
application_function: $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(inductive_decl,type,
inductive: $i > $o ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(asymmetric_decl,type,
asymmetric: $i > $i > $o ).
thf(rest_relation_decl,type,
rest_relation: $i ).
thf(first_decl,type,
first: $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(limit_ordinals_decl,type,
limit_ordinals: $i ).
thf(omega_decl,type,
omega: $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(well_ordering_decl,type,
well_ordering: $i > $i > $o ).
thf(recursion_decl,type,
recursion: $i > $i > $i > $i ).
thf(add_relation_decl,type,
add_relation: $i ).
thf(ordinal_multiply_decl,type,
ordinal_multiply: $i > $i > $i ).
thf(ordinal_add_decl,type,
ordinal_add: $i > $i > $i ).
thf(integer_of_decl,type,
integer_of: $i > $i ).
thf(segment_decl,type,
segment: $i > $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(not_well_ordering_decl,type,
not_well_ordering: $i > $i > $i ).
thf(connected_decl,type,
connected: $i > $i > $o ).
thf(symmetrization_of_decl,type,
symmetrization_of: $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(singleton_relation_decl,type,
singleton_relation: $i ).
thf(least_decl,type,
least: $i > $i > $i ).
thf(section_decl,type,
section: $i > $i > $i > $o ).
thf(transitive_decl,type,
transitive: $i > $i > $o ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(irreflexive_decl,type,
irreflexive: $i > $i > $o ).
thf(78,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(333,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(8,axiom,
! [B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
= A )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions4) ).
thf(176,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
= A )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(177,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
= A )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[176]) ).
thf(178,plain,
! [B: $i,A: $i] :
( ~ ( B @ recursion_equation_functions @ ( A @ member ) )
| ( ( A @ rest_of @ ( B @ compose ) )
= A ) ),
inference(lifteq,[status(thm)],[177]) ).
thf(143,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(472,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(1,negated_conjecture,
~ ( x @ successor @ ( x @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_successor_of_set_is_set3_1) ).
thf(160,plain,
~ ( x @ successor @ ( x @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(161,plain,
~ ( x @ successor @ ( x @ subclass ) ),
inference(polarity_switch,[status(thm)],[160]) ).
thf(1039,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[472,161]) ).
thf(1040,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1039]) ).
thf(1041,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( ( B @ ( A @ compose ) )
!= x ) ),
inference(simp,[status(thm)],[1040]) ).
thf(1198,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ rest_of @ ( B @ compose ) )
!= ( D @ ( C @ compose ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( A != x )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[178,1041]) ).
thf(1199,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( B != x )
| ~ ( A @ recursion_equation_functions @ ( B @ member ) ) ),
inference(pattern_uni,[status(thm)],[1198:[bind(A,$thf( E )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E @ rest_of ))]]) ).
thf(1231,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ~ ( A @ recursion_equation_functions @ ( x @ member ) ) ),
inference(simp,[status(thm)],[1199]) ).
thf(1393,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( A @ recursion_equation_functions @ ( x @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[333,1231]) ).
thf(1394,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( A @ recursion_equation_functions @ ( x @ member ) ) )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) ) ),
inference(simp,[status(thm)],[1393]) ).
thf(1411,plain,
! [A: $i] :
( ( ( A @ recursion_equation_functions )
!= universal_class )
| ( omega != x )
| ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) ) ),
inference(simp,[status(thm)],[1394]) ).
thf(141,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(468,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(33,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(234,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(87,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(352,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(92,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(365,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(35,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(238,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(239,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[238]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(340,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(341,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[340]) ).
thf(509,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[333,341]) ).
thf(510,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[509]) ).
thf(511,plain,
~ ( universal_class @ complement @ ( omega @ member ) ),
inference(pattern_uni,[status(thm)],[510:[bind(A,$thf( omega )),bind(B,$thf( universal_class ))]]) ).
thf(1147,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[239,511]) ).
thf(1148,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1147]) ).
thf(1180,plain,
! [A: $i] :
( ( A
!= ( universal_class @ complement ) )
| ( omega != null_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1148]) ).
thf(1190,plain,
( ( omega != null_class )
| ~ ( universal_class @ complement @ inductive ) ),
inference(simp,[status(thm)],[1180]) ).
thf(40,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(249,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)],[40]) ).
thf(136,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_class_definition1) ).
thf(459,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(626,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[459,161]) ).
thf(627,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ compose_class @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[626]) ).
thf(628,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( ( A @ compose_class )
!= x ) ),
inference(simp,[status(thm)],[627]) ).
thf(5,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(168,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(169,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[168]) ).
thf(170,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[169]) ).
thf(52,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(276,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_multiplication) ).
thf(307,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(308,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( A @ ( add_relation @ apply ) @ ( null_class @ recursion ) ) )
= ( B @ ( A @ ordinal_multiply ) ) ),
inference(lifteq,[status(thm)],[307]) ).
thf(131,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',section1) ).
thf(449,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(39,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(247,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)],[39]) ).
thf(127,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(440,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(101,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering1) ).
thf(385,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ connected ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(100,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_term_defn2) ).
thf(383,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(384,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ second )
= ( A @ single_valued2 ) ),
inference(lifteq,[status(thm)],[383]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(426,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(427,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[426]) ).
thf(428,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[427]) ).
thf(429,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[428]) ).
thf(547,plain,
! [A: $i] :
( ( ( A @ ( A @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[429,161]) ).
thf(548,plain,
! [A: $i] :
( ( A @ ( A @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[547]) ).
thf(549,plain,
! [A: $i] :
( ( A
!= ( x @ successor ) )
| ( A != x ) ),
inference(simp,[status(thm)],[548]) ).
thf(550,plain,
( ( x @ successor )
!= x ),
inference(simp,[status(thm)],[549]) ).
thf(7,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(174,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)],[7]) ).
thf(11,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(184,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)],[11]) ).
thf(185,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)],[184]) ).
thf(63,axiom,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers1) ).
thf(301,plain,
! [A: $i] :
( ( A @ ( element_relation @ well_ordering ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(72,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/sandbox/benchmark/theBenchmark.p',application_function_defn2) ).
thf(320,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)],[72]) ).
thf(124,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( B @ ( B @ ( A @ not_well_ordering ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering7) ).
thf(436,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( B @ ( B @ ( A @ not_well_ordering ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(56,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(286,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(1154,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[286,239]) ).
thf(1155,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1154]) ).
thf(1156,plain,
omega @ ( null_class @ member ),
inference(pattern_uni,[status(thm)],[1155:[bind(A,$thf( omega ))]]) ).
thf(1237,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1156,511]) ).
thf(1238,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1237]) ).
thf(1264,plain,
( ( ( universal_class @ complement )
!= omega )
| ( omega != null_class ) ),
inference(simp,[status(thm)],[1238]) ).
thf(137,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_of_range_map1) ).
thf(460,plain,
universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(115,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(416,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(74,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_term_defn1) ).
thf(324,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(325,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ not_subclass_element ) @ first )
= ( A @ single_valued1 ) ),
inference(lifteq,[status(thm)],[324]) ).
thf(70,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(316,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)],[70]) ).
thf(45,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(261,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)],[45]) ).
thf(3,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(164,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(9,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ compose ) )
= C )
| ~ ( composition_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_composition_function2) ).
thf(179,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)],[9]) ).
thf(42,axiom,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_relation3) ).
thf(253,plain,
! [A: $i] :
( ( rest_relation @ ( A @ rest_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(85,axiom,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetrization) ).
thf(348,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(353,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[352]) ).
thf(146,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(480,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(91,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(362,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(363,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[362]) ).
thf(364,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[363]) ).
thf(2177,plain,
! [A: $i] :
( ( A
!= ( x @ successor ) )
| ( null_class != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,550]) ).
thf(2178,plain,
( ( null_class != x )
| ( x @ successor @ ( x @ successor @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2177:[bind(A,$thf( x @ successor ))]]) ).
thf(25,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(216,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)],[25]) ).
thf(54,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ asymmetric ) )
| ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',asymmetric2) ).
thf(280,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ asymmetric ) )
| ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(133,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(453,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(105,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps1) ).
thf(396,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(113,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ transitive ) )
| ~ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitive2) ).
thf(411,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ transitive ) )
| ~ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(46,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(264,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(265,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[264]) ).
thf(20,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(206,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(59,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(292,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)],[59]) ).
thf(99,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( ( B @ ( A @ not_well_ordering ) )
!= null_class )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering6) ).
thf(380,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ well_ordering ) )
| ( ( B @ ( A @ not_well_ordering ) )
!= null_class )
| ~ ( B @ ( A @ connected ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(126,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',application_function_defn1) ).
thf(439,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(49,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(270,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)],[49]) ).
thf(544,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[460,161]) ).
thf(545,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( union_of_range_map @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[544]) ).
thf(546,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( union_of_range_map != x ) ),
inference(simp,[status(thm)],[545]) ).
thf(18,axiom,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions2) ).
thf(201,plain,
! [B: $i,A: $i] :
( ( A @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(27,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/sandbox/benchmark/theBenchmark.p',compose_class_definition3) ).
thf(220,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)],[27]) ).
thf(2952,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( x @ successor @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2178,341]) ).
thf(2953,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( x @ successor @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2952]) ).
thf(2954,plain,
( ~ ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
| ( null_class != x ) ),
inference(pattern_uni,[status(thm)],[2953:[bind(A,$thf( x @ successor @ regular )),bind(B,$thf( x @ successor ))]]) ).
thf(3652,plain,
( ( ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
!= ( x @ successor @ ( x @ successor @ regular @ member ) ) )
| ~ $true
| ( null_class != x )
| ( null_class != x ) ),
inference(paramod_ordered,[status(thm)],[2178,2954]) ).
thf(3653,plain,
( ( ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
!= ( x @ successor @ ( x @ successor @ regular @ member ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3652]) ).
thf(3704,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( ( x @ successor @ regular )
!= ( x @ successor @ regular ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3653]) ).
thf(3726,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[3704]) ).
thf(4166,plain,
! [A: $i] :
( ( null_class != null_class )
| ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,3726]) ).
thf(4167,plain,
! [A: $i] :
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4166:[]]) ).
thf(4172,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[4167]) ).
thf(507,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[333,341]) ).
thf(508,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( B @ complement @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[507]) ).
thf(513,plain,
! [B: $i,A: $i] :
( ( ( B @ complement )
!= universal_class )
| ( omega != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[508]) ).
thf(516,plain,
! [A: $i] :
( ( ( A @ complement )
!= universal_class )
| ~ ( A @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[513]) ).
thf(1251,plain,
! [A: $i] :
( ( ( A @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ( ( A @ complement )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1156,516]) ).
thf(1252,plain,
! [A: $i] :
( ( ( A @ ( omega @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1251]) ).
thf(1265,plain,
! [A: $i] :
( ( A != omega )
| ( omega != null_class )
| ( ( A @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1252]) ).
thf(1274,plain,
( ( omega != null_class )
| ( ( omega @ complement )
!= universal_class ) ),
inference(simp,[status(thm)],[1265]) ).
thf(79,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(334,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)],[79]) ).
thf(349,plain,
! [A: $i] :
( ( A @ inverse @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(lifteq,[status(thm)],[348]) ).
thf(148,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(483,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(559,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[483,161]) ).
thf(560,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[559]) ).
thf(561,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( successor_relation != x ) ),
inference(simp,[status(thm)],[560]) ).
thf(111,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(407,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(165,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[164]) ).
thf(123,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_of1) ).
thf(435,plain,
! [A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(623,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[435,161]) ).
thf(624,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ rest_of @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[623]) ).
thf(625,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( ( A @ rest_of )
!= x ) ),
inference(simp,[status(thm)],[624]) ).
thf(76,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(328,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(329,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[328]) ).
thf(138,axiom,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ compose ) @ subclass ) )
| ~ ( B @ ( A @ transitive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitive1) ).
thf(461,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ ( B @ ( B @ ( A @ restrict ) ) @ compose ) @ subclass ) )
| ~ ( B @ ( A @ transitive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(153,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',irreflexive2) ).
thf(493,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ irreflexive ) )
| ~ ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(147,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(482,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(522,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[482,161]) ).
thf(523,plain,
! [A: $i] :
( ( universal_class @ ( A @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[522]) ).
thf(524,plain,
! [A: $i] :
( ( ( x @ successor )
!= universal_class )
| ( A != x ) ),
inference(simp,[status(thm)],[523]) ).
thf(525,plain,
( ( x @ successor )
!= universal_class ),
inference(simp,[status(thm)],[524]) ).
thf(2510,plain,
! [A: $i] :
( ( A
!= ( x @ successor ) )
| ( universal_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,525]) ).
thf(2511,plain,
( ( universal_class != null_class )
| ( x @ successor @ ( x @ successor @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2510:[bind(A,$thf( x @ successor ))]]) ).
thf(3026,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( x @ successor @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != null_class ) ),
inference(paramod_ordered,[status(thm)],[2511,341]) ).
thf(3027,plain,
! [B: $i,A: $i] :
( ( ( x @ successor @ ( x @ successor @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[3026]) ).
thf(3028,plain,
( ~ ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
| ( universal_class != null_class ) ),
inference(pattern_uni,[status(thm)],[3027:[bind(A,$thf( x @ successor @ regular )),bind(B,$thf( x @ successor ))]]) ).
thf(31,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(231,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(673,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)],[231,341]) ).
thf(674,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)],[673]) ).
thf(675,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[674:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( universal_class ))]]) ).
thf(682,plain,
! [B: $i,A: $i] :
~ ( universal_class @ complement @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(simp,[status(thm)],[675]) ).
thf(24,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(215,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(2740,plain,
! [A: $i] :
( ( A != choice )
| ( null_class @ function )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,215]) ).
thf(2741,plain,
( ( null_class @ function )
| ( choice @ ( choice @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2740:[bind(A,$thf( choice ))]]) ).
thf(28,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/sandbox/benchmark/theBenchmark.p',application_function_defn4) ).
thf(224,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)],[28]) ).
thf(30,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(229,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)],[30]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(484,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(109,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ ( A @ least ) @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering5) ).
thf(403,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ ( A @ least ) @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(13,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(188,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(118,axiom,
! [A: $i] :
( ( ordinal_numbers @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers3) ).
thf(420,plain,
! [A: $i] :
( ( ordinal_numbers @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(15,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(193,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(151,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(489,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(194,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[193]) ).
thf(21,axiom,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',kind_1_ordinals) ).
thf(208,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(209,plain,
( ( ordinal_numbers @ ( successor_relation @ image ) @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(lifteq,[status(thm)],[208]) ).
thf(4053,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( ordinal_numbers @ ( successor_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[194,209]) ).
thf(4054,plain,
( ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ range_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[4053:[bind(A,$thf( successor_relation )),bind(B,$thf( ordinal_numbers ))]]) ).
thf(1255,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[1156,341]) ).
thf(1256,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1255]) ).
thf(1257,plain,
~ ( omega @ complement @ ( null_class @ member ) ),
inference(pattern_uni,[status(thm)],[1256:[bind(A,$thf( null_class )),bind(B,$thf( omega ))]]) ).
thf(1295,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[239,1257]) ).
thf(1296,plain,
! [A: $i] :
( ( ( A @ ( null_class @ member ) )
!= ( omega @ complement @ ( null_class @ member ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[1295]) ).
thf(1297,plain,
~ ( omega @ complement @ inductive ),
inference(pattern_uni,[status(thm)],[1296:[bind(A,$thf( omega @ complement ))]]) ).
thf(135,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps3) ).
thf(457,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(6,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(171,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)],[6]) ).
thf(172,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)],[171]) ).
thf(173,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)],[172]) ).
thf(95,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(371,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)],[95]) ).
thf(158,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(503,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(34,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(236,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(237,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[236]) ).
thf(47,axiom,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',limit_ordinals) ).
thf(266,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(90,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(360,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(68,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',integer_function2) ).
thf(311,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(106,axiom,
! [B: $i,A: $i] :
( ( A @ recursion_equation_functions @ ( B @ member ) )
| ( ( B @ rest_of @ ( A @ compose ) )
!= B )
| ~ ( ordinal_numbers @ ( B @ domain_of @ member ) )
| ~ ( B @ function )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions5) ).
thf(398,plain,
! [B: $i,A: $i] :
( ( A @ recursion_equation_functions @ ( B @ member ) )
| ( ( B @ rest_of @ ( A @ compose ) )
!= B )
| ~ ( ordinal_numbers @ ( B @ domain_of @ member ) )
| ~ ( B @ function )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(62,axiom,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_relation2) ).
thf(298,plain,
! [B: $i,A: $i] :
( ( ( A @ rest_of )
= B )
| ~ ( rest_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(121,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(430,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(23,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(213,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(214,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(lifteq,[status(thm)],[213]) ).
thf(807,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)],[265,214]) ).
thf(808,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ range_of ) ),
inference(pattern_uni,[status(thm)],[807:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(4115,plain,
! [A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ range_of ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[808,4054]) ).
thf(4116,plain,
( ( universal_class @ ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ cross_product ) @ flip @ domain_of @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[4115:[bind(A,$thf( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) ))]]) ).
thf(4,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(166,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)],[4]) ).
thf(61,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ ( A @ member ) )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_of2) ).
thf(296,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ ( A @ member ) )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(129,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(444,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)],[129]) ).
thf(97,axiom,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',integer_function1) ).
thf(375,plain,
! [A: $i] :
( ( ( A @ integer_of )
= A )
| ~ ( omega @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(108,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(402,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(1124,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)],[808,214]) ).
thf(1125,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of )
= ( A @ inverse @ domain_of ) ),
inference(pattern_uni,[status(thm)],[1124:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(114,axiom,
! [A: $i] :
( ( A = ordinal_numbers )
| ( ordinal_numbers @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers4) ).
thf(413,plain,
! [A: $i] :
( ( A = ordinal_numbers )
| ( ordinal_numbers @ ( A @ member ) )
| ~ ( A @ ( A @ sum_class @ subclass ) )
| ~ ( A @ ( element_relation @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(312,plain,
! [A: $i] :
( ( ( A @ integer_of )
= null_class )
| ( omega @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[311]) ).
thf(313,plain,
! [A: $i] :
( ( omega @ ( A @ member ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(lifteq,[status(thm)],[312]) ).
thf(156,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( A @ least ) @ ( C @ ( A @ segment ) ) )
= null_class )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering4) ).
thf(498,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( A @ least ) @ ( C @ ( A @ segment ) ) )
= null_class )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(335,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)],[334]) ).
thf(180,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)],[179]) ).
thf(181,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)],[180]) ).
thf(86,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(350,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(538,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[402,161]) ).
thf(539,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[538]) ).
thf(540,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( element_relation != x ) ),
inference(simp,[status(thm)],[539]) ).
thf(12,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(186,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)],[12]) ).
thf(187,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)],[186]) ).
thf(51,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/sandbox/benchmark/theBenchmark.p',definition_of_composition_function3) ).
thf(274,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)],[51]) ).
thf(117,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(419,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ segment ) ) )
= ( C @ singleton @ ( B @ ( A @ restrict ) ) @ domain_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',segment) ).
thf(318,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ segment ) ) )
= ( C @ singleton @ ( B @ ( A @ restrict ) ) @ domain_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(1042,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)],[237,682]) ).
thf(1043,plain,
! [A: $i] :
~ ( universal_class @ complement @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1042:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(64,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(303,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(1285,plain,
( ( ( omega @ complement @ ( null_class @ member ) )
!= ( omega @ ( null_class @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1156,1257]) ).
thf(1286,plain,
( ( omega @ complement @ ( null_class @ member ) )
!= ( omega @ ( null_class @ member ) ) ),
inference(simp,[status(thm)],[1285]) ).
thf(1301,plain,
( ( ( omega @ complement )
!= omega )
| ( null_class != null_class ) ),
inference(simp,[status(thm)],[1286]) ).
thf(1307,plain,
( ( omega @ complement )
!= omega ),
inference(simp,[status(thm)],[1301]) ).
thf(132,axiom,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_numbers2) ).
thf(451,plain,
! [A: $i] :
( ( A @ ( A @ sum_class @ subclass ) )
| ~ ( ordinal_numbers @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(107,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_composition_function1) ).
thf(401,plain,
universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(565,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[401,161]) ).
thf(566,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( composition_function @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[565]) ).
thf(567,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( composition_function != x ) ),
inference(simp,[status(thm)],[566]) ).
thf(38,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(245,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)],[38]) ).
thf(2115,plain,
! [A: $i] :
( ( A
!= ( x @ successor ) )
| ~ ( null_class @ ( x @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,161]) ).
thf(2116,plain,
( ~ ( null_class @ ( x @ subclass ) )
| ( x @ successor @ ( x @ successor @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2115:[bind(A,$thf( x @ successor ))]]) ).
thf(145,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(477,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)],[145]) ).
thf(102,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ compose ) )
= B )
| ~ ( C @ compose_class @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_class_definition2) ).
thf(387,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)],[102]) ).
thf(116,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(418,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(629,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[418,161]) ).
thf(630,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[629]) ).
thf(631,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= ( x @ successor ) )
| ( ( A @ flip )
!= x ) ),
inference(simp,[status(thm)],[630]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(198,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)],[17]) ).
thf(77,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ apply ) )
= C )
| ~ ( application_function @ ( C @ ( B @ ordered_pair ) @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',application_function_defn3) ).
thf(330,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)],[77]) ).
thf(84,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ well_ordering ) )
| ~ ( C @ ( B @ connected ) )
| ( ( A @ ( C @ ( B @ not_well_ordering ) @ ( B @ segment ) ) )
!= null_class )
| ~ ( C @ ( B @ not_well_ordering ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering8) ).
thf(345,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ well_ordering ) )
| ~ ( C @ ( B @ connected ) )
| ( ( A @ ( C @ ( B @ not_well_ordering ) @ ( B @ segment ) ) )
!= null_class )
| ~ ( C @ ( B @ not_well_ordering ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(93,axiom,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose_can_define_singleton) ).
thf(367,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(724,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[313,516]) ).
thf(725,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( B @ ( omega @ member ) ) )
| ( ( B @ complement )
!= universal_class )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[724]) ).
thf(726,plain,
( ( ( omega @ complement )
!= universal_class )
| ( ( omega @ integer_of )
= null_class ) ),
inference(pattern_uni,[status(thm)],[725:[bind(A,$thf( omega )),bind(B,$thf( omega ))]]) ).
thf(36,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(240,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(41,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(251,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)],[41]) ).
thf(518,plain,
( ( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[333,511]) ).
thf(519,plain,
( ( universal_class @ complement @ ( omega @ member ) )
!= ( universal_class @ ( omega @ member ) ) ),
inference(simp,[status(thm)],[518]) ).
thf(520,plain,
( ( ( universal_class @ complement )
!= universal_class )
| ( omega != omega ) ),
inference(simp,[status(thm)],[519]) ).
thf(521,plain,
( ( universal_class @ complement )
!= universal_class ),
inference(simp,[status(thm)],[520]) ).
thf(53,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(278,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(134,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(455,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(606,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[439,161]) ).
thf(607,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) @ ( application_function @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[606]) ).
thf(608,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( application_function != x ) ),
inference(simp,[status(thm)],[607]) ).
thf(104,axiom,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_domain_relation3) ).
thf(394,plain,
! [A: $i] :
( ( domain_relation @ ( A @ domain_of @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( A @ ( C @ restrict ) ) )
= B )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_of3) ).
thf(226,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( A @ ( C @ restrict ) ) )
= B )
| ~ ( C @ rest_of @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(267,plain,
( ( ordinal_numbers @ ( kind_1_ordinals @ complement @ intersection ) )
= limit_ordinals ),
inference(lifteq,[status(thm)],[266]) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(159,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function3) ).
thf(505,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)],[159]) ).
thf(125,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_relation1) ).
thf(438,plain,
universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(144,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(473,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(2,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(43,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(255,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)],[43]) ).
thf(83,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(342,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)],[83]) ).
thf(150,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ domain_of @ ( A @ maps ) ) )
| ~ ( B @ ( A @ range_of @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps4) ).
thf(487,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)],[150]) ).
thf(688,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[419,161]) ).
thf(689,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[688]) ).
thf(690,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) )
!= ( x @ successor ) )
| ( ( A @ rotate )
!= x ) ),
inference(simp,[status(thm)],[689]) ).
thf(112,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( B @ ( C @ ( A @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',section2) ).
thf(409,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( B @ ( C @ ( A @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ section ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(10,axiom,
! [B: $i,A: $i] :
( ( ordinal_numbers @ ( A @ domain_of @ member ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions3) ).
thf(182,plain,
! [B: $i,A: $i] :
( ( ordinal_numbers @ ( A @ domain_of @ member ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(441,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(lifteq,[status(thm)],[440]) ).
thf(139,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( A @ least ) @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering3) ).
thf(463,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( A @ least ) @ member ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(81,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(338,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(142,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(470,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(89,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(358,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(359,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[358]) ).
thf(80,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(336,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(2144,plain,
! [A: $i] :
( ( A != omega )
| ( universal_class @ ( null_class @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,333]) ).
thf(2145,plain,
( ( universal_class @ ( null_class @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2144:[bind(A,$thf( omega ))]]) ).
thf(69,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(314,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)],[69]) ).
thf(130,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( A @ least ) @ member ) )
| ( C = null_class )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',well_ordering2) ).
thf(446,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( C @ ( A @ least ) @ member ) )
| ( C = null_class )
| ~ ( B @ ( C @ subclass ) )
| ~ ( B @ ( A @ well_ordering ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(175,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)],[174]) ).
thf(154,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_domain_relation1) ).
thf(495,plain,
universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(26,axiom,
! [A: $i] :
( ( A @ single_valued2 @ ( A @ single_valued1 @ singleton @ ( A @ inverse @ image ) @ ( A @ domain ) ) )
= ( A @ single_valued3 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_term_defn3) ).
thf(218,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)],[26]) ).
thf(368,plain,
( ( element_relation @ ( identity_relation @ complement @ ( element_relation @ compose ) @ complement @ intersection ) )
= singleton_relation ),
inference(lifteq,[status(thm)],[367]) ).
thf(140,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(465,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)],[140]) ).
thf(19,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(203,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(351,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[350]) ).
thf(103,axiom,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ range_of @ sum_class )
!= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_of_range_map3) ).
thf(390,plain,
! [B: $i,A: $i] :
( ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ range_of @ sum_class )
!= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(4196,plain,
( ( ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
!= ( x @ successor @ ( x @ successor @ regular @ member ) ) )
| ~ $true
| ( universal_class != null_class )
| ( universal_class != null_class ) ),
inference(paramod_ordered,[status(thm)],[2511,3028]) ).
thf(4197,plain,
( ( ( x @ successor @ complement @ ( x @ successor @ regular @ member ) )
!= ( x @ successor @ ( x @ successor @ regular @ member ) ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[4196]) ).
thf(4240,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( ( x @ successor @ regular )
!= ( x @ successor @ regular ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[4197]) ).
thf(4250,plain,
( ( ( x @ successor @ complement )
!= ( x @ successor ) )
| ( universal_class != null_class ) ),
inference(simp,[status(thm)],[4240]) ).
thf(44,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(258,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)],[44]) ).
thf(235,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[234]) ).
thf(122,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(433,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(2329,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[364,286]) ).
thf(2330,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[2329:[bind(A,$thf( omega ))]]) ).
thf(50,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(272,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(1044,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)],[237,231]) ).
thf(1045,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[1044:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(58,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(290,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)],[58]) ).
thf(98,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(378,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(789,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)],[235,235]) ).
thf(790,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[789:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(800,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[790]) ).
thf(57,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(287,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)],[57]) ).
thf(1956,plain,
! [B: $i,A: $i] :
( ( ( A @ inverse )
!= ( B @ inverse ) )
| ( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( B @ union ) )
= ( B @ symmetrization_of ) ) ),
inference(paramod_ordered,[status(thm)],[265,349]) ).
thf(1957,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( A @ union ) )
= ( A @ symmetrization_of ) ),
inference(pattern_uni,[status(thm)],[1956:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(110,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ section ) ) )
| ~ ( A @ ( A @ ( B @ ( C @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',section3) ).
thf(405,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ section ) ) )
| ~ ( A @ ( A @ ( B @ ( C @ restrict ) ) @ domain_of @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(14,axiom,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',definition_of_domain_relation2) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( domain_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(183,plain,
! [B: $i,A: $i] :
( ( ordinal_numbers @ ( A @ domain_of @ member ) )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[182]) ).
thf(119,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(422,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)],[119]) ).
thf(273,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[272]) ).
thf(96,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(373,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(562,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[495,161]) ).
thf(563,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( domain_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[562]) ).
thf(564,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( domain_relation != x ) ),
inference(simp,[status(thm)],[563]) ).
thf(65,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(305,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)],[65]) ).
thf(22,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',maps2) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ domain_of )
= B )
| ~ ( C @ ( B @ ( A @ maps ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(75,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(326,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(60,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(294,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)],[60]) ).
thf(48,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(268,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(16,axiom,
! [B: $i,A: $i] :
( ( ( A @ range_of @ sum_class )
= B )
| ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_of_range_map2) ).
thf(195,plain,
! [B: $i,A: $i] :
( ( ( A @ range_of @ sum_class )
= B )
| ~ ( union_of_range_map @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(1001,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)],[351,235]) ).
thf(1002,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1001:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1033,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1002]) ).
thf(55,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(283,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)],[55]) ).
thf(37,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
= null_class )
| ~ ( B @ ( A @ asymmetric ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',asymmetric1) ).
thf(242,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ ( A @ inverse @ ( A @ intersection ) @ restrict ) ) )
= null_class )
| ~ ( B @ ( A @ asymmetric ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(712,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ~ $true
| ( ( A @ integer_of )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[313,511]) ).
thf(713,plain,
! [A: $i] :
( ( ( omega @ ( A @ member ) )
!= ( universal_class @ complement @ ( omega @ member ) ) )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[712]) ).
thf(736,plain,
! [A: $i] :
( ( ( universal_class @ complement )
!= omega )
| ( A != omega )
| ( ( A @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[713]) ).
thf(740,plain,
( ( ( universal_class @ complement )
!= omega )
| ( ( omega @ integer_of )
= null_class ) ),
inference(simp,[status(thm)],[736]) ).
thf(4117,plain,
! [A: $i] :
( ( ( A @ range_of )
!= ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ range_of ) )
| ( ( A @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ) ),
inference(paramod_ordered,[status(thm)],[214,4054]) ).
thf(4118,plain,
( ( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ singleton @ union ) )
= kind_1_ordinals ),
inference(pattern_uni,[status(thm)],[4117:[bind(A,$thf( universal_class @ ( ordinal_numbers @ ( successor_relation @ restrict ) ) ))]]) ).
thf(163,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[162]) ).
thf(277,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[276]) ).
thf(128,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ connected ) )
| ~ ( B @ symmetrization_of @ ( identity_relation @ union ) @ ( A @ ( A @ cross_product ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',connected2) ).
thf(442,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ connected ) )
| ~ ( B @ symmetrization_of @ ( identity_relation @ union ) @ ( A @ ( A @ cross_product ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(157,axiom,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',irreflexive1) ).
thf(501,plain,
! [B: $i,A: $i] :
( ( identity_relation @ complement @ ( B @ ( B @ ( A @ restrict ) ) @ subclass ) )
| ~ ( B @ ( A @ irreflexive ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(94,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(369,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(73,axiom,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',recursion_equation_functions1) ).
thf(322,plain,
! [B: $i,A: $i] :
( ( B @ function )
| ~ ( B @ recursion_equation_functions @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(802,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[265,353]) ).
thf(803,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[802:[bind(A,$thf( subset_relation ))]]) ).
thf(88,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ rest_of @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( universal_class @ ( A @ ( B @ restrict ) ) )
!= C )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rest_of4) ).
thf(354,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ rest_of @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( universal_class @ ( A @ ( B @ restrict ) ) )
!= C )
| ~ ( B @ domain_of @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(67,axiom,
! [B: $i,A: $i] :
( ( B @ ( union_of_range_map @ ( successor_relation @ ( A @ recursion ) ) @ apply ) )
= ( B @ ( A @ ordinal_add ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinal_addition) ).
thf(309,plain,
! [B: $i,A: $i] :
( ( B @ ( union_of_range_map @ ( successor_relation @ ( A @ recursion ) ) @ apply ) )
= ( B @ ( A @ ordinal_add ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(541,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[438,161]) ).
thf(542,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( rest_relation @ subclass ) )
!= ( x @ successor @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[541]) ).
thf(543,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= ( x @ successor ) )
| ( rest_relation != x ) ),
inference(simp,[status(thm)],[542]) ).
thf(152,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(491,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(155,axiom,
! [B: $i,A: $i] :
( ( A @ symmetrization_of @ ( identity_relation @ union ) @ ( B @ ( B @ cross_product ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',connected1) ).
thf(496,plain,
! [B: $i,A: $i] :
( ( A @ symmetrization_of @ ( identity_relation @ union ) @ ( B @ ( B @ cross_product ) @ subclass ) )
| ~ ( B @ ( A @ connected ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(32,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(232,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)],[32]) ).
thf(47151,plain,
$false,
inference(e,[status(thm)],[1411,468,234,352,365,1190,249,333,628,170,276,308,449,247,1041,440,511,385,384,550,472,340,174,185,301,320,436,184,1264,460,416,325,316,261,164,179,253,238,348,353,480,2178,216,280,453,396,411,265,206,292,380,439,270,546,201,220,4172,2954,307,1274,334,349,160,561,407,429,165,625,435,329,461,493,3028,682,2741,324,224,229,484,403,188,420,328,193,489,4054,1297,1231,457,173,371,503,237,266,360,311,398,298,430,4116,166,264,161,296,444,375,176,402,1125,286,413,313,3726,498,335,236,181,350,540,187,274,419,318,1043,303,208,1307,362,451,567,245,2116,477,387,631,483,198,330,345,367,213,726,240,251,521,278,455,608,394,226,267,167,505,438,473,426,162,255,516,342,194,209,487,525,690,1156,409,182,441,463,338,470,808,359,336,1257,2145,314,446,175,2511,495,218,368,465,231,482,203,351,390,186,401,171,418,4250,258,235,433,2330,272,383,1045,290,378,214,800,287,1957,168,405,190,183,422,273,373,564,305,210,326,294,341,268,358,195,1033,283,239,242,740,4118,459,163,178,277,442,501,369,322,803,354,309,543,215,491,496,364,232]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : NUM142-1 : TPTP v9.3.1. Bugfixed v2.1.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.15/0.42 % Computer : n003.cluster.edu
% 0.15/0.42 % Model : x86_64 x86_64
% 0.15/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.42 % Memory : 8046.5625MB
% 0.15/0.42 % OS : Linux 6.8.0-71-generic
% 0.15/0.42 % CPULimit : 300
% 0.15/0.42 % WCLimit : 300
% 0.15/0.42 % DateTime : Sat Sep 26 02:08:09 UTC 2026
% 0.15/0.42 % CPUTime :
% 0.15/0.42 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.84/0.85 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.47/1.13 % [INFO] Parsing done (278ms).
% 1.47/1.15 % [INFO] Running in sequential loop mode.
% 2.67/1.48 % [INFO] eprover registered as external prover.
% 2.67/1.49 % [INFO] Scanning for conjecture ...
% 3.22/1.61 % [INFO] Found a conjecture (or negated_conjecture) and 158 axioms. Running axiom selection ...
% 3.52/1.76 % [INFO] Axiom selection finished. Selected 158 axioms (removed 0 axioms).
% 4.33/1.95 % [INFO] Problem is propositional (TPTP CNF).
% 4.33/1.97 % [INFO] Type checking passed.
% 4.33/1.97 % [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 ...
% 170.34/39.97 % External prover 'e' found a proof!
% 170.34/39.97 % [INFO] Killing All external provers ...
% 170.34/39.97 % Time passed: 39449ms (effective reasoning time: 38822ms)
% 170.34/39.98 % Axioms used in derivation (158): range, recursion_equation_functions1, equal_implies_subclass1, rest_relation2, complement1, one_to_one2, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, maps4, range_of, transitive1, function3, recursion_equation_functions5, operation4, regularity1, symmetrization, singleton_set, identity_relation, union, flip3, element_relation2, unordered_pairs_in_universal, rest_of4, compose2, homomorphism3, compatible3, application_function_defn2, symmetric_difference, cartesian_product1, compose_can_define_singleton, image, definition_of_domain_relation1, replacement, well_ordering4, not_subclass_members1, compose_class_definition3, definition_of_composition_function2, successor, ordered_pair, successor_relation1, irreflexive1, rest_of2, domain, homomorphism1, rest_relation3, one_to_one3, single_valued_term_defn3, single_valued_class1, compose3, omega_is_inductive2, rotate1, choice1, section1, single_valued_term_defn1, power_class2, regularity2, equal_implies_subclass2, element_relation3, homomorphism6, recursion_equation_functions4, intersection1, well_ordering3, flip2, well_ordering6, transitive2, inverse, segment, asymmetric2, connected1, sum_class_definition, ordinal_multiplication, inductive2, compatible2, sum_class2, subclass_members, cartesian_product2, maps1, not_subclass_members2, ordinal_numbers2, inductive1, single_valued_term_defn2, successor_relation3, function1, maps3, domain2, compatible4, intersection3, limit_ordinals, kind_1_ordinals, ordinal_numbers4, single_valued_class2, compose_class_definition1, unordered_pair_member, apply, well_ordering7, element_relation1, operation1, application_function_defn1, asymmetric1, restriction1, union_of_range_map1, ordinal_addition, cartesian_product3, section2, definition_of_domain_relation3, flip1, homomorphism5, maps2, ordinal_numbers1, compatible1, unordered_pair2, application_function_defn4, recursion_equation_functions3, well_ordering2, complement2, operation2, connected2, union_of_range_map3, rotate3, well_ordering8, compose_class_definition2, one_to_one1, subclass_implies_equal, function2, rest_of1, cantor_class, successor_relation2, rest_relation1, domain1, restriction2, omega_in_universal, ordinal_numbers3, well_ordering1, diagonalisation, intersection2, class_elements_are_sets, integer_function2, section3, rest_of3, inductive3, integer_function1, irreflexive2, operation3, well_ordering5, application_function_defn3, definition_of_domain_relation2, choice2, definition_of_composition_function1, homomorphism2, power_class_definition, union_of_range_map2, definition_of_composition_function3, unordered_pair3, recursion_equation_functions2, subset_relation, compose1
% 170.34/39.98 % No. of inferences in proof: 513
% 170.34/39.98 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 39449 ms resp. 38822 ms w/o parsing
% 170.96/40.08 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 170.96/40.08 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------